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A356264 Partition triangle read by rows, counting reducible permutations, refining triangle A356265.

%I A356264 #11 Aug 23 2022 05:34:56
%S A356264 0,0,1,0,1,2,0,1,5,3,2,0,1,9,12,15,10,2,0,1,14,23,12,47,94,11,31,24,2,
%T A356264 0,1,20,38,48,113,293,154,137,183,409,78,63,54,2,0,1,27,60,87,49,227,
%U A356264 738,883,451,457,670,2157,1007,1580,79,605,1520,384,127,116,2,0
%N A356264 Partition triangle read by rows, counting reducible permutations, refining triangle A356265.
%H A356264 Peter Luschny, <a href="https://github.com/PeterLuschny/PermutationsWithLehmer/blob/main/PermutationsWithLehmer.ipynb">Permutations with Lehmer</a>, a SageMath Jupyter Notebook.
%e A356264 [0] 0;
%e A356264 [1] 0;
%e A356264 [2] 1, 0;
%e A356264 [3] 1, 2, 0;
%e A356264 [4] 1, [5, 3], 2, 0;
%e A356264 [5] 1, [9, 12], [15,  10],   2,  0;
%e A356264 [6] 1, [14, 23, 12], [ 47,  94, 11], [31,   24],   2,  0;
%e A356264 [7] 1, [20, 38, 48], [113, 293, 154, 137], [183, 409, 78], [63, 54], 2, 0;
%e A356264 Summing the bracketed terms reduces the triangle to A356265.
%o A356264 (SageMath)
%o A356264 import collections
%o A356264 def reducible(p) -> bool:  # p is a Sage-Permutation
%o A356264     return any(i for i in range(1, p.size())
%o A356264                   if all(p(j) < p(k)
%o A356264                       for j in range(1, i + 1)
%o A356264                           for k in range(i + 1, p.size() + 1) ) )
%o A356264 def void(L) -> bool: return True
%o A356264 def perm_red_stats(n: int, part_costraint, lehmer_constraint):
%o A356264     res = collections.defaultdict(int)
%o A356264     for p in Permutations(n):
%o A356264         if not part_costraint(p): continue
%o A356264         l: list[int] = p.to_lehmer_code()
%o A356264         if lehmer_constraint(l):
%o A356264             c: list[int] = [l.count(i) for i in range(len(p)) if i in l]
%o A356264             res[Partition(reversed(sorted(c)))] += 1
%o A356264     return sorted(res.items(), key=lambda x: len(x[0]))
%o A356264 @cache
%o A356264 def A356264_row(n: int) -> list[int]:
%o A356264     if n < 2: return [0]
%o A356264     return [v[1] for v in perm_red_stats(n, reducible, void)] + [0]
%o A356264 def A356264(n: int, k: int) -> int:
%o A356264     return A356264_row(n)[k]
%o A356264 for n in range(0, 8): print(A356264_row(n))
%Y A356264 Cf. A356265 (reduced), A356262, A356263, A356291 (row sums).
%K A356264 nonn,tabf
%O A356264 0,6
%A A356264 _Peter Luschny_, Aug 05 2022