
Andrew SillsGeorgia Southern University | GSU · Department of Mathematical Sciences
Andrew Sills
Ph.D.
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Introduction
Skills and Expertise
Publications
Publications (92)
In the literature, derivations of exact null distributions of rank-sum statistics is often avoided in cases where one or more ties exist in the data. By deriving the null distribution in the no-ties case with the aid of classical $q$-series results of Euler and Rothe, we demonstrate how a natural generalization of the method may be employed to deri...
We use sums over integer compositions analogous to generating functions in partition theory, to express certain partition enumeration functions as sums over compositions into parts that are k-gonal numbers; our proofs employ Ramanujan’s theta functions. We explore applications to lacunary q-series, and to a new class of composition-theoretic Dirich...
This is a lightly edited version of the author's unpublished master's essay, submitted in partial fulfillment of the requirements of the degree of Master of Arts at the Pennsylvania State University, dated June 1994, written under the supervision of Professor George E. Andrews. It was retyped by the author on November 23, 2022. Obvious typographica...
We use sums over {integer compositions} analogous to generating functions in {partition theory}, to express certain partition enumeration functions as sums over compositions into parts that are $k$-gonal numbers; our proofs employ Ramanujan's theta functions. We explore applications to lacunary $q$-series, and to a new class of composition-theoreti...
A case is made that researchers are interested in studying processes. Often the inferences they are interested in making are about the process and its associated population. On other occasions, a researcher may be interested in making an inference about the collection of individuals the process has generated. We will call the statistical methods em...
A generalization of the law of total covariance is presented and proved.
Let $d_S$ denote the arithmetic density of a subset $S \subseteq \mathbb N$. We derive a power series in $q\in \mathbb C$, $|q|<1$, with co\"efficients related to integer partitions and integer compositions, that yields $1/d_S$ in the limit as $q\to 1$ radially.
Define a non-unitary partition to be an integer partition with no part equal to one, and let $\nu(n)$ denote the number of non-unitary partitions of size $n$. In recent work, the sixth author proved a formula to compute $p(n)$ by enumerating only non-unitary partitions of size $n$, and recorded a number of conjectures. Here we refine and prove some...
q-analogs of special functions, including hypergeometric functions, play a central role in mathematics and have numerous applications in physics. In the theory of probability, q-analogs of various probability distributions have been introduced over the years, including the binomial distribution. Here, I propose a new refinement of the binomial dist...
MacMahon showed that the generating function for partitions into at most k parts can be decomposed into a partial fraction-type sum indexed by the partitions of k. In the present work, a generalization of MacMahon’s result is given, which in turn provides a full combinatorial explanation.
q$-analogs of special functions, including hypergeometric functions, play a central role in mathematics and have numerous applications in physics. In the theory of probability, $q$-analogs of various probability distributions have been introduced over the years, including the binomial distribution. Here, I propose a new refinement of the binomial d...
We examine “partition zeta functions” analogous to the Riemann zeta function but summed over subsets of integer partitions. We prove an explicit formula for a family of partition zeta functions already shown to have nice properties — those summed over partitions of fixed length — which yields complete information about analytic continuation, poles...
The derivation of the Hardy-Ramanujan-Rademacher formula for the number of partitions of $n$ is reviewed. Next, the steps for finding analogous formulas for certain restricted classes of partitions or overpartiitons is examined, bearing in mind how these calculations can be automated in a CAS. Finally, a number of new formulas of this type which we...
Two closely related discrete probability distributions are introduced. In each case the support is a set of vectors in Rn obtained from the partitions of the fixed positive integer n. These distributions arise naturally when considering equally-likely random permutations on the set of n letters. For one of the distributions, the expectation vector...
Two closely related discrete probability distributions are introduced. In each case the support is a set of vectors in $\mathbb{R}^n$ obtained from the partitions of the fixed positive integer $n$. These distributions arise naturally when considering equally-likely random permutations on the set of $n$ letters. For one of the distributions, the exp...
MacMahon showed that the generating function for partitions into at most k parts can be decomposed into a partial fraction-type sum indexed by the partitions of k. In the present work, a generalization of MacMahon’s result is given, which in turn provides a full combinatorial explanation.
We examine "partition zeta functions" analogous to the Riemann zeta function but summed over subsets of integer partitions. We prove an explicit formula for a family of partition zeta functions already shown to have nice properties -- those summed over partitions of fixed length -- which yields complete information about analytic continuation, pole...
In this article we study the norm of an integer partition, which is defined as the product of the parts. This partition-theoretic statistic has appeared here and there in the literature of the last century or so, and is at the heart of current research by both authors. We survey known results and give new results related to this all-but-overlooked...
Grubbs and Weaver (1947) suggest a minimum-variance unbiased estimator for the population standard deviation of a normal random variable, where a random sample is drawn and a weighted sum of the ranges of subsamples is calculated. The optimal choice involves using as many subsamples of size eight as possible. They verified their results numerically...
Polynomial generalizations of all 130 of the identities in Slater's list of identities of the Rogers-Ramanujan type are presented. Furthermore, duality relationships among many of the identities are derived. Some of the these polynomial identities were previously known but many are new. The author has implemented much of the finitization process in...
Using a pair of two variable series-product identities recorded by Ramanujan in the lost notebook as inspiration, we find some new identities of similar type. Each identity immediately implies an infinite family of Rogers-Ramanujan type identities, some of which are well-known identities from the literature. We also use these identities to derive s...
We provide finite analogs of a pair of two-variable $q$-series identities from Ramanujan's lost notebook and a companion identity.
Lucy Slater used Bailey's $_6\psi_6$ summation formula to derive the Bailey pairs she used to construct her famous list of 130 identities of the Rogers-Ramanujan type. In the present paper we apply the same techniques to Chu's $_{10}\psi_{10}$ generalization of Bailey's formula to produce quite general Bailey pairs. Slater's Bailey pairs are then r...
We give "hybrid" proofs of the $q$-binomial theorem and other identities. The proofs are "hybrid" in the sense that we use partition arguments to prove a restricted version of the theorem, and then use analytic methods (in the form of the Identity Theorem) to prove the full version. We prove three somewhat unusual summation formulae, and use these...
In this survey article, we present an expanded version of Lucy Slater's famous list of identities of the Rogers-Ramanujan type, including identities of similar type, which were discovered after the publication of Slater's papers, and older identities (such as those in Ramanujan's lost notebook) which were not included in Slater's papers. We attempt...
A pair of sequences $(\alpha_{n}(a,k,q),\beta_{n}(a,k,q))$ such that $\alpha_0(a,k,q)=1$ and \[ \beta_{n}(a,k,q) = \sum_{j=0}^{n} \frac{(k/a; q)_{n-j}(k; q)_{n+j}}{(q;q)_{n-j}(aq;q)_{n+j}}\alpha_{j}(a,k,q) \] is termed a \emph{WP-Bailey Pair}. Upon setting $k=0$ in such a pair we obtain a \emph{Bailey pair}. In the present paper we consider the pro...
We provide the missing member of a family of four $q$-series identities related to the modulus 36, the other members having been found by Ramanujan and Slater. We examine combinatorial implications of the identities in this family, and of some of the identities we considered in "Identities of the Ramanujan-Slater type related to the moduli 18 and 2...
We describe three computer searches (in PARI/GP, Maple, and Mathematica, respectively) which led to the discovery of a number of identities of Rogers-Ramanujan type and identities of false theta functions.
Dedicated to George E. Andrews on the occasion of his 70th birthday. Submitted to a special issue for this occasion. We use Andrews' notion of a `signed partition' (i.e. partition where some parts are allowed to be negative) to interpret the G\"ollnitz--Gordon sum, and then provide a bijective map between these signed partitions and the usual G\"ol...
I discuss the computational methods behind the formulation of some conjectures related to variants on Andrews' $q$-Dyson conjecture.
We present several new families of Rogers-Ramanujan type identities related to the moduli 18 and 24. A few of the identities were found by either Ramanujan, Slater, or Dyson, but most are believed to be new. For one of these families, we discuss possible connections with Lie algebras. We also present two families of related false theta function ide...
It is shown that (two-variable generalizations of) more than half of Slater's list of 130 Rogers-Ramanujan identities (L. J. Slater, Further identities of the Rogers-Ramanujan type, \emph{Proc. London Math Soc. (2)} \textbf{54} (1952), 147--167) can be easily derived using just three multiparameter Bailey pairs and their associated $q$-difference e...
Some examples of naturally arising multisum $q$-series which turn out to have representations as fermionic single sums are presented. The resulting identities are proved using transformation formulas from the theory of basic hypergeometric series.
MacMahon showed that the generating function for partitions into at most $k$ parts can be decomposed into a partial fractions-type sum indexed by the partitions of $k$. In this present work, a generalization of MacMahon's result is given, which in turn provides a full combinatorial explanation.
The fibbinary numbers are positive integers whose binary representation contains no consecutive ones. We prove the following result: If the $j$th odd fibbinary is the $n$th \emph{odd} fibbinary number, then $j = \lfloor n\phi^2 \rfloor - 1.
Grubbs and Weaver (JASA 42 (1947) 224--241) suggest a minimum-variance unbiased estimator for the population standard deviation of a normal random variable, where a random sample is drawn and a weighted sum of the ranges of subsamples is calculated. The optimal choice involves using as many subsamples of size eight as possible. They verified their...
This is a written expansion of the talk delivered by the author at the International Conference on Number Theory in Honor of Krishna Alladi for his 60th Birthday, held at the University of Florida, March 17--21, 2016. Here we derive Bailey pairs that give rise to Rogers--Ramanujan type identities which are the principally specialized character of t...
We derive a combinatorial multisum expression for the number $D(n,k)$ of partitions of $n$ with Durfee square of order $k$. An immediate corollary is therefore a combinatorial formula for $p(n)$, the number of partitions of $n$. We then study $D(n,k)$ as a quasipolynomial. We consider the natural polynomial approximation $\tilde{D}(n,k)$ to the qua...
We present some Euler-type recurrences for the partition function $p(n)$.
We find relationships between subword patterns and residue classes of parts in the set of integer compositions of a given weight. In particular, we show that it is always possible to express the total number of parts in compositions of n that are congruent to i modulo m as a linear combination of the total number of occurrences of subword patterns...
The Rogers--Ramanujan identities are a pair of infinite series-infinite product identities that were first discovered in 1894. Over the past several decades these identities, and identities of similar type, have found applications in number theory, combinatorics, Lie algebra and vertex operator algebra theory, physics (especially statistical mechan...
This is a written expansion of the talk delivered by the author at the International Conference on Number Theory in Honor of Krishna Alladi for his 60th Birthday, held at the University of Florida, March 17–21, 2016. Here, we derive Bailey pairs that give rise to Rogers–Ramanujan type identities, the product sides of which are known to be the princ...
We derive a combinatorial multisum expression for the number D(n, k) of partitions of n with Durfee square of order k. An immediate corollary is therefore a combinatorial formula for p(n), the number of partitions of n. We then study D(n, k) as a quasipolynomial. We consider the natural polynomial approximation \({\tilde{D}(n, k)}\) to the quasipol...
In this note, we consider the trees (caterpillars) that minimize the number of subtrees among trees with a given degree sequence. This is a question naturally related to the extremal structures of some distance based graph invariants. We first confirm the expected fact that the number of subtrees is minimized by some caterpillar. As with other grap...
We present what we call a "motivated proof" of the Andrews-Bressoud partition
identities for even moduli. A "motivated proof" of the Rogers-Ramanujan
identities was given by G. E. Andrews and R. J. Baxter, and this proof was
generalized to the odd-moduli case of Gordon's identities by J. Lepowsky and M.
Zhu. Recently, a "motivated proof" of the som...
The fibbinary numbers are positive integers whose binary representation contains no consecutive ones. We prove the following result: If the jth odd fibbinary is the nth odd fibbinary number, then j=⌊nϕ 2 ⌋-1.
A bijective proof is given for the following theorem: the number of
compositions of n into odd parts equals the number of compositions of n + 1
into parts greater than one. Some commentary about the history of partitions
and compositions is provided.
Using a pair of two variable series-product identities recorded by Ramanujan in the lost notebook as inspiration, we find some new identities of similar type. Each identity immediately implies an infinite family of Rogers-Ramanujan type identities, some of which are well-known identities from the literature. We also use these identities to derive s...
The Wiener index of a graph is the sum of the distances between all pairs of vertices. It has been one of the main descriptors that correlate a chemical compound’s molecular structure with experimentally gathered data regarding the compound’s characteristics. In 2008, Wang and Zhang independently characterized trees with specified degree sequence t...
In this survey article, we present an expanded version of Lucy Slater’s famous list of identities of the Rogers-Ramanujan type, including identities of similar type, which were discovered after the publication of Slater’s papers, and older identities (such as those in Ramanujan’s lost notebook) which were not included in Slater’s papers. We attempt...
In his book \emph{Topics in Analytic Number Theory}, Hans Rademacher
conjectured that the limits of certain sequences of coefficients that arise in
the ordinary partial fraction decomposition of the generating function for
partitions of integers into at most $N$ parts exist and equal particular values
that he specified. Despite being open for nearl...
The purpose of this short article is to announce, and briefly describe, a
Maple package, PARTITIONS, that (inter alia) completely automatically
discovers, and then proves, explicit expressions (as sums of quasi-polynomials)
for pm(n) for any desired m. We do this to demonstrate the power of "rigorous
guessing" as facilitated by the quasi-polynomial...
We provide finite analogs of a pair of two-variable q-series identi- ties from Ramanujan's lost notebook and a companion identity.
A collection of Hardy-Ramanujan-Rademacher type formulas for restricted partition and overpartition functions is presented,
framed by several biographical anecdotes.
KeywordsPartitions-Circle method-Rogers-Ramanujan identities
Mathematics Subject Classification (2000)11P82-11P85-05A19
A Rademacher-type convergent series formula which generalizes the Hardy-Ramanujan-Rademacher formula for the number of partitions of n and the Zuckerman formula for the Fourier coefficients of ϑ4(0∣τ)-1 is presented.
The derivation of the Hardy-Ramanujan-Rademacher formula for the number of partitions of n is reviewed. Next, the steps for finding anal-ogous formulas for certain restricted classes of partitions or overpartitions is examined, bearing in mind how these calculations can be automated in a CAS. Finally, a number of new formulas of this type which wer...
We give "hybrid" proofs of the q-binomial theorem and other identities. The proofs are "hybrid" in the sense that we use partition arguments to prove a restricted version of the theorem, and then use analytic methods (in the form of the Identity Theorem) to prove the full version. We prove three somewhat unusual summation formulae, and use these to...
Lucy Slater used Bailey’s 6 ψ 6 summation formula to derive the Bailey pairs she used to construct her famous list of 130 Rogers-Ramanujan type identities. In the present paper we apply the same techniques to Chu’s 10 ψ 10 generalization of Bailey’s formula to produce quite general Bailey pairs. Slater’s Bailey pairs are then recovered as special l...
We show that an identity of Gessel and Stanton (Trans. Amerc. Math Soc. 277 (1983), p. 197, Eq. (7.24)) can be viewed as a a symmetric version of a recent analytic variation of the little Göllnitz identities. This is significant, since the Göllnitz-Gordon identities are considered the usual symmetric counterpart to little Göllnitz theorems. Is it p...
We describe three computer searches (in PARI/GP, Maple, and Mathematica, respectively) which led to the discovery of a number of identities of Rogers-Ramanujan type and identities of false theta functions.
A pair of sequences (α n (a,k,q),β(a,k,q)) such that α 0 (a,k,q)=1 and β n (a,k,q)=∑ j=0 n (k/a;q) n-j (k;q) n+j (q;q) n-j (aq;q) n+j α j (a,k,q) is termed a WP-Bailey pair. Upon setting k=0 in such a pair we obtain a Bailey pair. In the present paper we consider the problem of “lifting” a Bailey pair to a WP-Bailey pair, and use some of the new WP...
We use generalized lecture hall partitions to discover a new pair of q-series identities. These identities are unusual in that they involve partitions into parts from asymmetric residue classes, much like the little Göllnitz partition theorems. We derive a two-parameter generalization of our identities that, surprisingly, gives new analytic counter...
We present several new families of Rogers-Ramanujan type identities related to the moduli 18 and 24. A few of the identities were found by either Ramanujan, Slater, or Dyson, but most are believed to be new. For one of these families, we discuss possible connections with Lie algebras. We also present two families of related false theta function ide...
In this survey article, we present an expanded version of Lucy Slater's famous list of identities of the Rogers-Ramanujan type, including identities of similar type, which were discovered after the publication of Slater's papers, and older identities (such as those in Ramanujan's lost notebook) which were not included in Slater's papers. We attempt...
We provide a bijective map from the partitions enumerated by the series side of the Rogers-Selberg mod 7 identities onto partitions associated with a special case of Basil Gordon's combinatorial generalization of the Rogers-Ramanujan identities. The implications of applying the same map to a special case of David Bressoud's even modulus analog of G...
Just as the authors anxiously waited for the identities of the Top Seven Songs of the week years ago, readers of this article must now be brimming with unbridled excitement to learn the identities of the Top Ten Most Fascinating Formulas from Ra-manujan's Lost Notebook. The choices for the Top Ten Formulas were made by the authors. However, motivat...
We provide the missing member of a family of four q-series identities related to the modulus 36, the other members having been found by Ramanujan and Slater. We examine combina-torial implications of the identities in this family, and of some of the identities we considered in "Identities of the Ramanujan-Slater type related to the moduli 18 and 24...
I discuss the computational methods behind the formulation of some conjectures related to variants on Andrews' q-Dyson conjecture.
Some examples of naturally arising multisum q-series which turn out to have representations as fermionic single sums are presented. The resulting identities are proved using transformation formulas from the theory of basic hypergeometric series.
In this we paper we prove several new identities of the Rogers-Ramanujan-Slater type. These identities were found as the result of computer searches. The proofs involve a variety of techniques, including series-series identities, Bailey pairs, a theorem of Watson on basic hypergeometric series, generating functions and miscellaneous methods.
In this we paper we prove several new identities of the Rogers-Ramanujan-Slater type. These identities were found as the result
of computer searches. The proofs involve a variety of techniques, including series-series identities, Bailey pairs, a theorem
of Watson on basic hypergeometric series, generating functions and miscellaneous methods.
A generalized Bailey pair, which contains several special cases considered by Bailey (Proc. London Math. Soc. (2), 50, 421–435 (1949)), is derived and used to find a number of new Rogers-Ramanujan type identities. Consideration of associated
q-difference equations points to a connection with a mild extension of Gordon’s combinatorial generalization...
We present a case study in experimental yet rigorous mathematics by describing an algorithm, fully implemented in both Mathematica and Maple, that automatically conjectures, and then automatically proves, closed-form expressions extending Dyson's celebrated constant-term conjecture.
A multiparameter generalization of the Bailey pair is defined in such a way as to include as special cases all Bailey pairs considered by W.N. Bailey in his paper [Identities of the Rogers–Ramanujan type, Proc. London Math. Soc. (2) 50 (1949) 421–435]. This leads to the derivation of a number of elegant new Rogers–Ramanujan type identities.
In his paper, On a partition function of Richard Stanley, George Andrews
proves a certain partition identity analytically and asks for a
combinatorial proof.This paper provides the requested
combinatorial proof.
Using Lie theory, Stefano Capparelli conjectured an interesting Rogers–Ramanujan type partition identity in his 1988 Rutgers PhD thesis. The first proof was given by George Andrews, using combinatorial methods. Later, Capparelli was able to provide a Lie theoretic proof.Most combinatorial Rogers–Ramanujan type identities (e.g., the Göllnitz–Gordon...
In a recent paper, I defined the "standard multiparameter Bailey pair" (SMPBP) and demon-strated that all of the classical Bailey pairs considered by W.N. Bailey in his famous paper (Proc. London Math. Soc. (2), 50 (1948), 1–10) arose as special cases of the SMPBP. Additionally, I was able to find a number of new Rogers-Ramanujan type identities. F...
In 1958, Richard Guy proved that the number of partitions of n into odd parts greater than one equals the number of partitions of n into distinct parts with no powers of 2 allowed, which is closely related to Euler's famous theorem that the number of partitions of n into odd parts equals the number of partitions of n into distinct parts. We conside...
The purpose of this paper is to introduce the RRtools and recpf Maple packages which were developed by the author to assist in the discovery and proof of finitizations of identities of the Rogers–Ramanujan type.
Polynomial generalizations of all 130 of the identities in Slater's list of identities of the Rogers-Ramanujan type are presented. Furthermore, duality relationships among many of the identities are derived. Some of the these polynomial identities were previously known but many are new. The author has implemented much of the nitization process in a...
We examine a pair of Rogers–Ramanujan type identities of Lebesgue, and give polynomial identities for which the original identities are limiting cases. The polynomial identities turn out to be q-analogs of the Pell sequence. Finally, we provide combinatorial interpretations for the identities.
Dyson's celebrated constant term conjecture (J. Math. Phys., 3 (1962): 140–156) states that the constant term in the expansion of 1i =jn (1 − x i /x j) a j is the multinomial coefficient (a 1 + a 2 + · · · + a n)!/(a 1 !a 2 ! · · · a n !). The definitive proof was given by I. In this paper, closed form expressions are given for the coefficients of...