A264082 Total number of inversions in all set partitions of [n].

0, 0, 0, 1, 10, 74, 504, 3383, 23004, 160444, 1154524, 8594072, 66243532, 528776232, 4369175522, 37343891839, 329883579768, 3008985817304, 28312886239136, 274561779926323, 2741471453779930, 28159405527279326, 297291626845716642, 3223299667111201702

Offset: 0

Author

Alois P. Heinz, Apr 03 2016

Keywords

• nonn

Comments

Each set partition is written as a sequence of blocks, ordered by the smallest elements in the blocks.

Examples

			a(3) = 1: one inversion in 13|2.
a(4) = 10: one inversion in each of 124|3, 13|24, 13|2|4, 1|24|3, and two inversions in each of 134|2, 14|23, 14|2|3.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..573
• Wikipedia, Inversion (discrete mathematics)
• Wikipedia, Partition of a set

Crossrefs

Cf. A001809, A125810, A189052, A211606, A216239, A240796, A271370.

Programs

Maple

b:= proc(n, t) option remember; `if`(n=0, [1, 0], add((p-> p+
      [0, p[1]*(j*t/2)])(b(n-j, t+j-1))*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n, 0)[2]:
seq(a(n), n=0..23);  # Alois P. Heinz, Feb 20 2025

Mathematica

b[n_, t_] := b[n, t] = If[n == 0, {1, 0}, Sum[Function[p, p+{0, p[[1]]*(j*t/2)}][b[n-j, t+j-1]]*Binomial[n-1, j-1], {j, 1, n}]];
a[n_] := b[n, 0][[2]];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, May 21 2025, after Alois P. Heinz *)

Formula

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