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

0,7

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

Example

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.

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 *)

Crossrefs

Cf. A000041.

Sequence in context: A277191 A335129 A035632 * A349464 A325260 A325325

Adjacent sequences: A114698 A114699 A114700 * A114702 A114703 A114704

Keyword

nonn

Author

Vladeta Jovovic, Feb 18 2006

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

Status

approved