Basic series identities and combinatorics

Analogous to P.A. MacMahon’s combinatorial interpretations of the Rogers–Ramanujan identities, we interpret two basic series identities combinatorially in two different ways—using split \((n+t)\)-color partitions and the modified lattice paths. This leads to two new 3-way combinatorial identities. We conclude by posing three significant open problems.     

On 3-way combinatorial identities

In this paper, we provide combinatorial meanings to two generalized basic series with the aid of associated lattice paths. These results produce two new classes of infinite 3-way combinatorial identities. Five particular cases are also discussed. These particular cases provide new combinatorial versions of Göllnitz–Gordon identities and Göllnitz identity. Seven q-identities of Slater and five q-identities of Rogers are further explored using the same combinatorial object. These results are an extension of the work of Goyal and Agarwal (Utilitas Math. 95 (2014) 141–148), Agarwal and Rana (Utilitas Math. 79 (2009) 145–155), and Agarwal (J. Number Theory 28 (1988) 299–305).     

Generalized Pleijel identity

A family of identities primarily associated with isoperimetric inequalities for planar convex domains was discovered by Pleijel in 1956. We call these identities classical Pleijel identities. R. V. Ambartzumian gave combinatorial proof of these identities and pointed out that they can be applied to find chord length distribution functions for convex domains. In the classical Pleijel identities integration is over the measure in the space \(\mathbb{G}\) of lines which is invariant with respect to the all Euclidean motions. In the present paper they are considered for any locally-finite measure in the space \(\mathbb{G}\). These identities are applied to find the so-called orientation-dependent chord length distribution (or density) functions for bounded convex domains.     

Duality of Graded Graphs

A graph is said to be graded if its vertices are divided into levels numbered by integers, so that the endpoints of any edge lie on consecutive levels. Discrete modular lattices and rooted trees are among the typical examples. The following three types of problems are of interest to us:(1) path counting in graded graphs, and related combinatorial identities;(2) bijective proofs of these identities;(3) design and analysis of algorithms establishing corresponding bijections.This article is devoted to (1); its sequel [7] is concerned with the problems (2)–(3). A simplified treatment of some of these results can be found in [8].In this article, R.P. Stanley's [26, 27] linear-algebraic approach to (1) is extended to cover a wide range of graded graphs. The main idea is to consider pairs of graded graphs with a common set of vertices and common rank function. Such graphs are said to be dual if the associated linear operators satisfy a certain commutation relation (e.g., the Heisenberg one). The algebraic consequences of these relations are then interpreted as combinatorial identities. (This idea is also implicit in [27].)[7] contains applications to various examples of graded graphs, including the Young, Fibonacci, Young-Fibonacci and Pascal lattices, the graph of shifted shapes, the r-nary trees, the Schensted graph, the lattice of finite binary trees, etc. Many enumerative identities (both known and unknown) are obtained. Abstract of [7]. These identities can also be derived in a purely combinatorial way by generalizing the Robinson-Schensted correspondence to the class of graphs under consideration (cf. [5]). The same tools can be applied to permutation enumeration, including involution counting and rook polynomials for Ferrers boards. The bijective correspondences mentioned above are naturally constructed by Schensted-type algorithms. A general approach to these constructions is given. As particular cases we rederive the classical algorithm of Robinson, Schensted, and Knuth [20, 12, 21], the Sagan-Worley [17, 32] and Haiman [11] algorithms, the algorithm for the Young-Fibonacci graph [5, 15], and others. Several new applications are given.     

Two infinite families of terminating binomial sums

We present a family of identities including both binomial coefficients and a power of a natural number \(m \ge 2\). We find a combinatorial interpretation of these identities, which provides bijective proof. Dual alternating sign identities are also presented.     

Combinatorial interpretations of Ramanujan’s tau function

We use a $q$-series identity by Ramanujan to give a combinatorial interpretation of Ramanujan’s tau function which involves $t$-cores and a new class of partitions which we call $(m,k)$-capsids. The same method can be applied in conjunction with other related identities yielding alternative combinatorial interpretations of the tau function.     

Rook theory, compositions, and zeta functions

A new family of Dirichlet series having interesting combinatorial properties is introduced. Although they have no functional equation or Euler product, under the Riemann Hypothesis it is shown that these functions have no zeros in . Some identities in the ring of formal power series involving rook theory and continued fractions are developed.

     

Supernomial Coefficients,Polynomial Identities and q-Series

q-Analogues of the coefficients of xa in the expansion of _j=1 ^N (1 + x + + x^j)^Lj are proposed. Useful properties, such as recursion relations, symmetries and limiting theorems of the q-supernomial coefficients are derived, and a combinatorial interpretation using generalized Durfee dissection partitions is given. Polynomial identities of boson-fermion-type, based on the continued fraction expansion of p/k and involving the q-supernomial coefficients, are proven. These include polynomial analogues of the Andrews-Gordon identities. Our identities unify and extend many of the known boson-fermion identities for one-dimensional configuration sums of solvable lattice models, by introducing multiple finitization parameters.     

Warnaar’s bijection and colored partition identities, II

In our previous paper (J. Comb. Theory Ser. A 120(1):28–38, 2013), we determined a unified combinatorial framework to look at a large number of colored partition identities, and studied the five identities corresponding to the exceptional modular equations of prime degree of the Schröter, Russell, and Ramanujan type. The goal of this paper is to use the master bijection of Sandon and Zanello (J. Comb. Theory Ser. A 120(1):28–38, 2013) to show combinatorially several new and highly nontrivial colored partition identities. We conclude by listing a number of further interesting identities of the same type as conjectures.     

Translation invariant valuations in the space of planes

The paper is related to the problem of measure generation in the space IE of planes in IR ³ by combinatorial, translation invariant valuations. General results concerning that problem have been derived in 1994 and 1996 (this journal vol. 31, no. 4. and vol. 33, no. 4, respectively). The purpose of the present article is to give a proof of two geometrical identities on which the theorem on valuations in the space IE can be based. The article consists of the motivational first part that contains the basic concepts from the theory of combinatorial valuations and measure generation in IE, and the second that gives a proof of the identities in question.     

Multi-poly-Bernoulli numbers and polynomials with a <Emphasis Type="Italic">q</Emphasis> parameter

We introduce themulti-poly-Bernoulli numbers and polynomialswith a q parameter, which are generalizations of the poly-Bernoulli numbers and polynomials with a q parameter, respectively.We give several combinatorial identities and properties of these new numbers and polynomials.     

A new proof of the Ramanujan congruences for the partition function

Let p(n) denote the number of partitions of n. Recall Ramanujan’s three congruences for the partition function,

The Lambert series factorization theorem

A factorization for partial sums of Lambert series is introduced in this paper. As corollaries, we derive some connections between partitions and divisors. These results can be easily used to discover and prove new combinatorial identities involving important functions from number theory: the Möbius function \(\mu (n)\), Euler’s totient \(\varphi (n)\), Jordan’s totient \(J_k(n)\), Liouville’s function \(\lambda (n)\), the von Mangoldt function \(\Lambda (n)\), and the divisor function \(\sigma _x(n)\). The fascinating feature of these identities is their common nature.     

Some Combinatorial and Analytical Identities

We give new proofs and explain the origin of several combinatorial identities of Fu and Lascoux, Dilcher, Prodinger, Uchimura, and Chen and Liu. We use the theory of basic hypergeometric functions, and generalize these identities. We also exploit the theory of polynomial expansions in the Wilson and Askey-Wilson bases to derive new identities which are not in the hierarchy of basic hypergeometric series. We demonstrate that a Lagrange interpolation formula always leads to very-well-poised basic hypergeometric series. As applications we prove that the Watson transformation of a balanced ${_{4}\phi_{3}}$ to a very-well-poised ${_{8}\phi_{7}}$ is equivalent to the Rodrigues-type formula for the Askey-Wilson polynomials. By applying the Leibniz formula for the Askey-Wilson operator we also establish the ${_{8}\phi_{7}}$ summation theorem.     

A generalization of Fibonacci and Lucas matrices

We define the matrix of type s, whose elements are defined by the general second-order non-degenerated sequence and introduce the notion of the generalized Fibonacci matrix , whose nonzero elements are generalized Fibonacci numbers. We observe two regular cases of these matrices (s=0 and s=1). Generalized Fibonacci matrices in certain cases give the usual Fibonacci matrix and the Lucas matrix. Inverse of the matrix is derived. In partial case we get the inverse of the generalized Fibonacci matrix and later known results from [Gwang-Yeon Lee, Jin-Soo Kim, Sang-Gu Lee, Factorizations and eigenvalues of Fibonaci and symmetric Fibonaci matrices, Fibonacci Quart. 40 (2002) 203–211; P. Staˇnicaˇ, Cholesky factorizations of matrices associated with r-order recurrent sequences, Electron. J. Combin. Number Theory 5 (2) (2005) #A16] and [Z. Zhang, Y. Zhang, The Lucas matrix and some combinatorial identities, Indian J. Pure Appl. Math. (in press)]. Correlations between the matrices , and the generalized Pascal matrices are considered. In the case a=0,b=1 we get known result for Fibonacci matrices [Gwang-Yeon Lee, Jin-Soo Kim, Seong-Hoon Cho, Some combinatorial identities via Fibonacci numbers, Discrete Appl. Math. 130 (2003) 527–534]. Analogous result for Lucas matrices, originated in [Z. Zhang, Y. Zhang, The Lucas matrix and some combinatorial identities, Indian J. Pure Appl. Math. (in press)], can be derived in the partial case a=2,b=1. Some combinatorial identities involving generalized Fibonacci numbers are derived.     

The eigenvalues of q-Kneser graphs

In this note, by proving some combinatorial identities, we obtain a simple form for the eigenvalues of $q$-Kneser graphs.     

An exotic shuffle relation for multiple zeta values

In this short note we will provide a new proof of the following exotic shuffle relation of multiple zeta values: This was proved by Zagier when n = 0, by Broadhurst when m = 0, and by Borwein, Bradley, and Broadhurst when m = 1. In general this was proved by Bowman and Bradley. Our new idea is to use the method of Borwein et al. to reduce the above general relation to some families of combinatorial identities which can be verified by Zeilberger’s algorithm [9, 10] that is part of the WZ method. Received: 27 November 2007 Revised: 28 June 2008     

Combinatorics of tenth-order mock theta functions

In this paper, we provide the combinatorial interpretations of two tenth-order mock theta functions which appeared in some identities given in Ramanujan’s lost notebook ((1988) Narosa Publishing House, New Delhi).     

On an extension of a combinatorial identity

Using Frobenius partitions we extend the main results of [4]. This leads to an infinite family of 4-way combinatorial identities. In some particular cases we get even 5-way combinatorial identities which give us four new combinatorial versions of Göllnitz-Gordon identities.