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A322439 Number of ordered pairs of integer partitions of n where no part of the first is greater than any part of the second.

%I A322439 #19 Apr 30 2023 12:56:50
%S A322439 1,1,3,5,11,15,33,42,82,114,195,258,466,587,954,1317,2021,2637,4124,
%T A322439 5298,7995,10565,15075,19665,28798,36773,51509,67501,93060,119299,
%U A322439 165589,209967,285535,366488,487536,622509,833998,1048119,1380410,1754520,2291406,2876454
%N A322439 Number of ordered pairs of integer partitions of n where no part of the first is greater than any part of the second.
%H A322439 Alois P. Heinz, <a href="/A322439/b322439.txt">Table of n, a(n) for n = 0..10000</a>
%F A322439 a(n) = Sum_{k = 1..n} A026820(n,k) * A026794(n,k).
%F A322439 a(n) = A000041(2n) - A362051(n) for n>=1. - _Alois P. Heinz_, Apr 27 2023
%e A322439 The a(5) = 15 pairs of integer partitions:
%e A322439       (5)|(5)
%e A322439      (41)|(5)
%e A322439      (32)|(5)
%e A322439     (311)|(5)
%e A322439     (221)|(5)
%e A322439     (221)|(32)
%e A322439    (2111)|(5)
%e A322439    (2111)|(32)
%e A322439   (11111)|(5)
%e A322439   (11111)|(41)
%e A322439   (11111)|(32)
%e A322439   (11111)|(311)
%e A322439   (11111)|(221)
%e A322439   (11111)|(2111)
%e A322439   (11111)|(11111)
%p A322439 g:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
%p A322439       g(n, i-1) +g(n-i, min(i, n-i)))
%p A322439     end:
%p A322439 b:= proc(n, i) option remember; `if`(n=0, 1,
%p A322439       `if`(i>n, 0, b(n, i+1)+b(n-i, i)))
%p A322439     end:
%p A322439 a:= proc(n) option remember; `if`(n=0, 1,
%p A322439       add(g(n, i)*b(n-i, i), i=1..n))
%p A322439     end:
%p A322439 seq(a(n), n=0..50);  # _Alois P. Heinz_, Dec 09 2018
%t A322439 Table[Length[Select[Tuples[IntegerPartitions[n],2],Max@@First[#]<=Min@@Last[#]&]],{n,20}]
%t A322439 (* Second program: *)
%t A322439 g[n_, i_] := g[n, i] = If[n == 0 || i == 1, 1, g[n, i - 1] + g[n - i, Min[i, n - i]]];
%t A322439 b[n_, i_] := b[n, i] = If[n == 0, 1, If[i>n, 0, b[n, i+1] + b[n-i, i]]];
%t A322439 a[n_] := a[n] = If[n == 0, 1, Sum[g[n, i]*b[n - i, i], {i, 1, n}]];
%t A322439 a /@ Range[0, 50] (* _Jean-François Alcover_, May 17 2021, after _Alois P. Heinz_ *)
%Y A322439 Cf. A026794, A026820, A265947, A285573, A317144, A318915, A322435, A322436, A322440, A322441, A322442, A362051.
%K A322439 nonn
%O A322439 0,3
%A A322439 _Gus Wiseman_, Dec 08 2018