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

0,8

Links

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

Formula

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

Example

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).

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 *)

Crossrefs

Cf. A000041, A297388, A369695, A369696.

Sequence in context: A054451 A309479 A352720 * A308775 A305335 A123102

Adjacent sequences: A369694 A369695 A369696 * A369698 A369699 A369700

Keyword

nonn

Author

Alois P. Heinz, Jan 29 2024

Status

approved