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%I #31 Jul 19 2024 15:33:16
%S 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,
%T 124,148,169,203,225,275,310,363,408,484,537,635,709,824,918,1075,
%U 1191,1379,1540,1767,1971,2269,2517,2889,3208,3656,4068,4629,5120,5813,6452,7280,8068,9113
%N 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.
%H Alois P. Heinz, <a href="/A114701/b114701.txt">Table of n, a(n) for n = 0..120</a> (first 71 terms from Jean-François Alcover)
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ConjugatePartition.html">Conjugate Partition</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Integer_partition#Conjugate_and_self-conjugate_partitions">Integer partition, Conjugate and self-conjugate partitions</a>
%e 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.
%p 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
%t ConjugatePartition[e_] := Length /@ Most[NestWhileList[Function[{s}, Select[s - 1, # > 0 &]], e, # =!= {} &]]; (* this ConjugatePartition code is due to Arnoud B. in MathWorld (see link) *)
%t 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]];
%t Table[Print[n, " ", a[n]]; a[n], {n, 0, 60}] (* _Jean-François Alcover_, Jul 18 2024 *)
%Y Cf. A000041.
%K nonn
%O 0,7
%A _Vladeta Jovovic_, Feb 18 2006
%E More terms from _Emeric Deutsch_, Apr 15 2006
%E a(0)=1 prepended and a(56)-a(60) added by _Alois P. Heinz_, Sep 28 2023