A369707 - OEIS

%I #17 Feb 29 2024 14:25:12

%S 0,0,0,1,3,6,17,28,62,107,201,316,607,909,1567,2444,4025,5979,9749,

%T 14250,22467,32950,50137,72295,109728,156182,230570,328089,477606,

%U 670213,968324,1346662,1917385,2658120,3736326,5139004,7183707,9798418,13546453,18414693

%N Number of pairs (p,q) of distinct partitions of n such that the set of parts in q is a subset of the set of parts in p.

%H Alois P. Heinz, <a href="/A369707/b369707.txt">Table of n, a(n) for n = 0..350</a>

%F a(n) = A369704(n) - A000041(n).

%e a(5) = 6: (2111, 11111), (2111, 221), (221, 11111), (221, 2111), (311, 11111), (41, 11111).

%e a(6) = 17: (21111, 111111), (21111, 2211), (21111, 222), (2211, 111111), (2211, 21111), (2211, 222), (3111, 111111), (321, 111111), (321, 21111), (321, 2211), (321, 222), (321, 3111), (3111, 33), (321, 33), (411, 111111), (42, 222), (51, 111111).

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

%p      `if`(m=0, 1, 0), `if`(i<1, 0, b(n, m, i-1)+add(

%p       add(b(n-i*j, m-i*h, i-1), h=0..m/i), j=1..n/i)))

%p     end:

%p a:= n-> b(n$3)-combinat[numbpart](n):

%p seq(a(n), n=0..42);

%t b[n_, m_, i_] := b[n, m, i] = If[n == 0,

%t     If[m == 0, 1, 0], If[i < 1, 0, b[n, m, i - 1] +

%t     Sum[Sum[b[n - i*j, m - i*h, i - 1], {h, 0, m/i}], {j, 1, n/i}]]];

%t a[n_] := b[n, n, n] - PartitionsP[n];

%t Table[a[n], {n, 0, 42}] (* _Jean-François Alcover_, Feb 29 2024, after _Alois P. Heinz_ *)

%Y Cf. A000041, A369704, A369910.

%K nonn

%O 0,5

%A _Alois P. Heinz_, Jan 29 2024