Partition Identities Research Articles

New Ramanujan-Kolberg type partition identities

In this article, we use functions studied by N. J. Fine and R. J. Evans to construct analogues of modular equations first discovered by S. Ramanujan. We then use these functions to construct new identities satisfied by � ∞=0 p(ln+k)q n , with odd prime l and 0 ≤ k ≤ (l − 1). Our new partition identities are inspired by the work of O. Kolberg and Ramanujan.

Gap Unification Theory: A Note on the Treatment of Partitions via Their Gaps

Recently, several authors have been treating partitions by considering the gaps between their consecutive parts. Many of the existing results involving “partitions with parts in the gaps” bear a striking resemblance to certain weighted partition identities that also arise from the gaps of partitions. In this paper, we show that this is not merely a coincidence, and we demonstrate that many of these weighted partition identities are equivalent to identities involving partitions with parts in the gaps. We intend that this brief note will more or less act as a Rosetta Stone for some of the literature on the gap-theoretic study of partitions.

Q-ENGEL SERIES EXPANSIONS AND SLATER'S IDENTITIES

Dedicated to the memory of John Knopfmacher (1937–1999) We describe the q-Engel series expansion for Laurent series discovered by John Knopfmacher and use this algorithm to shed new light on partition identities related to two entries from Slater's list. In our study Al-Salam/Ismail and Santos polynomials play a crucial rĉole.

N-graphs, modular Sidon and sum-free sets, and partition identities

Using a new graphical representation for partitions, the author obtains a family of partition identities associated with partitions into distinct parts of an arithmetic progression, or, more generally, with partitions into distinct parts of a set that is a finite union of arithmetic progressions associated with a modular sum-free Sidon set. Partition identities are also constructed for sets associated with modular sum-free sets.

Elementary Proofs of Identities for Schur Functions and Plane Partitions

We use elementary methods to prove formulas that represent sums of restricted classes of Schur functions as ratios of determinants. This includes recent formulas for sums over bounded partitions with even parts and sums over bounded partitions whose conjugates have only even parts. All of these formulas imply plane partition generating functions.

A Family of Partition Identities Proved Combinatorially

We present three infinite families of partitions that, for corresponding parameters, are equinumerous. This generalizes an identity proved by the first author (Pacific Journal of Mathematics, 77(1) (1978)) and two identities proved by the second and third authors (Relatorio Tecnico \({\text{n}}\underline {\text{o}} \) 08/99 IMECC-UNICAMP, 1999).

The Rogers-Ramanujan identities, the finite general linear groups, and the Hall-Littlewood polynomials

We connect Gordon's generalization of the Rogers-Ramanujan identities with the Hall-Littlewood polynomials and with generating functions which arise in a probabilistic setting in the finite general linear groups. This yields a Rogers-Ramanujan type product formula for the n -+ oo probability that an element of GL(n, q) or Mat(n, q) is semisimple. 1. BACKGROUND AND NOTATION The Rogers-Ramanujan identities are among the most remarkable partition identities in number theory and combinatorics. This paper will be concerned with the following generalization of the Rogers-Ramanujan identities, due to Gordon. Let (x)n denote (1 x)(1 x2)... (1 xn). Theorem 1 ([A, page 111]). For 1 2, and complex x with Ixl O )*'' r0,?i(mod 2k+l) where Nj = nj + + nk-1. Gordon's generalization of the Rogers-Ramanujan identities has been widely studied and appears in many places in mathematics and physics. Andrews [A] discusses combinatorial aspects of these identities. In an important series of papers, Lepowsky and Wilson [LW1], [LW2], [LW3] connect the Gordon identities with affine Lie algebras and structures that they called Z-algebras (later interpreted as parafermion algebras in conformal field theory). Meurman and Primc [MP] solve a problem left open in [LW3], proving the independence of a Z-algebra basis and obtaining a Z-algebra proof of the Gordon identities. Feigin and Frenkel [FF] interpret the Gordon identities as a character formula for the Virasoro algebra. Andrews, Baxter, and Forrester [ABF] and Warnaar [W] relate the Gordon identities with statistical mechanics. For some number theoretic connections see the conference proceedings [AABRR]. We use the following standard notation from the theory of partitions. Call A = (A1,A2,--.) a partition of n = AIX if A1 > A2 > > 0 where the Ai are Received by the editors March 6, 1998. 1991 Mathematics Subject Classification. Primary 20P05, 05E05. ()1999 American Mathematical Society 17 This content downloaded from 207.46.13.142 on Sat, 28 May 2016 06:40:23 UTC All use subject to http://about.jstor.org/terms

Identities over set partitions

Certain types of partition identities, interpreted as moments of stochastic processes in the exponential case, are extended to larger classes by way of Faa DiBruno's formula. Several special cases are then examined. A probabilistic interpretation is given to the formula associated to the geometric series by way of the gambler's ruin problem.

Extending Theorems of Göllnitz, A New Family of Partition Identities

A new family of partition identities is given which include as special cases two theorems of Gollnitz. We show, also, a relation between our result and a theorem given by Sylvester.

A generalization of Sylvester's identity

We consider a new generalization of Euler's and Sylvester's identities for partitions. Our proof is based on an explicit bijection.

A probabilistic approach to some of Euler’s number theoretic identities

Probabilistic proofs and interpretations are given for the q-binomial theorem, q-binomial series, two of Euler's fundamental partition identities, and for q-analogs of product expansions for the Riemann zeta and Euler phi functions. The underlying processes involve Bernoulli trials with variable probabilities. Also presented are several variations on the classical derangement problem inherent in the distributions considered.

Rogers-Ramanujan Type Identities for Partitions with Attached Odd Parts

In this paper we present a new infinite family of partition identities. The genesis of our work lies in two formulas of Lucy Slater related to the modulus 8. Hirschhorn, Agarwal and Subbarao have previously found intriguing interpretations for Slater's formula, but none has led to an infinite family of results.

Partition identities involving gaps and weights

We obtain interesting new identities connecting the famous partition functions of Euler, Gauss, Lebesgue, Rogers–Ramanujan and others by attaching weights to the gaps between parts. The weights are in general multiplicative. Some identities involve powers of 2 as weights and yield combinatorial information about some remarkable partition congruences modulo powers of 2.

The quintuple product identity and shifted partition functions

The purpose of this paper is two-fold. First, a new proof of the Quintuple Product Identity is given. Next, the method of this proof is used to establish certain shifted partition identities extending earlier results that the author had obtained by analyzing the Göllnitz-Gordon and Rogers-Ramanujan identities.

On the Equivalence of Ramanujan's Partition Identities and a Connection with the Rogers–Ramanujan Continued Fraction

A famous identity of Ramanujan connected with partitions modulo 5 is shown to be equivalent to another identity of Ramanujan. The latter identity is used to establish a differential equation for the Rogers–Ramanujan continued fraction found in Ramanujan's lost notebook. We also prove that two other identities of Ramanujan are equivalent, one of which is associated with Ramanujan's partition congruence modulo 7. Last, we give a new proof of the transformation formula for the Dedekind eta-function, which is used in our proofs of equivalence.

A refinement of a partition identity and blocks of some modular characters

A refinement of a partition identity and blocks of some modular characters

New Proofs of Ramanujan′s Partition Identities for Moduli 5 and 7

The purpose of this paper is to present new proofs of modular identities (1.1) and (1.2). These identities are closely related to Ramanujan′s partition identities of moduli 5 and 7. The common feature in these proofs is that they both begin with the trigonometric identity (1.3).

Generalisations of the Andrews-Olsson partition identity and applications

In (Bessenrodt, 1991) a combinatorial proof of a refinement of the Andrews-Olsson partition identity (Andrews and Olsson, 1991) was given. In this article we explore the methods of (Bessenrodt, 1991) further to deal also with partitions where repetitions are allowed and where the restrictions on the congruence set are omitted. We obtain a considerable generalisation of the earlier results, again giving bijective proofs of the partition identities in question. As an application, it is then shown how to deduce (refinements of) some old results by Sylvester and Schur as well as some recent results by Alladi and Gordon [1].

Some new observations on the Göllnitz-Gordon and Rogers-Ramanujan identities

Two new, short and elementary proofs of the Go1nitz-Gordon identities are presented by considering the odd-even split of the Euler Pentagonal Series and the Triangular Series of Gauss. Using this approach the equality of certain shifted partition functions are established. Next, the odd and even parts of the famous Rogers-Ramanujan series are shown to have interesting product representations (mod 80). From this, new shifted partition identities are derived.

On a theorem of Alladi and Gordon and the gaussian polynomials

Alladi and Gordon [ J. Combin. Theory Ser. A 63 (1993), 275–300] have recently proved a general partition identity which generalized earlier results by Bressoud and Göllnitz by analyzing a continued fraction introduced by Ramanujan. This result was the motivation for the investigation of the sets of partitions and bipartitions defined in this article. We construct explicit bijections between the sets of partitions and bipartitions under consideration, giving a new proof of the formula for the Gaussian polynomials and leading to a strong refinement of the theorem by Alladi and Gordon.