A332909 Number of entries in the fifth cycles of all permutations of [n] when cycles are ordered by increasing lengths.

1, 31, 442, 6441, 88909, 1253104, 18332744, 280902678, 4497959259, 75694569341, 1336697348846, 24765423361061, 480653174845257, 9764210398405166, 207238383834819974, 4591419670284107644, 106002478632623159679, 2547169063966472089803, 63617191700084723716234

Offset: 5

Author

Alois P. Heinz, Mar 02 2020

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 5..450
• Andrew V. Sills, Integer Partitions Probability Distributions, arXiv:1912.05306 [math.CO], 2019.
• Wikipedia, Permutation

Crossrefs

Column k=5 of A322383.

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, 5)[2]:
seq(a(n), n=5..23);

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, 0}, Sum[ Function[p, p + If[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[Table[i, {j}], n - i*j]]], {j, 0, n/i}]]];
a[n_] := b[n, 1, 5][[2]];
Table[a[n], {n, 5, 23}] (* Jean-François Alcover, Mar 14 2021, after Alois P. Heinz *)