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

0,4

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

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 *)

Crossrefs

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

Sequence in context: A227292 A179683 A179685 * A286072 A002830 A365654

Adjacent sequences: A286074 A286075 A286076 * A286078 A286079 A286080

Keyword

nonn

Author

Alois P. Heinz, May 01 2017

Status

approved