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

3,2

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

Formula

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

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 *)

Crossrefs

Column k=3 of A322384.

Cf. A350015.

Sequence in context: A365773 A067318 A072948 * A356827 A178962 A203267

Adjacent sequences: A332849 A332850 A332851 * A332853 A332854 A332855

Keyword

nonn

Author

Alois P. Heinz, Feb 26 2020

Status

approved