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 $n\to \infty$.     

The Laplacian spectral radius of trees and maximum vertex degree

Let $\Delta \left(T\right)$ and $\mu \left(T\right)$ denote the maximum degree and the Laplacian spectral radius of a tree $T$, respectively. In this paper we prove that for two trees $T_1$ and $T_2$ on $n(n\ge 21)$ vertices, if $\Delta \left(T_1\right)>\Delta \left(T_2\right)$ and $\Delta \left(T_1\right)\ge ?\frac{11n}{30}?+1$, then $\mu \left(T_1\right)>\mu \left(T_2\right)$, and the bound “$\Delta \left(T_1\right)\ge ?\frac{11n}{30}?+1$” is the best possible. We also prove that for two trees $T_1$ and $T_2$ on $2k(k\ge 4)$ vertices with perfect matchings, if $\Delta \left(T_1\right)>\Delta \left(T_2\right)$ and $\Delta \left(T_1\right)\ge ?\frac{k}{2}?+2$, then $\mu \left(T_1\right)>\mu \left(T_2\right)$.     

Congruences modulo powers of 3 for 3- and 9-colored generalized Frobenius partitions

Let $c{?}_k\left(n\right)$ be the number of $k$-colored generalized Frobenius partitions of $n$. We establish some infinite families of congruences for $c{?}_3\left(n\right)$ and $c{?}_9\left(n\right)$ modulo arbitrary powers of 3, which refine the results of Kolitsch. For example, for $k\ge 3$ and $n\ge 0$, we prove that

$c{?}_3\left(3^{2k}n+\frac{7?3^{2k}+1}{8}\right)\equiv 0(mod{3}^{4k+5}).$

We give two different proofs to the congruences satisfied by $c{?}_9\left(n\right)$. One of the proofs uses a relation between $c{?}_9\left(n\right)$ and $c{?}_3\left(n\right)$ due to Kolitsch, for which we provide a new proof in this paper.     

Extremal product-one free sequences in Dihedral and Dicyclic Groups

Let $G$ be a finite group, written multiplicatively. The Davenport constant of $G$ is the smallest positive integer $D\left(G\right)$ such that every sequence of $G$ with $D\left(G\right)$ elements has a non-empty subsequence with product $1$. Let $D_{2n}$ be the Dihedral Group of order $2n$ and $Q_{4n}$ be the Dicyclic Group of order $4n$. Zhuang and Gao (2005) showed that $D\left(D_{2n}\right)=n+1$ and Bass (2007) showed that $D\left(Q_{4n}\right)=2n+1$. In this paper, we give explicit characterizations of all sequences $S$ of $G$ such that $\left|S\right|=D\left(G\right)?1$ and $S$ is free of subsequences whose product is 1, where $G$ is equal to $D_{2n}$ or $Q_{4n}$ for some $n$.     

Sums of powers of Catalan triangle numbers

In this paper, we consider combinatorial numbers ${\left(C_{m,k}\right)}_{m\ge 1,k\ge 0}$, mentioned as Catalan triangle numbers where $C_{m,k}?\left(\genfrac{}{}{0ex}{}{m?1}{k}\right)?\left(\genfrac{}{}{0ex}{}{m?1}{k?1}\right)$. These numbers unify the entries of the Catalan triangles $B_{n,k}$ and $A_{n,k}$ for appropriate values of parameters $m$ and $k$, i.e., $B_{n,k}=C_{2n,n?k}$ and $A_{n,k}=C_{2n+1,n+1?k}$. In fact, these numbers are suitable rearrangements of the known ballot numbers and some of these numbers are the well-known Catalan numbers $C_n$ that is $C_{2n,n?1}=C_{2n+1,n}=C_n$.We present identities for sums (and alternating sums) of $C_{m,k}$, squares and cubes of $C_{m,k}$ and, consequently, for $B_{n,k}$ and $A_{n,k}$. In particular, one of these identities solves an open problem posed in Gutiérrez et al. (2008). We also give some identities between ${\left(C_{m,k}\right)}_{m\ge 1,k\ge 0}$ and harmonic numbers ${\left(H_n\right)}_{n\ge 1}$. Finally, in the last section, new open problems and identities involving ${\left(C_n\right)}_{n\ge 0}$ are conjectured.     

On the sum of distinct primes or squares of primes

In 1965 Erd?s introduced $f_2(s)$: $f_2(s)$ is the smallest integer such that every $l>f_2(s)$ is the sum of s distinct primes or squares of primes where a prime and its square are not both used. We prove that for all sufficiently large s, $f_2(s)?p_2+p_3+?+p_{s+1}+3106$, and the set of s with the equality has the density 1.     

Counting numerical semigroups by genus and even gaps

Let $n_g$ be the number of numerical semigroups of genus $g$. We present an approach to compute $n_g$ by using even gaps, and the question: Is it true that $n_{g+1}>n_g$? is investigated. Let $N_{\gamma }\left(g\right)$ be the number of numerical semigroups of genus $g$ whose number of even gaps equals $\gamma$. We show that $N_{\gamma }\left(g\right)=N_{\gamma }\left(3\gamma \right)$ for $\gamma \le ?g∕3?$ and $N_{\gamma }\left(g\right)=0$ for $\gamma >?2g∕3?$; thus the question above is true provided that $N_{\gamma }(g+1)>N_{\gamma }\left(g\right)$ for $\gamma =?g∕3?+1,\dots ,?2g∕3?$. We also show that $N_{\gamma }\left(3\gamma \right)$ coincides with $f_{\gamma }$, the number introduced by Bras-Amorós (2012) in connection with semigroup-closed sets. Finally, the stronger possibility $f_{\gamma }～{\phi }^{2\gamma }$ arises being $\phi =(1+\sqrt{5})∕2$ the golden number.     

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.