A332907 Number of entries in the third cycles of all permutations of [n] when cycles are ordered by increasing lengths.

1, 13, 101, 896, 7967, 78205, 827521, 9507454, 117211469, 1560454523, 22172178965, 336532052884, 5423997488041, 92726171603161, 1673203210233137, 31845893246619770, 636647098018469141, 13356074486442181999, 293166974869955073469, 6724854183662407594768

Offset: 3

Author

Alois P. Heinz, Mar 02 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 A322383.

Cf. A350016.

Programs

Maple

b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i>n, 0,
      add((p-> p+`if`(t>0 and t-j<1, [0, p[1]*i], 0))((i-1)!^j*
        b(n-i*j, i+1, max(0, t-j))/j!*combinat[multinomial]
         (n, i$j, n-i*j)), j=0..n/i)))
    end:
a:= n-> b(n, 1, 3)[2]:
seq(a(n), n=3..22);

Mathematica

multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, t_] := b[n, i, t] = If[n == 0, {1, 0}, If[i > n, 0, Sum[Function[ p, p + If[p =!= 0 && t > 0 && t - j < 1, {0, p[[1]]*i}, {0, 0}]][(i - 1)!^j*b[n - i*j, i + 1, Max[0, t - j]]/j!*multinomial[n, Append[Array[i&, j], n - i*j]]], {j, 0, n/i}]]];
a[n_] := b[n, 1, 3][[2]];
a /@ Range[3, 22] (* Jean-François Alcover, Apr 21 2020, after Alois P. Heinz *)

Formula

a(n) = Sum_{k=0..n-2} k * A350016(n,k). - Alois P. Heinz, Dec 12 2021