OFFSET
0,2
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).
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
FORMULA
EXAMPLE
For n = 2 the six pairs are (empty set,2), (1,2), (2,2), (empty set,11), (1,11), (11,11).
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 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Richard Stanley, Dec 29 2017
STATUS
approved