A332851 Number of entries in the second cycles of all permutations of [n] when cycles are ordered by decreasing lengths.

1, 4, 21, 131, 950, 7694, 70343, 709015, 7867174, 94945874, 1241471451, 17442660795, 262748183438, 4215551220698, 71873188059263, 1296535610096063, 24690833276595022, 494624740502491786, 10406110949475668267, 229244825016301188267, 5280217980442316367510

Offset

2,2

Links

Alois P. Heinz, Table of n, a(n) for n = 2..450

Andrew V. Sills, Integer Partitions Probability Distributions, arXiv:1912.05306 [math.CO], 2019.

Wikipedia, Permutation

Formula

a(n) = Sum_{k=0..floor(n/2)} k * A349979(n,k). - Alois P. Heinz, Dec 07 2021

Maple

b:= proc(n, l) option remember; `if`(n=0, l[2], add((j-1)!*b(n-j,
      sort([l[], j], `>`)[1..2])*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n, [0$2]):
seq(a(n), n=2..23);

Mathematica

b[n_, l_] := b[n, l] = If[n == 0, l[[2]], Sum[(j-1)! b[n-j, ReverseSort[ Append[l, j]][[1 ;; 2]]] Binomial[n - 1, j - 1], {j, 1, n}]];
a[n_] := b[n, {0, 0}];
a /@ Range[2, 23] (* Jean-François Alcover, Mar 01 2020, after Alois P. Heinz *)

Crossrefs

Column k=2 of A322384.

Cf. A349979.

Sequence in context: A090366 A273956 A131965 * A303563 A284816 A226067

Adjacent sequences: A332848 A332849 A332850 * A332852 A332853 A332854

Keyword

nonn

Author

Alois P. Heinz, Feb 26 2020

Status

approved