A322439 - OEIS

%I #19 Apr 30 2023 12:56:50

%S 1,1,3,5,11,15,33,42,82,114,195,258,466,587,954,1317,2021,2637,4124,

%T 5298,7995,10565,15075,19665,28798,36773,51509,67501,93060,119299,

%U 165589,209967,285535,366488,487536,622509,833998,1048119,1380410,1754520,2291406,2876454

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

%H Alois P. Heinz, <a href="/A322439/b322439.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n) = Sum_{k = 1..n} A026820(n,k) * A026794(n,k).

%F a(n) = A000041(2n) - A362051(n) for n>=1. - _Alois P. Heinz_, Apr 27 2023

%e The a(5) = 15 pairs of integer partitions:

%e       (5)|(5)

%e      (41)|(5)

%e      (32)|(5)

%e     (311)|(5)

%e     (221)|(5)

%e     (221)|(32)

%e    (2111)|(5)

%e    (2111)|(32)

%e   (11111)|(5)

%e   (11111)|(41)

%e   (11111)|(32)

%e   (11111)|(311)

%e   (11111)|(221)

%e   (11111)|(2111)

%e   (11111)|(11111)

%p g:= proc(n, i) option remember; `if`(n=0 or i=1, 1,

%p       g(n, i-1) +g(n-i, min(i, n-i)))

%p     end:

%p b:= proc(n, i) option remember; `if`(n=0, 1,

%p       `if`(i>n, 0, b(n, i+1)+b(n-i, i)))

%p     end:

%p a:= proc(n) option remember; `if`(n=0, 1,

%p       add(g(n, i)*b(n-i, i), i=1..n))

%p     end:

%p seq(a(n), n=0..50);  # _Alois P. Heinz_, Dec 09 2018

%t Table[Length[Select[Tuples[IntegerPartitions[n],2],Max@@First[#]<=Min@@Last[#]&]],{n,20}]

%t (* Second program: *)

%t g[n_, i_] := g[n, i] = If[n == 0 || i == 1, 1, g[n, i - 1] + g[n - i, Min[i, n - i]]];

%t b[n_, i_] := b[n, i] = If[n == 0, 1, If[i>n, 0, b[n, i+1] + b[n-i, i]]];

%t a[n_] := a[n] = If[n == 0, 1, Sum[g[n, i]*b[n - i, i], {i, 1, n}]];

%t a /@ Range[0, 50] (* _Jean-François Alcover_, May 17 2021, after _Alois P. Heinz_ *)

%Y Cf. A026794, A026820, A265947, A285573, A317144, A318915, A322435, A322436, A322440, A322441, A322442, A362051.

%K nonn

%O 0,3

%A _Gus Wiseman_, Dec 08 2018