[seqfan] Re: Modular Partitions
William Keith
william.keith at gmail.com
Sat May 3 07:19:54 CEST 2014
Chris' conjectures are accurate.
Suppose we are in the case n !==0 mod p. Diagram one of Jen's partitions
in Z/pZ into n parts with a Ferrers diagram, with parts of size 0 noted.
It will fit in a p-1 by n box. Choose any profile, to create a Ferrers
diagram; there are (n+p-1)-choose-n possible profiles.
If we add 1 to each part, rotating any new parts of size p to the end as
parts of size 0, we can do so p times and get back our original profile.
If p is prime and n !==0 mod p, then none of this group of p modpartitions
can possibly be the same: Suppose you had k parts of size 0, t additions
ago, then now you have k parts of size t. If this is the same partition as
the one t additions ago, then you must have had k parts of size t, so now
you have k parts of size 2t, etc, until you had k parts of each size to
begin with, i.e. p divides n.
So we can group the p resulting profiles. But adding 1 to each of n parts
adds n to the total mod p, so the residue of the profiles mod p are all
distinct, and hence exactly one of them is 0 mod p. Thus there are (1/p)
(n-1+p choose n) partitions of 0 mod p with n parts when p is prime, p not
dividing n.
When n = kp, then exactly the one partition listed above will be separate
from the groups; take (1/p)[(n+p-1 choose n) - 1] + 1. QED.
I think this diagram technique, and some sort of rotation counting, should
also be able to prove symmetry. I don't know anything about Molien
series. (Yet.) If something from that subject proves symmetry I'd be
interested in seeing it. But I would also like to see a map defined in
terms of the modpartitions themselves that gives symmetry in n and k.
William Keith
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