OFFSET
0,4
EXAMPLE
The a(5) = 16 pairs of integer partitions:
(51)|(6)
(42)|(6)
(411)|(6)
(33)|(6)
(321)|(6)
(3111)|(6)
(222)|(6)
(222)|(33)
(2211)|(6)
(2211)|(33)
(21111)|(6)
(21111)|(33)
(111111)|(6)
(111111)|(42)
(111111)|(33)
(111111)|(222)
MAPLE
g:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
g(n, i-1) +g(n-i, min(i, n-i)))
end:
b:= proc(n, i) option remember; `if`(n=0, 1,
`if`(i>n, 0, b(n, i+1)+b(n-i, i)))
end:
a:= proc(n) option remember; `if`(n=0, 1,
add(g(n-i, min(n-i, i))*b(n, i+1), i=1..n))
end:
seq(a(n), n=0..50); # Alois P. Heinz, Dec 09 2018
MATHEMATICA
Table[Length[Select[Tuples[IntegerPartitions[n], 2], Max@@First[#]<Min@@Last[#]&]], {n, 20}]
(* Second program: *)
g[n_, i_] := g[n, i] = If[n==0 || i==1, 1, g[n, i-1]+g[n-i, Min[i, n-i]]];
b[n_, i_] := b[n, i] = If[n==0, 1, If[i>n, 0, b[n, i+1] + b[n-i, i]]];
a[n_] := a[n] = If[n==0, 1, Sum[g[n-i, Min[n-i, i]]*b[n, i+1], {i, 1, n}]];
a /@ Range[0, 50] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 08 2018
STATUS
approved