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.

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, 1, 7, 148, 3851, 27808, 4683, 64, 1, 1, 8, 249, 14808, 567733, 2257888, 47293, 128, 1, 1, 9, 388, 44643, 5942608, 251790113, 369572160, 545835, 256, 1, 1, 10, 571, 113480, 40065301, 9546508128, 335313799327, 121459776768, 7087261, 512

Offset

0,6

Links

Alois P. Heinz, Antidiagonals n = 0..55, flattened

Wikipedia, Inversion

Wikipedia, Partition of a set

Formula

A(n,k) = Sum_{j=0..binomial(n,2)} k^j * A381299(n,j).

Example

Square array A(n,k) begins:
   1,    1,       1,         1,          1,            1,             1, ...
   1,    1,       1,         1,          1,            1,             1, ...
   2,    3,       4,         5,          6,            7,             8, ...
   4,   13,      36,        79,        148,          249,           388, ...
   8,   75,     696,      3851,      14808,        44643,        113480, ...
  16,  541,   27808,    567733,    5942608,     40065301,     199246816, ...
  32, 4683, 2257888, 251790113, 9546508128, 179833594207, 2099255895008, ...

Maple

b:= proc(o, u, t, k) option remember; `if`(u+o=0, 1, `if`(t=1,
      b(u+o, 0$2, k), 0)+add(k^(u+j-1)*b(o-j, u+j-1, 1, k), j=1..o))
    end:
A:= (n, k)-> b(n, 0$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..10);

Mathematica

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}]];
A[n_, k_] := b[n, 0, 0, k];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Apr 19 2025, after Alois P. Heinz *)

Crossrefs

Columns k=0-9 give: A011782, A000670, A289545, A347841, A347842, A347843, A385408, A347844, A347845, A347846.

Main diagonal gives A381427.

Cf. A381299, A381369.

Sequence in context: A104495 A093541 A336187 * A171881 A321877 A320251

Adjacent sequences: A381423 A381424 A381425 * A381427 A381428 A381429

Keyword

nonn,tabl

Author

Alois P. Heinz, Feb 23 2025

Status

approved