On L-functions of certain exponential sums

Let ${\mathbb{F}}_q$ denote the finite field of order q of characteristic p. We study the p-adic valuations for zeros of L-functions associated with exponential sums of the following family of Laurent polynomials$f(x)=a_1{x}_{n+1}(x_1+\frac{1}{x_1})+\cdots +a_n{x}_{n+1}(x_n+\frac{1}{x_n})+a_{n+1}{x}_{n+1}+\frac{1}{x_{n+1}}$ where $a_i\in {\mathbb{F}}_q^{⁎}$, $i=1,2,\dots ,n+1$. When $n=2$, the estimate of the associated exponential sum appears in Iwaniecʼs work on small eigenvalues of the Laplace–Beltrami operator acting on automorphic functions with respect to the group ${\Gamma }_0(p)$, and Adolphson and Sperber gave complex absolute values for zeros of the corresponding L-function. Using the decomposition theory of Wan, we determine the generic Newton polygon (q-adic values of the reciprocal zeros) of the L-function. Working on the chain level version of Dworkʼs trace formula and using Wanʼs decomposition theory, we are able to give an explicit Hasse polynomial for the generic Newton polygon in low dimensions, i.e., $n⩽3$.     

Hankel determinants of linear combinations of consecutive Catalan-like numbers

Let ${\left(a_n\right)}_{n\ge 0}$ be a sequence of the Catalan-like numbers. We evaluate Hankel determinants $det{[\lambda a_{i+j}+\mu a_{i+j+1}]}_{0\le i,j\le n}$ and $det{[\lambda a_{i+j+1}+\mu a_{i+j+2}]}_{0\le i,j\le n}$ for arbitrary coefficients $\lambda$ and $\mu$. Our results unify many known results of Hankel determinant evaluations for classic combinatorial counting coefficients, including the Catalan, Motzkin and Schröder numbers.     

On reconstruction of signed permutations distorted by reversal errors

The problem of reconstructing signed permutations on n elements from their erroneous patterns distorted by reversal errors is considered in this paper. A reversal is the operation of taking a segment of the signed permutation, reversing it, and flipping the signs of its elements. The reversal metric is defined as the least number of reversals transforming one signed permutation into another. It is proved that for any $n?2$ an arbitrary signed permutation is uniquely reconstructible from three distinct signed permutations at reversal distance at most one from the signed permutation. The proposed approach is based on the investigation of structural properties of a Cayley graph $G_{2n}$ whose vertices form a subgroup of the symmetric group ${\mathrm{Sym}}_{2n}$. It is also proved that an arbitrary signed permutation is reconstructible from two distinct signed permutations with probability $p_2～\frac{1}{3}$ as $n\to \infty$. In the case of at most two reversal errors it is shown that at least $n(n+1)$ distinct erroneous patterns are required in order to reconstruct an arbitrary signed permutation.     

Partitions of natural numbers with the same weighted representation functions

TextFor any given two positive integers $k_1$ and $k_2$, and any set A of nonnegative integers, let $r_{k_1,k_2}(A,n)$ denote the number of solutions of the equation $n=k_1{a}_1+k_2{a}_2$ with $a_1,a_2\in A$. In this paper, we determine all pairs $k_1,k_2$ of positive integers for which there exists a set $A?\mathbb{N}$ such that $r_{k_1,k_2}(A,n)=r_{k_1,k_2}(\mathbb{N}?A,n)$ for all $n?n_0$. We also pose several problems for further research.VideoFor a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=EnezEsJl0OY.     

Resolvable 4-cycle group divisible designs with two associate classes: Part size even

Let ${\lambda }_1{K}_a$ denote the graph on a vertices with ${\lambda }_1$ edges between every pair of vertices. Take p copies of this graph ${\lambda }_1{K}_a$, and join each pair of vertices in different copies with ${\lambda }_2$ edges. The resulting graph is denoted by $K(a,p;{\lambda }_1,{\lambda }_2)$, a graph that was of particular interest to Bose and Shimamoto in their study of group divisible designs with two associate classes. The existence of z-cycle decompositions of this graph have been found when $z\in \{3,4\}$. In this paper we consider resolvable decompositions, finding necessary and sufficient conditions for a 4-cycle factorization of $K(a,p;{\lambda }_1,{\lambda }_2)$ (when ${\lambda }_1$ is even) or of $K(a,p;{\lambda }_1,{\lambda }_2)$ minus a 1-factor (when ${\lambda }_1$ is odd) whenever a is even.     

Automorphism groups of Cayley graphs generated by block transpositions and regular Cayley maps

This paper deals with the Cayley graph $\mathrm{Cay}({\mathrm{Sym}}_n,T_n)$, where the generating set consists of all block transpositions. A motivation for the study of these particular Cayley graphs comes from current research in Bioinformatics. As the main result, we prove that $\text{Aut}\left(\mathrm{Cay}({\mathrm{Sym}}_n,T_n)\right)$ is the product of the left translation group and a dihedral group ${\mathsf{D}}_{n+1}$ of order $2(n+1)$. The proof uses several properties of the subgraph $\Gamma$ of $\mathrm{Cay}({\mathrm{Sym}}_n,T_n)$ induced by the set $T_n$. In particular, $\Gamma$ is a $2(n?2)$-regular graph whose automorphism group is ${\mathsf{D}}_{n+1},$ $\Gamma$ has as many as $n+1$ maximal cliques of size $2$, and its subgraph $\Gamma \left(V\right)$ whose vertices are those in these cliques is a 3-regular, Hamiltonian, and vertex-transitive graph. A relation of the unique cyclic subgroup of ${\mathsf{D}}_{n+1}$ of order $n+1$ with regular Cayley maps on ${\text{Sym}}_n$ is also discussed. It is shown that the product of the left translation group and the latter group can be obtained as the automorphism group of a non-$t$-balanced regular Cayley map on ${\text{Sym}}_n$.     

The smallest surface that contains all signed graphs on K4,n

Finding the smallest number of crosscaps that suffice to orientation-embed every edge signature of the complete bipartite graph $K_{m,n}$ is an open problem. In this paper that number for the complete bipartite graph $K_{4,n}$, $n\ge 4$, is determined by using diamond products of signed graphs. The number is $2?\frac{n?1}{2}?+1$, which is attained by $K_{4,n}$ with exactly 1 negative edge, except that when $n=4$, the number is 4, which is attained by $K_{4,4}$ with exactly 4 independent negative edges.     

Maximum average degree and relaxed coloring

We say a graph is $(d,d,\dots ,d,0,\dots ,0)$-colorable with $a$ of $d$’s and $b$ of $0$’s if $V\left(G\right)$ may be partitioned into $b$ independent sets $O_1,O_2,\dots ,O_b$ and $a$ sets $D_1,D_2,\dots ,D_a$ whose induced graphs have maximum degree at most $d$. The maximum average degree, $mad\left(G\right)$, of a graph $G$ is the maximum average degree over all subgraphs of $G$. In this note, for nonnegative integers $a,b$, we show that if $mad\left(G\right)<\frac{4}{3}a+b$, then $G$ is $(1_1,1_2,\dots ,1_a,0_1,\dots ,0_b)$-colorable.     

Star-critical Ramsey numbers for large generalized fans and books

In this paper, we show that for any fixed integers $m\ge 2$ and $t\ge 2$, the star-critical Ramsey number $r_1(K_1+n{K}_t,K_{m+1})=(m?1)tn+t$ for all sufficiently large $n$. Furthermore, for any fixed integers $p\ge 2$ and $m\ge 2$, $r_1(K_p+n{K}_1,K_{m+1})=(m?1+o\left(1\right))n$ as $n\to \infty$.     

Ramsey numbers of odd cycles versus larger even wheels

The generalized Ramsey number $R(G_1,G_2)$ is the smallest positive integer $N$ such that any red–blue coloring of the edges of the complete graph $K_N$ either contains a red copy of $G_1$ or a blue copy of $G_2$. Let $C_m$ denote a cycle of length $m$ and $W_n$ denote a wheel with $n+1$ vertices. In 2014, Zhang, Zhang and Chen determined many of the Ramsey numbers $R(C_{2k+1},W_n)$ of odd cycles versus larger wheels, leaving open the particular case where $n=2j$ is even and $k<j<3k∕2$. They conjectured that for these values of $j$ and $k$, $R(C_{2k+1},W_{2j})=4j+1$. In 2015, Sanhueza-Matamala confirmed this conjecture asymptotically, showing that $R(C_{2k+1},W_{2j})\le 4j+334$. In this paper, we prove the conjecture of Zhang, Zhang and Chen for almost all of the remaining cases. In particular, we prove that $R(C_{2k+1},W_{2j})=4j+1$ if $j?k\ge 251$, $k<j<3k∕2$, and $j\ge 212299$.