Related papers: The Method of Multiple Combinatorial Telescoping
A summation framework is developed that enhances Karr's difference field approach. It covers not only indefinite nested sums and products in terms of transcendental extensions, but it can treat, e.g., nested products defined over roots of…
This is an introduction to algebraic combinatorics, written for a quarter-long graduate course. It starts with a rigorous introduction to formal power series with some combinatorial applications, then discusses integer partitions (proving…
We prove some identities for the squares of generalized Tribonacci numbers. Various summation identities involving these numbers are derived.
Using the correspondence between a cycle up-down permutation and a pair of matchings, we give a combinatorial proof of the enumeration of alternating permutations according to the given peak set.
We give a direct deduction and proof of two identities in the theory of plane partitions. The first one is known to enumerate the traces of plane partitions. The second one comes without any combinatorial interpretation.
We give a new proof of a_4\phi_3 summation due to G.E. Andrews and confirm another_4\phi_3 summation conjectured by him recently. Some variations of these two_4\phi_3 summations are also given.
We provide combinatorial proofs of some of the q-series identities considered by Andrews, Jimenez-Urroz and Ono [q-series identities and values of certain $L$-functions. Duke Math. J. 108 (2001), no. 3, 395--419].
Islands are combinatorial objects that can be intuitively defined on a board consisting of a finite number of cells. Based on the neighbor relation of the cells, it is a fundamental property that two islands are either containing or…
Tensor polynomial identities generalize the concept of polynomial identities on $d \times d$ matrices to identities on tensor product spaces. Here we completely characterize a certain class of tensor polynomial identities in terms of their…
We provide a framework for which one can approach showing the integer decomposition property for symmetric polytopes. We utilize this framework to prove a special case which we refer to as $2$-partition maximal polytopes in the case where…
This paper presents an alternative approach to simplify the proofs of some important results related to polynomial mappings in Computational Algebraic Geometry such as Polynomial Implicitization, Image Closure and some properties of the…
Trace monoids and heaps of pieces appear in various contexts in combinatorics. They also constitute a model used in computer science to describe the executions of asynchronous systems. The design of a natural probabilistic layer on top of…
We compute the sum and the alternating sum of the reciprocals of triangular numbers using two standard methods from calculus: a telescoping series approach and a power series approach. We then extend these results to generalized…
The main results of this paper interpret mixed volumes of lattice polytopes as mixed multiplicities of ideals and mixed multiplicities of ideals as Samuel's multiplicities. In particular, we can give a purely algebraic proof of Bernstein's…
The article addresses the problem whether indefinite double sums involving a generic sequence can be simplified in terms of indefinite single sums. Depending on the structure of the double sum, the proposed summation machinery may provide…
We produce new combinatorial methods for approaching the tropical maximal rank conjecture, including inductive procedures for deducing new cases of the conjecture on graphs of increasing genus from any given case. Using explicit…
Mendes recently conjectured an identity simplifying the Poincar\'e series of the space of equivariant polynomial maps from $\mathbb{R}^{n}$ to a subrepresentation of $Sym^{2}(\mathbb{R}^{n})$. We show how to prove this identity using a…
We establish two binomial coefficient--generalized harmonic sum identities using the partial fraction decomposition method. These identities are a key ingredient in the proofs of numerous supercongruences. In particular, in other works of…
In 1968 and 1969, Andrews proved two partition theorems of the Rogers-Ramanujan type which generalise Schur's celebrated partition identity (1926). Andrews' two generalisations of Schur's theorem went on to become two of the most…
We present a subdivision method to solve systems of congruence equations. This method is inspired in a subdivision method, based on Bernstein forms, to solve systems of polynomial inequalities in several variables and arbitrary degrees. The…