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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.
3
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.
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
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