Related papers: A $q$-Queens Problem. III. Nonattacking Partial Qu…
In this work we study orthogonal polynomials via polynomial mappings in the framework of the $H_q-$semiclassical class. We consider two monic orthogonal polynomial sequences $\{p_n (x)\}_{n\geq0}$ and $\{q_n(x)\}_{n\geq0}$ such that $$…
An n-ary operation q:A^n->A is called an n-ary quasigroup of order |A| if in x_0=q(x_1,...,x_n) knowledge of any n elements of x_0,...,x_n uniquely specifies the remaining one. An n-ary quasigroup q is permutably reducible if…
The plane partition polynomial $Q_n(x)$ is the polynomial of degree $n$ whose coefficients count the number of plane partitions of $n$ indexed by their trace. Extending classical work of E.M. Wright, we develop the asymptotics of these…
The general position problem in graphs seeks the largest set of vertices such that no three vertices lie on a common geodesic. Its counting refinement, the general position polynomial $\psi(G)$, asks for all such possible sets. In this…
It is shown that generic N-party pure quantum states (with equidimensional subsystems) are uniquely determined by their reduced states of just over half the parties; in other words, all the information in almost all N-party pure states is…
Chess graphs encode the moves that a particular chess piece can make on an $m\times n$ chessboard. We study through these graphs through the lens of chip-firing games and graph gonality. We provide upper and lower bounds for the gonality of…
We consider the load balancing system under Poisson arrivals, exponential services, and homogeneous servers. Upon arrival, a job is to be routed to one of the servers, where it is queued until service. We consider the Power-of-$d$ choices…
Higher braiding gates, a new kind of quantum gate, are introduced. These are matrix solutions of the polyadic braid equations (which differ from the generalized Yang-Baxter equations). Such gates support a special kind of multi-qubit…
In the Maker-Breaker positional game, Maker and Breaker take turns picking vertices of a hypergraph $H$, and Maker wins if and only if she possesses all the vertices of some edge of $H$. Deciding the outcome (i.e. which player has a winning…
Let $\Gamma$ be a quiver on n vertices $v_1, v_2, ..., v_n$ with $g_{ij}$ edges between $v_i$ and $v_j$, and let $\alpha \in \N^n$. Hua gave a formula for $A_{\Gamma}(\alpha, q)$, the number of isomorphism classes of absolutely…
We describe an occupation-number-like picture of Fractional Quantum Hall (FQH) states in terms of polynomial wavefunctions characterized by a dominant occupation-number configuration. The bosonic variants of single-component abelian and…
We introduce the entangled quantum polynomial hierarchy $\mathsf{QEPH}$ as the class of problems that are efficiently verifiable given alternating quantum proofs that may be entangled with each other. We prove $\mathsf{QEPH}$ collapses to…
We study the number $P(n)$ of partitions of an integer $n$ into sums of distinct squares and derive an integral representation of the function $P(n)$. Using semi-classical and quantum statistical methods, we determine its asymptotic average…
Define $QC(n)$ to be the number of quasiplatonic topological actions of the cyclic group $C_n$ on surfaces of genus at least two. We use formulas of Benim and Wootton to give an explicit formula for $QC(n)$. In addition, we relate the…
We study the ring of quasisymmetric polynomials in $n$ anticommuting (fermionic) variables. Let $R_n$ denote the polynomials in $n$ anticommuting variables. The main results of this paper show the following interesting facts about…
The $q$-analogue of the binomial coefficient, known as a $q$-binomial coefficient, is typically denoted $\left[{n \atop k}\right]_q$. These polynomials are important combinatorial objects, often appearing in generating functions related to…
We present a polynomial partitioning theorem for finite sets of points in the real locus of an irreducible complex algebraic variety of codimension at most two. This result generalizes the polynomial partitioning theorem on the Euclidean…
We consider the problem of learning an unknown partition of an $n$ element universe using rank queries. Such queries take as input a subset of the universe and return the number of parts of the partition it intersects. We give a simple…
As a generalization of orbit-polynomial and distance-regular graphs, we introduce the concept of a quotient-polynomial graph. In these graphs every vertex $u$ induces the same regular partition around $u$, where all vertices of each cell…
We construct positional numeral systems that work natively over nonderived polyadic $\left( m,n\right) $-rings whose addition takes $m$ arguments and multiplication takes $n$. In such rings, the length of an admissible additive word and a…