Combinatorics (math.CO)

Abstract: We prove that every $n$-vertex complete simple topological graph generates at least $\Omega(n)$ pairwise disjoint $4$-faces. This improves upon a recent result by Hubard and Suk. As an immediate corollary, every $n$-vertex complete simple topological graph drawn in the unit square generates a $4$-face with area at most $O(1/n)$. This can be seen as a topological variant of Heilbronn's problem for $4$-faces. We construct examples showing that our result is asymptotically tight. We also discuss the similar problem for $k$-faces with arbitrary $k\geq 3$.

Abstract: We give uniform formulas for the number of full reflection factorizations of a parabolic quasi-Coxeter element in a Weyl group or complex reflection group, generalizing the formula for the genus-0 Hurwitz numbers. This paper is the culmination of a series of three.

Abstract: Let $G$ be an undirected graph of order $n$ and let $C_i$ be an $i$-cycle graph. $G$ is called pancyclic if $G$ contains a $C_i$ for any $i\in \{3,4,\ldots,n\}$. The Paley graph of order $q$ is a graph whose vertex set is the finite field $\mathbb{F}_q$, where $q\equiv 1\pmod{4}$. In this graph, two vertices, $a$ and $b$, are adjacent if and only if $a-b$ is nonzero square in $\mathbb{F}_q$. We prove that Paley graph of order $q$ is pancyclic for any $q \neq 5$.

Abstract: We introduce two generalizations of bracket vectors from binary trees to permutrees. These new vectors help describe algebraic and geometric properties of the rotation lattice of permutrees defined by Pilaud and Pons. The first generalization serves the role of an inversion vector for permutrees allowing us to define an explicit meet operation and provide a new constructive proof of the lattice property for permutree rotation lattices. The second generalization, which we call cubic vectors, allows for the construction of a cubic realization of these lattices which is proven to form a cubical embedding of the corresponding permutreehedra. These results specialize to those known about permutahedra and associahedra.

Abstract: Total positivity of matrices is deeply studied and plays an important role in various branches of mathematics. The main purpose of this paper is to study total positivity of a matrix $M=[M_{n,k}]_{n,k}$ generated by the weighted lattice paths in $\mathbb{N}^2$ from the origin $(0,0)$ to the point $(k,n)$ consisting of types of steps: $(0,1)$ and $(1,t+i)$ for $0\leq i\leq \ell$, where each step $(0,1)$ from height~$n-1$ gets the weight~$b_n(\textbf{y})$ and each step $(1,t+i)$ from height~$n-t-i$ gets the weight $a_n^{(i)}(\textbf{x})$. Using an algebraic method, we prove that the $\textbf{x}$-total positivity of the weight matrix $[a_i^{(i-j)}(\textbf{x})]_{i,j}$ implies that of $M$. Furthermore, using the Lindstr\"{o}m-Gessel-Viennot lemma, we obtain that both $M$ and the Toeplitz matrix of each row sequence of $M$ with $t\geq1$ are $\textbf{x}$-totally positive under the following three cases respectively: (1) $\ell=1$, (2) $\ell=2$ and restrictions for $a_n^{(i)}$, (3) general $\ell$ and both $a^{(i)}_n$ and $b_n$ are independent of $n$. In addition, for the case (3), we show that the matrix $M$ is a Riordan array, present its explicit formula and prove total positivity of the Toeplitz matrix of the each column of $M$. In particular, from the results for Toeplitz-total positivity, we also obtain the P\'olya frequency and log-concavity of the corresponding sequence. Finally, as applications, we in a unified manner establish total positivity and the Toeplitz-total positivity for many well-known combinatorial triangles, including the Pascal triangle, the Pascal square, the Delannoy triangle, the Delannoy square, the signless Stirling triangle of the first kind, the Legendre-Stirling triangle of the first kind, the Jacobi-Stirling triangle of the first kind, the Brenti's recursive matrix, and so on.

Abstract: Given an $n$-element set $C\subseteq\mathbb{R}^d$ and a (sufficiently generic) $k$-element multiset $V\subseteq\mathbb{R}^d$, we can order the points in $C$ by ranking each point $c\in C$ according to the sum of the distances from $c$ to the points of $V$. Let $\Psi_k(C)$ denote the set of orderings of $C$ that can be obtained in this manner as $V$ varies, and let $\psi^{\mathrm{max}}_{d,k}(n)$ be the maximum of $\lvert\Psi_k(C)\rvert$ as $C$ ranges over all $n$-element subsets of $\mathbb{R}^d$. We prove that $\psi^{\mathrm{max}}_{d,k}(n)=\Theta_{d,k}(n^{2dk})$ when $d \geq 2$ and that $\psi^{\mathrm{max}}_{1,k}(n)=\Theta_k(n^{4\lceil k/2\rceil -1})$. As a step toward proving this result, we establish a bound on the number of sign patterns determined by a collection of functions that are sums of radicals of nonnegative polynomials; this can be understood as an analogue of a classical theorem of Warren. We also prove several results about the set $\Psi(C)=\bigcup_{k\geq 1}\Psi_k(C)$; this includes an exact description of $\Psi(C)$ when $d=1$ and when $C$ is the set of vertices of a vertex-transitive polytope.

Abstract: Let $n_k(s)$ be the maximal length $n$ such that a quaternary additive $[n,k,n-s]_4$-code exists. We solve a natural asymptotic problem by determining the limsup $\lambda_k$ of $n_k(s)/s,$ and the smallest value of $s$ such that $n_k(s)/s=\lambda_k.$ The proof uses geometric language and is elementary.