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

1, 7, 46, 341, 2871, 26797, 275353, 3090544, 37652660, 495756306, 7015094802, 106125820737, 1710625964061, 29267936828691, 529655709670675, 10110999740354242, 203072647138681534, 4280118000323963708, 94470690960204259548, 2179212745888578818307

Offset: 3

Author

Alois P. Heinz, Feb 26 2020

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 3..450
• Andrew V. Sills, Integer Partitions Probability Distributions, arXiv:1912.05306 [math.CO], 2019.
• Wikipedia, Permutation

Crossrefs

Column k=3 of A322384.

Cf. A350015.

Programs

Maple

b:= proc(n, l) option remember; `if`(n=0, l[3], add((j-1)!*b(n-j,
      sort([l[], j], `>`)[1..3])*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n, [0$3]):
seq(a(n), n=3..23);

Mathematica

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}]];
a[n_] := b[n, {0, 0, 0}];
a /@ Range[3, 23] (* Jean-François Alcover, Mar 01 2020, after Alois P. Heinz *)

Formula

a(n) = Sum_{k=0..floor(n/3)} k * A350015(n,k). - Alois P. Heinz, Dec 12 2021