arXiv subjects
Publications and source records attributed to William Keith.
There has been recent interest in integer partitions whose parts satisfy parity restrictions: for example, those where all the odd parts are distinct, or those where all the even parts are larger than the odd parts. Often results about such partitions have been obtained by algebraic manipulation of generating functions. We show that a number of these identities can be proved in a bijective, and sometimes simpler, manner.
We consider simultaneous $(s,s+t,s+2t,\dots,s+pt)$-core partitions in the large-$p$ limit, or (when $s<t$), partitions in which no hook may be of length $s \pmod{t}$. We study generating functions, containment properties, and congruences when $s$ is not coprime to $t$. As a boundary case of the general study made by Cho, Huh and Sohn, we provide enumerations when $s$ is coprime to $t$, and answer positively a conjecture of Fayers on the polynomial behavior of the size of the set of simultaneous $(s,s+t,s+2t,\dots,s+pt)$-core partitions when $p$ grows arbitrarily large. Of particular interest throughout is the comparison to the behavior of simultaneous $(s,t)$-cores.