|
|
A137438
|
|
Number of conjugate-congruent partitions of n.
|
|
0
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
|
|
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
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,more
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|