A322440 Number of pairs of integer partitions of n where every part of the first is less than every part of the second.

1, 0, 1, 2, 5, 7, 16, 20, 40, 55, 97, 124, 235, 287, 482, 654, 1033, 1318, 2137, 2676, 4157, 5439, 7891, 10144, 15280, 19171, 27336, 35652, 49756, 63150, 89342, 111956, 154400, 197413, 264572, 336082, 456724, 568932, 756065, 959566, 1261803, 1576355, 2078267

Offset: 0

Author

Gus Wiseman, Dec 08 2018

Keywords

• nonn

Examples

			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)
		

Crossrefs

Cf. A265947, A317144, A318915, A322435, A322436, A322439.

Programs

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[#]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 *)

Formula

a(n) = Sum_{k=1..n-1} A026820(n, k) * A026794(n, k + 1).