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A297388 Number of pairs (p,q) of partitions such that q is a partition of n and p <= q (diagram containment).

%I A297388 #26 May 24 2018 20:44:48
%S A297388 1,2,6,13,30,58,120,219,413,730,1296,2201,3766,6206,10241,16500,26502,
%T A297388 41748,65600,101417,156264,237741,360146,539838,806030,1192365,
%U A297388 1756766,2568418,3739724,5408247,7791474,11156601,15916288,22585112,31933166,44932450,63010688
%N A297388 Number of pairs (p,q) of partitions such that q is a partition of n and p <= q (diagram containment).
%C A297388 For fixed q, the number of p is given by a determinant due to MacMahon (the case mu=empty set and n=1 of Exercise 3.149 of the reference below).
%D A297388 R. Stanley, Enumerative Combinatorics, vol. 1, second ed., Cambridge Univ. Press, 2012.
%H A297388 Alois P. Heinz, <a href="/A297388/b297388.txt">Table of n, a(n) for n = 0..1000</a>
%F A297388 a(n) = A000041(n) + Sum_{k=1..n} A259478(n,k). - _Alois P. Heinz_, Jan 10 2018
%e A297388 For n = 2 the six pairs are (empty set,2), (1,2), (2,2), (empty set,11), (1,11), (11,11).
%p A297388 b:= proc(n, i, t) option remember; `if`(n=0 or i=1, 1+
%p A297388       `if`(t=0, 0, n), b(n, i-1, min(i-1, t))+ add(
%p A297388        b(n-i, min(i, n-i), min(j, n-i)), j=0..t))
%p A297388     end:
%p A297388 a:= n-> b(n$3):
%p A297388 seq(a(n), n=0..40);  # _Alois P. Heinz_, Dec 29 2017
%t A297388 b[n_, i_, t_] := b[n, i, t] = If[n == 0 || i == 1, 1 + If[t == 0, 0, n], b[n, i - 1, Min[i - 1, t]] + Sum[b[n - i, Min[i, n - i], Min[j, n - i]], {j, 0, t}]];
%t A297388 a[n_] := b[n, n, n];
%t A297388 Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Apr 30 2018, after _Alois P. Heinz_ *)
%Y A297388 Cf. A000041, A259478, A305023.
%K A297388 nonn,easy
%O A297388 0,2
%A A297388 _Richard Stanley_, Dec 29 2017