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.     

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.     

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 $j\ge 212299$.     

Proper connection and size of graphs

An edge-coloured graph $G$ is called properly connected if any two vertices are connected by a path whose edges are properly coloured. The proper connection number of a connected graph $G,$ denoted by $\mathrm{pc}\left(G\right)$, is the smallest number of colours that are needed in order to make $G$ properly connected. Our main result is the following: Let $G$ be a connected graph of order $n$ and $k\ge 2$. If $\left|E\left(G\right)\right|\ge \left(\genfrac{}{}{0ex}{}{n?k?1}{2}\right)+k+2$, then $\mathrm{pc}\left(G\right)\le k$ except when $k=2$ and $G\in \{G_1,G_2\},$ where $G_1=K_1\vee (2K_1+K_2)$ and $G_2=K_1\vee (K_1+2K_2).$     

On the distance between linear codes

Let V be an n-dimensional vector space over the finite field consisting of q elements and let ${\mathrm{\Gamma }}_k(V)$ be the Grassmann graph formed by k-dimensional subspaces of V, $1<k<n-1$. Denote by $\mathrm{\Gamma }{(n,k)}_q$ the restriction of ${\mathrm{\Gamma }}_k(V)$ to the set of all non-degenerate linear ${[n,k]}_q$ codes. We show that for any two codes the distance in $\mathrm{\Gamma }{(n,k)}_q$ coincides with the distance in ${\mathrm{\Gamma }}_k(V)$ only in the case when $n<{(q+1)}^2+k-2$, i.e. if n is sufficiently large then for some pairs of codes the distances in the graphs ${\mathrm{\Gamma }}_k(V)$ and $\mathrm{\Gamma }{(n,k)}_q$ are distinct. We describe one class of such pairs.     

On topological and geometric (194) configurations

An $\left(n_k\right)$ configuration is a set of  $n$ points and  $n$ lines such that each point lies on  $k$ lines while each line contains  $k$ points. The configuration is geometric, topological, or combinatorial depending on whether lines are considered to be straight lines, pseudolines, or just combinatorial lines. The existence and enumeration of $\left(n_k\right)$ configurations for a given  $k$ has been subject to active research. A current front of research concerns geometric $\left(n_4\right)$ configurations: it is now known that geometric $\left(n_4\right)$ configurations exist for all  $n\ge 18$, apart from sporadic exceptional cases. In this paper, we settle by computational techniques the first open case of $\left(19_4\right)$ configurations: we obtain all topological $\left(19_4\right)$ configurations among which none are geometrically realizable.     

Constructions for optimal cyclic ternary constant-weight codes of weight four and distance six

A cyclic ${(n,d,w)}_q$ code is a $q$-ary cyclic code of length $n$, minimum Hamming distance $d$ and weight $w$. In this paper, we investigate cyclic ${(n,6,4)}_3$ codes. A new upper bound on $C{A}_3(n,6,4)$, the largest possible number of codewords in a cyclic ${(n,6,4)}_3$ code, is given. Two new constructions for optimal cyclic ${(n,6,4)}_3$ codes based on cyclic $(n,4,1)$ difference packings are presented. As a consequence, the exact value of $C{A}_3(n,6,4)$ is determined for any positive integer $n\equiv 0,6,18(mod24)$. We also obtain some other infinite classes of optimal cyclic ${(n,6,4)}_3$ codes.     

The sizes of optimal (n,4,λ,3) optical orthogonal codes

Let $\Phi (n,k,{\lambda }_a,{\lambda }_c)$ denote the largest possible size among all $(n,k,{\lambda }_a,{\lambda }_c)$-OOCs. An $(n,k,{\lambda }_a,{\lambda }_c)$-OOC with $\Phi (n,k,{\lambda }_a,{\lambda }_c)$ codewords is said to be optimal. In this paper, the exact value of $\Phi (n,4,\lambda ,3)$ is determined. Equivalently, the size of an optimal $(n,4,\lambda ,3)$ optical orthogonal code is calculated.