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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 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[#]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

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Last modified June 20 13:53 EDT 2021. Contains 345165 sequences. (Running on oeis4.)