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A356116 Triangle read by row. The reduced triangle of the partition_triangle A355776.

%I A356116 #13 Aug 23 2022 05:47:09
%S A356116 0,0,0,0,1,0,0,5,5,0,0,16,46,16,0,0,42,252,252,42,0,0,99,1086,2241,
%T A356116 1086,99,0,0,219,4097,15129,15129,4097,219,0,0,466,14272,87058,154426,
%U A356116 87058,14272,466,0,0,968,47300,452672,1305062,1305062,452672,47300,968,0
%N A356116 Triangle read by row. The reduced triangle of the partition_triangle A355776.
%C A356116 By a partition triangle, we understand an irregular triangle where each row corresponds to a mapping of Partitions(n) -> ZZ. We assume a fixed order of the partitions given. Here we will use the ordering defined in A080577. Examples are A355776, A355777, and A134264. 'Reducing' then means summing the values corresponding to the partitions of n with length k. The 'reduced partition triangle' then is a regular triangle with T(n, k) with 1 <= k <= n.
%C A356116 Conversely, A355776, the statistic of permutations whose Lehmer code is nonmonotonic, can be seen as a refinement of this triangle, which in turn is a refinement of the sequence A056986, the number of permutations on [n] containing any given pattern alpha in the symmetric group S_3.
%H A356116 FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/StatisticsDatabase/St000002/">The number of occurrences of the pattern [1,2,3] inside a permutation of length at least 3</a>.
%H A356116 Peter Luschny, <a href="https://github.com/PeterLuschny/PermutationsWithLehmer/blob/main/PermutationsWithLehmer.ipynb">Permutations with Lehmer</a>, a SageMath Jupyter Notebook.
%e A356116 Triangle T(n, k) starts:
%e A356116 [1] [0]
%e A356116 [2] [0,   0]
%e A356116 [3] [0,   1,     0]
%e A356116 [4] [0,   5,     5,      0]
%e A356116 [5] [0,  16,    46,     16,       0]
%e A356116 [6] [0,  42,   252,    252,      42,       0]
%e A356116 [7] [0,  99,  1086,   2241,    1086,      99,      0]
%e A356116 [8] [0, 219,  4097,  15129,   15129,    4097,    219,     0]
%e A356116 [9] [0, 466, 14272,  87058,  154426,   87058,  14272,   466,   0]
%e A356116 [10][0, 968, 47300, 452672, 1305062, 1305062, 452672, 47300, 968, 0]
%e A356116 .
%e A356116 Row 6 of the partition triangle A355776 is:
%e A356116 [0, [10, 20, 12], [61,  162, 29], [102, 150], 42, 0]
%e A356116 Adding the bracketed terms reduces this row to row 6 of the above triangle.
%o A356116 (SageMath)
%o A356116 from functools import cache
%o A356116 @cache
%o A356116 def Pn(n: int, k: int) -> int:
%o A356116     if k == 0: return 0
%o A356116     if n == 0 or k == 1: return 1
%o A356116     return Pn(n, k - 1) + Pn(n - k, k) if k <= n else Pn(n, k - 1)
%o A356116 def reduce_parts(fun, n: int) -> list[int]:
%o A356116     funn: list[int] = fun(n)
%o A356116     return [sum(funn[Pn(n, k):Pn(n, k + 1)]) for k in range(n)]
%o A356116 def reduce_partition_triangle(fun, n: int) -> list[list[int]]:
%o A356116     return [reduce_parts(fun, k) for k in range(1, n)]
%o A356116 reduce_partition_triangle(A355776_row, 6)
%Y A356116 A002662 (column 1), A056986 (row sums), A355776 (refinement).
%K A356116 nonn,tabl
%O A356116 1,8
%A A356116 _Peter Luschny_, Jul 28 2022