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Number of permutations of [n] with decreasing cycle sizes.
8

%I #8 May 24 2018 02:56:40

%S 1,1,1,4,12,60,340,2280,17220,151872,1459584,15624000,182318400,

%T 2316837600,31596570720,465582237120,7283287851840,121620647715840,

%U 2149774858183680,40196871701360640,790002144844738560,16364478334463078400,354458730544573132800

%N Number of permutations of [n] with decreasing cycle sizes.

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

%H Alois P. Heinz, <a href="/A286073/b286073.txt">Table of n, a(n) for n = 0..450</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>

%p b:= proc(n, i) option remember;

%p `if`(n>i*(i+1)/2, 0, `if`(n=0, 1, b(n, i-1)+

%p `if`(i>n, 0, b(n-i, i-1)*(i-1)!*binomial(n-1, i-1))))

%p end:

%p a:= n-> b(n$2):

%p seq(a(n), n=0..30);

%t b[n_, i_] := b[n, i] = If[n > i*(i + 1)/2, 0, If[n == 0, 1, b[n, i - 1] + If[i > n, 0, b[n - i, i - 1]*(i - 1)!*Binomial[n - 1, i - 1]]]];

%t a[n_] := b[n, n];

%t Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, May 24 2018, translated from Maple *)

%Y Cf. A275309, A286071, A286072, A286074, A286075, A286076, A286077.

%K nonn

%O 0,4

%A _Alois P. Heinz_, May 01 2017