Searcharxiv⌕ Search

arXiv subjects

Cristina Ballantine

Publications and source records attributed to Cristina Ballantine.

At least 19 recordsLinked to original sources
arXiv2026

Identities involving the number of missing integers in partitions - combinatorial proofs

A missing integer in a partition $λ$ is a positive integer less than the largest part of $λ$ that does not appear as a part in $λ$. In the recent paper On the number of missing integers in partitions, Bhoria, Eyyunni, and Santra examined the number of partitions (overpartitions) of $n$ with a fixed number of missing integers and established generating functions, identities, and congruences for them. In this article, we provide combinatorial proofs for several of their results.

math.CO↗
arXiv2026

Franklin's identity for $n$-color partitions and companion Beck-type identities

We show that some classical identities valid for ordinary partitions have precise analogues for $n$-color partitions, that is partitions in which a part of size $n\geq 1$ can occur in colors $1, 2, \ldots, n$. For $r \ge 2$ and $j \ge 0$, we write $\mathcal{O}_{j,r}(m)$ and $\mathcal{D}_{j,r}(m)$ for the sets of $n$-color partitions of $m$ with, respectively, exactly $j$ different parts whose size and color are divisible by $r$, and exactly $j$ different parts occurring at least $r$ times. We prove an $n$-color version of Franklin's theorem, $|\mathcal{O}_{j,r}(m)| = |\mathcal{D}_{j,r}(m)|$, along with two Beck-type identities. We give both analytic and combinatorial proofs for all theorems.

math.CO↗
arXiv2026

Reciprocals of Subsum Polynomials

We introduce the subsum polynomial of a partition $λ=(λ_1, λ_2, \ldots, λ_k)$ defined by $\mathrm{sp}(λ, x)=\prod_{i=1}^k(1+x^{λ_i})$. We study the sum of reciprocals of $\mathrm{sp}(λ, x)$ over all partitions of $n$. We prove arithmetic properties of related polynomials and offer connections to other combinatorial objects.

math.NT↗
arXiv2025

On partitions associated with elementary symmetric polynomials

The elementary symmetric partition function is a map on the set of partitions. It sends a partition lambda to the partition whose parts are the summands in the evaluation of the elementary symmetric function on the parts of lambda. These elementary symmetric partition functions have been studied before, and are related to plethysm. In this note, we study properties of the elementary symmetric partition functions, particularly related to injectivity and the number of parts appearing in their image partitions.

math.CO↗
arXiv2025

Lambert series and double Lambert series

We consider relationships between classical Lambert series, multiple Lambert series and classical $q$-series of the Rogers-Ramanujan type. We conclude with a contemplation on the Andrews-Dixit-Schultz-Yee conjecture.

math.NT↗
arXiv2024

Elementary symmetric partitions

Let e_k(x_1,...,x_l) be an elementary symmetric polynomial and let mu = (mu_1,...,mu_l) be an integer partition. Define pre_k(mu) to be the partition whose parts are the summands in the evaluation e_k(mu_1,...,mu_l). The study of such partitions was initiated by Ballantine, Beck, and Merca who showed (among other things) that pre_2 is injective as a map on binary partitions of n. In the present work we derive a host of identities involving the sequences which count the number of parts of a given value in the image of pre_2. These include generating functions, explicit expressions, and formulas for forward differences. We generalize some of these to d-ary partitions and explore connections with color partitions. Our techniques include the use of generating functions and bijections on rooted partitions. We end with a list of conjectures and a direction for future research.

math.CO↗
arXiv2024

Partitions and elementary symmetric polynomials -- an experimental approach

Given a partition $λ$, we write $e_j(λ)$ for the $j^{\textrm{th}}$ elementary symmetric polynomial $e_j$ evaluated at the parts of $λ$ and $e_jp_A(n)$ for the sum of $e_j(λ)$ as $λ$ ranges over the set of partitions of $n$ with parts in $A$. For $e_jp_A(n)$, we prove analogs of the classical formula for the partition function, $p(n)=1/n \sum_{k=0}^{n-1}σ_1(n-k)p(k)$, where $σ_1$ is the sum of divisors function. We prove several congruences for $e_2p_4(n)$, the sum of $e_2$ over the set of partitions of $n$ into four parts. Define the function $\textrm{pre}_j(λ)$ to be the multiset of monomials in $e_j(λ)$, which is itself a partition. If $\mathcal A$ is a set of partitions, we define $\textrm{pre}_j(\mathcal A)$ to be the set of partitions $\textrm{pre}_j(λ)$ as $λ$ ranges over $\mathcal A$. If $\mathcal P(n)$ is the set of all partitions of $n$, we conjecture that the number of odd partitions in $\textrm{pre}_2(\mathcal P(n))$ is at least the number of distinct partitions. We prove some results about $\textrm{pre}_2(\mathcal B(n))$, where $\mathcal B(n)$ is the set of binary partitions of $n$. We conclude with conjectures on the log-concavity of functions related to $e_jp(n)$, the sum of $e_j(λ)$ for all $λ\in \mathcal P(n)$.

math.CO↗
arXiv2024

Combinatorial proofs of inequalities involving the number of partitions with parts separated by parity

We consider the number of various partitions of $n$ with parts separated by parity and prove combinatorially several inequalities between these numbers. For example, we show that for $n\geq 5$ we have $p_{od}^{eu}(n)<p_{ed}^{ou}(n)$, where $p_{od}^{eu}(n)$ is the number of partitions of $n$ with odd parts distinct and even parts unrestricted and all odd parts less than all even parts and $p_{ed}^{ou}(n)$ is the number of partitions of $n$ with even parts distinct and odd parts unrestricted and all even parts less than all odd parts. We also prove a conjectural inequality of Fu and Tang involving partitions with parts separated by parity with restrictions on the multiplicity of parts.

math.CO↗
arXiv2024

Truncated Theta Series Related to the Jacobi Triple Product Identity

The work of Andrews and Merca on the truncated Euler's pentagonal number theorem led to a resurgence in research on truncated theta series identities. In particular, Yee proved a truncated version of the Jacobi Triple Product (JTP) identity. Recently, Merca conjectured a stronger form of the truncated JTP identity. In this article we prove the first three cases of the conjecture and several related truncated identities. We prove combinatorially an identity related to the JTP identity which in particular cases reduces to identities conjectured by Merca and proved analytically by Krattenthaler, Merca and Radu. Moreover, we introduce a new combinatorial interpretation for the number of distinct 5-regular partitions of n.

math.CO↗
arXiv2023

Hook length biases and general linear partition inequalities

Motivated in part by hook-content formulas for certain restricted partitions in representation theory, we consider the total number of hooks of fixed length in odd versus distinct partitions. We show that there are more hooks of length $2$, respectively $3$, in all odd partitions of $n$ than in all distinct partitions of $n$, and make the analogous conjecture for arbitrary hook length $t \geq 2$. We also establish additional bias results on the number of gaps of size $1,$ respectively $2$, in all odd versus distinct partitions of $n$. We conjecture similar biases and asymptotics, as well as congruences for the number of hooks of fixed length in odd distinct partitions versus self-conjugate partitions. An integral component of the proof of our bias result for hooks of length $3$ is a linear inequality involving $q(n)$, the number of distinct partitions of $n$. In this article we also establish effective linear inequalities for $q(n)$ in great generality, a result which is of independent interest. Our methods are both analytic and combinatorial, and our results and conjectures intersect the areas of representation theory, analytic number theory, partition theory, and $q$-series. In particular, we use a Rademacher-type exact formula for $q(n),$ Wright's circle method, modularity, $q$-series transformations, asymptotic methods, and combinatorial arguments.

math.CO↗
arXiv2023

PED and POD partitions: combinatorial proofs of recurrence relations

PED partitions are partitions with even parts distinct while odd parts are unrestricted. Similarly, POD partitions have distinct odd parts while even parts are unrestricted. Merca proved several recurrence relations analytically for the number of PED partitions of $n$. They are similar to the recurrence relation for the number of partitions of $n$ given by Euler's pentagonal number theorem. We provide combinatorial proofs for all of these theorems and also for the pentagonal number theorem for PED partitions proved analytically by Fink, Guy, and Krusemeyer. Moreover, we prove combinatorially a recurrence for POD partitions given by Ballantine and Merca, Beck-type identities involving PED and POD partitions, and several other results about PED and POD partitions.

math.CO↗
arXiv2023

Generalizations of POD and PED partitions

Partitions with even (respectively odd) parts distinct and all other parts unrestricted are often referred to as PED (respectively POD) partitions. In this article, we generalize these notions and study sets of partitions in which parts with fixed residue(s) modulo r are distinct while all other parts are unrestricted. We also study partitions in which parts divisible by r (respectively congruent to r modulo 2r) must occur with multiplicity greater than one.

math.CO↗
arXiv2023

Mock theta functions and related combinatorics

In this paper we add to the literature on the combinatorial nature of the mock theta functions, a collection of curious $q$-hypergeometric series introduced by Ramanujan in his last letter to Hardy in 1920, which we now know to be important examples of mock modular forms. Our work is inspired by Beck's conjecture, now a theorem of Andrews, related to Euler's identity: the excess of the number of parts in all partitions of $n$ into odd parts over the number of partitions of $n$ into distinct parts is equal to the number of partitions with only one (possibly repeated) even part and all other parts odd. We establish Beck-type identities associated to partition identities due to Andrews, Dixit, and Yee for the third order mock theta functions $ω(q), ν(q)$, and $ϕ(q)$. Our proofs are both analytic and combinatorial in nature, and involve mock theta generating functions and combinatorial bijections.

math.CO↗
arXiv2023

Partitions enumerated by self-similar sequences

The Fibonacci numbers are the prototypical example of a recursive sequence, but grow too quickly to enumerate sets of integer partitions. The same is true for the other classical sequences $a(n)$ defined by Fibonacci-like recursions: the tribonacci, Padovan, Pell, Narayana's cows, and Lucas sequences. For each sequence $a(n)$, however, we can define a related sequence $\textrm{sa}(n)$ by defining $\textrm{sa}(n)$ to have the same recurrence and initial conditions as $a(n)$, except that $\textrm{sa}(2n)=\textrm{sa}(n)$. Growth is no longer a problem: for each $n$ we construct recursively a set $\mathcal{SA}(n)$ of partitions of $n$ such that the cardinality of $\mathcal{SA}(n)$ is $\textrm{sa}(n)$. We study the properties of partitions in $\mathcal{SA}(n)$ and in each case we give non-recursive descriptions. We find congruences for $\textrm{sa}(n)$ and also for $\textrm{psa}(n)$, the total number of parts in all partitions in $\mathcal{SA}(n)$.

math.CO↗
arXiv2023

On the number of parts in all partitions enumerated by the Rogers-Ramanujan identities

The celebrated Rogers-Ramanujan identities equate the number of integer partitions of $n$ ($n\in\mathbb N_0$) with parts congruent to $\pm 1 \pmod{5}$ (respectively $\pm 2 \pmod{5}$) and the number of partitions of $n$ with super-distinct parts (respectively super-distinct parts greater than $1$). In this paper, we establish companion identities to the Rogers-Ramanujan identities on the number of parts in all partitions of $n$ of the aforementioned types, in the spirit of earlier work by Andrews and Beck on a partition identity of Euler.

math.NT↗
arXiv2023

$6$-regular partitions: new combinatorial properties, congruences, and linear inequalities

We consider the number of the $6$-regular partitions of $n$, $b_6(n)$, and give infinite families of congruences modulo $3$ (in arithmetic progression) for $b_6(n)$. We also consider the number of the partitions of $n$ into distinct parts not congruent to $\pm 2$ modulo $6$, $Q_2(n)$, and investigate connections between $b_6(n)$ and $Q_2(n)$ providing new combinatorial interpretations for these partition functions. In this context, we discover new infinite families of linear inequalities involving Euler's partition function $p(n)$. Infinite families of linear inequalities involving the $6$-regular partition function $b_6(n)$ and the distinct partition function $Q_2(n)$ are proposed as open problems.

math.NT↗
arXiv2022

Parity of 3-regular partition numbers and Diophantine equations

Let $b_3(n)$ be the number of $3$-regular partitions of $n$. Recently, W. J. Keith and F. Zanello discovered infinite families of Ramanujan type congruences modulo $2$ for $b_3(2n)$ involving every prime $p$ with $p \equiv 13, 17, 19, 23 \pmod {24}$, and O. X. M. Yao provided new infinite families of Ramanujan type congruences modulo $2$ for $b_3(2n)$ involving every prime $p\geqslant 5$. In this paper, we introduce new infinite Ramanujan type congruences modulo $2$ for $b_3(2n)$. They complement naturally the results of Keith-Zanello and Yao and involve primes in $\mathcal P=\{p \text{ prime } : \exists \, j\in \{1,4,8\},\, x, y \in \mathbb Z,\, \gcd(x,y)=1 \text { with } x^2+216y^2=jp\}$ whose Dirichlet density is $1/6$. As a key ingredient in our proof we show that of the number of primitive solutions for $x^2+216y^2=pm$, $p \in \mathcal P$, $p\nmid m$ and $pm\equiv 1\pmod{24}$, is divisible by $8$. Here, the difficulty arises from the fact that $216$ is not idoneal. We also give a conjectural exact formula for the number of solutions for this Diophantine equation. In the second part of the article, we study reversals of Euler-type identities. These are motivated by recent work of the second author on a reversal of Schur's identity which involves $3$-regular partitions weighted by the parity of their length.

math.NT↗