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James A. Sellers

Publications and source records attributed to James A. Sellers.

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arXiv2021

Combinatorial Perspectives on the Crank and Mex Partition Statistics

Several authors have recently considered the smallest positive part missing from an integer partition, known as the minimum excludant or mex. In this work, we revisit and extend connections between Dyson's crank statistics, the mex, and Frobenius symbols, with a focus on combinatorial proof techniques. One highlight is a generating function expression for the number of partitions with a bounded crank that does not include an alternating sum. This leads to a combinatorial interpretation involving types of Durfee rectangles.

math.CO↗
arXiv2020

On the parity of the number of partitions with odd multiplicities

Recently, Hirschhorn and the first author considered the parity of the function $a(n)$ which counts the number of integer partitions of $n$ wherein each part appears with odd multiplicity. They derived an effective characterization of the parity of $a(2m)$ based solely on properties of $m.$ In this note, we quickly reprove their result, and then extend it to an explicit characterization of the parity of $a(n)$ for all $n\not\equiv 7 \pmod{8}.$ We also exhibit some infinite families of congruences modulo 2 which follow from these characterizations. We conclude by discussing the case $n\equiv 7 \pmod{8}$, where, interestingly, the behavior of $a(n)$ modulo 2 appears to be entirely different. In particular, we conjecture that, asymptotically, $a(8m+7)$ is odd precisely $50\%$ of the time. This conjecture, whose broad generalization to the context of eta-quotients will be the topic of a subsequent paper, remains wide open.

math.CO↗
arXiv2020

Dyson's Crank and the Mex of Integer Partitions

Andrews and Newman have recently introduced the notion of the mex of a partition, the smallest positive integer that is not a part. The concept has been used since at least 2011, though, with connections to Frobenius symbols. Recently the parity of the mex has been associated to the crank statistic named by Dyson in 1944. In this note, we extend and strengthen the connection between the crank and mex (along with a new generaliztion of the mex) by proving a number of properties that naturally relate these partition statistics.

math.CO↗
arXiv2020

Congruences for the coefficients of the Gordon and McIntosh mock theta function $ξ(q)$

Recently Gordon and McIntosh introduced the third order mock theta function $ξ(q)$ defined by $$ ξ(q)=1+2\sum_{n=1}^{\infty}\frac{q^{6n^2-6n+1}}{(q;q^6)_{n}(q^5;q^6)_{n}}. $$ Our goal in this paper is to study arithmetic properties of the coefficients of this function. We present a number of such properties, including several infinite families of Ramanujan--like congruences.

math.NT↗
arXiv2020

Sequentially congruent partitions and partitions into squares

In recent work, M. Schneider and the first author studied a curious class of integer partitions called "sequentially congruent" partitions: the $m$th part is congruent to the $(m+1)$th part modulo $m$, with the smallest part congruent to zero modulo the number of parts. Let $p_{\mathcal S}(n)$ be the number of sequentially congruent partitions of $n,$ and let $p_{\square}(n)$ be the number of partitions of $n$ wherein all parts are squares. In this note we prove bijectively, for all $n\geq 1,$ that $p_{\mathcal S}(n) = p_{\square}(n).$ Our proof naturally extends to show other exotic classes of partitions of $n$ are in bijection with certain partitions of $n$ into $k$th powers.

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arXiv2020

Simultaneous core partitions with nontrivial common divisor

A tremendous amount of research has been done in the last two decades on $(s,t)$-core partitions when $s$ and $t$ are positive integers with no common divisor. Here we change perspective slightly and explore properties of $(s,t)$-core and $(\bar{s},\bar{t})$-core partitions for $s$ and $t$ with nontrivial common divisor $g$. We begin by revisiting work by D. Aukerman, D. Kane and L. Sze on $(s,t)$-core partitions for nontrivial $g$ before obtaining a generating function for the number of $(\bar{s},\bar{t})$-core partitions of $n$ under the same conditions. Our approach, using the $g$-core, $g$-quotient and bar-analogues, allows for new results on $t$-cores and self-conjugate $t$-cores that are {\it not} $g$-cores and $\bar{t}$-cores that are {\it not} $\bar{g}$-cores, thus strengthening positivity results of K. Ono and A. Granville, J. Baldwin et. al., and I. Kiming. We then detail a new bijection between self-conjugate $(s,t)$-core and $(\bar{s},\bar{t})$-core partitions for $s$ and $t$ odd with odd, nontrivial common divisor $g$. Here the core-quotient construction fits remarkably well with certain lattice-path labelings due to B. Ford, H. Mai, and L. Sze and C. Bessenrodt and J. Olsson. Along the way we give a new proof of a correspondence of J. Yang between self-conjugate $t$-core and $\bar{t}$-core partitions when $t$ is odd and positive. We end by noting $(s,t)$-core and $(\bar{s}, \bar{t})$-core partitions inherit Ramanujan-type congruences from those of $g$-core and $\bar{g}$-core partitions.

math.CO↗
arXiv2020

Parity Considerations for Mex-Related Partition Functions of Andrews and Newman

In a recent paper, Andrews and Newman extended the mex-function to integer partitions and proved many partition identities connected with these functions. In this paper, we present parity considerations of one of the families of functions they studied, namely $p_{t,t}(n)$. Among our results, we provide complete parity characterizations of $p_{1,1}(n)$ and $p_{3,3}(n)$.

math.NT↗
arXiv2016

Abaci structures of $(s,ms\pm1)$-core partitions

We develop a geometric approach to the study of $(s,ms-1)$-core and $(s,ms+1)$-core partitions through the associated $ms$-abaci. This perspective yields new proofs for results of H. Xiong and A. Straub (originally proposed by T. Amdeberhan) on the enumeration of $(s, s+1)$ and $(s,ms-1)$-core partitions with distinct parts. It also enumerates the $(s, ms+1)$-cores with distinct parts. Furthermore, we calculate the weight of the $(s, ms-1,ms+1)$-core partition with the largest number of parts. Finally we use 2-core partitions to enumerate self-conjugate core partitions with distinct parts. The central idea is that the $ms$-abaci of maximal $(s,ms\pm1)$-cores can be built up from $s$-abaci of $(s,s\pm 1)$-cores in an elegant way.

math.CO↗
arXiv2015

A periodic approach to plane partition congruences

Ramanujan's celebrated congruences of the partition function $p(n)$ have inspired a vast amount of results on various partition functions. Kwong's work on periodicity of rational polynomial functions yields a general theorem used to establish congruences for restricted plane partitions. This theorem provides a novel proof of several classical congruences and establishes two new congruences. We additionally prove several new congruences which do not fit the scope of the theorem, using only elementary techniques, or a relationship to existing multipartition congruences.

math.NT↗
arXiv2015

A combinatorial proof of a relationship between maximal $(2k-1,2k+1)$ and $(2k-1,2k,2k+1)$-cores

Integer partitions which are simultaneously $t$--cores for distinct values of $t$ have attracted significant interest in recent years. When $s$ and $t$ are relatively prime, Olsson and Stanton have determined the size of the maximal $(s,t)$-core $κ_{s,t}$. When $k\geq 2$, a conjecture of Amdeberhan on the maximal $(2k-1,2k,2k+1)$-core $κ_{2k-1,2k,2k+1}$ has also recently been verified by numerous authors. In this work, we analyze the relationship between maximal $(2k-1,2k+1)$-cores and maximal $(2k-1,2k,2k+1)$-cores. In previous work, the first author noted that, for all $k\geq 1,$ $$ \vert \, κ_{2k-1,2k+1}\, \vert = 4\vert \, κ_{2k-1,2k,2k+1}\, \vert $$ and requested a combinatorial interpretation of this unexpected identity. Here, using the theory of abaci, partition dissection, and elementary results relating triangular numbers and squares, we provide such a combinatorial proof.

math.CO↗
arXiv2015

Congruences for spin characters of the double covers of the symmetric and alternating groups

Let $p$ be an odd prime. The bar partitions with sign and $p$-bar-core partitions with sign respectively label the spin characters and $p$-defect zero spin characters of the double cover of the symmetric group, and by restriction, those of the alternating group. The generating functions for these objects have been determined by J. Olsson. We study these functions from an arithmetic perspective, using classical analytic tools and elementary generating function manipulation to obtain many Ramanujan-like congruences.

math.RT↗
arXiv2014

Arithmetic Properties of Andrews' Singular Overpartitions

In a very recent work, G. E. Andrews defined the combinatorial objects which he called {\it singular overpartitions} with the goal of presenting a general theorem for overpartitions which is analogous to theorems of Rogers--Ramanujan type for ordinary partitions with restricted successive ranks. As a small part of his work, Andrews noted two congruences modulo 3 which followed from elementary generating function manipulations. In this work, we prove that Andrews' results modulo 3 are two examples of an infinite family of congruences modulo 3 which hold for that particular function. We also expand the consideration of such arithmetic properties to other functions which are part of Andrews' framework for singular overpartitions.

math.NT↗
arXiv2014

Congruences for the Fishburn Numbers

The Fishburn numbers, $ξ(n),$ are defined by a formal power series expansion $$ \sum_{n=0}^\infty ξ(n)q^n = 1 + \sum_{n=1}^\infty \prod_{j=1}^n (1-(1-q)^j). $$ For half of the primes $p$, there is a non--empty set of numbers $T(p)$ lying in $[0,p-1]$ such that if $j\in T(p),$ then for all $n\geq 0,$ $$ ξ(pn+j)\equiv 0 \pmod{p}. $$

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arXiv2013

Congruences for Generalized Frobenius Partitions with an Arbitrarily Large Number of Colors

In his 1984 AMS Memoir, George Andrews defined the family of $k$--colored generalized Frobenius partition functions. These are denoted by $cϕ_k(n)$ where $k\geq 1$ is the number of colors in question. In that Memoir, Andrews proved (among many other things) that, for all $n\geq 0,$ $cϕ_2(5n+3) \equiv 0\pmod{5}.$ Soon after, many authors proved congruence properties for various $k$--colored generalized Frobenius partition functions, typically with a small number of colors. Work on Ramanujan--like congruence properties satisfied by the functions $cϕ_k(n)$ continues, with recent works completed by Baruah and Sarmah as well as the author. Unfortunately, in all cases, the authors restrict their attention to small values of $k.$ This is often due to the difficulty in finding a "nice" representation of the generating function for $cϕ_k(n)$ for large $k.$ Because of this, no Ramanujan--like congruences are known where $k$ is large. In this note, we rectify this situation by proving several infinite families of congruences for $cϕ_k(n)$ where $k$ is allowed to grow arbitrarily large. The proof is truly elementary, relying on a generating function representation which appears in Andrews' Memoir but has gone relatively unnoticed.

math.NT↗
arXiv2013

An Unexpected Congruence Modulo 5 for 4--Colored Generalized Frobenius Partitions

In his 1984 AMS Memoir, George Andrews defined the family of $k$--colored generalized Frobenius partition functions. These are denoted by $cϕ_k(n)$ where $k\geq 1$ is the number of colors in question. In that Memoir, Andrews proved (among many other things) that, for all $n\geq 0,$ $cϕ_2(5n+3) \equiv 0\pmod{5}.$ Soon after, many authors proved congruence properties for various $k$--colored generalized Frobenius partition functions, typically with a small number of colors. In 2011, Baruah and Sarmah proved a number of congruence properties for $cϕ_4$, all with moduli which are powers of 4. In this brief note, we add to the collection of congruences for $cϕ_4$ by proving this function satisfies an unexpected result modulo 5. The proof is elementary, relying on Baruah and Sarmah's results as well as work of Srinivasa Ramanujan.

math.NT↗
arXiv2003

On Non-Squashing Partitions

A partition n = p_1 + p_2 + ... + p_k with 1 <= p_1 <= p_2 <= ... <= p_k is called non-squashing if p_1 + ... + p_j <= p_{j+1} for 1 <= j <= k-1. Hirschhorn and Sellers showed that the number of non-squashing partitions of n is equal to the number of binary partitions of n. Here we exhibit an explicit bijection between the two families, and determine the number of non-squashing partitions with distinct parts, with a specified number of parts, or with a specified maximal part. We use the results to solve a certain box-stacking problem.

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