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A381426 A(n,k) is the sum over all ordered partitions of [n] of k^j for an ordered partition with j inversions; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A381426 #25 Jun 27 2025 18:26:23
%S A381426 1,1,1,1,1,2,1,1,3,4,1,1,4,13,8,1,1,5,36,75,16,1,1,6,79,696,541,32,1,
%T A381426 1,7,148,3851,27808,4683,64,1,1,8,249,14808,567733,2257888,47293,128,
%U A381426 1,1,9,388,44643,5942608,251790113,369572160,545835,256,1,1,10,571,113480,40065301,9546508128,335313799327,121459776768,7087261,512
%N A381426 A(n,k) is the sum over all ordered partitions of [n] of k^j for an ordered partition with j inversions; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H A381426 Alois P. Heinz, <a href="/A381426/b381426.txt">Antidiagonals n = 0..55, flattened</a>
%H A381426 Wikipedia, <a href="https://en.wikipedia.org/wiki/Inversion_(discrete_mathematics)">Inversion</a>
%H A381426 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A381426 A(n,k) = Sum_{j=0..binomial(n,2)} k^j * A381299(n,j).
%e A381426 Square array A(n,k) begins:
%e A381426    1,    1,       1,         1,          1,            1,             1, ...
%e A381426    1,    1,       1,         1,          1,            1,             1, ...
%e A381426    2,    3,       4,         5,          6,            7,             8, ...
%e A381426    4,   13,      36,        79,        148,          249,           388, ...
%e A381426    8,   75,     696,      3851,      14808,        44643,        113480, ...
%e A381426   16,  541,   27808,    567733,    5942608,     40065301,     199246816, ...
%e A381426   32, 4683, 2257888, 251790113, 9546508128, 179833594207, 2099255895008, ...
%p A381426 b:= proc(o, u, t, k) option remember; `if`(u+o=0, 1, `if`(t=1,
%p A381426       b(u+o, 0$2, k), 0)+add(k^(u+j-1)*b(o-j, u+j-1, 1, k), j=1..o))
%p A381426     end:
%p A381426 A:= (n, k)-> b(n, 0$2, k):
%p A381426 seq(seq(A(n, d-n), n=0..d), d=0..10);
%t A381426 b[o_, u_, t_, k_] := b[o, u, t, k] = If[u + o == 0, 1, If[t == 1, b[u + o, 0, 0, k], 0] + Sum[If[k == u + j - 1 == 0, 1, k^(u + j - 1)]*b[o - j, u + j - 1, 1, k], {j, 1, o}]];
%t A381426 A[n_, k_] := b[n, 0, 0, k];
%t A381426 Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* _Jean-François Alcover_, Apr 19 2025, after _Alois P. Heinz_ *)
%Y A381426 Columns k=0-9 give: A011782, A000670, A289545, A347841, A347842, A347843, A385408, A347844, A347845, A347846.
%Y A381426 Main diagonal gives A381427.
%Y A381426 Cf. A381299, A381369.
%K A381426 nonn,tabl
%O A381426 0,6
%A A381426 _Alois P. Heinz_, Feb 23 2025