A286071 Number of permutations of [n] with nonincreasing cycle sizes.

1, 1, 2, 5, 19, 85, 496, 3229, 25117, 215225, 2100430, 22187281, 261228199, 3284651245, 45163266604, 659277401525, 10380194835601, 172251467909809, 3057368096689690, 56867779157145769, 1122474190194034555, 23137433884903034501, 502874858021076645784

Offset: 0

Author

Alois P. Heinz, May 01 2017

Keywords

• nonn

Comments

Each cycle is written with the smallest element first and cycles are arranged in increasing order of their first elements.

a(n) is even if and only if n in { A016825 }.

Examples

			a(3) = 5: (123), (132), (12)(3), (13)(2), (1)(2)(3).
a(4) = 19: (1234), (1243), (1324), (1342), (1423), (1432), (123)(4), (132)(4), (124)(3), (142)(3), (12)(34), (12)(3)(4), (134)(2), (143)(2), (13)(24), (13)(2)(4), (14)(23), (14)(2)(3), (1)(2)(3)(4).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..450
• Wikipedia, Permutation

Crossrefs

Cf. A016825, A275310, A286072, A286073, A286074, A286075, A286076, A286077.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, add((j-1)!*
      b(n-j, j)*binomial(n-1, j-1), j=1..min(n, i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, 1, Sum[(j - 1)!*b[n - j, j]*Binomial[n - 1, j - 1], {j, 1, Min[n, i]}]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 24 2018, translated from Maple *)