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

1, 2, 6, 13, 30, 58, 120, 219, 413, 730, 1296, 2201, 3766, 6206, 10241, 16500, 26502, 41748, 65600, 101417, 156264, 237741, 360146, 539838, 806030, 1192365, 1756766, 2568418, 3739724, 5408247, 7791474, 11156601, 15916288, 22585112, 31933166, 44932450, 63010688

Offset: 0

Author

Richard Stanley, Dec 29 2017

Keywords

• nonn
• easy

Comments

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).

Examples

			For n = 2 the six pairs are (empty set,2), (1,2), (2,2), (empty set,11), (1,11), (11,11).
		

References

• R. Stanley, Enumerative Combinatorics, vol. 1, second ed., Cambridge Univ. Press, 2012.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000041, A259478, A305023.

Programs

Maple

b:= proc(n, i, t) option remember; `if`(n=0 or i=1, 1+
      `if`(t=0, 0, n), b(n, i-1, min(i-1, t))+ add(
       b(n-i, min(i, n-i), min(j, n-i)), j=0..t))
    end:
a:= n-> b(n$3):
seq(a(n), n=0..40);  # Alois P. Heinz, Dec 29 2017

Mathematica

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}]];
a[n_] := b[n, n, n];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Apr 30 2018, after Alois P. Heinz *)

Formula

a(n) = A000041(n) + Sum_{k=1..n} A259478(n,k). - Alois P. Heinz, Jan 10 2018