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

0,4

Links

Table of n, a(n) for n=0..42.

Formula

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

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

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

Sequence in context: A032141 A032045 A207035 * A067580 A325210 A181447

Adjacent sequences: A322437 A322438 A322439 * A322441 A322442 A322443

Keyword

nonn

Author

Gus Wiseman, Dec 08 2018

Status

approved