62 research outputs found
On the density of the odd values of the partition function
The purpose of this note is to introduce a new approach to the study of one
of the most basic and seemingly intractable problems in partition theory,
namely the conjecture that the partition function is equidistributed
modulo 2.
Our main result will relate the densities, say , of the odd values
of the -multipartition functions , for several integers . In
particular, we will show that if for some , then (assuming it exists) ; that is,
itself is odd with positive density. Notice that, currently, the best
unconditional result does not even imply that is odd for
values of . In general, we conjecture that for all
odd, i.e., that similarly to the case of , all multipartition functions
are in fact equidistributed modulo 2.
Our arguments will employ a number of algebraic and analytic methods, ranging
from an investigation modulo 2 of some classical Ramanujan identities and
several other eta product results, to a unified approach that studies the
parity of the Fourier coefficients of a broad class of modular form identities
recently introduced by Radu.Comment: Several changes with respect to the 2015 version. 18 pages. To appear
in the Annals of Combinatoric
Congruences for the dots bracelet partition functions
By finding the congruent relations between the generating function of the 5
dots bracelet partitions and that of the 5-regular partitions, we get some new
congruences modulo 2 for the 5 dots bracelet partition function. Moreover, for
a given prime , we study the arithmetic properties modulo of the
dots bracelet partitions.Comment: 10 page
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