Searcharxiv⌕ Search

arXiv subjects

William J. Keith

Publications and source records attributed to William J. Keith.

At least 19 recordsLinked to original sources
arXiv2026

Parity of the coefficients of certain eta-quotients, III: two special classes

We continue a series of papers studying the parity of families of eta-quotients, which provide implications for the parity of the partition function as well as an overarching conjecture on related $q$-series. The present article focuses on two classes. One consists of eta-quotients of the form $f_t^3/f_1$, a distinguished case of Andrews' singular overpartitions that has recently attracted attention among researchers. In addition, we investigate the parity of certain pure eta-powers $f_1^t$, appending new results to known density theorems.

math.CO↗
arXiv2026

On a conjecture of Andrews and Bachraoui

Recently, Andrews and Bachraoui considered a generating function $F_{k,m}(q)$ associated with certain two-color partitions, and conjectured that this function has non-negative coefficients for $m=1$. They showed this property for $1 \leq k \leq 4$. In this note, we prove that $F_{k,1}(q)$ has non-negative coefficients for $5 \leq k \leq 10$. Moreover, we show that, as $k\to\infty$, $F_{k,1}(q)$ is related to Ramanujan's third order mock theta function $ω(q)$ and to quotients of certain $q$-binomial coefficients.

math.NT↗
arXiv2025

Combinatorial proof of a congruence for partitions into two sizes of part

Previous work showed that, for $ν_2(n)$ the number of partitions of $n$ into exactly two part sizes, one has $ν_2(16n + 14) \equiv 0 \pmod{4}$. The earlier proof required the technology of modular forms, and a combinatorial proof was desired. This article provides the requested proof, in the process refining divisibility to finer subclasses. Some of these subclasses have counts closely related to the divisor function $d(16n + 14)$, and we offer a conjecture on a potential rank statistic.

math.CO↗
arXiv2025

Reciprocals of false theta functions

We investigate reciprocals of false theta functions, producing results such as congruences, simple asymptotic bounds, and combinatorial identities. Of particular interest is a connection between $1/Ψ(-q^2,q)$ and the truncated pentagonal number theorem of Andrews and Merca. We record a useful dissection identity analogous to the known theta function dissection.

math.CO↗
arXiv2024

Partitions with fixed points in the sequence of first-column hook lengths

Recently, Blecher and Knopfmacher applied the notion of fixed points to integer partitions. This has already been generalized and refined in various ways such as $h$-fixed points for an integer parameter $h$ by Hopkins and Sellers. Here, we consider the sequence of first column hook lengths in the Young diagram of a partition and corresponding fixed hooks. We enumerate these, using both generating function and combinatorial proofs, and find that they match occurrences of part sizes equal to their multiplicity. We establish connections to work of Andrews and Merca on truncations of the pentagonal number theorem and classes of partitions partially characterized by certain minimal excluded parts (mex).

math.CO↗
arXiv2023

Parity of the coefficients of certain eta-quotients, II: The case of even-regular partitions

We continue our study of the density of the odd values of eta-quotients, here focusing on the $m$-regular partition functions $b_m$ for $m$ even. Based on extensive computational evidence, we propose an elegant conjecture which, in particular, completely classifies such densities: Let $m = 2^j m_0$ with $m_0$ odd. If $2^j < m_0$, then the odd density of $b_m$ is $1/2$; moreover, such density is equal to $1/2$ on every (nonconstant) subprogression $An+B$. If $2^j > m_0$, then $b_m$, which is already known to have density zero, is identically even on infinitely many non-nested subprogressions. This and all other conjectures of this paper are consistent with our ''master conjecture'' on eta-quotients presented in the previous work. In general, our results on $b_m$ for $m$ even determine behaviors considerably different from the case of $m$ odd. Also interesting, it frequently happens that on subprogressions $An+B$, $b_m$ matches the parity of the multipartition functions $p_t$, for certain values of $t$. We make a suitable use of Ramanujan-Kolberg identities to deduce a large class of such results; as an example, $b_{28}(49n+12) \equiv p_3(7n+2) \pmod{2}$. Additional consequences are several ''almost always congruences'' for various $b_m$, as well as new parity results specifically for $b_{11}$. We wrap up our work with a much simpler proof of the main result of a recent paper by Cherubini-Mercuri, which fully characterized the parity of $b_8$.

math.CO↗
arXiv2022

A bijection for partitions simultaneously $s$-regular and $t$-distinct

In this note a bijection is constructed between the set of partitions of n simultaneously s-regular and t-distinct, and those simultaneously t-regular and s-distinct. Some implications of the map are discussed. As a generalized version of Glaisher's bijection, the map may be widely useful in other partition combinatorics. A previous conjecture concerning iterations of Glaisher's bijection is given a counterexample.

math.CO↗
arXiv2022

Schmidt-type theorems for partitions with uncounted parts

Schmidt's theorem is significantly generalized, to partitions in which periodic but otherwise arbitrary subsets of parts are counted or uncounted. The identification of such sets of partitions with colored partitions satisfying certain specifications may be a generally useful tool for establishing sum-product $q$-series identities, examples of which are given.

math.CO↗
arXiv2021

Parity of the coefficients of certain eta-quotients

We investigate the parity of the coefficients of certain eta-quotients, extensively examining the case of $m$-regular partitions. Our theorems concern the density of their odd values, in particular establishing lacunarity modulo 2 for specified coefficients; self-similarities modulo 2; and infinite families of congruences in arithmetic progressions. For all $m \leq 28$, we either establish new results of these types where none were known, extend previous ones, or conjecture that such results are impossible. All of our work is consistent with a new, overarching conjecture that we present for arbitrary eta-quotients, greatly extending Parkin-Shanks' classical conjecture for the partition function. We pose several other open questions throughout the paper, and conclude by suggesting a list of specific research directions for future investigations in this area.

math.CO↗
arXiv2020

Restricted k-color partitions, II

We consider $(k,j)$-colored partitions, partitions in which $k$ colors exist but at most $j$ colors may be chosen per size of part. In particular these generalize overpartitions. Advancing previous work, we find new congruences, including in previously unexplored cases where $k$ and $j$ are not coprime, as well as some noncongruences. As a useful aside, we give the apparently new generating function for the number of partitions in the $N \times M$ box with a given number of part sizes, and extend to multiple colors a conjecture of Dousse and Kim on unimodality in overpartitions.

math.CO↗
arXiv2019

Binary compositions and semi-Pell compositions

In analogy with the semi-Fibonacci partitions studied recently by Andrews, we define semi-Pell compositions and semi-$m$-Pell compositions. We find that these are in bijection with certain weakly unimodal $m$-ary compositions. We give generating functions, bijective proofs, and a number of unexpected congruences for these objects.

math.CO↗
arXiv2019

Partitions into parts simultaneously regular, distinct, and/or flat

We explore partitions that lie in the intersection of several sets of classical interest: partitions with parts indivisible by $m$, appearing fewer than $m$ times, or differing by less than $m$. We find results on their behavior and generating functions: more results for those simultaneously regular and distinct, fewest for those distinct and flat. We offer some conjectures in the area.

math.CO↗
arXiv2018

Colored partitions and the hooklength formula: partition statistic identities

We give relations between the joint distributions of multiple hook lengths and of frequencies and part sizes in partitions, extending prior work in this area. These results are discovered by investigating truncations of the Han/Nekrasov-Okounkov hooklength formula and of (k,j)-colored partitions, a unification of k-colored partitions and overpartitions. We establish the observed relations at the constant and linear terms for all n, and for j=2 in their quadratic term, with the associated hook/frequency identities. Further results of this type seem likely.

math.CO↗
arXiv2018

Families of major index distributions: closed forms and unimodality

Closed forms for $f_{λ,i} (q) := \sum_{τ\in SYT(λ) : des(τ) = i} q^{maj(τ)}$, the distribution of the major index over standard Young tableaux of given shapes and specified number of descents, are established for a large collection of $λ$ and $i$. Of particular interest is the family that gives a positive answer to a question of Sagan and collaborators. All formulas established in the paper are unimodal, most by a result of Kirillov and Reshetikhin. Many can be identified as specializations of Schur functions via the Jacobi-Trudi identities. If the number of arguments is sufficiently large, it is shown that any finite principal specialization of any Schur function $s_λ(1,q,q^2,\dots,q^{n-1})$ has a combinatorial realization as the distribution of the major index over a given set of tableaux.

math.CO↗
arXiv2017

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 $δ_t$, of the odd values of the $t$-multipartition functions $p_t(n)$, for several integers $t$. In particular, we will show that if $δ_t>0$ for some $t\in \{5,7,11,13,17,19,23,25\}$, then (assuming it exists) $δ_1>0$; that is, $p(n)$ itself is odd with positive density. Notice that, currently, the best unconditional result does not even imply that $p(n)$ is odd for $\sqrt{x}$ values of $n\le x$. In general, we conjecture that $δ_t=1/2$ for all $t$ odd, i.e., that similarly to the case of $p(n)$, 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.

math.CO↗
arXiv2017

Major index over descent for pattern-avoiding permutations

An open conjecture in pattern avoidance theory is that the distribution of the major index among 321-avoiding permutations is distributed unimodally. We construct a formula for this distribution, and in the case of 2 descents prove unimodality, with unimodality for 3 through 5 descents likely being little more complicated. The formula refines the $q$-analogue of the Frame-Robinson-Thrall hooklength formula for two-rowed partitions, and in the latter part of the paper we discuss another theorem of the same type, and further exploration toward this question. We also give observations on the analogous behaviors for other permutation patterns of length 3.

math.CO↗