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A286074 Number of permutations of [n] with nondecreasing cycle sizes.

%I A286074 #7 May 25 2018 04:31:55
%S A286074 1,1,2,4,13,45,250,1342,10085,76165,715588,6786108,78636601,896672473,
%T A286074 12112535378,163963519810,2534311844905,39211836764457,
%U A286074 688584972407680,12003902219337760,234324625117308533,4571805253649981173,98183221386947058286
%N A286074 Number of permutations of [n] with nondecreasing cycle sizes.
%C A286074 Each cycle is written with the smallest element first and cycles are arranged in increasing order of their first elements.
%C A286074 a(n) mod 2 = A133872(n).
%H A286074 Alois P. Heinz, <a href="/A286074/b286074.txt">Table of n, a(n) for n = 0..450</a>
%H A286074 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%p A286074 b:= proc(n, i) option remember; `if`(n=0, 1, add(
%p A286074       (j-1)!*b(n-j, j)*binomial(n-1, j-1), j=i..n))
%p A286074     end:
%p A286074 a:= n-> b(n, 1):
%p A286074 seq(a(n), n=0..30);
%t A286074 b[n_, i_] := b[n, i] = If[n == 0, 1, Sum[(j - 1)!*b[n - j, j]*Binomial[n - 1, j - 1], {j, i, n}]];
%t A286074 a[n_] := b[n, 1];
%t A286074 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, May 25 2018, translated from Maple *)
%Y A286074 Cf. A133872, A275311, A286071, A286072, A286073, A286075, A286076, A286077.
%K A286074 nonn
%O A286074 0,3
%A A286074 _Alois P. Heinz_, May 01 2017