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

5,2

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>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 *)

Crossrefs

Column k=5 of A322383.

Sequence in context: A278964 A010836 A022723 * A084234 A187622 A187630

Adjacent sequences: A332906 A332907 A332908 * A332910 A332911 A332912

Keyword

nonn

Author

Alois P. Heinz, Mar 02 2020

Status

approved