A381427 Sum over all ordered partitions of [n] of n^j for an ordered partition with j inversions.

1, 1, 4, 79, 14808, 40065301, 2099255895008, 2651651342949844915, 96254339565438079064819328, 116387990444553949414146511586296381, 5327195120249449992420082364255283659438679552, 10333056290045508772052838892223597279253890797441054043823

Offset: 0

Author

Alois P. Heinz, Feb 23 2025

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..36
• Wikipedia, Inversion
• Wikipedia, Partition of a set

Crossrefs

Main diagonal of A381426.

Cf. A062173, A135528, A381299, A381373.

Programs

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-> b(n, 0$2, n):
seq(a(n), n=0..11);

Formula

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

a(n) = A381426(n,n).

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

a(n) mod 2 = A135528(n+1).