A332908 Number of entries in the fourth cycles of all permutations of [n] when cycles are ordered by increasing lengths.

1, 21, 226, 2612, 29261, 346453, 4338214, 57819554, 815225643, 12234293579, 194294281572, 3264124624256, 57826690252441, 1079032037759257, 21142347350725466, 434563256137908638, 9344589765620199919, 209952915324112384719, 4919186923210370523448

Offset: 4

Author

Alois P. Heinz, Mar 02 2020

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 4..450
• Andrew V. Sills, Integer Partitions Probability Distributions, arXiv:1912.05306 [math.CO], 2019.
• Wikipedia, Permutation

Crossrefs

Column k=4 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, 4)[2]:
seq(a(n), n=4..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, 4][[2]];
a /@ Range[4, 22] (* Jean-François Alcover, Apr 21 2020, after Alois P. Heinz *)