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

5,2

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

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 *)

Crossrefs

Column k=5 of A322384.

Sequence in context: A093060 A153885 A016226 * A154240 A081679 A154249

Adjacent sequences: A332851 A332852 A332853 * A332855 A332856 A332857

Keyword

nonn

Author

Alois P. Heinz, Feb 26 2020

Status

approved