OFFSET
1,3
COMMENTS
See reference for precise definition.
Let P be a partition of n and let Q denote its conjugate partition. Then P is said to be conjugate-congruent if there is an integer m>1 such that both P and Q give the same set R(P,m) of residues when their parts are reduced modulo m, where R(P,m) contains less than m elements. - Augustine O. Munagi, Dec 18 2008
LINKS
A. O. Munagi, Pairing conjugate partitions by residue classes, Discrete Math., 308 (2008), 2492-2501.
Eric Weisstein's World of Mathematics, Conjugate Partition.
EXAMPLE
a(8) = 12: the 12 conjugate-congruent partitions of 8 are shown below, in conjugate pairs followed by their common residues. 8/1+1+1+1+1+1+1+1 by 1 mod 7, 1+7/1+1+1+1+1+1+2 by 1,2 mod 5, 2+6/1+1+1+1+2+2 by 1,2 mod 5, 4+4/2+2+2+2 by 0 mod 2, 1+1+6/1+1+1+1+1+3 by 0,1 mod 3, 2+3+3/2+3+3 by 0,2 mod 3, 1+1+2+4/1+1+2+4 by 1,2 mod 3. - Augustine O. Munagi, Dec 18 2008
MAPLE
with(combinat): isconjcong:=proc(P::partition) local m; option remember; if P[ -1]>=conjpart(P)[ -1] then for m from 2 to P[ -1]+1 do if {op(P mod m)}={op(conjpart(P) mod m)} and nops({op(P mod m)})<m then return true; end if; end do; else for m from 2 to conjpart(P)[ -1]+1 do if {op(P mod m)}={op(conjpart(P) mod m)} and nops({op(P mod m)})<m then return true; end if; end do; end if; false; end proc: seq(nops(select(isconjcong, partition(n))), n=1..30); # Augustine O. Munagi, Dec 18 2008
MATHEMATICA
ConjugatePartition[e_List] := Length /@ Most[NestWhileList[Function[{s}, Select[s - 1, # > 0 &]], e, # =!= {} &]]; (* this ConjugatePartition code is due to Arnoud B. in MathWorld (see link) *)
isconjcong[P_] := isconjcong[P] = Module[{m, Q = ConjugatePartition[P]}, If[P[[1]] >= Q[[1]], For[m = 2, m <= P[[ 1]] + 1, m++, If[Union@Mod[P, m] == Union@Mod[Q, m] && Length[Union@Mod[P, m]] < m, Return[True]]], For[m = 2, m <= Q[[1]] + 1, m++, If[Union@Mod[P, m] == Union@Mod[Q, m] && Length[Union@Mod[P, m]] < m, Return[True]]]]; False];
a[n_] := a[n] = Length[Select[IntegerPartitions[n], isconjcong]];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 60}] (* Jean-François Alcover, Jul 19 2024, after Maple code *)
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane May 07 2008
EXTENSIONS
a(36)-a(40) from Augustine O. Munagi, Dec 18 2008
a(41)-a(60) from Jean-François Alcover, Jul 19 2024
STATUS
approved