A108796 - OEIS

A108796

Number of unordered pairs of partitions of n (into distinct parts) with empty intersection.

1, 0, 0, 1, 1, 3, 4, 7, 9, 16, 21, 33, 46, 68, 95, 140, 187, 266, 372, 507, 683, 948, 1256, 1692, 2263, 3003, 3955, 5248, 6824, 8921, 11669, 15058, 19413, 25128, 32149, 41129, 52578, 66740, 84696, 107389, 135310, 170277, 214386, 268151, 335261, 418896, 521204

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,6

COMMENTS

Counted as orderless pairs since intersection is commutative.

LINKS

Vaclav Kotesovec Table of n, a(n) for n = 0..1050 (terms 0..700 from Alois P. Heinz)

FORMULA

a(n) = ceiling(1/2 * [(x*y)^n] Product_{j>0} (1+x^j+y^j)). - Alois P. Heinz, Mar 31 2017

a(n) = ceiling(A365662(n)/2). - Gus Wiseman, Oct 07 2023

EXAMPLE

Of the partitions of 12 into different parts, the partition (5+4+2+1) has an empty intersection with only (12) and (9+3).

From Gus Wiseman, Oct 07 2023: (Start)

The a(6) = 4 pairs are:

((6),(5,1))

((6),(4,2))

((6),(3,2,1))

((5,1),(4,2))

(End)

MATHEMATICA

using DiscreteMath`Combinatorica`and ListPartitionsQ[n_Integer]:= Flatten[ Reverse /@ Table[(Range[m-1, 0, -1]+#1&)/@ TransposePartition/@ Complement[Partitions[ n-m* (m-1)/2, m], Partitions[n-m*(m-1)/2, m-1]], {m, -1+Floor[1/2*(1+Sqrt[1+8*n])]}], 1]; Table[Plus@@Flatten[Outer[If[Intersection[Flatten[ #1], Flatten[ #2]]==={}, 1, 0]&, ListPartitionsQ[k], ListPartitionsQ[k], 1]], {k, 48}]/2

nmax = 50; p = 1; Do[p = Expand[p*(1 + x^j + y^j)]; p = Select[p, (Exponent[#, x] <= nmax) && (Exponent[#, y] <= nmax) &], {j, 1, nmax}]; p = Select[p, Exponent[#, x] == Exponent[#, y] &]; Table[Coefficient[p, x^n*y^n]/2, {n, 1, nmax}] (* Vaclav Kotesovec, Apr 07 2017 *)

Table[Length[Select[Subsets[Select[IntegerPartitions[n], UnsameQ@@#&], {2}], Intersection@@#=={}&]], {n, 15}] (* Gus Wiseman, Oct 07 2023 *)

PROG

(PARI) a(n) = {my(A=1 + O(x*x^n) + O(y*y^n)); polcoef(polcoef(prod(k=1, n, A + x^k + y^k), n), n)/2} \\ Andrew Howroyd, Oct 10 2023

CROSSREFS

Column k=2 of A258280.

Cf. A079122, A086737.

Main diagonal of A284593 times (1/2).

This is the strict case of A260669.

The ordered version is A365662 = strict case of A054440.

This is the disjoint case of A366132, with twins A366317.

A000041 counts integer partitions, strict A000009.

A002219 counts biquanimous partitions, strict A237258, ordered A064914.