diff --git a/posts/permutations/2/graph_data/__init__.py b/posts/permutations/2/graph_data/__init__.py
new file mode 100644
index 0000000..e69de29
diff --git a/posts/permutations/2/graph_data/base.py b/posts/permutations/2/graph_data/base.py
new file mode 100644
index 0000000..78bc3d0
--- /dev/null
+++ b/posts/permutations/2/graph_data/base.py
@@ -0,0 +1,135 @@
+from dataclasses import dataclass
+import sympy
+from sympy.abc import x
+
+
+def display_integral_root(
+    root: int, multiplicity: int, pad_to: int | None = None
+) -> str:
+    return (
+        "("
+        # denote sign
+        + ("\\pm" if root != 0 else "")
+        + str(root)
+        + ")^{"
+        + str(multiplicity)
+        # pad multiplicity
+        + ("\\phantom{" + ("0" * pad_to) + "}" if pad_to is not None else "")
+        + "}"
+    )
+
+
+@dataclass
+class SymmetricSpectrum:
+    """
+    Dataclass for symmetric spectra -- those for which, if a spectrum contains a root,
+    it also contains its negative.
+
+    If a root polynomial is not symmetric, then its symmetric counterpart will *not* be specified.
+    """
+
+    # Either the integral root or a polynomial with the root, and the multiplicity
+    roots: list[tuple[sympy.Expr | int, int]]
+    not_shown_count: int = 0
+
+    def to_markdown(self) -> str:
+        """
+        Display a spectrum as a markdown string
+        """
+        integer_roots = [
+            (root, multiplicity)
+            for root, multiplicity in self.roots
+            if isinstance(root, int)
+        ]
+        # symmetrify polynomials
+        polynomial_roots = [
+            (root, multiplicity)
+            for base_root, multiplicity in self.roots
+            if isinstance(base_root, sympy.Expr)
+            for root in set(
+                [base_root, (-1) ** (sympy.degree(base_root)) * base_root.subs(x, -x)]
+            )
+        ]
+
+        shown_roots = len(polynomial_roots) > 0 or len(integer_roots) > 0
+        not_shown_string = (
+            (
+                f"Not shown: {self.not_shown_count} other roots"
+                if shown_roots
+                else f"Not shown: all {self.not_shown_count} roots"
+            )
+            if self.not_shown_count > 0
+            else ""
+        )
+
+        # strictly integral spectra
+        if len(polynomial_roots) == 0 and len(integer_roots) != 0:
+            longest_integer = max(
+                len(str(multiplicity)) for _, multiplicity in integer_roots
+            )
+            return (
+                (
+                "$$\\begin{gather*}"
+                + " \\\\ ".join(
+                    display_integral_root(*rm, pad_to=longest_integer)
+                    for rm in integer_roots
+                )
+                + "\\end{gather*}$$"
+            )
+                if shown_roots
+                else ""
+            ) + not_shown_string
+
+        # conditionally split integral roots onto multiple lines
+        integral_roots_lines = (
+            " ".join(display_integral_root(*rm) for rm in integer_roots)
+            if len(integer_roots) < 5
+            else (
+                " ".join(
+                    display_integral_root(*rm)
+                    for rm in integer_roots[: len(integer_roots) // 2]
+                )
+                + "\\\\"
+                + " ".join(
+                    display_integral_root(*rm)
+                    for rm in integer_roots[len(integer_roots) // 2 :]
+                )
+            )
+        )
+
+        return ((
+            "$$\\begin{gather*}"
+            + integral_roots_lines
+            + " \\\\ "
+            + " \\\\ ".join(
+                sympy.latex(root**multiplicity)
+                for root, multiplicity in polynomial_roots
+            )
+            + "\\end{gather*}$$"
+        )
+            if shown_roots
+            else ""
+        ) + not_shown_string
+
+
+def count_roots_spectrum(spectrum: SymmetricSpectrum) -> int:
+    return (
+        sum(
+            [
+                (
+                    (multiplicity * 2 if root != 0 else multiplicity)
+                    if isinstance(root, int)
+                    else (int(sympy.degree(root, x)) * multiplicity)
+                )
+                for root, multiplicity in spectrum.roots
+            ]
+        )
+        + spectrum.not_shown_count
+    )
+
+@dataclass
+class GraphData:
+    vertex_count: int
+    edge_count: int
+    distance_classes: list[int]
+    spectrum: SymmetricSpectrum
diff --git a/posts/permutations/2/graph_data/complete.py b/posts/permutations/2/graph_data/complete.py
new file mode 100644
index 0000000..73dc702
--- /dev/null
+++ b/posts/permutations/2/graph_data/complete.py
@@ -0,0 +1,50 @@
+from .base import SymmetricSpectrum
+
+# TODO: other non-spectral data
+complete_spectra = {
+    3: SymmetricSpectrum([(0, 4), (3, 1)]),
+    4: SymmetricSpectrum([(0, 4), (2, 9), (6, 1)]),
+    5: SymmetricSpectrum(
+        [
+            (0, 36),
+            (2, 25),
+            (5, 16),
+            (10, 1),
+        ]
+    ),
+    6: SymmetricSpectrum(
+        [
+            (0, 256),
+            (3, 125),
+            (5, 81),
+            (9, 25),
+            (15, 1),
+        ]
+    ),
+    7: SymmetricSpectrum(
+        [
+            (0, 400),
+            (1, 441),
+            (3, 1225),
+            (6, 196),
+            (7, 225),
+            (9, 196),
+            (14, 36),
+            (21, 1),
+        ]
+    ),
+    # TODO: missing a root (and its negative) of multiplicity 400
+    8: SymmetricSpectrum(
+        [
+            (0, 9864),
+            (2, 3136),
+            (4, 6125),
+            (7, 4096),
+            (8, 196),
+            (10, 784),
+            (12, 441),
+            (20, 49),
+            (28, 1),
+        ]
+    ),
+}
diff --git a/posts/permutations/2/graph_data/path.py b/posts/permutations/2/graph_data/path.py
new file mode 100644
index 0000000..f2cdd66
--- /dev/null
+++ b/posts/permutations/2/graph_data/path.py
@@ -0,0 +1,50 @@
+from .base import SymmetricSpectrum, x
+
+path_spectra = {
+    3: SymmetricSpectrum(
+        [(1, 2), (2, 1)]
+    ),
+    4: SymmetricSpectrum(
+        [
+            (1, 3),
+            (3, 1),
+            (x**2 - 3, 2),
+            (x**2 - 2 * x - 1, 3),
+            (x**2 + 2 * x - 1, 3),
+        ]
+    ),
+    5: SymmetricSpectrum(
+        [
+            (0, 12),
+            (1, 6),
+            (4, 1),
+            (x**2 - 5, 6),
+            (x**2 + 5 * x + 5, 4),
+            (x**2 + 3 * x + 1, 4),
+            (x**2 + 2 * x - 1, 5),
+            (x**3 + 2 * x**2 - 5 * x - 4, 5),
+        ]
+    ),
+    6: SymmetricSpectrum(
+        [
+            (0, 20),
+            (1, 25),
+            (2, 15),
+            (3, 5),
+            (4, 5),
+            (5, 1),
+            (x**2 - 3, 20),
+        ],
+        not_shown_count=558,
+    ),
+    7: SymmetricSpectrum(
+        [
+            (0, 35),
+            (1, 20),
+            (2, 45),
+            (6, 1),
+        ],
+        not_shown_count=4873,
+    ),
+    8: SymmetricSpectrum([], not_shown_count=40320),
+}
diff --git a/posts/permutations/2/graph_data/star.py b/posts/permutations/2/graph_data/star.py
new file mode 100644
index 0000000..ba180e0
--- /dev/null
+++ b/posts/permutations/2/graph_data/star.py
@@ -0,0 +1,55 @@
+from .base import SymmetricSpectrum
+
+star_spectra = {
+    3: SymmetricSpectrum([(1, 2), (2, 1)]),
+    4: SymmetricSpectrum(
+        [
+            (0, 4),
+            (1, 3),
+            (2, 6),
+            (3, 1),
+        ]
+    ),
+    5: SymmetricSpectrum(
+        [
+            (0, 30),
+            (1, 4),
+            (2, 28),
+            (3, 12),
+            (4, 1),
+        ]
+    ),
+    6: SymmetricSpectrum(
+        [
+            (0, 168),
+            (1, 30),
+            (2, 120),
+            (3, 105),
+            (4, 20),
+            (5, 1),
+        ]
+    ),
+    7: SymmetricSpectrum(
+        [
+            (0, 840),
+            (1, 468),
+            (2, 495),
+            (3, 830),
+            (4, 276),
+            (5, 30),
+            (6, 1),
+        ]
+    ),
+    8: SymmetricSpectrum(
+        [
+            (0, 3960),
+            (1, 5691),
+            (2, 2198),
+            (3, 6321),
+            (4, 3332),
+            (5, 595),
+            (6, 42),
+            (7, 1),
+        ]
+    ),
+}
diff --git a/posts/permutations/2/index.qmd b/posts/permutations/2/index.qmd
index ccca29a..9a8db38 100644
--- a/posts/permutations/2/index.qmd
+++ b/posts/permutations/2/index.qmd
@@ -19,6 +19,8 @@ categories:
 
 from IPython.display import Markdown
 from tabulate import tabulate
+
+from graph_data import complete, path, star
 ```
 
 
@@ -234,51 +236,44 @@ In the case of these graphs, they are a partition of the factorial numbers.
 
 ```{python}
 #| echo: false
+#| classes: plain
+
+mahonian = [
+  [1, 2, 2, 1],
+  [1, 3, 5, 6, 5, 3, 1],
+  [1, 4, 9, 15, 20, 22, 20, 15, 9, 4, 1],
+  [1, 5, 14, 29, 49, 71, 90, 101, 101, 90, 71, 49, 29, 14, 5, 1],
+  [1, 6, 20, 49, 98, 169, 259, 359, 455, 531, 573, 573, 531, 455, 359, 259, 169, 98, 49, 20, 6, 1],
+]
+
+whitney_second_kind = [
+  [1, 2, 2, 1],
+  [1, 3, 6, 9, 5],
+  [1, 4, 12, 30, 44, 26, 3],
+  [1, 5, 20, 70, 170, 250, 169, 35],
+  [1, 6, 30, 135, 460, 1110, 1689, 1254, 340, 15],
+]
+
+stirling_second_kind = [
+  [1, 3, 2],
+  [1, 6, 11, 6],
+  [1, 10, 35, 50, 24],
+  [1, 15, 85, 225, 274, 120],
+  [1, 21, 175, 735, 1624, 1764, 720],
+]
+
+half_break = lambda x: str(x[:len(x) // 2])[:-1] + "<br>" + str(x[len(x) // 2:])[1:]
 
 Markdown(tabulate(
   [
-    [
-      3,
-      "[1, 2, 2, 1]",
-      "[1, 2, 2, 1]",
-      "[1, 3, 2]"
-    ],
-    [
-      4,
-      "[1, 3, 5, 6, 5, 3, 1]",
-      "[1, 2, 2, 1]",
-      "[1, 6, 11, 6]"
-    ],
-    [
-      5,
-      "[1, 4, 9, 15, 20, 22, 20, 15, 9, 4, 1]",
-      "[1, 4, 12, 30, 44, 26, 3]",
-      "[1, 10, 35, 50, 24]"
-    ],
-    [
-      6,
-      "[1, 5, 14, 29, 49, 71, 90, 101,<br>101, 90, 71, 49, 29, 14, 5, 1]",
-      "[1, 5, 20, 70, 170, 250, 169, 35]",
-      "[1, 15, 85, 225, 274, 120]"
-    ],
-    [
-      7,
-      "[1, 6, 20, 49, 98, 169, 259, 359, 455, 531, 573,<br>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]"
-    ],
-    [
-      "Rule",
-      "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)"
-    ],
+    [i, j if i <= 5 else half_break(j), k, l]
+    for i, j, k, l in zip(range(3,8), mahonian, whitney_second_kind, stirling_second_kind)
   ],
   headers=[
     "*n*",
-    r"$dists(\exp( P_n ))$",
-    r"$dists(\exp( \bigstar_n ))$",
-    r"$dists(\exp( K_n ))$"
+    r"$\text{dists}(\exp( P_n ))$",
+    r"$\text{dists}(\exp( \bigstar_n ))$",
+    r"$\text{dists}(\exp( K_n ))$"
   ]
 ))
 ```
@@ -297,130 +292,20 @@ Unfortunately, eigenvalues are not necessarily integers, and therefore not easil
 
 ```{python}
 #| echo: false
-
-# TODO: less raw latex, more data converted to latex
-
-f = lambda x: x.replace("\n", "")
+#| classes: plain
 
 Markdown(tabulate(
   [
-    [3, r"$(\pm 1)^2 (\pm 2)$", r"$(\pm 1)^2 (\pm 2)$", r"$(0)^4 (\pm 3)$"],
-    [4, f(r"""$$
-\begin{gather*}
-  (\pm 1)^3 (\pm 3) \\
-  (x^2 -\ 3)^2 \\
-  (x^2 -\ 2x -\ 1)^3
-  (x^2 + 2x -\ 1)^3
-\end{gather*}$$"""), f(r"""$$
-\begin{gather*}
-  (0)^4 \\
-  (\pm 1)^3 \\
-  (\pm 2)^6 \\
-  (\pm 3)^{\phantom{0}}
-\end{gather*}$$"""), f(r"""$$
-\begin{gather*}
-  (0)^4 \\
-  (\pm 2)^9 \\
-  (\pm 6)^{\phantom{0}}
-\end{gather*}$$""")],
-    [5, f(r"""$$
-\begin{gather*}
-  (0)^{12} (\pm 1)^6 (\pm 4) \\
-  (x^2 -\ 5)^6 \\
-  (x^2 -\ 5x + 5)^4
-  (x^2 + 5x + 5)^4 \\
-  (x^2 -\ 3x + 1)^4
-  (x^2 + 3x + 1)^4 \\
-  (x^2 -\ 2x -\ 1)^5
-  (x^2 + 2x -\ 1)^5 \\
-  (x^3 -\ 2x^2 -\ 5x + 4)^5 \\
-  (x^3 + 2x^2 -\ 5x -\ 4)^5
-\end{gather*}$$"""), f(r"""$$
-\begin{gather*}
-  (0)^{30} \\
-  (\pm 1)^{4\phantom{0}} \\
-  (\pm 2)^{28} \\
-  (\pm 3)^{12} \\
-  (\pm 4)^{\phantom{00}}
-\end{gather*}$$"""), f(r"""$$
-\begin{gather*}
-  (0)^{36} \\
-  (\pm 2)^{25} \\
-  (\pm 5)^{16} \\
-  (\pm 10)^{\phantom{00}}
-  \end{gather*}$$""")],
-      [6, f(r"""$$\begin{gather*}
-  (0)^{20} (\pm 1)^{25} (\pm 2)^{15} \\
-  (\pm 3)^5 (\pm 4)^5 (\pm 5) \\
-  (x^2 -\ 3)^{20} \\
-\end{gather*}
-$$ <br> *Not shown: 558 other roots*"""), f(r"""$$
-\begin{gather*}
-  (0)^{168} \\
-  (\pm 1)^{30\phantom{0}} \\
-  (\pm 2)^{120} \\
-  (\pm 3)^{105} \\
-  (\pm 4)^{20\phantom{0}} \\
-  (\pm 5)^{\phantom{000}}
-\end{gather*}$$"""), f(r"""$$
-\begin{gather*}
-  (0)^{256} \\
-  (\pm 3)^{125} \\
-  (\pm 5)^{81\phantom{0}} \\
-  (\pm 9)^{25\phantom{0}} \\
-  (\pm 15)^{\phantom{000}}
-\end{gather*}$$""")],
-    [7, f(r"""
-    $(0)^{35} (\pm 1)^{20} (\pm 2)^{45} (6)$ <br>
-*Not shown: 4873 other roots*
-    """), f(r"""$$
-\begin{gather*}
-  (0)^{840} \\
-  (\pm 1)^{468} \\
-  (\pm 2)^{495} \\
-  (\pm 3)^{830} \\
-  (\pm 4)^{276} \\
-  (\pm 5)^{30\phantom{0}} \\
-  (\pm 6)^{\phantom{000}}
-\end{gather*}$$"""), f(r"""$$
-\begin{gather*}
-  (0)^{400\phantom{0}} \\
-  (\pm 1)^{441\phantom{0}} \\
-  (\pm 3)^{1225} \\
-  (\pm 6)^{196\phantom{0}} \\
-  (\pm 7)^{225\phantom{0}} \\
-  (\pm 9)^{196\phantom{0}} \\
-  (\pm 14)^{36\phantom{00}} \\
-  (\pm 21)^{\phantom{0000}}
-\end{gather*}$$""")],
-    [8, f(r"""*Not shown: all 40320 roots*"""), f(r"""$$
-\begin{gather*}
-  (0)^{3960} \\
-  (\pm 1)^{5691} \\
-  (\pm 2)^{2198} \\
-  (\pm 3)^{6321} \\
-  (\pm 4)^{3332} \\
-  (\pm 5)^{595\phantom{0}} \\
-  (\pm 6)^{42\phantom{00}} \\
-  (\pm 7)^{\phantom{0000}}
-\end{gather*}$$"""), f(r"""$$
-\begin{gather*}
-  (0)^{9864} \\
-  (\pm 2)^{3136} \\
-  (\pm 4)^{6125} \\
-  (\pm 7)^{4096} \\
-  (\pm 8)^{196\phantom{0}} \\
-  (\pm 10)^{784\phantom{0}} \\
-  (\pm 12)^{441\phantom{0}} \\
-  (\pm 20)^{49\phantom{00}} \\
-  (\pm 28)^{\phantom{0000}}
-\end{gather*}$$""")
-    ],
+    [i, pathG.to_markdown(), starG.to_markdown(), completeG.to_markdown()]
+    for i in range(3, 9)
+    for pathG, starG, completeG in (
+      [path.path_spectra[i], star.star_spectra[i], complete.complete_spectra[i]],
+    )
   ],
   headers=[
     "*n*",
     r"$\text{Spec}(\exp( P_n ))$",
-    r"$\text{Spec}(\exp( \bigstar_n)$",
+    r"$\text{Spec}(\exp( \bigstar_n )$",
     r"$\text{Spec}(\exp( K_n ))$"
   ]
 ))