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+ "markdown": "---\ntitle: \"Generating Polynomials, Part 1: Regular Constructibility\"\ndescription: |\n What kinds of regular polygons are constructible with compass and straightedge?\nformat:\n html:\n html-math-method: katex\ndate: \"2021-08-18\"\ndate-modified: \"2025-06-17\"\ncategories:\n - geometry\n - generating functions\n - algebra\n - python\n---\n\n\n\n\n\n[Recently](/posts/misc/platonic-volume), I used coordinate-free geometry to derive\n the volumes of the Platonic solids, a problem which was very accessible to the ancient Greeks.\nOn the other hand, they found certain problems regarding which figures can be constructed via\n compass and straightedge to be very difficult. For example, they struggled with problems\n like [doubling the cube](https://en.wikipedia.org/wiki/Doubling_the_cube)\n or [squaring the circle](https://en.wikipedia.org/wiki/Squaring_the_circle),\n which are known (through circa 19th century mathematics) to be impossible.\nHowever, before even extending planar geometry by a third dimension or\n calculating the areas of circles, a simpler problem becomes apparent.\nNamely, what kinds of regular polygons are constructible?\n\n\nRegular Geometry and a Complex Series\n-------------------------------------\n\nWhen constructing a regular polygon, one wants a ratio between the length of a edge\n and the distance from a vertex to the center of the figure.\n\n{.wide}\n\nIn a convex polygon, the total central angle is always one full turn, or 2π radians.\nThe central angle of a regular *n*-gon is ${2\\pi \\over n}$ radians,\n and the green angle above (which we'll call *θ*) is half of that.\nThis means that the ratio we're looking for is $\\sin(\\theta) = \\sin(\\pi / n)$.\nWe can multiply by *n* inside the function on both sides to give\n $\\sin(n\\theta) = \\sin(\\pi) = 0$.\nTherefore, constructing a polygon is actually equivalent to solving this equation,\n and we can rephrase the question as how to express $\\sin(n\\theta)$ (and $\\cos(n\\theta)$).\n\n\n### Complex Recursion\n\nThanks to [Euler's formula](https://en.wikipedia.org/wiki/Euler%27s_formula)\n and [de Moivre's formula](https://en.wikipedia.org/wiki/De_Moivre%27s_formula),\n the expressions we're looking for can be phrased in terms of the complex exponential.\n\n$$\n\\begin{align*}\n e^{i\\theta}\n &= \\text{cis}(\\theta) = \\cos(\\theta) + i\\sin(\\theta)\n & \\text{ Euler's formula}\n \\\\\n \\text{cis}(n \\theta) = e^{i(n\\theta)}\n &= e^{(i\\theta)n} = {(e^{i\\theta})}^n = \\text{cis}(\\theta)^n\n \\\\\n \\cos(n \\theta) + i\\sin(n \\theta)\n &= (\\cos(\\theta) + i\\sin(\\theta))^n\n & \\text{ de Moivre's formula}\n\\end{align*}\n$$\n\nDe Moivre's formula for $n = 2$ gives\n\n$$\n\\begin{align*}\n \\text{cis}(\\theta)^2\n &= (\\text{c} + i\\text{s})^2\n \\\\\n &= \\text{c}^2 + 2i\\text{cs} - \\text{s}^2 + (0 = \\text{c}^2 + \\text{s}^2 - 1)\n \\\\\n &= 2\\text{c}^2 + 2i\\text{cs} - 1\n \\\\\n &= 2\\text{c}(\\text{c} + i\\text{s}) - 1\n \\\\\n &= 2\\cos(\\theta)\\text{cis}(\\theta) - 1\n\\end{align*}\n$$\n\nThis can easily be massaged into a recurrence relation.\n\n$$\n\\begin{align*}\n \\text{cis}(\\theta)^2\n &= 2\\cos(\\theta)\\text{cis}(\\theta) - 1\n \\\\\n \\text{cis}(\\theta)^{n+2}\n &= 2\\cos(\\theta)\\text{cis}(\\theta)^{n+1} - \\text{cis}(\\theta)^n\n \\\\\n \\text{cis}((n+2)\\theta)\n &= 2\\cos(\\theta)\\text{cis}((n+1)\\theta) - \\text{cis}(n\\theta)\n\\end{align*}\n$$\n\nRecurrence relations like this one are powerful.\nThrough some fairly straightforward summatory manipulations,\n the sequence can be interpreted as the coefficients in a Taylor series,\n giving a [generating function](https://en.wikipedia.org/wiki/Generating_function).\nCall this function *F*. Then,\n\n$$\n\\begin{align*}\n \\sum_{n=0}^\\infty \\text{cis}((n+2)\\theta)x^n\n &= 2\\cos(\\theta) \\sum_{n=0}^\\infty \\text{cis}((n+1)\\theta) x^n\n - \\sum_{n=0}^\\infty \\text{cis}(n\\theta) x^n\n \\\\\n {F(x; \\text{cis}(\\theta)) - 1 - x\\text{cis}(\\theta) \\over x^2}\n &= 2\\cos(\\theta) {F(x; \\text{cis}(\\theta)) - 1 \\over x}\n - F(x; \\text{cis}(\\theta))\n \\\\[10pt]\n F - 1 - x\\text{cis}(\\theta)\n &= 2\\cos(\\theta) x (F - 1)\n - x^2 F\n \\\\\n F - 2\\cos(\\theta) x F + x^2 F\n &= 1 + x(\\text{cis}(\\theta) - 2\\cos(\\theta))\n \\\\[10pt]\n F(x; \\text{cis}(\\theta))\n &= {1 + x(\\text{cis}(\\theta) - 2\\cos(\\theta)) \\over\n 1 - 2\\cos(\\theta)x + x^2}\n\\end{align*}\n$$\n\nSince $\\text{cis}$ is a complex function, we can separate *F* into real and imaginary parts.\nConveniently, these correspond to $\\cos(n\\theta)$ and $\\sin(n\\theta)$, respectively.\n\n$$\n\\begin{align*}\n \\Re[ F(x; \\text{cis}(\\theta)) ]\n &= {1 + x(\\cos(\\theta) - 2\\cos(\\theta)) \\over 1 - 2\\cos(\\theta)x + x^2}\n \\\\\n &= {1 - x\\cos(\\theta) \\over 1 - 2\\cos(\\theta)x + x^2} = A(x; \\cos(\\theta))\n \\\\\n \\Im[ F(x; \\text{cis}(\\theta)) ]\n &= {x \\sin(\\theta) \\over 1 - 2\\cos(\\theta)x + x^2} = B(x; \\cos(\\theta))\\sin(\\theta)\n\\end{align*}\n$$\n\nIn this form, it becomes obvious that the even though the generating function *F* was originally\n parametrized by $\\text{cis}(\\theta)$, *A* and *B* are parametrized only by $\\cos(\\theta)$.\nExtracting the coefficients of *x* yields an expression for $\\cos(n\\theta)$ and $\\sin(n\\theta)$\n in terms of $\\cos(\\theta)$ (and in the latter case, a common factor of $\\sin(\\theta)$).\n\nIf $\\cos(\\theta)$ in *A* and *B* is replaced with the parameter *z*, then all trigonometric functions\n are removed from the equation, and we are left with only polynomials[^1].\nThese polynomials are [*Chebyshev polynomials*](https://en.wikipedia.org/wiki/Chebyshev_polynomial)\n *of the first (A) and second (B) kind*.\nIn actuality, the polynomials of the second kind are typically offset by 1\n (the x in the numerator of *B* is omitted).\nHowever, retaining this term makes indexing consistent between *A* and *B*\n (and will make things clearer later).\n\n[^1]:\n This can actually be observed as early as the recurrence relation.\n\n $$\n \\begin{align*}\n \\text{cis}(\\theta)^{n+2}\n &= 2\\cos(\\theta)\\text{cis}(\\theta)^{n+1} - \\text{cis}(\\theta)^n\n \\\\\n a_{n+2}\n &= 2 z a_{n+1} - a_n\n \\\\\n \\Re[ a_0 ]\n &= 1,~~ \\Im[ a_0 ] = 0\n \\\\\n \\Re[ a_1 ]\n &= z,~~ \\Im[ a_1 ] = 1 \\cdot \\sin(\\theta)\n \\end{align*}\n $$\n\n\nWe were primarily interested in $\\sin(n\\theta)$, so let's tabulate\n the first few polynomials of the second kind (at $z / 2$).\n\n::: {#tbl-chebyshevu .cell .plain tbl-cap='[OEIS A049310](http://oeis.org/A049310)' execution_count=3}\n\n::: {.cell-output .cell-output-display .cell-output-markdown execution_count=2}\n*n* $[x^n]B(x; z / 2) = U_{n - 1}(z / 2)$ Factored\n----- --------------------------------------------- -------------------------------------------------------------------------------------------------\n0 $0$ $0$\n1 $1$ $1$\n2 $z$ $z$\n3 $z^{2} - 1$ $\\left(z - 1\\right) \\left(z + 1\\right)$\n4 $z^{3} - 2 z$ $z \\left(z^{2} - 2\\right)$\n5 $z^{4} - 3 z^{2} + 1$ $\\left(z^{2} - z - 1\\right) \\left(z^{2} + z - 1\\right)$\n6 $z^{5} - 4 z^{3} + 3 z$ $z \\left(z - 1\\right) \\left(z + 1\\right) \\left(z^{2} - 3\\right)$\n7 $z^{6} - 5 z^{4} + 6 z^{2} - 1$ $\\left(z^{3} - z^{2} - 2 z + 1\\right) \\left(z^{3} + z^{2} - 2 z - 1\\right)$\n8 $z^{7} - 6 z^{5} + 10 z^{3} - 4 z$ $z \\left(z^{2} - 2\\right) \\left(z^{4} - 4 z^{2} + 2\\right)$\n9 $z^{8} - 7 z^{6} + 15 z^{4} - 10 z^{2} + 1$ $\\left(z - 1\\right) \\left(z + 1\\right) \\left(z^{3} - 3 z - 1\\right) \\left(z^{3} - 3 z + 1\\right)$\n10 $z^{9} - 8 z^{7} + 21 z^{5} - 20 z^{3} + 5 z$ $z \\left(z^{2} - z - 1\\right) \\left(z^{2} + z - 1\\right) \\left(z^{4} - 5 z^{2} + 5\\right)$\n:::\n:::\n\n\nEvaluating the polynomials at $z / 2$ cancels the 2 in the denominator (and recurrence),\n making these expressions much simpler.\nThis evaluation has an interpretation in terms of the previous diagram --\n recall we used *half* the length of a side as a leg of the right triangle.\nFor a unit circumradius, the side length itself is then $2\\sin( {\\pi / n} )$.\nTo compensate for this doubling, the Chebyshev polynomial must be evaluated at half its normal argument.\n\n\n### Back on the Plane\n\nThe constructibility criterion is deeply connected to the Chebyshev polynomials.\nIn compass and straightedge constructions, one only has access to linear forms (lines)\n and quadratic forms (circles).\nThis means that a figure is constructible if and only if the root can be expressed using\n normal arithmetic (which is linear) and square roots (which are quadratic).\n\n\n#### Pentagons\n\nLet's look at a regular pentagon.\nThe relevant polynomial is\n\n$$\n[x^5]B ( x; z / 2 )\n = z^4 - 3z^2 + 1\n = (z^2 - z - 1) (z^2 + z - 1)\n$$\n\nAccording to how we derived this series, when $z = 2\\cos(\\theta)$, the roots of this polynomial\n correspond to when $\\sin(5\\theta) / \\sin(\\theta) = 0$.\nThis relation itself is true when $\\theta = \\pi / 5$, since $\\sin(5 \\pi / 5) = 0$.\n\nOne of the factors must therefore be the minimal polynomial of $2\\cos(\\pi / 5 )$.\nThe former happens to be correct correct, since $2\\cos( \\pi / 5 ) = \\varphi$, the golden ratio.\nNote that the second factor is the first evaluated at -*z*.\n\n\n#### Heptagons\n\nAn example of where constructability fails is for $2\\cos( \\pi / 7 )$.\n\n$$\n\\begin{align*}\n [x^7]B ( x; z / 2 )\n &= z^6 - 5 z^4 + 6 z^2 - 1\n \\\\\n &= ( z^3 - z^2 - 2 z + 1 ) ( z^3 + z^2 - 2 z - 1 )\n\\end{align*}\n$$\n\nWhichever is the minimal polynomial (the former), it is a cubic, and constructing\n a regular heptagon is equivalent to solving it for *z*.\nBut there are no (nondegenerate) cubics that one can produce via compass and straightedge,\n and all constructions necessarily fail.\n\n\n#### Decagons\n\nOne might think the same of $2\\cos(\\pi /10 )$\n\n$$\n\\begin{align*}\n [x^{10}]B ( x; z / 2 )\n &= z^9 - 8 z^7 + 21 z^5 - 20 z^3 + 5 z\n \\\\\n &= z ( z^2 - z - 1 )( z^2 + z - 1 )( z^4 - 5 z^2 + 5 )\n\\end{align*}\n$$\n\nThis expression also contains the polynomials for $2\\cos( \\pi / 5 )$.\nThis is because a regular decagon would contain two disjoint regular pentagons,\n produced by connecting every other vertex.\n\n\n\nThe polynomial which actually corresponds to $2\\cos( \\pi / 10 )$ is the quartic,\n which seems to suggest that it will require a fourth root and somehow decagons are not constructible.\nHowever, it can be solved by completing the square...\n\n$$\n\\begin{align*}\n z^4 - 5z^2 &= -5\n \\\\\n z^4 - 5z^2 + (5/2)^2 &= -5 + (5/2)^2\n \\\\\n ( z^2 - 5/2)^2 &= {25 - 20 \\over 4}\n \\\\\n ( z^2 - 5/2) &= {\\sqrt 5 \\over 2}\n \\\\\n z^2 &= {5 \\over 2} + {\\sqrt 5 \\over 2}\n \\\\\n z &= \\sqrt{ {5 + \\sqrt 5 \\over 2} }\n\\end{align*}\n$$\n\n...and we can breathe a sigh of relief.\n\n\nThe Triangle behind Regular Polygons\n------------------------------------\n\nPreferring *z* to be halved in $B(x; z/2)$ makes something else more evident.\nObserve these four rows of the Chebyshev polynomials\n\n::: {#273cfec0 .cell .plain execution_count=4}\n\n::: {.cell-output .cell-output-display .cell-output-markdown execution_count=3}\n*n* $[x^n]B(x; z / 2)$ *k* $[z^{k}][x^n]B(x; z / 2)$\n----- ------------------------------- ----- ---------------------------\n4 $z^{3} - 2 z$ 3 1\n5 $z^{4} - 3 z^{2} + 1$ 2 -3\n6 $z^{5} - 4 z^{3} + 3 z$ 1 3\n7 $z^{6} - 5 z^{4} + 6 z^{2} - 1$ 0 -1\n:::\n:::\n\n\nThe last column looks like an alternating row of Pascal's triangle\n (namely, ${n - \\lfloor {k / 2} \\rfloor - 1 \\choose k}(-1)^k$).\nThis resemblance can be made more apparent by listing the coefficients of the polynomials in a table.\n\n::: {#92d420a6 .cell .plain execution_count=5}\n\n::: {.cell-output .cell-output-display .cell-output-markdown execution_count=4}\n n $z^9$ $z^8$ $z^7$ $z^6$ $z^5$ $z^4$ $z^3$ $z^2$ $z$ $1$\n--- ------------------------------ ------------------------------------ ------------------------------------- ----------------------------------- ----------------------------------- ----------------------------------- ------------------------------------ ------------------------------------- ------------------------------------- -------------------------------------\n 1 1\n 2 1 0\n 3 1 0 -1\n 4 1 0 -2 0\n 5 1 0 -3 0 1\n 6 1 0 -4 0 3 0\n 7 1 0 -5 0 6 0 -1\n 8 1 0 -6 0 10 0 -4 0\n 9 1 0 -7 0 15 0 -10 0 1\n 10 1 0 -8 0 21 0 -20 0 5 0\n:::\n:::\n\n\nThough they alternate in sign, the rows of Pascal's triangle appear along diagonals,\n which I have marked in rainbow.\nMeanwhile, alternating versions of the naturals (1, 2, 3, 4...),\n the triangular numbers (1, 3, 6, 10...),\n the tetrahedral numbers (1, 4, 10, 20...), etc.\n are present along the columns, albeit spaced out by 0's.\n\nThe relationship of the Chebyshev polynomials to the triangle is easier to see if\n the coefficient extraction of $B(x; z / 2)$ is reversed.\nIn other words, we extract *z* before extracting *x*.\n\n$$\n\\begin{align*}\n B(x; z / 2) &= {x \\over 1 - zx + x^2}\n = {x \\over 1 + x^2 - zx}\n = {x \\over 1 + x^2}\n \\cdot {1 \\over {1 + x^2 \\over 1 + x^2} - z{x \\over 1 + x^2}}\n \\\\[10pt]\n [z^n]B(x; z / 2) &= {x \\over 1 + x^2} [z^n] {1 \\over 1 - z{x \\over 1 + x^2}}\n = {x \\over 1 + x^2} \\left( {x \\over 1 + x^2} \\right)^n\n \\\\\n &= \\left( {x \\over 1 + x^2} \\right)^{n+1}\n = x^{n+1} (1 + x^2)^{-n - 1}\n \\\\\n &= x^{n+1} \\sum_{k=0}^\\infty {-n - 1 \\choose k}(x^2)^k\n \\quad \\text{Binomial theorem}\n\\end{align*}\n$$\n\nWhile the use of the binomial theorem is more than enough to justify\n the appearance of Pascal's triangle (along with explaining the 0's),\n I'll simplify further to explicitly show the alternating signs.\n\n$$\n\\begin{align*}\n {(-n - 1)_k} &= (-n - 1)(-n - 2) \\cdots (-n - k)\n \\\\\n &= (-1)^k (n + k)(n + k - 1) \\cdots (n + 1)\n \\\\\n &= (-1)^k (n + k)_k\n \\\\\n \\implies {-n - 1 \\choose k}\n &= {n + k \\choose k}(-1)^k\n \\\\[10pt]\n [z^n]B(x; z / 2)\n &= x^{n+1} \\sum_{k=0}^\\infty {n + k \\choose k} (-1)^k x^{2k}\n\\end{align*}\n$$\n\nSquinting hard enough, the binomial coefficient is similar to the earlier\n which gave the third row of Pascal's triangle.\nIf k is fixed, then this expression actually generates the antidiagonal entries\n of the coefficient table, which are the columns with uniform sign.\nThe alternation instead occurs between antidiagonals (one is all positive,\n the next is 0's, the next is all negative, etc.).\nThe initial $x^{n+1}$ lags these sequences so that they reproduce the triangle.\n\n\n### Imagined Transmutation\n\nThe generating function of the Chebyshev polynomials resembles other two term recurrences.\nFor example, the Fibonacci numbers have generating function\n\n$$\n\\sum_{n = 0}^\\infty \\text{Fib}_n x^n = {1 \\over 1 - x - x^2}\n$$\n\nThis resemblance can be made explicit with a simple algebraic manipulation.\n\n$$\n\\begin{align*}\n B(ix; -iz / 2)\n &= {1 \\over 1 -\\ (-i z)(ix) + (ix)^2}\n = {1 \\over 1 -\\ (-i^2) z x + (i^2)(x^2)}\n \\\\\n &= {1 \\over 1 -\\ z x -\\ x^2}\n\\end{align*}\n$$\n\nIf $z = 1$, these two generating functions are equal.\nThe same can be said for $z = 2$ with the generating function of the Pell numbers,\n and so on for higher recurrences (corresponding to metallic means) for higher integral *z*.\n\nIn terms of the Chebyshev polynomials, this series manipulation removes the alternation in\n the coefficients of $U_n$, restoring Pascal's triangle to its nonalternating form.\nRelated to the previous point, it is possible to find the Fibonacci numbers (Pell numbers, etc.)\n in Pascal's triangle, which you can read more about\n [here](http://users.dimi.uniud.it/~giacomo.dellariccia/Glossary/Pascal/Koshy2011.pdf).\n\n\nManipulating the Series\n-----------------------\n\nLook back to the table of $U_{n - 1}(z / 2)$ (@tbl-chebyshevu).\nWhen I brought up $U_{10 - 1}(z / 2)$ and decagons, I pointed out their relationship to pentagons\n as an explanation for why $U_{5 -\\ 1}(z / 2)$ appears as a factor.\nConveniently, $U_{2 -\\ 1}(z / 2) = z$ is also a factor, and 2 is likewise a factor of 10.\n\nThis pattern is present throughout the table; $n = 6$ contains factors for\n $n = 2 \\text{ and } 3$ and the prime numbers have no smaller factors.\nIf this observation is legitimate, call the newest term $f_n(z)$\n and denote $p_n(z) = U_{n -\\ 1}( z / 2 )$.\n\n\n### Factorization Attempts\n\nThe relationship between $p_n$ and the intermediate $f_d$, where *d* is a divisor of *n*,\n can be made explicit by a [Möbius inversion](https://en.wikipedia.org/wiki/M%C3%B6bius_inversion_formula).\n\n$$\n\\begin{align*}\n p_n(z) &= \\prod_{d|n} f_n(z)\n \\\\\n \\log( p_n(z) )\n &= \\log \\left( \\prod_{d|n} f_d(z) \\right)\n = \\sum_{d|n} \\log( f_d(z) )\n \\\\\n \\log( f_n(z) ) &= \\sum_{d|n} { \\mu \\left({n \\over d} \\right)}\n \\log( p_d(z) )\n \\\\\n f_n(z) &= \\prod_{d|n} p_d(z)^{ \\mu (n / d) }\n \\\\[10pt]\n f_6(z) = g_6(z)\n &= p_6(z)^{\\mu(1)}\n p_3(z)^{\\mu(2)}\n p_2(z)^{\\mu(3)}\n \\\\\n &= {p_6(z) \\over p_3(z) p_2(z)}\n\\end{align*}\n$$\n\nUnfortunately, it's difficult to apply this technique across our whole series.\nMöbius inversion over series typically uses more advanced generating functions such as\n [Dirichlet series](https://en.wikipedia.org/wiki/Dirichlet_series#Formal_Dirichlet_series)\n or [Lambert series](https://en.wikipedia.org/wiki/Lambert_series).\nHowever, naively reaching for these fails for two reasons:\n\n- We built our series of polynomials on a recurrence relation, and these series\n are opaque to such manipulations.\n- To do a proper Möbius inversion, we need these kinds of series over the *logarithm*\n of each polynomial (*B* is a series over the polynomials themselves).\n\nIgnoring these (and if you're in the mood for awful-looking math) you may note\n the Lambert equivalence[^2]:\n\n[^2]:\n This equivalence applies to other polynomial series obeying the same factorization rule\n such as the [cyclotomic polynomials](https://en.wikipedia.org/wiki/Cyclotomic_polynomial).\n\n$$\n\\begin{align*}\n \\log( p_n(z) )\n &= \\sum_{d|n} \\log( f_d(z) )\n \\\\\n \\sum_{n = 1}^\\infty \\log( p_n ) x^n\n &= \\sum_{n = 1}^\\infty \\sum_{d|n} \\log( f_d ) x^n\n \\\\\n &= \\sum_{k = 1}^\\infty \\sum_{m = 1}^\\infty \\log( f_m ) x^{m k}\n \\\\\n &= \\sum_{m = 1}^\\infty \\log( f_m ) \\sum_{k = 1}^\\infty (x^m)^k\n \\\\\n &= \\sum_{m = 1}^\\infty \\log( f_m ) {x^m \\over 1 - x^m}\n\\end{align*}\n$$\n\nEither way, the number-theoretic properties of this sequence are difficult to ascertain\n without advanced techniques.\nIf research has been done, it is not easily available in the OEIS.\n\n\n### Total Degrees\n\nIt can be also be observed that the new term is symmetric ($f(z) = f(-z)$), and is therefore\n either irreducible or the product of polynomial and its reflection (potentially negated).\nFor example,\n\n$$\np_9(z) = \\left\\{\n\\begin{matrix}\n (z - 1)(z + 1)\n & \\cdot\n & (z^3 - 3z - 1)(z^3 - 3z + 1)\n \\\\\n \\shortparallel && \\shortparallel\n \\\\\n f_3(z)\n & \\cdot\n & f_9(z)\n \\\\\n \\shortparallel && \\shortparallel\n \\\\\n g_3(z) \\cdot g_3(-z)\n & \\cdot\n & g_9(z) \\cdot -g_9(-z)\n\\end{matrix}\n\\right.\n$$\n\nThese factor polynomials $g_n$ are the minimal polynomials of $2\\cos( \\pi / n )$.\n\nMultiplying these minimal polynomials by their reflection can be observed in the Chebyshev polynomials\n for $n = 3, 5, 7, 9$, strongly implying that it occurs on the odd terms.\nAssuming this is true, we have\n\n$$\nf_n(z) = \\begin{cases}\n g_n(z) & \\text{$n$ is even}\n \\\\\n g_n(z)g_n(-z)\n & \\text{$n$ is odd and ${\\deg(f_n) \\over 2}$ is even}\n \\\\\n -g_n(z)g_n(-z)\n & \\text{$n$ is odd and ${\\deg(f_n) \\over 2}$ is odd}\n\\end{cases}\n$$\n\nWithout resorting to any advanced techniques, the degrees of $f_n$ are\n not too difficult to work out.\nThe degree of $p_n(z)$ is $n -\\ 1$, which is also the degree of $f_n(z)$ if *n* is prime.\nIf *n* is composite, then the degree of $f_n(z)$ is $n -\\ 1$ minus the degrees\n of the divisors of $n -\\ 1$.\nThis leaves behind how many numbers less than *n* are coprime to *n*.\nTherefore $\\deg(f_n) = \\phi(n)$, the\n [Euler totient function](https://en.wikipedia.org/wiki/Euler_totient_function) of the index.\n\nThe totient function can be used to examine the parity of *n*.\nIf *n* is odd, it is coprime to 2 and all even numbers.\nThe introduced factor of 2 to 2*n* removes the evens from the totient, but this is compensated by\n the addition of the odd multiples of old numbers coprime to *n* and new primes.\nThis means that $\\phi(2n) = \\phi(n)$ for odd *n* (other than 1).\n\nThe same argument can be used for even *n*: there are as many odd numbers from 0 to *n* as there are\n from *n* to 2*n*, and there are an equal number of numbers coprime to 2*n* in either interval.\nTherefore, $\\phi(2n) = 2\\phi(n)$ for even *n*.\n\nThis collapses all cases of the conditional factorization of $f_n$ into one,\n and the degrees of $g_n$ are\n\n$$\n\\begin{align*}\n \\deg( g_n(z) )\n &= \\begin{cases}\n \\deg( f_n(z) )\n = \\phi(n)\n & n \\text{ is even} & \\implies \\phi(n) = \\phi(2n) / 2\n \\\\\n \\deg( f_n(z) ) / 2\n = \\phi(n) / 2\n & n \\text{ is odd} & \\implies \\phi(n) / 2 = \\phi(2n) / 2\n \\end{cases}\n \\\\\n &= \\varphi(2n) / 2\n\\end{align*}\n$$\n\nThough they were present in the earlier Chebyshev table,\n the $g_n$ themselves are presented again, along with the expression for their degree\n\n::: {#835a5369 .cell .plain execution_count=6}\n\n::: {.cell-output .cell-output-display .cell-output-markdown execution_count=5}\nn $\\varphi(2n)/2$ $g_n(z)$ Coefficient list, rising powers\n--- --------------------------------------- ------------------------- ---------------------------------------\n2 1 $z$ [0, 1]\n3 1 $z - 1$ [-1, 1]\n4 2 $z^{2} - 2$ [-2, 0, 1]\n5 2 $z^{2} - z - 1$ [-1, -1, 1]\n6 2 $z^{2} - 3$ [-3, 0, 1]\n7 3 $z^{3} - z^{2} - 2 z + 1$ [1, -2, -1, 1]\n8 4 $z^{4} - 4 z^{2} + 2$ [2, 0, -4, 0, 1]\n9 3 $z^{3} - 3 z - 1$ [-1, -3, 0, 1]\n9 3 $z^{3} - 3 z + 1$ [1, -3, 0, 1]\n10 4 $z^{4} - 5 z^{2} + 5$ [5, 0, -5, 0, 1]\n- [OEIS A055034](http://oeis.org/A055034) - [OEIS A187360](http://oeis.org/A187360)\n:::\n:::\n\n\nClosing\n-------\n\nMy initial jumping off point for writing this article was completely different.\nHowever, in the process of writing, its share of the article shrank and shrank until its\n introduction was only vaguely related to what preceded it.\nBut alas, the introduction via geometric constructions flows better coming off my\n [post about the Platonic solids](/posts/misc/platonic-volume).\nAlso, it reads better if I rely less on \"if you search for this sequence of numbers\"\n and more on how to interpret the definition.\n\nConsider reading [the follow-up](../2) to this post if you're interested in another way\n one can obtain the Chebyshev polynomials.\n\nDiagrams created with GeoGebra.\n\n\n\n",
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+ "markdown": "---\ntitle: \"A Game of Permutations, Part 1\"\ndescription: |\n Some basic, interesting connections between graph theory and permutation groups.\nformat:\n html:\n html-math-method: katex\njupyter: haskell\ndate: \"2022-01-18\"\ndate-modified: \"2025-07-04\"\ncategories:\n - graph theory\n - group theory\n - sorting algorithms\n---\n\n\n\nIn the time since [my last post](../chebyshev/2) discussing graphs,\n I have been spurred on to continue playing with them, with a slight focus on abstract algebra.\nThis post will primarily focus on some fundamental concepts before launching into some constructions which will make the journey down this road more manageable.\nHowever, I will still assume you are already familiar with what both a\n [group](https://mathworld.wolfram.com/Group.html) and a\n [graph](https://mathworld.wolfram.com/Graph.html) are.\n\n\nThe Symmetric Group\n-------------------\n\nSome of the most important finite groups are the symmetric groups of degree *n* ($S_n$).\nThey are the groups formed by permutations of lists containing *n* items.\nThere is a group element for each permutation, so the order of the group is the\n same as the number of permutations, *n!*.\n\n\n On lists versus sets
\n\n The typical definition of the symmetric group uses sets, which are fundamentally unordered.\n This is potentially confusing, since it can appear as though an object after applying\n a permutation is the same object as before.\n\n For example, let *p* be the permutation swapping \"1\" and \"3\".\n Then,\n\n $$\n \\begin{align*}\n p(\\{ 1, 2, 3 \\}) = \\{ p(1), p(2), p(3) \\} = \\{ 3, 2, 1 \\}\n \\\\\n p([ 1, 2, 3 ]) = [ p(1), p(2), p(3) ] = [ 3, 2, 1 ]\n \\end{align*}\n $$\n\n After applying *p*, the set is unchanged, since all of the elements are the same.\n On the other hand, the lists differ in the first element and cannot be equal.\n\n Sets are still useful as a container.\n For example, the elements of a group are unordered.\n To keep vocabulary simple, I will do my best to refer to objects in a group as\n \"group elements\" and the objects in a list as \"items\".\n \n\nThere are many ways to denote elements of the symmetric group.\nI will take the liberty of explaining some common notations, each of which are useful in different ways.\nMore information about them can be found [elsewhere](https://mathworld.wolfram.com/Permutation.html)\n [online](https://en.wikipedia.org/wiki/Permutation#Notations)\n as well as any adequate group theory text.\n\n\n### Naive List Notation\n\nArguably the simplest way to do things is to denote the permutation by the result\n of applying it to a canonical list.\nTake the element of $S_3$ which can be described as the action\n \"assign the first item to the second index, the second to the third, and the third to the first\".\nOur canonical list in this case is $[1, 2, 3]$, matching the degree of the group.\nThis results in $[3, 1, 2]$, since \"1\" is in now in position two, and similarly for the other items.\n\nUnfortunately, this choice is too result-oriented.\nThis choice makes it difficult to compose group elements in a meaningful way, since all\n of the information about the permutation is in the position of items in the list,\n rather than the items of the list themselves.\nFor example, under the same rule, $[c, b, a]$ is mapped to $[a, c, b]$.\n\n\n### True List Notations (Two- and One-line Notation)\n\nInstead, let's go back to the definition of the element.\nAll we have to do is list out every index on one line, then the destination\n of every index on the next.\nThis is known as *two-line notation*.\n\n$$\n\\begin{pmatrix}\n 1 & 2 & 3 \\\\\n p(1) & p(2) & p(3)\n\\end{pmatrix}\n= \\begin{pmatrix}\n 1 & 2 & 3 \\\\\n 2 & 3 & 1\n\\end{pmatrix}\n$$\n\nFor simplicity, the first row is kept as a list whose items match their indices.\n\nThis notation makes it easier to identify the inverse of a group element.\nAll we have to do is sort the columns of the permutation by the second row,\n then swap the two rows.\n\n$$\n\\begin{pmatrix}\n 1 & 2 & 3 \\\\\n 2 & 3 & 1\n\\end{pmatrix}^{-1}\n= \\begin{pmatrix}\n 3 & 1 & 2 \\\\\n 1 & 2 & 3\n\\end{pmatrix}^{-1}\n= \\begin{pmatrix}\n 1 & 2 & 3 \\\\\n 3 & 1 & 2\n\\end{pmatrix}\n$$\n\nNote that the second row is now the same as the result from the naive notation.\n\nSince the first row will be the same in all cases, we can omit it, which results in\n *one-line notation*.\nThe *n*th item in the list now describes the position in which *n* can be found after\n the permutation.\n\n$$\n\\begin{pmatrix}\n 1 & 2 & 3 \\\\\n 2 & 3 & 1\n\\end{pmatrix}\n\\equiv [\\![2, 3, 1]\\!]\n$$\n\nDouble brackets are used to distinguish this as a permutation and not an ordinary list.\n\nThese notations make it straightforward to encode symmetric group elements on a computer.\nAfter all, we only have to read the items of a list by the indices in another.\nHere's a compact definition in Haskell:\n\n::: {#2e6870e0 .cell execution_count=2}\n``` {.haskell .cell-code}\n-- convenient wrapper type for below\nnewtype Permutation = P { unP :: [Int] }\n\napply :: Permutation -> [a] -> [a]\napply = flip (\\xs ->\n map ( -- for each item of the permutation, map it to...\n (xs !!) -- the nth item of the first list\n . (+(-1)) -- (indexed starting with 1)\n ) . unP) -- (after undoing the type wrapping)\n-- written in a non-point free form\napply' (P xs) ys = map ( \\n -> ys !! (n-1) ) xs\n\nprint $ P [2,3,1] `apply` [1,2,3]\n```\n\n::: {.cell-output .cell-output-display}\n```\n[2,3,1]\n```\n:::\n:::\n\n\nNote that this means `P [2,3,1]` is actually equivalent to $[\\![2, 3, 1]\\!]^{-1}$,\n since we don't get $[3, 1, 2]$.\n\nWhile these notations are fairly explicit and easy to describe to a computer,\n it's easy to misinterpret an element as its inverse.\nThere is also some redundancy: $[\\![2, 1]\\!]$ and $[\\![2, 1, 3]\\!]$ both describe a group element\n which swaps the first two items of a list.\nOn one hand, the $S_n$ each belongs to is explicit, but on the other,\n every element of a smaller symmetric group also belongs to a larger one.\nThe verbosity of these notations also makes composing group elements difficult[^1].\n\n[^1]: Composition is relatively easy to describe in two-line notation.\n Recall that we reordered columns when finding an inverse.\n We can do so to match rows of two elements, then compose a new element\n by looking at the first and last rows.\n For example, with group inverses:\n $$\n \\begin{pmatrix}\n 1 & 2 & 3 \\\\\n 2 & 3 & 1\n \\end{pmatrix}\n \\begin{pmatrix}\n 1 & 2 & 3 \\\\\n 2 & 3 & 1\n \\end{pmatrix}^{-1}\n = \\begin{matrix}\n \\begin{pmatrix}\n 1 & 2 & 3 \\\\\n \\cancel{2} & \\cancel{3} & \\cancel{1}\n \\end{pmatrix}\n \\\\\n \\begin{pmatrix}\n \\cancel{2} & \\cancel{3} & \\cancel{1} \\\\\n 1 & 2 & 3\n \\end{pmatrix}\n \\end{matrix}\n = \\begin{pmatrix}\n 1 & 2 & 3 \\\\\n 1 & 2 & 3\n \\end{pmatrix}\n $$\n\n\n### Cycle Notation\n\n*Cycle notation* addresses all of these issues, but gets rid of the transparency with respect to lists.\nLet's try phrasing the element we've been describing differently.\n\n> assign the first item to the second index, the second to the third, and the third to the first\n\nWe start at index 1 and follow it to index 2, and from there follow it to index 3.\nContinuing from index 3, we return to index 1, and from then we'd loop forever.\nThis describes a *cycle*, denoted as $(1 ~ 2 ~ 3)$.\n\nCycle notation is much more delicate than list notation, since the notation is nonunique:\n\n- Naturally, the elements of a cycle may be cycled to produce an equivalent one.\n - $(1 ~ 2 ~ 3) = (3 ~ 1 ~ 2) = (2 ~ 3 ~ 1)$\n- Cycles which have no common elements (i.e., are disjoint) commute,\n since they act on separate parts of the list.\n - $(1 ~ 2 ~ 3)(4 ~ 5) = (4 ~ 5)(1 ~ 2 ~ 3)$\n\n\n#### Cycle Algebra\n\nThe true benefit of cycles is that they are easy to manipulate algebraically.\nFor some reason, [Wikipedia](https://en.wikipedia.org/wiki/Permutation#Cycle_notation)\n does not elaborate on the composition rules for cycles,\n and the text which I read as an introduction to group theory simply listed it as an exercise.\nWhile playing around with them and deriving these rules oneself *is* a good idea,\n I will list the most important here:\n\n- Cycles can be inverted by reversing their order.\n - $(1 ~ 2 ~ 3)^{-1} = (3 ~ 2 ~ 1) = (1 ~ 3 ~ 2)$\n- Cycles may be composed if the last element in the first is the first index on the right.\n Inversely, cycles may also be decomposed by partitioning on an index and duplicating.\n - $(1 ~ 2 ~ 3) = (1 ~ 2)(2 ~ 3)$\n- If an index in a cycle is repeated twice, it may be omitted from the cycle.\n - $(1 ~ 2 ~ 3)(1 ~ 3) = (1 ~ 2 ~ 3)(3 ~ 1) = (1 ~ 2 ~ 3 ~ 1) = (1 ~ 1 ~ 2 ~ 3) = (2 ~ 3)$\n\nGoing back to $(1 ~ 2 ~ 3)$, if we apply this permutation to the list $[1, 2, 3]$:\n\n$$\n(1 ~ 2 ~ 3) \\left( \\vphantom{0^{0^0}} [1, 2, 3] \\right)\n = (1 ~ 2)(2 ~ 3) \\left( \\vphantom{0^{0^0}} [1, 2, 3] \\right)\n = (1 ~ 2) \\left( \\vphantom{0^{0^0}} [1, 3, 2] \\right)\n = [3, 1, 2]\n$$\n\nWhich is exactly what we expected with our naive notation.\n\n\nGenerators, Permutation Groups, and Sorting\n-------------------------------------------\n\nIf we have a group *G*, then we can select a set of elements\n $\\langle g_1, g_2, g_3, ... \\rangle$ as *generators*.\nIf we form all possible products -- not only the pairwise ones $g_1 g_2$,\n but also $g_1 g_2 g_3$ and all powers of any $g_n$ -- then the products form a subgroup of *G*.\nNaturally, such a set is called a *generating set*.\n\nSymmetric groups are of primary interest because of their subgroups, also known as permutation groups.\n[Cayley's theorem](https://en.wikipedia.org/wiki/Cayley%27s_theorem),\n a fundamental result of group theory, states that all finite groups\n are isomorphic to one of these subgroups.\nThis means that we can encode effectively any group using elements of the symmetric group.\n\nFor example, consider the generating set $\\langle (1 ~ 2) \\rangle$, which contains a single element\n that swaps the first two items of a list.\nIts square is the identity, meaning that its inverse is itself.\nThe identity and $(1 ~ 2)$ are the only two elements of the generated group,\n which is isomorphic to $C_2$, the cyclic group of order 2.\nSimilarly, the 3-cycle in the generating set $\\langle (1 ~ 2 ~ 3) \\rangle$ generates $C_3$,\n the cyclic group of order 3.\n\nHowever, the generating set $\\langle (1 ~ 2), (1 ~ 2 ~ 3)\\rangle$, which contains both of these permutations, generates the entirety of $S_3$.\nIn fact, [\n every symmetric group can be generated by two elements\n ](https://groupprops.subwiki.org/wiki/Symmetric_group_on_a_finite_set_is_2-generated):\n a permutation which cycles all elements once, and the permutation which swaps the first two elements.\n\n$$\nS_n = \\langle (1 ~ 2), (1 ~ 2 ~ 3 ~ 4 ~ ... ~ n) \\rangle\n$$\n\n\n### Sorting and 2-Cycles\n\nThe proof linked to above, that every symmetric group can be generated by two elements,\n uses the (somewhat obvious) result that one can produce any permutation of a list by picking two items,\n swapping them, and repeating until the list is in the desired order.\n\nThis is reminiscent of how sorting algorithms are able to sort a list by only comparing and swapping items.\nAs mentioned earlier when finding inverses using in two-line notation, sorting is almost the inverse of permuting.\nHowever, not all 2-cycles are necessary to generate the whole symmetric group.\n\n\n### Bubble Sort\n\nConsider [bubble sort](https://en.wikipedia.org/wiki/Bubble_sort).\nIn this algorithm, we swap two items when the latter is less than the former,\n looping over the list until it is sorted.\nUntil the list is sorted, the algorithm finds all such adjacent inversions.\nIn the worst case, it will swap every pair of adjacent items, some possibly multiple times.\nThis corresponds to the generating set\n $\\langle (1 ~ 2), (2 ~ 3), (3 ~ 4), (4 ~ 5), …, (n-1 ~\\ ~ n) \\rangle$.\n\n{.narrow}\n\n\n### Selection Sort\n\nAnother method, [selection sort](https://en.wikipedia.org/wiki/Selection_sort),\n searches through the list for the smallest item and swaps it to the beginning.\nIf this is the final item of the list, this results in the permutation $(1 ~ n)$.\nSupposing that process is continued with the second item and we also want it\n to swap with the final item, then the next swap corresponds to $(2 ~ n)$.\nContinuing until the last item, this gives the generating set\n $\\langle (1 ~ n), (2 ~ n), (3 ~ n), (4 ~ n), …, (n-1 ~\\ ~ n) \\rangle$.\n\n{.narrow}\n\nThis behavior for selection sort is uncommon, and this animation omits the selection of a swap candidate.\nThe animation below shows a more destructive selection sort, in which the\n candidate least item is placed at the end of the list (position 5).\nOnce the algorithm hits the end of the list, the candidate is swapped to the least unsorted position,\n and the algorithm continues on the rest of the list.\n\n{.narrow}\n\n\nSwap Diagrams\n-------------\n\nGiven a set of 2-cycles, it would be nice to know at a glance if the entire group is generated.\nIn cycle notation, a 2-cycle is an unordered pair of natural numbers which swap items of an *n*-list.\nSimilarly, the edges of an undirected graph on *n* vertices (labelled from 1 to *n*)\n may be interpreted as an unordered pair of the vertices it connects.\n\nIf we treat the two objects as the same, then we can convert between graphs and sets of 2-cycles.\nGoing from the latter to the former, we start on an empty graph on n vertices\n (labelled from 1 to *n*).\nThen, we connect two vertices with an edge when the set includes the permutation swapping\n the indices labelled by the vertices.\n\nReturning to the generating sets we identified with sorting algorithms,\n we identify with each a graph family.\n\n- Bubble sort: $\\langle (1 ~ 2), (2 ~ 3), (3 ~ 4), (4 ~ 5), …, (n-1 ~~ n) \\rangle$\n - The path graphs ($P_n$), which are precisely as they sound:\n an unbranching path formed by *n* vertices.\n- \"Selection\" sort: $\\langle (1 ~ n), (2 ~ n), (3 ~ n), (4 ~ n), …, (n-1 ~~ n) \\rangle$\n - The star graphs ($\\bigstar_n$, as $S_n$ means the symmetric group),\n one vertex connected to all others.\n- Every 2-cycle in $S_n$\n - The complete graphs ($K_n$), which connect every vertex to every other vertex.\n\n\n\nThis interpretation of these objects doesn't have proper name, but I think the name \"swap diagram\" fits.\nThey allow us to answer at least one question about the generating set from a graph theory perspective.\n\n\n### Connected Graphs\n\nA graph is connected if a path exists between all pairs of vertices.\nThe simplest possible path is simply a single edge, which we already know to be an available 2-cycle.\n\nThe next simplest case is a path consisting of two edges.\nSome cycle algebra shows that we can produce a third cycle which corresponds\n to an edge connecting the two distant vertices.\n\n:::: {layout-ncol=\"2\"}\n\n\n::: {}\n$$\n\\begin{align*}\n &(m ~ n) (n ~ o) (m ~ n) \\\\\n &= (m ~ n ~ o) (m ~ n) \\\\\n &= (n ~ o ~ m) (m ~ n) \\\\\n &= (n ~ o ~ m ~ n) \\\\\n &= (o ~ m) = (m ~ o)\n\\end{align*}\n$$\nNote that this is just the conjugation of $(n ~ o)$ by $(m ~ n)$\n:::\n::::\n\nIn other words, if we have have two adjacent edges, the new edge corresponds to\n a product of elements from the generating set.\nGraph theory has a name for this operation: when we produce *all* new edges by linking vertices\n that were separated by a distance of 2, the result is called the *square of that graph*.\nIn fact, higher [graph powers](https://en.wikipedia.org/wiki/Graph_power) will reflect connections\n induced by more conjugations of adjacent edges.\n\n\n\nIf our graph is connected, then repeating this operation will tend toward a complete graph.\nComplete graphs contain every possible edge, and so correspond to all possible 2-cycles,\n which trivially generate the symmetric group.\nConversely, if a graph has *n* vertices, then for it to be connected, it must have at least $n - 1$ edges.\nThus, a generating set of 2-cycles must have at least $n - 1$ items to generate the symmetric group.\n\nPicking a different vertex labelling will correspond to a different generating set.\nFor example, in the image of $P_4$ above, if the edge connecting vertices 1 and 2\n is replaced with an edge connecting 1 and 4, then the resulting graph\n is still $P_4$, even though it describes a different generating set.\nWe can ignore these extra cases entirely -- either way, the graph power argument shows\n that a connected graph corresponds to a set generating the whole symmetric group.\n\n\n### Disconnected Graphs\n\nA disconnected graph is the\n [disjoint union](https://en.wikipedia.org/wiki/Disjoint_union_of_graphs) of connected graphs.\nUnder graph powers, we know that each connected graph tends toward a complete graph,\n meaning a disconnected graph as a whole tends toward a disjoint union of complete graphs,\n or [cluster graph](https://en.wikipedia.org/wiki/Cluster_graph).\nBut what groups do cluster graphs correspond to?\n\nThe simplest case to consider is what happens when the graph is $P_2 \\oplus P_2$.\nIf there is an edge connecting vertices 1 and 2 and an edge connecting vertices 3 and 4,\n it corresponds to the generating set $\\langle (1 ~ 2), (3 ~ 4) \\rangle$.\nThis is a pair of disjoint cycles, and the group they generate is\n\n$$\n\\{e, (1 ~ 2), (3 ~ 4), (1 ~ 2)(3 ~ 4) \\}\n \\cong S_2 \\times S_2\n \\cong C_2 \\times C_2\n$$\n\nOne way to look at this is by considering paths on each component:\n we can either cross an edge on the first component (corresponding to $(1 ~ 2)$),\n the second component (corresponding to $(3 ~ 4)$),\n or both at the same time.\nThis independence means that one group's structure is duplicated over the other's,\n or more succinctly, gives the direct product.\nIn general, if we denote *γ* as the map which \"runs\" the swap diagram and produces the group, then\n\n$$\n\\gamma( A \\oplus B ) = S_{|A|} \\times S_{|B|},\n ~ A, B \\text{ connected}\n$$\n\nwhere $|A|$ is the number of vertices in *A*.\n\n*γ* has the interesting property of mapping a sum-like object onto a product-like object.\nIf we express a disconnected graph *U* as the disjoint union of its connected components $V_i$, then\n\n$$\n\\begin{gather*}\n U = \\bigsqcup_i V_i\n \\\\\n \\gamma( U ) = \\gamma \\left( \\bigsqcup_i V_i \\right) = \\prod_i S_{|V_i|}\n\\end{gather*}\n$$\n\nThis describes *γ* for every simple graph.\nIt also shows that we're rather limited in the kinds of groups which can be expressed by a swap diagram.\n\n\nClosing\n-------\n\nThis concludes the dry introduction to some investigations of mine into symmetric groups.\nWhile I could have omitted the sections about permutation notation and generators,\n I wanted to be thorough and tie it to concepts which were useful to my understanding.\nThe notion of a graph encoding a generating set in particular will be fairly important going forward.\n\nOriginally, this post was half of a single, sprawling, meandering article.\nI hope I've improved the organization by keeping the digression about sorting algorithms\n to this initial article.\nThe [next post](../2) will cover some interesting structures which fill Euclidean space\n and incredibly large graphs.\n\nSorting and graph diagrams made with GeoGebra.\n\n",
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+ "markdown": "---\ntitle: \"A Game of Permutations, Part 2\"\ndescription: |\n Notes on an operation which makes some very large graphs.\nformat:\n html:\n html-math-method: katex\njupyter: python3\ndate: \"2022-01-18\"\ndate-modified: \"2025-07-06\"\ncategories:\n - graph theory\n - group theory\n---\n\n\n\n\n\nThis post assumes you have read (or at least skimmed over parts of) the [first post](../1),\n which talks about graphs and the symmetric group.\nThis post will contain some more \"empirical\" results, since I'm not an expert on graph theory.\nHowever, one hardly needs to be an expert to learn or to make computations, observations, and predictions.\n\nWe left off talking about producing a group from a graph, so we begin now by considering how to do the reverse.\n\n\nCayley Graphs\n-------------\n\nFor a given generating set, we can assign every element in the group it generates to a vertex in a graph.\nStarting with each of the generators, we draw edges from one vertex to another when\n the product of the initial vertex and a generator (in that order) is the product vertex.\nThis process continues until there are no more arrows to draw.\nThe resulting figure is known as a [Cayley graph](https://mathworld.wolfram.com/CayleyGraph.html)[^1].\n\n[^1]: The construction implies a labelling of edges by its generating set and a labelling\n of vertices by the generated elements.\n It is also common to describe an unlabelled graph as \"Cayley\" if it could\n be generated by this procedure.\n\nCayley graphs depend on the generating set used, so they can take a wide variety of shapes.\nHere are a few examples of Cayley graphs made from elements of $S_4$:\n\n\n\nOwing to the way in which they are defined, Cayley graphs have a few useful properties as graphs.\nAt every vertex, we have as many outward edges as we do generators in the generating set,\n so the outward (and in fact, inward) degree of each vertex is the same.\nIn other words, it is a regular graph.\nMore than that, it is [vertex-transitive](https://mathworld.wolfram.com/Vertex-TransitiveGraph.html),\n since labelling a single vertex's outward edges will label that of the entire graph.\n\nIn general, the Cayley graph is a directed graph.\nHowever, if for every member of the generating set, we also include its inverse,\n every directed edge will be matched by an edge in the opposite direction,\n and the Cayley graph may be considered undirected.\n\n\nGraphs to Graphs\n----------------\n\nAll 2-cycles are their own inverse, so generating sets which include only them\n produce undirected Cayley graphs.\nSince this kind of generating set can itself be thought of as a graph,\n we may consider an operation on graphs that maps a swap diagram to its Cayley graph.\n\n\n\nIt seems to be the case that $\\exp( A \\oplus B ) = \\exp( A ) \\times \\exp( B )$,\n where $\\oplus$ signifies the disjoint uinion and $\\times$ signifies the\n [Cartesian (box) product of graphs](https://en.wikipedia.org/wiki/Cartesian_product_of_graphs)[^2].\nUnlike *γ* from the previous post, both the input and output of this operation are graphs.\nBecause of this and the sum/product relationship, I've taken to calling this operation the\n \"graph exponential\"[^3].\n\n[^2]: Graphs have many product structures, such as the tensor product and strong product.\n The Cartesian product is (categorically) more natural when paired with disjoint unions.\n\n[^3]: Originally, I called this operation the \"graph factorial\", since it involves permutations\n and the number of vertices in the resulting graph grows factorially.\n\nThis operation is my own invention, so I am unsure whether or not\n it constitutes anything useful.\nIn fact, the possible graphs grow so rapidly that computing anything about the exponential\n of order 8 graphs starts to overwhelm a single computer.\nIt is, however, interesting, as I will hopefully be able to convince.\n\nA random graph will not generally correspond to an interesting generating set,\n and therefore, will also generally have an uninteresting exponential graph.\nHence, I will continue with the examples used previously: paths, stars, and complete graphs.\nThey are among the simplest graphs one can consider, and as we will see shortly,\n have exponentials which appear to have natural correspondences to other graph families.\n\n\nSome Small Exponential Graphs\n-----------------------------\n\nBecause of [the difficulty in determining graph isomorphism](\n https://en.wikipedia.org/wiki/Graph_isomorphism_problem\n ), it is challenging for a computer to find a graph in an encyclopedia.\nComputers think of graphs as a list of vertices and their outward edges,\n but this implementation faces inherent labelling issues.\nThese persist even if the graph is described as a list of (un)ordered pairs,\n an adjacency matrix, or an incidence matrix,\n the latter two of which have very large memory footprints[^4].\n\n[^4]: I was able to locate a project named the\n [Encyclopedia of Finite Graphs](https://github.com/thoppe/Encyclopedia-of-Finite-Graphs),\n but it is only able to build a database simple connected graphs which\n can be queried by invariants (and is outdated since it uses Python 2).\n\nHowever, as visual objects, humans can compare graphs fairly easily\n -- the name means \"drawing\" after all.\nExponentials of order 3 and order 4 graphs are neither so small as to be uninteresting\n nor so big as to be unparsable by humans.\n\n\n### Order 3\n\n{.narrow}\n\nAt this stage, we only really have two graphs to consider, since $P_3 = \\bigstar_3$.\nImmediately, one can see that $\\exp( P_3 ) = \\exp( \\bigstar_3 ) = C_6$,\n the 6-cycle graph (or hexagonal graph).\nIt is also apparent that $\\exp( K_3 )$ is the utility graph, $K_{3,3}$.\n\n\n\nHere, we can again demonstrate the sum rule of the graph exponential with $\\exp( P_3 \\oplus P_2 )$.\nSimplifying, since we know $\\exp( P_3 ) = C_6$, the result is $C_6 \\times P_2 = \\text{Prism}_6$,\n the hexagonal prism graph.\n\n\n### Order 4 (and beyond)\n\n::: {layout=\"[[1,1],[1]]\"}\n\n\n\n\n\n:::\n\nWith some effort, $\\exp( P_4 )$ can be imagined as a projection of a 3D object,\n the [truncated octahedron](https://en.wikipedia.org/wiki/Truncated_octahedron).\nBecause of its correspondence to a 3D solid, this graph is planar.\nBoth the hexagon and this solid belong to a class of polytopes called\n [*permutohedra*](https://en.wikipedia.org/wiki/Permutohedron), which are figures\n that are also formed by permutations of the coordinate (1, 2, 3, ..., *n*) in Euclidean space[^5].\nTechnically, there is a distinction between the Cayley graphs and permutohedra\n since their labellings differ.\nBoth have edges generated by swaps, but in the latter case, the connected vertices are expected to be\n separated by a certain distance.\nMore information about the distinction can be found at this article on [Wikimedia](\n https://commons.wikimedia.org/wiki/Category:Permutohedron_of_order_4_%28raytraced%29#Permutohedron_vs._Cayley_graph\n )[^6].\n\n[^5]: In fact, these figures are able to completely tessellate the $n-1$ dimensional subspace of\n $\\mathbb{R}^n$ where the coordinates sum to the $n-1$th triangular number.\n Note also that the previous graph in the sequence of $\\exp(P_n)$, the hexagonal graph,\n is visible in the truncated octahedron.\n This corresponds to the projection $(x,y,z,w) \\mapsto (x,y,z)$ over\n the coordinates of the permutohedra.\n\n[^6]: Actually, if one considers a *right* Cayley graph, where each generator is right-multiplied\n to the permutation at a node rather than left-multiplied, then a true correspondence is obtained,\n at least for order 4.\n\nMeanwhile, $\\exp( \\bigstar_4 )$ is more difficult to identify, at least without rearranging its vertices.\nIt turns out to be isomorphic to the [Nauru graph](https://mathworld.wolfram.com/NauruGraph.html),\n a graph with many strange properties.\nNotably, whereas the graph isomorphic to the permutohedron is obviously a spherical polyhedron,\n the Nauru graph can be topologically embedded on a torus.\nThe Nauru graph also belongs to the family of\n [permutation star graphs](https://mathworld.wolfram.com/PermutationStarGraph.html)\n $PS_n$ (*n* = 4), which also includes the hexagonal graph (*n* = 3).\nThe MathWorld article confirms some kind of correspondence, stating graphs of this form\n are generated by pairwise swaps.\n\nMy attempts at finding a graph isomorphic to $\\exp( K_4 )$ have thus far ended in failure.\nIt is certainly *not* isomorphic to $K_{4,4}$, since this graph has 8 vertices,\n as opposed to 24 in $\\exp( K_4 )$.\n\n\nGraph Invariants\n----------------\n\nWhile I have managed to identify the families to which some of these graphs belong,\n I am rather fond of computing (and conjecturing) sequences from objects.\nNot only is it much easier to consult something like [the OEIS](https://oeis.org/) for these quantities,\n but after finding a matching sequence, there are ample articles to consult for more information.\nBy linking to their respective entries, I hope you'll consider reading more there.\n\nEven though I have obtained these values empirically, I am certain that the sequences for\n $\\exp( P_n )$ and $\\exp( \\bigstar_n )$ match the corresponding OEIS entries.\nI also have great confidence in the sequences I found for $\\exp( K_n )$.\n\n\n### Edge Counts\n\nDespite knowing how many vertices there are (*n*!, the order of the symmetric group),\n we don't necessarily know how many edges there are.\n\n::: {#a412d7a0 .cell .plain execution_count=3}\n\n::: {.cell-output .cell-output-display .cell-output-markdown execution_count=7}\n*n* $\\#E(\\exp( P_n ))$ $\\#E(\\exp( \\bigstar_n ))$ $\\#E(\\exp( K_n ))$\n-------- ------------------------------------------------------------------------- --------------------------- --------------------------------------\n3 6 6 9\n4 36 36 72\n5 240 240 600\n6 1800 1800 5400\n7 15120 15120 52920\nSequence Second column of Lah numbers
[OEIS A001286](http://oeis.org/A001286) Same as previous [OEIS 001809](http://oeis.org/A001809)\nRule $L(n,2) = n!{(n-1)(n-2) \\over 4}$ $n!{n(n-1) \\over 4}$\n:::\n:::\n\n\n### Radius and Distance Classes\n\nThe radius of a graph is the smallest possible distance which separates two maximally-separated vertices.\nDue to vertex transitivity, the greatest distance between two vertices is the same for every vertex.\n\n::: {#8a1cc334 .cell .plain execution_count=4}\n\n::: {.cell-output .cell-output-display .cell-output-markdown execution_count=8}\n*n* $r(\\exp( P_n ))$ $r(\\exp( \\bigstar_n ))$ $r(\\exp( K_n ))$\n-------- --------------------------------------------------------------- -------------------------------------------------------------------------------- ------------------\n3 4 4 3\n4 7 5 4\n5 11 7 5\n6 16 8 6\n7 22 10 7\nSequence Triangular numbers
[OEIS A000217](http://oeis.org/A000217) Integers not congruent to 2 (mod 3)
[OEIS A032766](http://oeis.org/A032766) *n* - 1\nRule $\\Delta_{n-1} = {n(n-1) \\over 2}$ $\\lfloor {n-1 \\over 2} \\rfloor + n -\\ 1$\n:::\n:::\n\n\nThese quantities are somewhat reductive.\nIf a vertex is distinguished, the remaining vertices may be partitioned into classes by their\n distance from it.\nIncluding the vertex itself (which is distance 0 away), there will be $r + 1$ such classes,\n where *r* is the radius.\nThese classes are the same for every vertex due to transitivity.\n\nIn the case of these graphs, they are a partition of *n*!.\n\n::: {#b3698cfa .cell .plain execution_count=5}\n\n::: {.cell-output .cell-output-display .cell-output-markdown execution_count=9}\n*n* $\\text{dists}(\\exp( P_n ))$ $\\text{dists}(\\exp( \\bigstar_n ))$ $\\text{dists}(\\exp( K_n ))$\n-------- -------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------\n3 [1, 2, 2, 1] [1, 2, 2, 1] [1, 3, 2]\n4 [1, 3, 5, 6, 5, 3, 1] [1, 3, 6, 9, 5] [1, 6, 11, 6]\n5 [1, 4, 9, 15, 20, 22, 20, 15, 9, 4, 1] [1, 4, 12, 30, 44, 26, 3] [1, 10, 35, 50, 24]\n6 [1, 5, 14, 29, 49, 71, 90, 101
101, 90, 71, 49, 29, 14, 5, 1] [1, 5, 20, 70, 170, 250, 169, 35] [1, 15, 85, 225, 274, 120]\n7 [1, 6, 20, 49, 98, 169, 259, 359, 455, 531, 573
573, 531, 455, 359, 259, 169, 98, 49, 20, 6, 1] [1, 6, 30, 135, 460, 1110, 1689, 1254, 340, 15] [1, 21, 175, 735, 1624, 1764, 720]\nSequence Mahonian numbers
[OEIS A008302](http://oeis.org/A008302) Whitney numbers of the second kind (star poset)
[OEIS A007799](http://oeis.org/A007799) Stirling numbers of the first kind
[OEIS A132393](http://oeis.org/A132393)\n:::\n:::\n\n\nI am certain that the appearance of the Stirling numbers here is legitimate,\n since these numbers count the number of permutations of *n* objects with *k* disjoint cycles.\nObviously, the identity element is distance 1 from all 2-cycles since they are all in the generating set;\n likewise, all 3-cycles are distance 2 from the identity (but distance 1 from the 2-cycles),\n and so on until the entire graph has been mapped.\nThe shapes induced by these classes were used to create the diagrams\n of $\\exp( K_3 )$ and $\\exp( K_4 )$ above.\n\n\n#### Spectrum\n\nThe eigenvalues of the adjacency matrix of a graph can be interesting\n and sometimes help in identifying a graph.\nUnfortunately, eigenvalues are not necessarily integers, and therefore not easily\n found in the OEIS (though they are always real for graphs).\n\n::: {#1ce3c98e .cell .plain execution_count=6}\n\n::: {.cell-output .cell-output-display .cell-output-markdown execution_count=10}\n *n* $\\text{Spec}(\\exp( P_n ))$ $\\text{Spec}(\\exp( \\bigstar_n )$ $\\text{Spec}(\\exp( K_n ))$\n----- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------\n 3 $$\\begin{gather*}(\\pm1)^{2\\phantom{0}} \\\\ (\\pm2)^{1\\phantom{0}}\\end{gather*}$$ $$\\begin{gather*}(\\pm1)^{2\\phantom{0}} \\\\ (\\pm2)^{1\\phantom{0}}\\end{gather*}$$ $$\\begin{gather*}(0)^{4\\phantom{0}} \\\\ (\\pm3)^{1\\phantom{0}}\\end{gather*}$$\n 4 $$\\begin{gather*}(\\pm1)^{3} (\\pm3)^{1} \\\\ \\left(x^{2} - 3\\right)^{2} \\\\ \\left(x^{2} + 2 x - 1\\right)^{3} \\\\ \\left(x^{2} - 2 x - 1\\right)^{3} \\\\ \\left(x^{2} + 2 x - 1\\right)^{3} \\\\ \\left(x^{2} - 2 x - 1\\right)^{3}\\end{gather*}$$ $$\\begin{gather*}(0)^{4\\phantom{0}} \\\\ (\\pm1)^{3\\phantom{0}} \\\\ (\\pm2)^{6\\phantom{0}} \\\\ (\\pm3)^{1\\phantom{0}}\\end{gather*}$$ $$\\begin{gather*}(0)^{4\\phantom{0}} \\\\ (\\pm2)^{9\\phantom{0}} \\\\ (\\pm6)^{1\\phantom{0}}\\end{gather*}$$\n 5 $$\\begin{gather*}(0)^{12} (\\pm1)^{6} (\\pm4)^{1} \\\\ \\left(x^{2} - 5\\right)^{6} \\\\ \\left(x^{2} - 5 x + 5\\right)^{4} \\\\ \\left(x^{2} + 5 x + 5\\right)^{4} \\\\ \\left(x^{2} - 3 x + 1\\right)^{4} \\\\ \\left(x^{2} + 3 x + 1\\right)^{4} \\\\ \\left(x^{2} + 2 x - 1\\right)^{5} \\\\ \\left(x^{2} - 2 x - 1\\right)^{5} \\\\ \\left(x^{3} + 2 x^{2} - 5 x - 4\\right)^{5} \\\\ \\left(x^{3} - 2 x^{2} - 5 x + 4\\right)^{5}\\end{gather*}$$ $$\\begin{gather*}(0)^{30\\phantom{00}} \\\\ (\\pm1)^{4\\phantom{00}} \\\\ (\\pm2)^{28\\phantom{00}} \\\\ (\\pm3)^{12\\phantom{00}} \\\\ (\\pm4)^{1\\phantom{00}}\\end{gather*}$$ $$\\begin{gather*}(0)^{36\\phantom{00}} \\\\ (\\pm2)^{25\\phantom{00}} \\\\ (\\pm5)^{16\\phantom{00}} \\\\ (\\pm10)^{1\\phantom{00}}\\end{gather*}$$\n 6 $$\\begin{gather*}(0)^{20} (\\pm1)^{25} (\\pm2)^{15}\\\\(\\pm3)^{5} (\\pm4)^{5} (\\pm5)^{1} \\\\ \\left(x^{2} - 3\\right)^{20}\\end{gather*}$$Not shown: 558 other roots $$\\begin{gather*}(0)^{168\\phantom{000}} \\\\ (\\pm1)^{30\\phantom{000}} \\\\ (\\pm2)^{120\\phantom{000}} \\\\ (\\pm3)^{105\\phantom{000}} \\\\ (\\pm4)^{20\\phantom{000}} \\\\ (\\pm5)^{1\\phantom{000}}\\end{gather*}$$ $$\\begin{gather*}(0)^{256\\phantom{000}} \\\\ (\\pm3)^{125\\phantom{000}} \\\\ (\\pm5)^{81\\phantom{000}} \\\\ (\\pm9)^{25\\phantom{000}} \\\\ (\\pm15)^{1\\phantom{000}}\\end{gather*}$$\n 7 $$\\begin{gather*}(0)^{35\\phantom{00}} \\\\ (\\pm1)^{20\\phantom{00}} \\\\ (\\pm2)^{45\\phantom{00}} \\\\ (\\pm6)^{1\\phantom{00}}\\end{gather*}$$Not shown: 4873 other roots $$\\begin{gather*}(0)^{840\\phantom{000}} \\\\ (\\pm1)^{468\\phantom{000}} \\\\ (\\pm2)^{495\\phantom{000}} \\\\ (\\pm3)^{830\\phantom{000}} \\\\ (\\pm4)^{276\\phantom{000}} \\\\ (\\pm5)^{30\\phantom{000}} \\\\ (\\pm6)^{1\\phantom{000}}\\end{gather*}$$ $$\\begin{gather*}(0)^{400\\phantom{0000}} \\\\ (\\pm1)^{441\\phantom{0000}} \\\\ (\\pm3)^{1225\\phantom{0000}} \\\\ (\\pm6)^{196\\phantom{0000}} \\\\ (\\pm7)^{225\\phantom{0000}} \\\\ (\\pm9)^{196\\phantom{0000}} \\\\ (\\pm14)^{36\\phantom{0000}} \\\\ (\\pm21)^{1\\phantom{0000}}\\end{gather*}$$\n 8 Not shown: all 40320 roots $$\\begin{gather*}(0)^{3960\\phantom{0000}} \\\\ (\\pm1)^{5691\\phantom{0000}} \\\\ (\\pm2)^{2198\\phantom{0000}} \\\\ (\\pm3)^{6321\\phantom{0000}} \\\\ (\\pm4)^{3332\\phantom{0000}} \\\\ (\\pm5)^{595\\phantom{0000}} \\\\ (\\pm6)^{42\\phantom{0000}} \\\\ (\\pm7)^{1\\phantom{0000}}\\end{gather*}$$ $$\\begin{gather*}(0)^{9864\\phantom{0000}} \\\\ (\\pm2)^{3136\\phantom{0000}} \\\\ (\\pm4)^{6125\\phantom{0000}} \\\\ (\\pm7)^{4096\\phantom{0000}} \\\\ (\\pm8)^{196\\phantom{0000}} \\\\ (\\pm10)^{784\\phantom{0000}} \\\\ (\\pm12)^{441\\phantom{0000}} \\\\ (\\pm20)^{49\\phantom{0000}} \\\\ (\\pm28)^{1\\phantom{0000}}\\end{gather*}$$\n:::\n:::\n\n\nFrom what I have been able to identify, the spectrum of an exponential graph is symmetric about 0,\n by which I mean that the characteristic polynomial is even.\nThis has been the case for all graphs I have tried testing, even outside these graph families.\n\nSince all eigenvalues of $\\exp( \\bigstar_n )$ calculated are integers,\n it appears they are integral graphs, a fact of which I am reasonably sure\n because of the aforementioned correspondence to permutation star graphs.\nAdditionally, the eigenvalues are very conveniently the integers up to $n-1$ and down to $-n+1$.\nUnfortunately, despite the ease of reading the eigenvalues themselves,\n there isn't an OEIS entry for the multiplicities.\nI was able to identify the multiplicity of the 0 eigenvalue with\n [OEIS A217213](http://oeis.org/A217213), which counts orderings on Dyck paths.\nIf this is truly the sequence being generated, it means there is a 1:1 correspondence between\n these orderings and a basis of the nullspace of the adjacency matrix.\n\nIt seems to be the case that $\\exp( K_n )$ are also integral graphs.\nPerplexingly, the multiplicities for each of the eigenvalues appear to mostly be perfect powers.\nThis is the case until n = 8, which ruins the pattern because neither of\n $9864 = 2^3 \\cdot 3^2 \\cdot 137$ or $6125 = 5^3 \\cdot 7^2$ are perfect powers.\nI find both this[^7] and the fact that such a large prime appears among the factorization of the former\n rather creepy since all other primes which appear here are small -- 2, 3, 5, and 7.\n\n[^7]: Some physicists are fond of 137 for its closeness to the reciprocal\n of the fine structure constant (a bit of mostly-harmless numerology).\n\n\n### Notes about Spectral Computation\n\nFor *n* = 3 through 6, exactly computing the spectrum\n (or more accurately, the characteristic polynomial)\n is possible, although it takes upwards of 10 minutes for *n* = 6 on my machine using `sympy`.\nThe spectra of *n* = 7, 8 are marked with an asterisk because they were computed numerically,\n which still took nearly 8 hours in the case of the latter.\nIn fact, these graphs grow so quickly that it becomes nearly impossible to compute\n the spectrum without an explicit formula.\n\nFor *n* = 8, even storing the adjacency matrix in memory is a problem.\nAssuming the use of single-precision floating point, this behemoth of a matrix is\n $(8!)^2 \\cdot 4 \\text{ bytes} = 6.5\\text{GB}$.\nThis doesn't even factor in additional space requirements for eigenvalue algorithms,\n and is the reason I certainly won't be attempting to compute the spectrum for *n* = 9.\n\n\nGallery of Adjacency Matrices\n-----------------------------\n\nThe patterns in adjacency matrices depend on an enumeration of $S_n$ so that the vertices\n can be labelled from 1 to *n*!.\nWhile this is a fascinating topic unto itself, this post is already long enough,\n and I feel comfortable with just sharing the pictures.\n\n\n### [Plain Changes](https://en.wikipedia.org/wiki/Steinhaus%E2%80%93Johnson%E2%80%93Trotter_algorithm)\n\n::: {layout-ncol=\"3\"}\n\n\n\n\n\n\n\n\n\n\n\n:::\n\n\n### [Heap's Algorithm](https://en.wikipedia.org/wiki/Heap%27s_algorithm)\n\n:::: {}\n::: {layout-ncol=\"3\"}\n\n\n\n\n\n\n\n\n\n\n\n:::\n\nNote: GHC's `Data.List.permutations` is slightly different from Heap's algorithm as displayed on Wikipedia\n::::\n\n\nClosing\n-------\n\nAs previously stated, I am only mostly sure of the validity of the exponential law for graphs.\nIt *seems* too good to be true, but testing it directly on some graphs by comparing the spectra\n of the exponential of the sum against the product of the exponentials shows that they are at least cospectral.\nTry it yourself, preferably with a better tool than [the ones I made in Haskell](https://github.com/queue-miscreant/SymmetricGraph).\n\nFrom the articles I was able to find, permutation star graphs have applications to parallel computing,\n which is somewhat ironic considering how little care I had for the topic when writing this article.\nIf I needed ruthless efficiency, I probably could have used a library with GPU algorithms\n (or taken a stab at writing a shader myself).\nHowever, I *was* able to use this as a learning experience regarding mutable objects in Haskell.\nWith only immutable objects (and enough garbage to create an island in the Pacific),\n I was running out of memory even with 16GB of RAM and 16GB of swap.\nIntroducing mutability not only brought improvements in space, but also a great deal of speedup,\n enough to make rendering adjacency matrix images of order 8 graphs just barely doable within\n a reasonable time span.\n\nSaid images are cursed.\nRemember, as raw bitmaps, these files are on the order of *gigabytes* big.\nOn a much weaker computer than I used to render the images, merely opening my file explorer\n to the folder containing *the folder containing* the images\n caused its all-too-eager thumbnailer to run.\nThis started consuming all of my system resources, crashed all of my browser's tabs,\n distorted audio, and finally locked up the computer.\nDespite this, PNG is a wonderful format that is able to compress them down to just 4MB,\n which demonstrates just how sparse these matrices are.\n\nDespite everything I was able to find about permutation star graphs and permutohedra,\n I was surprised that there is no information about the Cayley graphs generated by *all*\n 2-cycles (or at least information which is easy to find).\nThis is especially disappointing considering the phantom pattern which gets destroyed by 137,\n and I would love to know more about why this happens in the first place.\n\nGraph diagrams made with GeoGebra and NetworkX (GraphViz).\n\n\n### Additional Links\n\n- [Whitney Numbers of the Second Kind for the Star Poset (\n Paper from ScienceDirect\n )](https://www.sciencedirect.com/science/article/pii/S0195669813801278)\n - This article includes a section about representing a list of generators as a graph,\n making me wonder if someone has tried defining this operation before\n- [Whitney Numbers (Josh Cooper's Mathpages)](https://people.math.sc.edu/cooper/graph.html)\n- [\n The Many Faces of the Nauru Graph (Blogpost by David Eppstein)\n ](https://11011110.github.io/blog/2007/12/12/many-faces-of.html)\n\n",
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