Even–odd partition identities of Rogers–Ramanujan type

In a recent paper, Frechette and Locus examined and found expressions for the infinite product [math] in terms of products of [math] -series of the Rogers–Ramanujan type coming from Hall–Littlewood polynomials, when [math] . These [math] -series were originally discovered in 2014 by Griffin, Ono, and Warnaar in their work on the framework of the Rogers–Ramanujan identities. Extending this framework, Rains and Warnaar also recently discovered more [math] -series and their corresponding infinite products. Frechette and Locus left open the case where [math] . Here we find such an expression for the infinite products for [math] by making use of the new [math] -series obtained by Rains and Warnaar.

On Modular Relations for the Göllnitz–Gordon Functions with Applications to Partitions

Basis partition polynomials, overpartitions and the Rogers–Ramanujan identities

In this paper, we research theta function identity involving Rogers– Ramanujan identity and establish a Rogers–Ramanujan–Slater type theta function identity related to G(q) and φ(q).

In this paper we interpret Macdonald's unspecialized identities as multivariable vector partition theorems and we relate the well-known Rogers-Ramanujan partition identities to the Weyl-Kac character formula for an infinite-dimensional Euclidean generalized Cartan matrix Lie algebra.

In this paper, we give the generalization of MacMahon's type combinatorial identities. A generalized $q$-series is interpreted as the generating function of two different combinatorial objects, viz., restricted $n$-color partitions and weighted lattice paths which give entirely new Rogers–Ramanujan–MacMahon type combinatorial identities. This result yields an infinite class of 2-way combinatorial identities which further extends the work of Agarwal and Goyal. We also discuss the bijective proof of the main result. Forbye, eight particular cases are also discussed which give a combinatorial interpretation of eight entirely new Rogers–Ramanujan type identities.

We give an explicit recursive description of the Hilbert series and Grobner bases for the family of quadratic ideals defining the jet schemes of a double point. We relate these recursions to the Rogers–Ramanujan identity and prove a conjecture of the second author, Oblomkov and Rasmussen.

We present a new partition identity and give a combinatorial proof of our result. This generalizes a result of Andrews in which he considers the generating function for partitions with respect to size, number of odd parts, and number of odd parts of the conjugate.

Introduction.Let Pd(n) denote the number of partitions of into positive integers of the form = A1+ +bs with -b + 1 ^ d where strict inequality holds if d\b.Px(n) appears in the first Rogers-Ramanujan identity [5, p. 291].P2(n) appears in the first of the Gllnitz-Gordon partition theorems [2, p. 945], and P3() appears in a partition theorem of I. J. Schur [6, p. 489].For d>3, we have the following result due to H. L. Alder [1, p. 713].(1.1) Let S be any fixed set of positive integers.Then Pd(n) is not identically equal to the number of partitions of into parts taken from S if d> 3.In a different approach, B. Gordon [4, p. 394] has made the following generalization of the Rogers-Ramanujan identities.(1.2) Let a and k be integers with 0<a^k.Let Akia(n) denote the number of partitions of into parts ^0, a (mod 2k +1).Let Bka(n) denote the number of partitions of of the form = >1+ +bs with ^>i + 1, , -i + fc_i^2, and with 1 appearing as a summand at most a-1 times.Then Aka(n) = Bk(1(n).The Gllnitz-Gordon identities have also been generalized in this manner [2, p. 946].(1.3) Let a and k be integers with 0<a/c.Let A^a(n) denote the number of partitions of into parts which are neither =2 (mod 4) nor =0, (2a-1) (mod 4k).Let B_a(n) denote the number of partitions of of the form n = by+ +bs, with b(bi+1, no odd parts are repeated, -i+fc_1^2 with strict inequality if 2[> and at most a-1 parts not exceeding 2 appear.Then Ak*,a(n) = Bffa(n).This leads us to define the following partition function.Let BAik<a(n) denote the number of partitions of of the form n = bx+ +bs, with btbt+i, no parts ^0 (modA+1) are repeated, bi-bi+k_x = A+l with strict inequality if X+l\bt and at most a-I parts not exceeding A + l appear.We note that B0ka(n) appears in (1.2); Bx,k,a(n) appears in (1.3), and 77d_1>2>2()=Pd().Question.If k is sufficiently large with respect to A, does B^ka(n) appear in a partition theorem of the Rogers-Ramanujan typelWe answer this question affirmatively with the following result.Theorem 1.Let O^X^a^k be integers, and 2X-l^k.If X is even, define

Providing an algebraic proof that φ is indeed a bijection is an instructive exercise. By exploiting the multiplicative structure of the codomain, we can construct a map ψ : Z+ × Z+ → Z+ which is immediately recognized as a bijection. (No need to resort to algebraic calculation or a pictorial argument with diagonals.) For each pair of positive integers m and n, let ψ(m, n) = 2m−1(2n − 1). Bijectivity of ψ is equivalent to the fact that every positive integer has a unique representation as the product of an odd positive integer and a non-negative integer power of 2. As one referee noted, this fact is also key to Glaisher’s bijection between partitions of a positive integer into odd parts and partitions with distinct parts [1, Ex. 2.2.6; 4, p. 12].

Combinatorial proofs and generalizations of conjectures related to Euler’s partition theorem

This article explores the history of the two results in integer partitions known as Stanley’s theorem and Elder’s theorem. While history has credited Richard Stanley with the discovery of the results, we note that Nathan Fine had established these results among a host of other partition identities over a decade earlier. In tribute to Fine, analogues in the sets of odd partitions and distinct partitions are presented.

The trinomial coefficients are defined centrally by Σ j = − m ∞ ( j m ) 2 x j = ( 1 + x + x − 1 ) m \Sigma _{j = - m}^\infty {(_j^m)_2}{x^j} = {(1 + x + {x^{ - 1}})^m} . Euler observed that for − 1 ≤ m ≤ 7 - 1 \leq m \leq 7 , 3 ( 0 m + 1 ) 2 − ( 0 m + 2 ) 2 = F m ( F m + 1 ) 3{(_{\,\;0}^{m + 1})_2} - {(_{\,\;0}^{m + 2})_2} = {F_m}({F_m} + 1) , where F m {F_m} is the m m th Fibonacci number. The assertion is false for m &gt; 7 m &gt; 7 . We prove general identities—one of which reduces to Euler’s assertion for m ≤ 7 m \leq 7 . Our main object is to analyze q q -analogs extending Euler’s observation. Among other things we are led to finite versions of dissections of the Rogers-Ramanujan identities into even and odd parts.

Corteel, Lovejoy, and Mallet concluded their paper “An extension to overpartitions of the Rogers–Ramanujan identities for even moduli” with an open question of investigating the combinatorial properties of a q-series with two additional parameters. We settle their question, unfortunately in the negative, by showing that the series yields only the known results in overpartitions. However, when one annihilates one of the parameters, the resulting series have nice integer partitions interpretations. Those series appeared in another publication as well. In particular, Corteel, Lovejoy, and Mallet’s series involve an index d. This index unifies two classes of overpartition identities for \(d = 1\) and \(d = 2\), but does not give additional overpartition identities for \(d \ge 3\). Upon setting one of the parameters zero, one does get regular partition identities for all d. The proofs are conventional, formal verifications for brevity, but we show how to make the proofs constructive.

The Abel’s lemma on summation by parts is employed to review identities of Rogers–Ramanujan type. Twenty examples are illustrated including several new RR identities.