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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.

%I A322440 #11 May 10 2021 07:40:57
%S A322440 1,0,1,2,5,7,16,20,40,55,97,124,235,287,482,654,1033,1318,2137,2676,
%T A322440 4157,5439,7891,10144,15280,19171,27336,35652,49756,63150,89342,
%U A322440 111956,154400,197413,264572,336082,456724,568932,756065,959566,1261803,1576355,2078267
%N A322440 Number of pairs of integer partitions of n where every part of the first is less than every part of the second.
%F A322440 a(n) = Sum_{k=1..n-1} A026820(n, k) * A026794(n, k + 1).
%e A322440 The a(5) = 16 pairs of integer partitions:
%e A322440       (51)|(6)
%e A322440       (42)|(6)
%e A322440      (411)|(6)
%e A322440       (33)|(6)
%e A322440      (321)|(6)
%e A322440     (3111)|(6)
%e A322440      (222)|(6)
%e A322440      (222)|(33)
%e A322440     (2211)|(6)
%e A322440     (2211)|(33)
%e A322440    (21111)|(6)
%e A322440    (21111)|(33)
%e A322440   (111111)|(6)
%e A322440   (111111)|(42)
%e A322440   (111111)|(33)
%e A322440   (111111)|(222)
%p A322440 g:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
%p A322440       g(n, i-1) +g(n-i, min(i, n-i)))
%p A322440     end:
%p A322440 b:= proc(n, i) option remember; `if`(n=0, 1,
%p A322440       `if`(i>n, 0, b(n, i+1)+b(n-i, i)))
%p A322440     end:
%p A322440 a:= proc(n) option remember; `if`(n=0, 1,
%p A322440       add(g(n-i, min(n-i, i))*b(n, i+1), i=1..n))
%p A322440     end:
%p A322440 seq(a(n), n=0..50);  # _Alois P. Heinz_, Dec 09 2018
%t A322440 Table[Length[Select[Tuples[IntegerPartitions[n],2],Max@@First[#]<Min@@Last[#]&]],{n,20}]
%t A322440 (* Second program: *)
%t A322440 g[n_, i_] := g[n, i] = If[n==0 || i==1, 1, g[n, i-1]+g[n-i, Min[i, n-i]]];
%t A322440 b[n_, i_] := b[n, i] = If[n==0, 1, If[i>n, 0, b[n, i+1] + b[n-i, i]]];
%t A322440 a[n_] := a[n] = If[n==0, 1, Sum[g[n-i, Min[n-i, i]]*b[n, i+1], {i, 1, n}]];
%t A322440 a /@ Range[0, 50] (* _Jean-François Alcover_, May 10 2021, after _Alois P. Heinz_ *)
%Y A322440 Cf. A265947, A317144, A318915, A322435, A322436, A322439.
%K A322440 nonn
%O A322440 0,4
%A A322440 _Gus Wiseman_, Dec 08 2018