A334623 Sum of the n-th powers of the descent set statistics for permutations of [n].

1, 1, 2, 18, 1576, 2675100, 128235838496, 265039489112493900, 31306198216486969509375104, 278983981168082455883720325976751040, 235157286166918393786165504356030195355598048512, 23075317400822150539572583950910707053701314350537805923757600

Offset: 0

Author

Alois P. Heinz, Sep 09 2020

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..24 (terms 0..21 from Alois P. Heinz)
• R. Ehrenborg and A. Happ, On the powers of the descent set statistic, arXiv:1709.00778 [math.CO], 2017.

Crossrefs

Main diagonal of A334622.

Cf. A060351, A335546.

Programs

Maple

b:= proc(u, o, t) option remember; expand(`if`(u+o=0, 1,
      add(b(u-j, o+j-1, t+1)*x^floor(2^(t-1)), j=1..u)+
      add(b(u+j-1, o-j, t+1), j=1..o)))
    end:
a:= n-> (p-> add(coeff(p, x, i)^n, i=0..degree(p)))(b(n, 0$2)):
seq(a(n), n=0..12);

Mathematica

b[u_, o_, t_] := b[u, o, t] = Expand[If[u + o == 0, 1,
    Sum[b[u - j, o + j - 1, t + 1]*x^Floor[2^(t - 1)], {j, 1, u}] +
    Sum[b[u + j - 1, o - j, t + 1], {j, 1, o}]]];
a[n_] := Function[p, Sum[Coefficient[p, x, i]^n, {i, 0, Exponent[p, x]}]][ b[n, 0, 0]];
a /@ Range[0, 12] (* Jean-François Alcover, Dec 20 2020, after Alois P. Heinz *)

Formula

a(n) = A334622(n,n).

a(n) = Sum_{j=0..ceiling(2^(n-1))-1} A060351(n,j)^n.