A381373 Sum over all partitions of [n] of n^j for a partition with j inversions.

1, 1, 2, 7, 72, 3276, 915848, 2011878835, 42723411900032, 10608257527069388539, 35808039364308986083608352, 1828963737334508176477805993389490, 1618534282345584818909121118371843799592960, 28472613161534902071627567919297331348486838233018341

Offset: 0

Author

Alois P. Heinz, Feb 21 2025

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..41
• Wikipedia, Inversion
• Wikipedia, Partition of a set

Crossrefs

Main diagonal of A381369.

Cf. A062173, A120325, A125810, A381427.

Programs

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-> b(n, 0$2, n):
seq(a(n), n=0..15);

Mathematica

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_] := b[n, 0, 0, n];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Mar 15 2025, after Alois P. Heinz *)

Formula

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

a(n) = A381369(n,n).

a(n) mod n = A062173(n) for n>=1.

a(n) mod 2 = A120325(n+1) for n>=1.