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

3,2

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..n-2} k * A350016(n,k). - Alois P. Heinz, Dec 12 2021

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 *)

Crossrefs

Column k=3 of A322383.

Cf. A350016.

Sequence in context: A075604 A142297 A131021 * A087595 A117653 A004635

Adjacent sequences: A332904 A332905 A332906 * A332908 A332909 A332910

Keyword

nonn

Author

Alois P. Heinz, Mar 02 2020

Status

approved