A114701 Number of sets {p, p'}, where p is a partition of n and p' is conjugate partition of p such that p and p' have no common parts.

1, 0, 1, 1, 1, 1, 2, 2, 4, 5, 5, 7, 9, 9, 13, 15, 18, 22, 30, 32, 41, 48, 57, 65, 82, 88, 111, 124, 148, 169, 203, 225, 275, 310, 363, 408, 484, 537, 635, 709, 824, 918, 1075, 1191, 1379, 1540, 1767, 1971, 2269, 2517, 2889, 3208, 3656, 4068, 4629, 5120, 5813, 6452, 7280, 8068, 9113

Offset: 0

Author

Vladeta Jovovic, Feb 18 2006

Keywords

• nonn

Examples

			a(6)=2 because the pairs of conjugate partitions of 6 are {[6], [1, 1, 1, 1, 1, 1]}, {[3, 3], [2, 2, 2]}, {[5, 1], [2, 1, 1, 1, 1]}, {[4, 2], [2, 2, 1, 1]}, {[3, 2, 1], [3, 2, 1]}, {[3, 1, 1, 1], [4, 1, 1]} and only in the first two pairs there are no common parts.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..120 (first 71 terms from Jean-François Alcover)
• Eric Weisstein's World of Mathematics, Conjugate Partition.
• Wikipedia, Integer partition, Conjugate and self-conjugate partitions

Crossrefs

Cf. A000041.

Programs

Maple

with(combinat): a:=proc(n) local P,ct,j: P:=partition(n): ct:=0: for j from 1 to numbpart(n) do if convert(P[j],set) intersect convert(conjpart(P[j]),set) = {} then ct:=ct+1 else fi: od: ceil(ct/2): end: seq(a(n),n=0..55); # for 55 terms execution takes hours - Emeric Deutsch, Apr 15 2006

Mathematica

ConjugatePartition[e_] := Length /@ Most[NestWhileList[Function[{s}, Select[s - 1, # > 0 &]], e, # =!= {} &]]; (* this ConjugatePartition code is due to Arnoud B. in MathWorld (see link) *)
a[n_] := a[n] = Module[{P, ct, j}, P = IntegerPartitions[n]; ct = 0; For[j = 1, j <= PartitionsP[n], j++, If[P[[j]] ~Intersection~ ConjugatePartition[ P[[j]]] == {}, ct = ct + 1]]; Ceiling[ct/2]];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 60}] (* Jean-François Alcover, Jul 18 2024 *)

Extensions

More terms from Emeric Deutsch, Apr 15 2006

a(0)=1 prepended and a(56)-a(60) added by Alois P. Heinz, Sep 28 2023