arXiv subjects

Louis Kao

Publications and source records attributed to Louis Kao.

4 recordsLinked to original sources

arXiv2026

Zero-Run Spectra of the $(3,2)$ Raney numbers Modulo Primes

We study the zero-run structure of the $(3,2)$-Raney numbers modulo a prime $p$. For every prime $p$, we determine the left-to-right maxima of the zero-run lengths, the complete zero-run spectrum, and the exact number of nonzero entries in $0\le n<p^m$. Interestingly, these results fall into three cases: $p=2$, $p=3$, and $p\geq5$, and the behaviors in these three cases are very different. For $p=2$, we characterize exactly the odd terms and determine the positions of the left-to-right maxima and the results involve Fibbinary integers, Fibonacci numbers, and Jacobsthal numbers. For $p=3$, we characterize exactly the nonzero terms and determine their residues. For $p\ge 5$, the zero runs are governed by a multiscale system of residue intervals modulo powers of $p$, from which both the record values and the complete zero-run spectrum are obtained.

math.CO↗

arXiv2023

Three New Refined Arnold Families

The Springer numbers, introduced by Arnold, are generalizations of Euler numbers in the sense of Coxeter groups. They appear as the row sums of a double triangular array $(v_{n,k})$ of integers, $1\leq|k|\leq n$, defined recursively by a boustrophedon algorithm. We say a sequence of combinatorial objects $(X_{n,k})$ is an Arnold family if $X_{n,k}$ is counted by $v_{n,k}$. A polynomial refinement $V_{n,k}(t)$ of $v_{n,k}$, together with the combinatorial interpretations in several combinatorial structures was introduced by Eu and Fu recently. In this paper, we provide three new Arnold families of combinatorial objects, namely the cycle-up-down permutations, the valley signed permutations and Knuth's flip equivalences on permutations. We shall find corresponding statistics to realize the refined polynomial arrays.

math.CO↗

arXiv2021

Sharp bounds of the $A_α$-spectral radii of mixed trees

A mixed tree is a tree in which both directed arcs and undirected edges may exist. Let $T$ be a mixed tree with $n$ vertices and $m$ arcs, where an undirected edge is counted twice as arcs. Let $A$ be the adjacency matrix of $T$. For $α\in[0,1]$, the matrix $A_α$ of $T$ is defined to be $αD^++(1-α)A$, where $D^+$ is the the diagonal out-degree matrix of $T$. The $A_α$-spectral radius of $T$ is the largest real eigenvalue of $A_α$. We will give a sharp upper bound and a sharp lower bound of the $A_α$-spectral radius of $T$.

math.CO↗

arXiv2020

The relation between Hamiltonian and $1$-tough properties of the Cartesian product graphs

The relation between Hamiltonicity and toughness of a graph is a long standing research problem. The paper studies the Hamiltonicity of the Cartesian product graph $G_1\square G_2$ of graphs $G_1$ and $G_2$ satisfying that $G_1$ is traceable and $G_2$ is connected with a path factor. Let Pn be the path of order $n$ and $H$ be a connected bipartite graph. With certain requirements of $n$, we show that the following three statements are equivalent: (i) $P_n\square H$ is Hamiltonian; (ii) $P_n\square H$ is $1$-tough; and (iii) $H$ has a path factor.

math.CO↗