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A332907 Number of entries in the third cycles of all permutations of [n] when cycles are ordered by increasing lengths.

%I A332907 #10 Dec 12 2021 10:01:07
%S A332907 1,13,101,896,7967,78205,827521,9507454,117211469,1560454523,
%T A332907 22172178965,336532052884,5423997488041,92726171603161,
%U A332907 1673203210233137,31845893246619770,636647098018469141,13356074486442181999,293166974869955073469,6724854183662407594768
%N A332907 Number of entries in the third cycles of all permutations of [n] when cycles are ordered by increasing lengths.
%H A332907 Alois P. Heinz, <a href="/A332907/b332907.txt">Table of n, a(n) for n = 3..450</a>
%H A332907 Andrew V. Sills, <a href="https://arxiv.org/abs/1912.05306">Integer Partitions Probability Distributions</a>, arXiv:1912.05306 [math.CO], 2019.
%H A332907 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%F A332907 a(n) = Sum_{k=0..n-2} k * A350016(n,k). - _Alois P. Heinz_, Dec 12 2021
%p A332907 b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i>n, 0,
%p A332907       add((p-> p+`if`(t>0 and t-j<1, [0, p[1]*i], 0))((i-1)!^j*
%p A332907         b(n-i*j, i+1, max(0, t-j))/j!*combinat[multinomial]
%p A332907          (n, i$j, n-i*j)), j=0..n/i)))
%p A332907     end:
%p A332907 a:= n-> b(n, 1, 3)[2]:
%p A332907 seq(a(n), n=3..22);
%t A332907 multinomial[n_, k_List] := n!/Times @@ (k!);
%t A332907 b[n_, i_, t_] := b[n, i, t] = If[n == 0, {1, 0}, If[i > n, 0, Sum[Function[ p, p + If[p =!= 0 && 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[Array[i&, j], n - i*j]]], {j, 0, n/i}]]];
%t A332907 a[n_] := b[n, 1, 3][[2]];
%t A332907 a /@ Range[3, 22] (* _Jean-François Alcover_, Apr 21 2020, after _Alois P. Heinz_ *)
%Y A332907 Column k=3 of A322383.
%Y A332907 Cf. A350016.
%K A332907 nonn
%O A332907 3,2
%A A332907 _Alois P. Heinz_, Mar 02 2020