Truncated Series with Nonnegative Coefficients from the Jacobi Triple Product

Andrews and Merca investigated a truncated version of Euler's pentagonal number theorem and showed that the coefficients of the truncated series are nonnegative. They also considered the truncated series arising from Jacobi's triple product identity, and they conjectured that its coefficients are nonnegative. This conjecture was posed by Guo and Zeng independently and confirmed by Mao and Yee using different approaches. In this paper, we provide a new combinatorial proof of their nonnegativity result related to Euler's pentagonal number theorem. Meanwhile, we find an analogous result for a truncated series arising from Jacobi's triple product identity in a different manner.

Proof of the Bessenrodt–Ono Inequality by Induction

In 2016 Bessenrodt–Ono discovered an inequality addressing additive and multiplicative properties of the partition function. Generalizations by several authors have been given; on partitions with rank in a given residue class by Hou–Jagadeesan and Males, on k-regular partitions by Beckwith–Bessenrodt, on k-colored partitions by Chern–Fu–Tang, and Heim–Neuhauser on their polynomization, and Dawsey–Masri on the Andrews spt-function. The proofs depend on non-trivial asymptotic formulas related to the circle method on one side, or a sophisticated combinatorial proof invented by Alanazi–Gagola–Munagi. We offer in this paper a new proof of the Bessenrodt–Ono inequality, which is built on a well-known recursion formula for partition numbers. We extend the proof to the result by Chern–Fu–Tang and its polynomization. Finally, we also obtain a new result.

Undergraduate Students’ Combinatorial Proof of Binomial Identities

Combinatorial proof serves both as an important topic in combinatorics and as a type of proof with certain properties and constraints. We report on a teaching experiment in which undergraduate students (who were novice provers) engaged in combinatorial reasoning as they proved binomial identities. We highlight ways of understanding that were important for their success with establishing combinatorial arguments; in particular, the students demonstrated referential symbolic reasoning within an enumerative representation system, and as the students engaged in successful combinatorial proof, they had to coordinate reasoning within algebraic and enumerative representation systems. We illuminate features of the students’ work that potentially contributed to their successes and highlight potential issues that students may face when working with binomial identities.

On the structure of solution-sets to regular word equations

For quadratic word equations, there exists an algorithm based on rewriting rules which generates a directed graph describing all solutions to the equation. For regular word equations – those for which each variable occurs at most once on each side of the equation – we investigate the properties of this graph, such as bounds on its diameter, size, and DAG-width, as well as providing some insights into symmetries in its structure. As a consequence, we obtain a combinatorial proof that the problem of deciding whether a regular word equation has a solution is in NP.

Computation of Al-Salam Carlitz and Askey-Wilson Moments using Motzkin Paths

In this paper we study the moments of polynomials from the Askey scheme, and we focus on Askey-Wilson polynomials. More precisely, we give a combinatorial proof for the case where $d=0$. Their values have already been computed by Kim and Stanton in 2015, however, the proof is not completely combinatorial, which means that an explicit bijection has not been exhibited yet. In this work, we use a new combinatorial approach for the simpler case of Al-Salam-Carlitz, using a sign reversing involution that directly operates on Motzkin path. We then generalize this method to Askey-Wilson polynomials with $d=0$ only, providing the first fully combinatorial proof for that case.

Channels, Billiards, and Perfect Matching 2-Divisibility

Let $m_G$ denote the number of perfect matchings of the graph $G$. We introduce a number of combinatorial tools for determining the parity of $m_G$ and giving a lower bound on the power of 2 dividing $m_G$. In particular, we introduce certain vertex sets called channels, which correspond to elements in the kernel of the adjacency matrix of $G$ modulo $2$. A result of Lovász states that the existence of a nontrivial channel is equivalent to $m_G$ being even. We give a new combinatorial proof of this result and strengthen it by showing that the number of channels gives a lower bound on the power of $2$ dividing $m_G$ when $G$ is planar. We describe a number of local graph operations which preserve the number of channels. We also establish a surprising connection between 2-divisibility of $m_G$ and dynamical systems by showing an equivalency between channels and billiard paths. We exploit this relationship to show that $2^{\frac{\gcd(m+1,n+1)-1}{2}}$ divides the number of domino tilings of the $m\times n$ rectangle. We also use billiard paths to give a fast algorithm for counting channels (and hence determining the parity of the number of domino tilings) in simply connected regions of the square grid.

The action of Young subgroups on the partition complex

We study the restrictions, the strict fixed points, and the strict quotients of the partition complex $|\Pi_{n}|$ | Π n | , which is the $\Sigma_{n}$ Σ n -space attached to the poset of proper nontrivial partitions of the set $\{1,\ldots,n\}$ { 1 , … , n } .We express the space of fixed points $|\Pi_{n}|^{G}$ | Π n | G in terms of subgroup posets for general $G\subset \Sigma_{n}$ G ⊂ Σ n and prove a formula for the restriction of $|\Pi_{n}|$ | Π n | to Young subgroups $\Sigma_{n_{1}}\times \cdots\times \Sigma_{n_{k}}$ Σ n 1 × ⋯ × Σ n k . Both results follow by applying a general method, proven with discrete Morse theory, for producing equivariant branching rules on lattices with group actions.We uncover surprising links between strict Young quotients of $|\Pi_{n}|$ | Π n | , commutative monoid spaces, and the cotangent fibre in derived algebraic geometry. These connections allow us to construct a cofibre sequence relating various strict quotients $|\Pi_{n}|^{\diamond} \mathbin {\operatorname* {\wedge }_{\Sigma_{n}}^{}} (S^{\ell})^{\wedge n}$ | Π n | ⋄ ∧ Σ n ( S ℓ ) ∧ n and give a combinatorial proof of a splitting in derived algebraic geometry.Combining all our results, we decompose strict Young quotients of $|\Pi_{n}|$ | Π n | in terms of “atoms” $|\Pi_{d}|^{\diamond} \mathbin {\operatorname* {\wedge }_{\Sigma_{d}}^{}} (S^{\ell})^{\wedge d}$ | Π d | ⋄ ∧ Σ d ( S ℓ ) ∧ d for $\ell$ ℓ odd and compute their homology. We thereby also generalise Goerss’ computation of the algebraic André-Quillen homology of trivial square-zero extensions from $\mathbf {F}_{2}$ F 2 to $\mathbf {F}_{p}$ F p for $p$ p an odd prime.