A286077 Number of permutations of [n] with a strongly unimodal cycle size list.

1, 1, 1, 5, 16, 80, 468, 3220, 24436, 218032, 2114244, 22759788, 267150264, 3413938512, 46668380592, 690881123856, 10841100147072, 181434400544160, 3215124610986240, 60280035304993920, 1186176116251848960, 24624604679704053120, 534223121657911528320

Offset: 0

Author

Alois P. Heinz, May 01 2017

Keywords

• nonn

Comments

Each cycle is written with the smallest element first and cycles are arranged in increasing order of their first elements.

Strongly unimodal means strictly increasing then strictly decreasing.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..450
• Wikipedia, Permutation

Crossrefs

Cf. A275389, A286071, A286072, A286073, A286074, A286075, A286076.

Programs

Maple

b:= proc(n, i, t) option remember; `if`(t=0 and n>i*(i-1)/2, 0,
      `if`(n=0, 1, add(b(n-j, j, 0)*binomial(n-1, j-1)*
      (j-1)!, j=1..min(n, i-1))+`if`(t=1, add(b(n-j, j, 1)*
       binomial(n-1, j-1)*(j-1)!, j=i+1..n), 0)))
    end:
a:= n-> b(n, 0, 1):
seq(a(n), n=0..30);

Mathematica

b[n_, i_, t_] := b[n, i, t] = If[t == 0 && n > i*(i-1)/2, 0, If[n == 0, 1, Sum[b[n-j, j, 0]*Binomial[n-1, j-1]*(j-1)!, {j, 1, Min[n, i-1]}] + If[t == 1, Sum[b[n-j, j, 1]*Binomial[n-1, j-1]*(j-1)!, {j, i+1, n}], 0]]];
a[n_] := b[n, 0, 1];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 28 2018, from Maple *)