A344532 Number of cycle-up-down permutations of [n^2] having n cycles.

1, 1, 7, 14698, 51629528080, 914192102910317528125, 199979553262025879510473132453855232, 1131253316618666789979709230473744963049785439771172168, 309491168658231587025767619097898747214052900521443034546657433273562730332160

Offset: 0

Author

Alois P. Heinz, May 22 2021

Keywords

• nonn

Comments

For the definition of cycle-up-down permutations see A186366.

Examples

			a(2) = 7: (1)(243), (143)(2), (142)(3), (132)(4), (12)(34), (13)(24), (14)(23).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..22
• Wikipedia, Permutation

Crossrefs

Cf. A186366, A218141, A344445.

Programs

Maple

b:= proc(u, o) option remember; `if`(u+o=0, 1,
      add(b(o-1+j, u-j), j=1..u))
    end:
g:= proc(n) option remember; expand(`if`(n=0, 1,
      add(g(n-j)*binomial(n-1, j-1)*x*b(j-1, 0), j=1..n)))
    end:
a:= n-> coeff(g(n^2), x, n):
seq(a(n), n=0..9);

Mathematica

b[u_, o_] := b[u, o] = If[u+o == 0, 1, Sum[b[o-1+j, u-j], {j, 1, u}]];
g[n_] := g[n] = Expand[If[n == 0, 1,
     Sum[g[n-j]*Binomial[n-1, j-1]*x*b[j-1, 0], {j, 1, n}]]];
a[n_] := Coefficient[g[n^2], x, n];
a /@ Range[0, 9] (* Jean-François Alcover, Jun 10 2021, after Alois P. Heinz *)

Formula

a(n) = (n^2)! * [x^(n^2) y^n] 1/(1-sin(x))^y.

a(n) = A186366(n^2,n).