A332854 Number of entries in the fifth cycles of all permutations of [n] when cycles are ordered by decreasing lengths.

1, 16, 197, 2311, 27568, 343909, 4541329, 63719723, 949770615, 15010838233, 250997692441, 4430433962701, 82376202579421, 1610014961936672, 33010385435710028, 708642421376230354, 15899009565671538930, 372166745683645768206, 9074749796104015627262

Offset: 5

Author

Alois P. Heinz, Feb 26 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 A322384.

Programs

Maple

b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
      add((p-> p+`if`(t>0 and t-j<1, [0, p[1]*i], 0))((i-1)!^j*
        b(n-i*j, min(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$2, 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<1, {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, Min[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, n, 5][[2]];
Table[a[n], {n, 5, 23}] (* Jean-François Alcover, Mar 13 2021, after Alois P. Heinz *)