A332852 - OEIS

%I #14 Dec 12 2021 10:04:58

%S 1,7,46,341,2871,26797,275353,3090544,37652660,495756306,7015094802,

%T 106125820737,1710625964061,29267936828691,529655709670675,

%U 10110999740354242,203072647138681534,4280118000323963708,94470690960204259548,2179212745888578818307

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

%H Alois P. Heinz, <a href="/A332852/b332852.txt">Table of n, a(n) for n = 3..450</a>

%H Andrew V. Sills, <a href="https://arxiv.org/abs/1912.05306">Integer Partitions Probability Distributions</a>, arXiv:1912.05306 [math.CO], 2019.

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

%F a(n) = Sum_{k=0..floor(n/3)} k * A350015(n,k). - _Alois P. Heinz_, Dec 12 2021

%p b:= proc(n, l) option remember; `if`(n=0, l[3], add((j-1)!*b(n-j,

%p       sort([l[], j], `>`)[1..3])*binomial(n-1, j-1), j=1..n))

%p     end:

%p a:= n-> b(n, [0$3]):

%p seq(a(n), n=3..23);

%t 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 a[n_] := b[n, {0, 0, 0}];

%t a /@ Range[3, 23] (* _Jean-François Alcover_, Mar 01 2020, after _Alois P. Heinz_ *)

%Y Column k=3 of A322384.

%Y Cf. A350015.

%K nonn

%O 3,2

%A _Alois P. Heinz_, Feb 26 2020