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A322440 Number of pairs of integer partitions of n where every part of the first is less than every part of the second. 6
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 (list; graph; refs; listen; history; text; internal format)
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}]

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

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Last modified July 12 00:27 EDT 2020. Contains 335658 sequences. (Running on oeis4.)