A137438 Number of conjugate-congruent partitions of n.

1, 0, 3, 3, 3, 7, 9, 12, 14, 22, 30, 39, 41, 57, 86, 87, 121, 179, 164, 225, 300, 362, 433, 571, 624, 846, 968, 1134, 1391, 1902, 1992, 2407, 3043, 3688, 4321, 5145, 5811, 7277, 8627, 10234, 11895, 14730, 16091, 19571, 24026, 27312, 31490, 37119, 43197, 52256, 59349, 68981, 79711, 94935, 108360, 126301, 147204, 169964, 193594, 227147

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

Table of n, a(n) for n=1..60.

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

Sequence in context: A268127 A200076 A342335 * A098524 A143015 A295671

Adjacent sequences: A137435 A137436 A137437 * A137439 A137440 A137441

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