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Cristina Ballantine

Publications and source records attributed to Cristina Ballantine.

At least 37 records · Page 2Linked to original sources
arXiv2022

$4$-Regular partitions and the pod function

The partition function $pod(n)$ enumerates the partitions of $n$ wherein odd parts are distinct and even parts are unrestricted. Recently, a number of properties for $pod(n)$ have been established. In this paper, for $k\in\{0,2\}$ we consider the partitions of $n$ into distinct parts not congruent to $k$ modulo $4$ and the $4$-regular partitions of $n$ in order to obtain new properties for $pod(n)$. In this context, we derive two new infinite families of linear inequalities involving the function $pod(n)$ and obtain new identities of Watson type.

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arXiv2022

Hook length and symplectic content in partitions

The dimension of an irreducible representation of $GL(n,\mathbb{C})$, $Sp(2n)$, or $SO(n)$ is given by the respective hook-length and content formulas for the corresponding partition. The first author, inspired by the Nekrasov-Okounkov formula, conjectured combinatorial interpretations of analogous expressions involving hook-lengths and symplectic/orthogonal contents. We prove special cases of these conjectures. In the process, we show that partitions of $n$ with all symplectic contents non-zero are equinumerous with partitions of $n$ into distinct even parts. We also present Beck-type companions to this identity. In this context, we give the parity of the number of partitions into distinct parts with odd (respectively, even) rank. We study the connection between the sum of hook-lengths and the sum of inversions in the binary representation of a partition. In addition, we introduce a new partition statistic, the $x$-ray list of a partition, and explore its connection with distinct partitions as well as partitions maximally contained in a given staircase partition.

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arXiv2022

Refinements of Beck-type partition identities

Franklin's identity generalizes Euler's identity and states that the number of partitions of $n$ with $j$ different parts divisible by $r$ equals the number of partitions of $n$ with $j$ repeated parts. In this article, we give a refinement of Franklin's identity when $j=1$. We prove Franklin's identity when $j=1$, $r=2$ for partitions with fixed perimeter, i.e., fixed largest hook. We also derive a Beck-type identity for partitions with fixed perimeter: the excess in the number of parts in all partitions into odd parts with perimeter $M$ over the number of parts in all partitions into distinct parts with perimeter $M$ equals the number of partitions with perimeter $M$ whose set of even parts is a singleton. We provide analytic and combinatorial proofs of our results.

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arXiv2022

Dyson's crank and unimodal compositions

The crank is a partition statistic requested by Dyson in 1944 in order to combinatorially prove a Ramanujan congruence of Euler's partition function $p(n)$. In this paper, we provide connections between Dyson's crank and unimodal compositions. Somewhat unrelated, we give a combinatorial proof of a new truncated Euler pentagonal number theorem due to Xia and Zhao.

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arXiv2022

On a Partition Identity of Lehmer

Euler's identity equates the number of partitions of any non-negative integer n into odd parts and the number of partitions of n into distinct parts. Beck conjectured and Andrews proved the following companion to Euler's identity: the excess of the number of parts in all partitions of n into odd parts over the number of parts in all partitions of n into distinct parts equals the number of partitions of n with exactly one even part (possibly repeated). Beck's original conjecture was followed by generalizations and so-called "Beck-type" companions to other identities. In this paper, we establish a collection of Beck-type companion identities to the following result mentioned by Lehmer at the 1974 International Congress of Mathematicians: the excess of the number of partitions of n with an even number of even parts over the number of partitions of n with an odd number of even parts equals the number of partitions of n into distinct, odd parts. We also establish various generalizations of Lehmer's identity, and prove related Beck-type companion identities. We use both analytic and combinatorial methods in our proofs.

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arXiv2021

Beck-type companion identities for Franklin's identity

The original Beck conjecture, now a theorem due to Andrews, states that the difference in the number of parts in all partitions into odd parts and the number of parts in all strict partitions is equal to the number of partitions whose set of even parts has one element, and also to the number of partitions with exactly one part repeated. This is a companion identity to Euler's identity. The theorem has been generalized by Yang to a companion identity to Glaisher's identity. Franklin generalized Glaisher's identity, and in this article, we provide a Beck-type companion identity to Franklin's identity and prove it both analytically and combinatorially. Andrews' and Yang's respective theorems fit naturally into this very general frame. We also discuss a generalization to Franklin's identity of the second Beck-type companion identity proved by Andrews and Yang in their respective work.

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arXiv2020

Beck-type identities: new combinatorial proofs and a theorem for parts congruent to $t$ mod $r$

Let $\mathcal O_r(n)$ be the set of $r$-regular partitions of $n$, $\mathcal D_r(n)$ the set of partitions of $n$ with parts repeated at most $r-1$ times, $\mathcal O_{1,r}(n)$ the set of partitions with exactly one part (possibly repeated) divisible by $r$, and let $\mathcal D_{1,r}(n)$ be the set of partitions in which exactly one part appears at least $r$ times. If $E_{r, t}(n)$ is the excess in the number of parts congruent to $t \pmod r$ in all partitions in $\mathcal O_r(n)$ over the number of different parts appearing at least $t$ times in all partitions in $\mathcal D_r(n)$, then $E_{r, t}(n) = |\mathcal O_{1,r}(n)| = |\mathcal D_{1,r}(n)|$. We prove this analytically and combinatorially using a bijection due to Xiong and Keith. As a corollary, we obtain the first Beck-type identity, i.e., the excess in the number of parts in all partitions in $\mathcal{O}_r(n)$ over the number of parts in all partitions in $\mathcal{D}_r(n)$ equals $(r - 1)|\mathcal{O}_{1,r}(n)|$ and also $(r - 1)|\mathcal{D}_{1,r}(n)|$. Our work provides a new combinatorial proof of this result that does not use Glaisher's bijection. We also give a new combinatorial proof based of the Xiong-Keith bijection for a second Beck-Type identity that has been proved previously using Glaisher's bijection.

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arXiv2020

Beck-type identities for Euler pairs of order $r$

Partition identities are often statements asserting that the set $\mathcal P_X$ of partitions of $n$ subject to condition $X$ is equinumerous to the set $\mathcal P_Y$ of partitions of $n$ subject to condition $Y$. A Beck-type identity is a companion identity to $|\mathcal P_X|=|\mathcal P_Y|$ asserting that the difference $b(n)$ between the number of parts in all partitions in $\mathcal P_X$ and the number of parts in all partitions in $\mathcal P_Y$ equals a $c|\mathcal P_{X'}|$ and also $c|\mathcal P_{Y'}|$, where $c$ is some constant related to the original identity, and $X'$, respectively $Y'$, is a condition on partitions that is a very slight relaxation of condition $X$, respectively $Y$. A second Beck-type identity involves the difference $b'(n)$ between the total number of different parts in all partitions in $\mathcal P_X$ and the total number of different parts in all partitions in $\mathcal P_Y$. We extend these results to Beck-type identities accompanying all identities given by Euler pairs of order $r$ (for any $r\geq 2$). As a consequence, we obtain many families of new Beck-type identities. We give analytic and bijective proofs of our results.

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arXiv2020

Combinatorial Proof of the Minimal Excludant Theorem

The minimal excludant of a partition $λ$, $\rm{mex}(λ)$, is the smallest positive integer that is not a part of $λ$. For a positive integer $n$, $ σ\, \rm{mex}(n)$ denotes the sum of the minimal excludants of all partitions of $n$. Recently, Andrews and Newman obtained a new combinatorial interpretations for $σ\, \rm{mex}(n)$. They showed, using generating functions, that $σ\, \rm{mex}(n)$ equals the number of partitions of $n$ into distinct parts using two colors. In this paper, we provide a purely combinatorial proof of this result and new properties of the function $σ\, \rm{mex}(n)$. We generalize this combinatorial interpretation to $σ_r\, \rm{mex}(n)$, the sum of least $r$-gaps in all partitions of $n$. The least $r$-gap of a partition $λ$ is the smallest positive integer that does not appear at least $r$ times as a part of $λ$.

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arXiv2018

Combinatorial proofs of two Euler type identities due to Andrews

We prove combinatorially some identities related to Euler's partition identity (the number of partitions of $n$ into distinct parts equals the number of partitions of $n$ into odd parts). They were conjectured by Beck and proved by Andrews via generating functions. Let $a(n)$ be the number of partitions of $n$ such that the set of even parts has exactly one element, $b(n)$ be the difference between the number of parts in all odd partitions of $n$ and the number of parts in all distinct partitions of $n$, and $c(n)$ be the number of partitions of $n$ in which exactly one part is repeated. Then, $a(n)=b(n)=c(n)$. The identity $a(n)=c(n)$ was proved combinatorially (in greater generality) by Fu and Tang. We prove combinatorially that $a(n)=b(n)$ and $b(n)=c(n)$. Our proof relies on bijections between a set and a multiset, where the partitions in the multiset are decorated with bit strings. Let $c_1(n)$ be the number of partitions of $n$ such that there is exactly one part occurring three times while all other parts occur only once and let $b_1(n)$ to be the difference between the total number of parts in the partitions of $n$ into distinct parts and the total number of different parts in the partitions of $n$ into odd parts. We prove combinatorially that $c_1(n)=b_1(n)$. In addition to these results by Andrews, we prove combinatorially that $b_1(n)=a_1(n)$, where $a_1(n)$ counts partitions of $n$ such that the set of even parts has exactly one element and satisfying some additional conditions. We also treat the case when exactly one part occurs twice while all other parts occur only once.

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arXiv2017

Quasisymmetric Power Sums

In the 1995 paper entitled "Noncommutative symmetric functions," Gelfand, et. al. defined two noncommutative symmetric function analogues for the power sum basis of the symmetric functions, along with analogues for the elementary and the homogeneous bases. They did not consider the noncommutative symmetric power sum duals in the quasisymmetric functions, which have since been explored only in passing by Derksen and Malvenuto-Reutenauer. These two distinct quasisymmetric power sum bases are the topic of this paper. In contrast to the simplicity of the symmetric power sums, or the other well known bases of the quasisymmetric functions, the quasisymmetric power sums have a more complex combinatorial description. As a result, although symmetric function proofs often translate directly to quasisymmetric analogues, this is not the case for quasisymmetric power sums. Neither is there a model for working with the quasisymmetric power sums in the work of Gelfand, et. al., which relies heavily on quasi-determinants (which can only be exploited by duality for our purposes) and is not particularly combinatorial in nature. This paper therefore offers a first glimpse at working with these two relatively unstudied quasisymmetric bases, avoiding duality where possible to encourage a previously unexplored combinatorial understanding.

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arXiv2017

Bisected theta series, least $r$-gaps in partitions, and polygonal numbers

The least $r$-gap, $g_r(λ)$, of a partition $λ$ is the smallest part of $λ$ appearing less than $r$ times. In this article we introduce two new partition functions involving least $r$-gaps. We consider a bisection of a classical theta identity and prove new identities relating Euler's partition function $p(n)$, polygonal numbers, and the new partition functions. To prove the results we use an interplay of combinatorial and $q$-series methods. We also give a combinatorial interpretation for $$\sum_{n=0}^\infty (\pm 1)^{k(k+1)/2} p(n-r\cdot k(k+1)/2).$$

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arXiv2015

Explicit construction of Ramanujan bigraphs

We construct explicitly an infinite family of Ramanujan graphs which are bipartite and biregular. Our construction starts with the Bruhat-Tits building of an inner form of $SU_3(\mathbb Q_p)$. To make the graphs finite, we take successive quotients by infinitely many discrete co-compact subgroups of decreasing size.

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arXiv2014

Stability of coefficients in the Kronecker product of a hook and a rectangle

We use recent work of Jonah Blasiak (2012) to prove a stability result for the coefficients in the Kronecker product of two Schur functions: one indexed by a hook partition and one indexed by a rectangle partition. We also give bounds for the size of the partition starting with which the Kronecker coefficients are stable. Moreover, we show that once the bound is reached, no new Schur functions appear in the decomposition of Kronecker product, thus allowing one to recover the decomposition from the smallest case in which the stability holds.

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arXiv2013

Schur-positivity in a Square

Determining if a symmetric function is Schur-positive is a prevalent and, in general, a notoriously difficult problem. In this paper we study the Schur-positivity of a family of symmetric functions. Given a partition λ, we denote by λ^c its complement in a square partition (m^m). We conjecture a Schur-positivity criterion for symmetric functions of the form s_{μ'}s_{μ^c}-s_{λ'}s_{λ^c}, where λis a partition of weight |μ|-1 contained in μand the complement of μis taken in the same square partition as the complement of λ. We prove the conjecture in many cases.

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arXiv2012

Powers of the Vandermonde determinant, Schur Functions, and recursive formulas

Since every even power of the Vandermonde determinant is a symmetric polynomial, we want to understand its decomposition in terms of the basis of Schur functions. We investigate several combinatorial properties of the coefficients in the decomposition. In particular, we give recursive formulas for the coefficient of the Schur function $s_{\m}$ in the decomposition of an even power of the Vandermonde determinant in $n + 1$ variables in terms of the coefficient of the Schur function $s_ł$ in the decomposition of the same even power of the Vandermonde determinant in $n$ variables if the Young diagram of $\m$ is obtained from the Young diagram of $ł$ by adding a tetris type shape to the top or to the left. An extended abstract containing the statement of the results presented here appeared in the Proceedings of FPSAC11

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