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A381369 A(n,k) is the sum over all partitions of [n] of k^j for a partition with j inversions; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A381369 #23 Mar 15 2025 18:45:03
%S A381369 1,1,1,1,1,2,1,1,2,4,1,1,2,5,8,1,1,2,6,15,16,1,1,2,7,28,52,32,1,1,2,8,
%T A381369 47,204,203,64,1,1,2,9,72,628,2344,877,128,1,1,2,10,103,1552,17327,
%U A381369 43160,4140,256,1,1,2,11,140,3276,84416,1022983,1291952,21147,512
%N A381369 A(n,k) is the sum over all partitions of [n] of k^j for a partition with j inversions; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H A381369 Alois P. Heinz, <a href="/A381369/b381369.txt">Antidiagonals n = 0..60, flattened</a>
%H A381369 Wikipedia, <a href="https://en.wikipedia.org/wiki/Inversion_(discrete_mathematics)">Inversion</a>
%H A381369 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A381369 A(n,k) = Sum_{j>=0} k^j * A125810(n,j).
%e A381369 Square array A(n,k) begins:
%e A381369    1,   1,    1,     1,     1,      1,      1, ...
%e A381369    1,   1,    1,     1,     1,      1,      1, ...
%e A381369    2,   2,    2,     2,     2,      2,      2, ...
%e A381369    4,   5,    6,     7,     8,      9,     10, ...
%e A381369    8,  15,   28,    47,    72,    103,    140, ...
%e A381369   16,  52,  204,   628,  1552,   3276,   6172, ...
%e A381369   32, 203, 2344, 17327, 84416, 307867, 915848, ...
%p A381369 b:= proc(o, u, t, k) option remember;
%p A381369      `if`(u+o=0, 1, `if`(t>0, b(u+o, 0$2, k), 0)+add(k^(u+j-1)*
%p A381369         b(o-j, u+j-1, min(2, t+1), k), j=`if`(t=0, 1, 1..o)))
%p A381369     end:
%p A381369 A:= (n, k)-> b(n, 0$2, k):
%p A381369 seq(seq(A(n, d-n), n=0..d), d=0..10);
%t A381369 Unprotect[Power]; 0^0 = 1; Protect[Power];
%t A381369 b[o_, u_, t_, k_] := b[o, u, t, k] =
%t A381369    If[u + o == 0, 1, If[t > 0, b[u + o, 0, 0, k], 0] + Sum[k^(u + j - 1)*
%t A381369    b[o - j, u + j - 1, Min[2, t + 1], k], {j, If[t == 0, {1}, Range[o]]}]];
%t A381369 A[n_, k_] := b[n, 0, 0, k];
%t A381369 Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* _Jean-François Alcover_, Mar 15 2025, after _Alois P. Heinz_ *)
%Y A381369 Columns k=0-5 give: A011782, A000110, A125812, A125813, A125814, A125815.
%Y A381369 Main diagonal gives A381373.
%Y A381369 Cf. A125810, A381426.
%K A381369 nonn,tabl
%O A381369 0,6
%A A381369 _Alois P. Heinz_, Feb 21 2025