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A332852 Number of entries in the third cycles of all permutations of [n] when cycles are ordered by decreasing lengths.

%I A332852 #14 Dec 12 2021 10:04:58
%S A332852 1,7,46,341,2871,26797,275353,3090544,37652660,495756306,7015094802,
%T A332852 106125820737,1710625964061,29267936828691,529655709670675,
%U A332852 10110999740354242,203072647138681534,4280118000323963708,94470690960204259548,2179212745888578818307
%N A332852 Number of entries in the third cycles of all permutations of [n] when cycles are ordered by decreasing lengths.
%H A332852 Alois P. Heinz, <a href="/A332852/b332852.txt">Table of n, a(n) for n = 3..450</a>
%H A332852 Andrew V. Sills, <a href="https://arxiv.org/abs/1912.05306">Integer Partitions Probability Distributions</a>, arXiv:1912.05306 [math.CO], 2019.
%H A332852 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%F A332852 a(n) = Sum_{k=0..floor(n/3)} k * A350015(n,k). - _Alois P. Heinz_, Dec 12 2021
%p A332852 b:= proc(n, l) option remember; `if`(n=0, l[3], add((j-1)!*b(n-j,
%p A332852       sort([l[], j], `>`)[1..3])*binomial(n-1, j-1), j=1..n))
%p A332852     end:
%p A332852 a:= n-> b(n, [0$3]):
%p A332852 seq(a(n), n=3..23);
%t A332852 b[n_, l_] := b[n, l] = If[n == 0, l[[3]], Sum[(j-1)! b[n-j, ReverseSort[ Append[l, j]][[1 ;; 3]]] Binomial[n - 1, j - 1], {j, 1, n}]];
%t A332852 a[n_] := b[n, {0, 0, 0}];
%t A332852 a /@ Range[3, 23] (* _Jean-François Alcover_, Mar 01 2020, after _Alois P. Heinz_ *)
%Y A332852 Column k=3 of A322384.
%Y A332852 Cf. A350015.
%K A332852 nonn
%O A332852 3,2
%A A332852 _Alois P. Heinz_, Feb 26 2020