William J. Keith

Distribution of the full rank in residue classes for odd moduli

The distribution of values of the full ranks of marked Durfee symbols is examined in prime and nonprime arithmetic progressions. The relative populations of different residues for the same modulus are determined: the primary result is that k-marked Durfee symbols of n equally populate the residue classes a and bmod2k+1 if gcd(a,2k+1)=gcd(b,2k+1). These are used to construct a few congruences. The general procedure is illustrated with a particular theorem on 4-marked symbols for multiples of 3.

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 p(n) is equidistributed modulo 2.Our main result will relate the densities, say,

The part-frequency matrices of a partition

{\delta_t}

Proof of a conjectured q,t-Schröder identity

δt, of the odd values of the t-multipartition functions

A Bijective Toolkit for Signed Partitions

{p_t(n)}

Partitions with prescribed hooksets

pt(n), for several integers t. In particular, we will show that if