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A275389
Number of set partitions of [n] with a strongly unimodal block size list.
3
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
0,4
COMMENTS
Strongly unimodal means strictly increasing then strictly decreasing.
LINKS
EXAMPLE
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.
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 *)
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jul 26 2016
STATUS
approved