arXiv subjects

James Sellers

Publications and source records attributed to James Sellers.

3 recordsLinked to original sources

arXiv2025

On simultaneous $(s, s+t, s+2t, \dots)$-core partitions

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.

math.CO↗

arXiv2025

A refined view of a curious identity for partitions into odd parts with designated summands

In 2002, Andrews, Lewis, and Lovejoy introduced the combinatorial objects which they called partitions with designated summands. These are constructed by taking unrestricted integer partitions and designating exactly one of each occurrence of a part. In the same work, they also considered the restricted partitions with designated summands wherein all parts must be odd, and they denoted the corresponding function by $\mathrm{PDO}(n)$.

math.CO↗

arXiv2021

Partitions Associated to Class Groups of Imaginary Quadratic Number Fields

We investigate properties of attainable partitions of integers, where a partition $(n_1,n_2, \dots, n_r)$ of $n$ is attainable if $\sum (3-2i)n_i\geq 0$. Conjecturally, under an extension of the Cohen and Lenstra heuristics by Holmin et. al., these partitions correspond to abelian $p$-groups that appear as class groups of imaginary quadratic number fields for infinitely many odd primes $p$. We demonstrate a connection to partitions of integers into triangular numbers, construct a generating function for attainable partitions, and determine the maximal length of attainable partitions.

math.NT↗