We consider an identity relating Fibonacci numbers to Pascal's triangle discovered by G. E. Andrews. Several authors provided proofs of this identity, all of them rather involved or else relying on sophisticated number theoretical arguments. We not only give a simple and elementary proof, but also show the identity generalizes to arrays other than Pascal's triangle. As an application we obtain identities relating trinomial coefficients and Catalan's triangle to Fibonacci numbers.
Following the method of combinatorial telescoping for alternating sums given by Chen, Hou and Mu, we present a combinatorial telescoping approach to partition identities on sums of positive terms. By giving a classification of the combinatorial objects corresponding to a sum of positive terms, we establish bijections that lead a telescoping relation. We illustrate this idea by giving a combinatorial telescoping relation for a classical identity of MacMahon. Recently, Andrews posed a problem of finding a combinatorial proof of an identity on the q-little Jacobi polynomials which was derived bas…
Recently, Andrews and Yee studied two-variable generalizations of two identities involving partition functions $p_ω(n)$ and $p_ν(n)$ introduced by Andrews, Dixit and Yee. In this paper, we present a combinatorial proof of an interesting identity in their work.
In this paper, by applying a range of classic summation and transformation formulas for basic hypergeometric series, we obtain a three-term identity for partial theta functions. It extends the Andrews-Warnaar partial theta function identity, and also unifies several results on partial theta functions due to Ramanujan, Lovejoy and Kim. We also establish a two-term version of the extension, which can be used to derive identities for partial and false theta functions. Finally, we present a relation between the big $q$-Jacobi polynomials and the Andrews-Warnaar partial theta function identity.
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.
We construct an evidently positive multiple series as a generating function for partitions satisfying the multiplicity condition in Schur's partition theorem. Refinements of the series when parts in the said partitions are classified according to their parities or values mod 3 are also considered. Direct combinatorial interpretations of the series are provided.
We present a dual of a family of partition identities of Andrews involving partitions with no repeated odd parts (among other conditions), along with an overpartition generalization that encapsulates both families. These were discovered during the course of research for an upcoming article by the authors along with Debajyoti Nandi. The proof uses Appell's comparison theorem.
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 ide…
In this article, a finite analogue of the generalized sum-of-tails identity of Andrews and Freitas is obtained. We derive several interesting results as special cases of this analogue, in particular, a recent identity of Dixit, Eyyyunni, Maji and Sood. We derive a new extension of Abel's lemma with the help of which we obtain a one-parameter generalization of a sum-of-tails identity of Andrews, Garvan and Liang, an identity of Ramanujan as well as two new results - one for Ramanujan's function $σ(q)$ and another for the function recently introduced by Andrews and Ballantine. Later we introduce…
Bhoria, Eyyunni and Maji recently obtained a four-parameter $q$-series identity which gives as special cases not only all five entries of Ramanujan on pages 354 and 355 of his second notebook but also allows them to obtain an analytical proof of a result of Bressoud and Subbarao. Here, we obtain a finite analogue of their identity which naturally gives finite analogues of Ramanujan's results. Using one of these finite analogues, we deduce an identity for a finite sum involving a ${}_2ϕ_1$. This identity is then applied to obtain a generalization of the generating function version of Andrews' f…
We consider an identity relating Fibonacci numbers to Pascal's triangle discovered by G. E. Andrews. Several authors provided proofs of this identity, most of them rather involved or else relying on sophisticated number theoretical arguments. We present a new proof, quite simple and based on a Riordan array argument. The main point of the proof is the construction of a new Riordan array from a given Riordan array, by the elimination of elements. We extend the method and as an application we obtain other identities, some of which are new. An important feature of our construction is that it esta…
We find an involution as a combinatorial proof of a Ramanujan's partial theta identity. Based on this involution, we obtain a Franklin type involution for squares in the sense that the classical Franklin involution provides a combinatorial interpretation of Euler's pentagonal number theorem. This Franklin type involution can be considered as a solution to a problem proposed by Pak concerning the parity of the number of partitions of n into distinct parts with the smallest part being odd. Using a weighted form of our involution, we give a combinatorial proof of a weighted partition theorem deri…
We extend the Andrews-Olsson identity to two-colored partitions. Regarding the sets of proper Young walls of quantum affine algebras $\g_n=A^{(2)}_{2n}$, $A^{(2)}_{2n-1}$, $B^{(1)}_{n}$, $D^{(1)}_{n}$ and $D^{(2)}_{n+1}$ as the sets of two-colored partitions, the extended Andrews-Olsson identity implies that the generating functions of the sets of reduced Young walls have very simple formulae: \begin{center} $\prod^{\infty}_{i=1}(1+t^i)^{κ_i}$ where $κ_i=0$, $1$ or $2$, and $κ_i$ varies periodically. \end{center} Moreover, we generalize the Bessenrodt's algorithms to prove the extended Andrews…
According to the method of series rearrangement, we establish two generalizations of Andrews' curious $q$-series identity with an extra integer parameter. The limiting cases of them produce two extensions of Andrews' curious $_3F_2(\frac{3}{4})$-series identity with an additional integer parameter. Meanwhile, several related results are also given.
In 2010, Andrews considers a variety of parity questions connected to classical partition identities of Euler, Rogers, Ramanujan and Gordon. As a large part in his paper, Andrews considered the partitions by restricting the parity of occurrences of even numbers or odd numbers in the Rogers-Ramanujan-Gordon type. The Rogers-Ramanujan-Gordon type partition was defined by Gordon in 1961 as a combinatorial generalization of the Rogers-Ramaujan identities with odd moduli. In 1974, Andrews derived an identity which can be considered as the generating function counterpart of the Rogers-Ramanujan-Gord…
Using jagged overpartitions, we give three generalizations of a weighted word version of Capparelli's identity due to Andrews, Alladi, and Gordon and present several corollaries.
George Andrews and Ae Ja Yee recently established beautiful results involving bivariate generalizations of the third order mock theta functions $ω(q)$ and $ν(q)$, thereby extending their earlier results with the second author. Generalizing the Andrews-Yee identities for trivariate generalizations of these mock theta functions remained a mystery, as pointed out by Li and Yang in their recent work. We partially solve this problem and generalize these identities. Several new as well as well-known results are derived. For example, one of our two main theorems gives, as a corollary, a special case …
George Andrews's recent challenge to automated identity-proving and the WZ method is dealt with. It is argued that the rivalry between the classical and automated approaches to hypergeometric sums is beneficial to both.
I revisit an automated proof of Andrews' pentagonal number theorem found by Riese. I uncover a simple polynomial identity hidden behind his proof. I explain how to use this identity to prove Andrews' result along with a variety of new formulas of similar type. I reveal an interesting relation between the tri-pentagonal theorem and items (19), (20), (94), (98) on the celebrated Slater list. Finally, I establish a new infinite family of multiple series identities.
In a recent study of sign-balanced, labelled posets Stanley, introduced a new integral partition statistic srank(pi) = O(pi) - O(pi'), where O(pi) denotes the number of odd parts of the partition pi and pi' is the conjugate of pi. Andrews proved the following refinement of Ramanujan's partition congruence mod 5: p[0](5n +4) = p[2](5n + 4) = 0 (mod 5), p(n) = p[0](n) + p[2](n), where p[i](n) (i = 0; 2) denotes the number of partitions of n with srank = i (mod 4) and p(n) is the number of unrestricted partitions of n. Andrews asked for a partition statistic that would divide the partitions enume…
A generalization of a beautiful $q$-series identity found in the unorganized portion of Ramanujan's second and third notebooks is obtained. As a consequence, we derive a new three-parameter identity which is a rich source of partition-theoretic information. In particular, we use this identity to obtain a generalization of a recent result of Andrews, Garvan and Liang, which itself generalizes the famous result of Fokkink, Fokkink and Wang. This three-parameter identity also leads to several new weighted partition identities as well as a natural proof of a recent result of Garvan. This natural p…
We prove a new four parameter q-hypergeometric series identity from which the three parameter key identity for the Goellnitz theorem due to Alladi, Andrews, and Gordon, follows as a special case by setting one of the parameters equal to 0. The new identity is equivalent to a four parameter partition theorem which extends the deep theorem of Goellnitz and thereby settles a problem raised by Andrews thirty years ago. Some consequences including a quadruple product extension of Jacobi's triple product identity, and prospects of future research are briefly discussed.
Andrews, Lewis and Lovejoy introduced the partition function PD(n) as the number of partitions of $n$ with designated summands, where we assume that among parts with equal size, exactly one is designated. They proved that PD(3n+2) is divisible by 3. We obtain a Ramanujan type identity for the generating function of PD(3n+2) which implies the congruence of Andrews, Lewis and Lovejoy. For PD(3n), Andrews, Lewis and Lovejoy showed that the generating function can be expressed as an infinite product of powers of $(1-q^{2n+1})$ times a function $F(q^2)$. We find an explicit formula for $F(q^2)$, wh…
In 2003, Alladi, Andrews and Berkovich proved an identity for partitions where parts occur in eleven colors: four primary colors, six secondary colors, and one quaternary color. Their work answered a longstanding question of how to go beyond a classical theorem of Göllnitz, which uses three primary and three secondary colors. Their main tool was a deep and difficult four parameter $q$-series identity. In this paper we take a different approach. Instead of adding an eleventh quaternary color, we introduce forbidden patterns and give a bijective proof of a ten-colored partition identity lying be…
The object of this paper is to propose and prove a new generalization of the Andrews-Gordon identities, extending a recent result of Garrett, Ismail and Stanton. We also give a combinatorial discussion of the finite form of their result, which appeared in the work of Andrews, Knopfmacher, and Paule.
In $2003$, Alladi, Andrews and Berkovich proved a four-parameter partition identity lying beyond a celebrated identity of Göllnitz. Since then it has been an open problem to extend their work to five or more parameters. In part I of this pair of papers, we took a first step in this direction by giving a bijective proof of a reformulation of their result. We introduced forbidden patterns, bijectively proved a ten-colored partition identity, and then related, by another bijection, our identity to the Alladi-Andrews-Berkovich identity. In this second paper, we state and bijectively prove an $\fra…
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,…
Recent works of Andrews--Newman and Hopkins--Sellers unveil an interesting relation between two partition statistics, the crank and the mex. They state that, for a positive integer $n$, there are as many partitions of $n$ with non-negative crank as partitions of $n$ with odd mex. In this paper, we give a bijective proof of a generalization of this identity provided by Hopkins, Sellers, and Stanton. Our method uses an alternative definition of the Durfee decomposition, whose combinatorial link to the crank was recently studied by Hopkins, Sellers, and Yee.
We present a new partition identity and give a combinatorial proof of our result. This generalizes a result of Andrew's in which he considers the generation function for partitions with respect to size, number of odd parts, and number of odd parts of the conjugate.
Recently, George Andrews has given a Glaisher style proof of a finite version of Euler's partition identity. We generalise this result by giving a finite version of Glaisher's partition identity. Both the generating function and bijective proofs are presented.
The Göllnitz-Gordon identities were found by Göllnitz and Gordon independently. In 1967, Andrews obtained a combinatorial generalization of the Göllnitz-Gordon identities, called the Andrews-Göllnitz-Gordon theorem. In 1980, Bressoud extended the Andrews-Göllnitz-Gordon theorem to even moduli, called the Bressoud-Göllnitz-Gordon theorem. Furthermore, Bressoud gave the generating functions for the generalizations of the Göllnitz-Gordon identities. In this article, we will give new companions to the generalizations of the Göllnitz-Gordon identities.
The notion of the spt-crank of a vector partition, or an $S$-partition, was introduced by Andrews, Garvan and Liang. Let $N_S(m,n)$ denote the number of $S$-partitions of $n$ with spt-crank $m$. Andrews, Dyson and Rhoades conjectured that $\{N_S(m,n)\}_m$ is unimodal for any $n$, and they showed that this conjecture is equivalent to an inequality between the rank and the crank of ordinary partitions. They obtained an asymptotic formula for the difference between the rank and the crank of ordinary partitions, which implies $N_S(m,n)\geq N_S(m+1,n)$ for sufficiently large $n$ and fixed $m$. In t…
We start with a bijective proof of Schur's theorem due to Alladi and Gordon and describe how a particular iteration of it leads to some very general theorems on colored partitions. These theorems imply a number of important results, including Schur's theorem, Bressoud's generalization of a theorem of Göllnitz, two of Andrews' generalizations of Schur's theorem, and the Andrews-Olsson identities.
We define two-parameter generalizations of two combinatorial constructions of Andrews: the kth symmetrized rank moment and the k-marked Durfee symbol. We prove that three specializations of the associated generating functions are so-called quasimock theta functions, while a fourth specialization gives quasimodular forms. We then define a two-parameter generalization of Andrews' smallest parts function and note that this leads to quasimock theta functions as well. The automorphic properties are deduced using q-series identities relating the relevant generating functions to known mock theta func…
We prove a constant term conjecture of Robbins and Zeilberger (J. Combin. Theory Ser. A 66 (1994), 17-27), by translating the problem into a determinant evaluation problem and evaluating the determinant. This determinant generalizes the determinant that gives the number of all totally symmetric self-complementary plane partitions contained in a $(2n)\times(2n)\times(2n)$ box and that was used by Andrews (J. Combin. Theory Ser. A 66 (1994), 28-39) and Andrews and Burge (Pacific J. Math. 158 (1993), 1-14) to compute this number explicitly. The evaluation of the generalized determinant is indepen…
Euler's classical identity states that the number of partitions of an integer into odd parts and distinct parts are equinumerous. Franklin gave a generalization by considering partitions with exactly $j$ different multiples of $r$, for a positive integer $r$. We prove an analogue of Franklin's identity by studying the number of partitions with $j$ multiples of $r$ in total and in the process, discover a natural generalization of the minimal excludant (mex) which we call the $r$-chain mex. Further, we derive the generating function for $σ_{rc} \textup{mex}(n)$, the sum of $r$-chain mex taken ov…
The two partition functions $p_ω(n)$ and $p_ν(n)$ were introduced by Andrews, Dixit and Yee, which are related to the third order mock theta functions $ω(q)$ and $ν(q)$, respectively. Recently, Andrews and Yee analytically studied two identities that connect the refinements of $p_ω(n)$ and $p_ν(n)$ with the generalized bivariate mock theta functions $ω(z;q)$ and $ν(z;q)$, respectively. However, they stated these identities cried out for bijective proofs. In this paper, we first define the generalized trivariate mock theta functions $ω(y,z;q)$ and $ν(y,z;q)$. Then by utilizing odd Ferrers graph…
We announce a new four parameter partition theorem from which the (big) theorem of Gollnitz follows by setting any one of the parameters equal to 0. This settles a problem of Andrews who asked whether there exists a result that goes beyond the partition theorem of Gollnitz. We state a four parameter q-series identity (key identity) which is the generating function form of this theorem. In a subsequent paper, the proof of the new four parameter key identity will be given.
In a recent work, Andrews gave analytic proofs of two conjectures concerning some variations of two combinatorial identities between partitions of a positive integer into odd parts and partitions into distinct parts discovered by Beck. Subsequently, using the same method as Andrews, Chern presented the analytic proof of another Beck's conjecture relating the gap-free partitions and distinct partitions with odd length. However, the combinatorial interpretations of these conjectures are still unclear and required. In this paper, motivated by Glaisher's bijection, we give the combinatorial proofs…
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…
We examine an identity originally stated in Ramanujan's ``lost notebook'' and first proven algebraically by Andrews and combinatorially by Kim. We give two independent combinatorial proofs and interpretations of this identity, which also extends an identity recently proven by Pak and Waarnar related to the product of partial theta functions: First, we give a direct combinatorial proof, using the involution principle, of a special case of the identity, and extend this into a direct combinatorial proof of the full identity as written. Second, we show that the identity can be rewritten, using min…
We give a new generalization of the spt-function of G.E. Andrews, namely $\textup{Spt}_j(n)$, and give its combinatorial interpretation in terms of successive lower-Durfee squares. We then generalize the higher order spt-function $\textup{spt}_k(n)$, due to F.G. Garvan, to ${}_j\textup{spt}_{k}(n)$, thus providing a two-fold generalization of $\textup{spt}(n)$, and give its combinatorial interpretation.
In this paper we prove the identity that generalizes the Andrews-Gordon identity. Also we discuss the relation of our formula to the geometry of affine flag varieties and to the geometry of polyhedra.
George Andrews recently proved a new identity between the cardinalities of the set of Semi-Fibonacci partitions and the set of partitions into powers of two with all parts appearing an odd number of times. This paper extends the identity to the set of Semi-$m$-Fibonacci partitions of $n$ and the set of partitions of $n$ into powers of $m$ in which all parts appear with multiplicity not divisible by $m$.
