We prove $K$-theoretic and shifted $K$-theoretic analogues of the bijection of Stanton and White between domino tableaux and pairs of semistandard tableaux. As a result, we obtain product formulas for certain pairs of stable Grothendieck polynomials and certain pairs of $K$-theoretic $Q$-Schur functions.

Let $K$ be an extension of the $p$-adic numbers with uniformizer $\pi$. Let $\phi$ and $\psi$ be Eisenstein polynomials over $K$ of degree $n$ that generate isomorphic extensions. We show that if the cardinality of the residue class field of $K$ divides $n\left(n-1\right)$, then $v\left(disc\left(\phi \right)-disc\left(\psi \right)\right)>v\left(disc\left(\phi \right)\right)$. This makes the first (nonzero) digit of the $\pi$-adic expansion of $disc\left(\phi \right)$ an invariant of the extension generated by $\phi$. Furthermore we find that noncyclic extensions of degree $p$ of the field of $p$-adic numbers are uniquely determined by this invariant.

An $m$-pseudo progression is an increasing list of numbers for which there are at most $m$ distinct differences between consecutive terms. This object generalizes the notion of an arithmetic progression. We give two counts for the number of $k$-term $m$-pseudo progressions in $\left\{1,2,\dots ,n\right\}$. We also provide computer-generated tables of values which agree with both counts and graphs that display the growth rates of these functions. Finally, we present a generating function which counts $k$-term progressions in $\left\{1,2,\dots ,n\right\}$ whose differences are all distinct, and we discuss further directions in Ramsey theory.

Given a graph $\mathrm{\Gamma }$, one can associate a right-angled Coxeter group $W$ and a cube complex $\mathrm{\Sigma }$ on which $W$ acts. By identifying $W$ with the vertex set of $\mathrm{\Sigma }$, one obtains a growth series for $W$ defined as $W\left(t\right)={\sum }_{w\in W}{t}^{\ell \left(w\right)}$, where $\ell \left(w\right)$ denotes the minimum length of an edge path in $\mathrm{\Sigma }$ from the vertex $1$ to the vertex $w$. The series $W\left(t\right)$ is known to be a rational function. We compute some examples and investigate the poles and zeros of this function.

Peg solitaire is a classical one-person game that has been played in various countries on different types of boards. Numerous studies have focused on the solvability of the games on these traditional boards and more recently on mathematical graphs. In this paper, we go beyond traditional peg solitaire and explore the solvability on graphs with pegs of more than one color and arrive at results that differ from previous works on the subject. This paper focuses on classifying the solvability of peg solitaire in three colors on several different types of common mathematical graphs, including the path, complete bipartite, and star. We also consider the solvability of peg solitaire on the Cartesian products of graphs.

We first describe an observation based on an analysis of data regarding the outcomes of decisions in cases considered by the United States Supreme Court. Based on this observation, we propose a simple model aiming toward producing an objective notion of an ideology index. As an initial step in justifying this concept we produce explicit formulas for the highest-energy eigenvectors of reversible Markov chains with rank-2 transition matrices.

Let $R$ be a ring. A nonempty subset $S$ of $R$ is a subring of $R$ if $S$ is closed under negatives, addition, and multiplication. We determine the rings $R$ for which every subring $S$ of $R$ has a multiplicative identity (which need not be the identity of $R$).

We prove that every simple graph of order 12 which has minimum degree 6 contains a $K_6$ minor, thus proving Jørgensen’s conjecture for graphs of order 12. In the process, we establish several lemmata linking the existence of $K_6$ minors for graphs to their size or degree sequence, by means of their clique sum structure. We also establish an upper bound for the order of graphs where the 6-connected condition is necessary for Jørgensen’s conjecture.

An important invariant of a chemical reaction network is its maximum number of positive steady states. This number, however, is in general difficult to compute. Nonetheless, there is an upper bound on this number — namely, a network’s mixed volume — that is easy to compute. Moreover, recent work has shown that, for certain biological signaling networks, the mixed volume does not greatly exceed the maximum number of positive steady states. Continuing this line of research, we further investigate this overcount and also compute the mixed volumes of small networks, those with only a few species or reactions.

We study the enumeration of profiles, which are outlines that occur when tiling a rectangular board with squares, dominoes, and trominoes. Profiles of length $m$ correspond to a special subset of the set ${\left\{0,1,2,3\right\}}^m$, called profile strings. Profiles and their corresponding strings first appeared in the enumeration of the tilings of rectangular $2\times n$ and $3\times n$ boards with squares, dominoes, and trominoes. Profiles also play a role in enumerating the tilings of an $m\times n$ board for any fixed $m\ge 2$. We describe how profiles arise when enumerating tilings, and we prove that the number of profile strings of length $m$ equals $m\cdot 3^{m-1}$.

A regular algebra of global dimension $n+1$ is often called a quantum ${\mathbb{P}}^n$. In 2011, Nafari, Vancliff and Zhang showed that graded skew Clifford algebras (GSCAs) could be used to classify most quadratic quantum ${\mathbb{P}}^2$s. Some time later, Chandler, Tomlin and Vancliff used their work with certain families of GSCAs to develop a conjecture on the quantum space of a generic quadratic quantum ${\mathbb{P}}^3$. These results suggest that (the isomorphism classes of) GSCAs are likely to play a fundamental role in the classification of quadratic quantum ${\mathbb{P}}^3$s. In this article, we will discuss some of these results on GSCAs and discuss new results on isomorphisms between GSCAs.

From the modularity theorem proven by Wiles, Taylor, Conrad, Diamond, and Breuil, we know that all elliptic curves are modular. It has been shown by Martin and Ono exactly which are represented by eta-quotients, and some examples of elliptic curves represented by modular forms that are linear combinations of eta-quotients have been given by Pathakjee, RosnBrick, and Yoong.

In this paper, we first show that eta-quotients which are modular for any congruence subgroup of level $N$ coprime to $6$ can be viewed as modular for ${\mathrm{\Gamma }}_0\left(N\right)$. We then categorize when even-weight eta-quotients can exist in $M_k\left({\mathrm{\Gamma }}_1\left(p\right)\right)$ and $M_k\left({\mathrm{\Gamma }}_1\left(pq\right)\right)$ for distinct primes $p,q$. We conclude by providing some new examples of elliptic curves whose corresponding modular forms can be written as a linear combination of eta-quotients, and describe an algorithmic method for finding additional examples.