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.

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, 47, 204, 203, 64, 1, 1, 2, 9, 72, 628, 2344, 877, 128, 1, 1, 2, 10, 103, 1552, 17327, 43160, 4140, 256, 1, 1, 2, 11, 140, 3276, 84416, 1022983, 1291952, 21147, 512

Offset

0,6

Links

Alois P. Heinz, Antidiagonals n = 0..60, flattened

Wikipedia, Inversion

Wikipedia, Partition of a set

Formula

A(n,k) = Sum_{j>=0} k^j * A125810(n,j).

Example

Square array A(n,k) begins:
   1,   1,    1,     1,     1,      1,      1, ...
   1,   1,    1,     1,     1,      1,      1, ...
   2,   2,    2,     2,     2,      2,      2, ...
   4,   5,    6,     7,     8,      9,     10, ...
   8,  15,   28,    47,    72,    103,    140, ...
  16,  52,  204,   628,  1552,   3276,   6172, ...
  32, 203, 2344, 17327, 84416, 307867, 915848, ...

Maple

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

Mathematica

Unprotect[Power]; 0^0 = 1; Protect[Power];
b[o_, u_, t_, k_] := b[o, u, t, k] =
   If[u + o == 0, 1, If[t > 0, b[u + o, 0, 0, k], 0] + Sum[k^(u + j - 1)*
   b[o - j, u + j - 1, Min[2, t + 1], k], {j, If[t == 0, {1}, Range[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, Mar 15 2025, after Alois P. Heinz *)

Crossrefs

Columns k=0-5 give: A011782, A000110, A125812, A125813, A125814, A125815.

Main diagonal gives A381373.

Cf. A125810, A381426.

Sequence in context: A274859 A137855 A113143 * A181802 A371823 A373778

Adjacent sequences: A381366 A381367 A381368 * A381370 A381371 A381372

Keyword

nonn,tabl

Author

Alois P. Heinz, Feb 21 2025

Status

approved