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%I #9 May 28 2018 10:22:42
%S 1,1,1,5,16,80,468,3220,24436,218032,2114244,22759788,267150264,
%T 3413938512,46668380592,690881123856,10841100147072,181434400544160,
%U 3215124610986240,60280035304993920,1186176116251848960,24624604679704053120,534223121657911528320
%N Number of permutations of [n] with a strongly unimodal cycle size list.
%C Each cycle is written with the smallest element first and cycles are arranged in increasing order of their first elements.
%C Strongly unimodal means strictly increasing then strictly decreasing.
%H Alois P. Heinz, <a href="/A286077/b286077.txt">Table of n, a(n) for n = 0..450</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%p b:= proc(n, i, t) option remember; `if`(t=0 and n>i*(i-1)/2, 0,
%p `if`(n=0, 1, add(b(n-j, j, 0)*binomial(n-1, j-1)*
%p (j-1)!, j=1..min(n, i-1))+`if`(t=1, add(b(n-j, j, 1)*
%p binomial(n-1, j-1)*(j-1)!, j=i+1..n), 0)))
%p end:
%p a:= n-> b(n, 0, 1):
%p seq(a(n), n=0..30);
%t 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]]];
%t a[n_] := b[n, 0, 1];
%t Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, May 28 2018, from Maple *)
%Y Cf. A275389, A286071, A286072, A286073, A286074, A286075, A286076.
%K nonn
%O 0,4
%A _Alois P. Heinz_, May 01 2017
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