A332853 Number of entries in the fourth cycles of all permutations of [n] when cycles are ordered by decreasing lengths.

1, 11, 101, 932, 9185, 98317, 1141614, 14297174, 192303794, 2767778378, 42482344932, 692989979996, 11977186926515, 218693683265139, 4207257180352118, 85067513019575426, 1803695291827381964, 40022639769763695724, 927602835889551674104, 22416055484058795043176

Offset: 4

Author

Alois P. Heinz, Feb 26 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 A322384.

Programs

Maple

b:= proc(n, l) option remember; `if`(n=0, l[4], add((j-1)!*b(n-j,
      sort([l[], j], `>`)[1..4])*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n, [0$4]):
seq(a(n), n=4..24);

Mathematica

b[n_, l_] := b[n, l] = If[n == 0, l[[4]], Sum[(j-1)! b[n-j, ReverseSort[ Append[l, j]][[1 ;; 4]]] Binomial[n - 1, j - 1], {j, 1, n}]];
a[n_] := b[n, {0, 0, 0, 0}];
a /@ Range[4, 24] (* Jean-François Alcover, Mar 01 2020, after Alois P. Heinz *)