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A356261 Partition triangle read by rows, counting irreducible permutations with weakly decreasing Lehmer code, refining triangle A119308.

%I A356261 #9 Aug 21 2022 14:09:53
%S A356261 1,1,0,1,0,2,1,0,2,1,5,1,0,2,2,7,7,9,1,0,2,2,1,9,18,3,16,24,14,1,0,2,
%T A356261 2,2,11,22,11,11,25,75,25,30,60,20,1,0,2,2,2,1,13,26,26,13,13,36,108,
%U A356261 54,108,9,55,220,110,50,125,27,1
%N A356261 Partition triangle read by rows, counting irreducible permutations with weakly decreasing Lehmer code, refining triangle A119308.
%H A356261 Peter Luschny, <a href="https://github.com/PeterLuschny/PermutationsWithLehmer/blob/main/PermutationsWithLehmer.ipynb">Permutations with Lehmer</a>, a SageMath Jupyter Notebook.
%e A356261 Partition table T(n, k) begins:
%e A356261 [0] 1;
%e A356261 [1] 1;
%e A356261 [2] 0, 1;
%e A356261 [3] 0, 2, 1;
%e A356261 [4] 0, [2, 1],  5,  1;
%e A356261 [5] 0, [2, 2], [7,  7],   9,  1;
%e A356261 [6] 0, [2, 2,  1], [9,   18, 3], [16, 24], 14,    1;
%e A356261 [7] 0, [2, 2,  2], [11,  22, 11, 11], [25, 75,  25], [30, 60],  20, 1;
%e A356261 [8] 0, [2, 2, 2, 1],[13, 26, 26, 13, 13],[36, 108, 54, 108,9],[55, 220, 110],[50, 125], 27, 1;
%e A356261 Summing the bracketed terms reduces the triangle to A119308.
%o A356261 (SageMath) # using function perm_red_stats and reducible from A356264
%o A356261 def weakly_decreasing(L: list[int]) -> bool:
%o A356261     return all(x >= y for x, y in zip(L, L[1:]))
%o A356261 @cache
%o A356261 def A356261_row(n: int) -> list[int]:
%o A356261     if n < 2: return [1]
%o A356261     return [0] + [v[1] for v in perm_red_stats(n, irreducible, weakly_decreasing)]
%o A356261 def A356261(n: int, k: int) -> int:
%o A356261     return A356261_row(n)[k]
%o A356261 for n in range(8):
%o A356261     print([n], A356261_row(n))
%Y A356261 Cf. A356264, A119308 (reduced), A071724 (row sums).
%K A356261 nonn,tabf
%O A356261 0,6
%A A356261 _Peter Luschny_, Aug 16 2022