September Morn Update

Task/Paraffins/Python/paraffins-3.py

@@ -0,0 +1,32 @@
+#!/usr/bin/python3
+
+from functools import lru_cache
+
+def Z_S(n, f, k):
+    """
+    The cycle index of the symmetric group has recurrence
+        Z(S_n, f(x)) = 1/n \sum_{i=1}^n f(x^i) Z(S_{n-i}, f(x)).
+    This function finds the coefficient of x^k in Z(S_n, f(x))
+    """
+    # Special case to avoid division by zero
+    if n == 0:
+        return 1 if k == 0 else 0
+    # Special case as a speed optimisation
+    if n == 1:
+        return f(k)
+    return sum(
+        sum(f(ij // i) * Z_S(n-i, f, k - ij) for ij in range(0, k+1, i))
+        for i in range(1, n+1)
+    ) // n
+
+@lru_cache(maxsize=None)
+def A000598(k): return 1 if k == 0 else Z_S(3, A000598, k-1)
+
+@lru_cache(maxsize=None)
+def A000642(k): return Z_S(2, A000598, k)
+
+def A000631(k): return Z_S(2, A000642, k)
+
+def A000602(k): return A000642(k) + (A000642((k-1) // 2) if k % 2 == 1 else 0) - A000631(k-1)
+
+for k in range(500): print(k, A000602(k))