A369697 Number of unordered pairs (p,q) of distinct partitions of n such that the set of parts in q is equal to the set of parts in p.

0, 0, 0, 0, 0, 1, 1, 4, 5, 12, 19, 35, 53, 91, 136, 225, 325, 505, 741, 1107, 1590, 2340, 3313, 4748, 6682, 9412, 13091, 18241, 25080, 34478, 47118, 64069, 86698, 117012, 157121, 210189, 280385, 372309, 493279, 650905, 856913, 1123675, 1471196, 1918293, 2497470

Offset: 0

Author

Alois P. Heinz, Jan 29 2024

Keywords

• nonn

Examples

			a(5) = 1: (221, 2111).
a(6) = 1: (2211, 21111).
a(7) = 4: (22111, 211111), (2221, 211111), (2221, 22111), (331, 31111).
a(8) = 5: (221111, 2111111), (22211, 2111111), (22211, 221111), (3221, 32111), (3311, 311111).
a(9) = 12: (2211111, 21111111), (222111, 21111111), (222111, 2211111), (22221, 21111111), (22221, 2211111), (22221, 222111), (32211, 321111), (33111, 3111111), (3321, 321111), (3321, 32211), (4221, 42111), (441, 411111).
a(10) = 19: (22111111, 211111111), (2221111, 211111111), (2221111, 22111111), (222211, 211111111), (222211, 22111111), (222211, 2221111), (322111, 3211111), (32221, 3211111), (32221, 322111), (331111, 31111111), (33211, 3211111), (33211, 322111), (33211, 32221), (3331, 31111111), (3331, 331111), (42211, 421111), (4411, 4111111), (442, 4222), (5221, 52111).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..350

Crossrefs

Cf. A000041, A297388, A369695, A369696.

Programs

Maple

b:= proc(n, m, i) option remember; `if`(n=0,
     `if`(m=0, 1, 0), `if`(i<1, 0, b(n, m, i-1)+add(add(
      b(sort([n-i*j, m-i*h])[], i-1), h=1..m/i), j=1..n/i)))
    end:
a:= n-> (b(n$3)-combinat[numbpart](n))/2:
seq(a(n), n=0..50);

Mathematica

b[n_, m_, i_] := b[n, m, i] = If[n == 0, If[m == 0, 1, 0], If[i < 1, 0, b[n, m, i-1] + Sum[Sum[b[Sequence @@ Sort[{n-i*j, m-i*h}], i-1], {h, 1, m/i}], { j, 1, n/i}]]];
a[n_] := (b[n, n, n] - PartitionsP[n])/2;
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Feb 29 2024, after Alois P. Heinz *)

Formula

a(n) = (A369695(n) - A000041(n))/2.