A275389 Number of set partitions of [n] with a strongly unimodal block size list.

1, 1, 1, 4, 7, 19, 71, 219, 759, 2697, 12395, 47477, 231950, 1040116, 4851742, 26690821, 131478031, 736418510, 4262619682, 24680045903, 145629814329, 935900941506, 5778263418232, 37626913475878, 257550263109475, 1782180357952449, 12526035635331581

Offset: 0

Author

Alois P. Heinz, Jul 26 2016

Keywords

• nonn

Comments

Strongly unimodal means strictly increasing then strictly decreasing.

Examples

			a(3) = 4: 123, 12|3, 13|2, 1|23.
a(4) = 7: 1234, 123|4, 124|3, 134|2, 1|234, 1|23|4, 1|24|3.
a(5) = 19: 12345, 1234|5, 1235|4, 123|45, 1245|3, 124|35, 125|34, 12|345, 1345|2, 134|25, 135|24, 13|245, 145|23, 14|235, 15|234, 1|2345, 1|234|5, 1|235|4, 1|245|3.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..677
• Wikipedia, Partition of a set

Crossrefs

Cf. A007837, A038041, A059618, A275309, A275310, A275311, A275312, A275313, A286077.

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..min(n, i-1))
     +`if`(t=1, add(b(n-j, j, 1)*binomial(n-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, Min[n, i-1]}] + If[t==1, Sum[b[n-j, j, 1]*Binomial[n-1, j-1], {j, i+1, n}], 0]]]; a[n_] := b[n, 0, 1]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 07 2017, translated from Maple *)