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A332909 Number of entries in the fifth cycles of all permutations of [n] when cycles are ordered by increasing lengths.

%I A332909 #8 Mar 14 2021 07:46:51
%S A332909 1,31,442,6441,88909,1253104,18332744,280902678,4497959259,
%T A332909 75694569341,1336697348846,24765423361061,480653174845257,
%U A332909 9764210398405166,207238383834819974,4591419670284107644,106002478632623159679,2547169063966472089803,63617191700084723716234
%N A332909 Number of entries in the fifth cycles of all permutations of [n] when cycles are ordered by increasing lengths.
%H A332909 Alois P. Heinz, <a href="/A332909/b332909.txt">Table of n, a(n) for n = 5..450</a>
%H A332909 Andrew V. Sills, <a href="https://arxiv.org/abs/1912.05306">Integer Partitions Probability Distributions</a>, arXiv:1912.05306 [math.CO], 2019.
%H A332909 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%p A332909 b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i>n, 0,
%p A332909       add((p-> p+`if`(t>0 and t-j<1, [0, p[1]*i], 0))((i-1)!^j*
%p A332909         b(n-i*j, i+1, max(0, t-j))/j!*combinat[multinomial]
%p A332909          (n, i$j, n-i*j)), j=0..n/i)))
%p A332909     end:
%p A332909 a:= n-> b(n, 1, 5)[2]:
%p A332909 seq(a(n), n=5..23);
%t A332909 multinomial[n_, k_List] := n!/Times @@ (k!);
%t A332909 b[n_, i_, t_] := b[n, i, t] = If[n == 0, {1, 0}, If[i > n, {0, 0}, Sum[ Function[p, p + If[t > 0 && t - j < 1, {0, p[[1]]*i}, {0, 0}]][(i - 1)!^j*b[n - i*j, i + 1, Max[0, t - j]]/j!*multinomial[n, Append[Table[i, {j}], n - i*j]]], {j, 0, n/i}]]];
%t A332909 a[n_] := b[n, 1, 5][[2]];
%t A332909 Table[a[n], {n, 5, 23}] (* _Jean-François Alcover_, Mar 14 2021, after _Alois P. Heinz_ *)
%Y A332909 Column k=5 of A322383.
%K A332909 nonn
%O A332909 5,2
%A A332909 _Alois P. Heinz_, Mar 02 2020