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A108796 Number of pairs of partitions of n (into different parts) with empty intersection. 5
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

1,5

COMMENTS

Counted as orderless pairs since intersection is commutative.

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 1..1050 (terms 1..700 from Alois P. Heinz)

FORMULA

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

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).

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 *)

CROSSREFS

Column k=2 of A258280.

Cf. A086737.

Main diagonal of A284593 times (1/2).

Sequence in context: A281734 A086336 A237258 * A048849 A076211 A167186

Adjacent sequences:  A108793 A108794 A108795 * A108797 A108798 A108799

KEYWORD

easy,nonn

AUTHOR

Wouter Meeussen, Jul 09 2005

STATUS

approved

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Last modified March 19 11:10 EDT 2019. Contains 321329 sequences. (Running on oeis4.)