Context-free grammars,generating functions and combinatorial arrays

Let ${\left[R_{n,k}\right]}_{n,k\ge 0}$ be an array of nonnegative numbers satisfying the recurrence relation $R_{n,k}=(a_{1}n+a_{2}k+a_3)R_{n-1,k}+(b_{1}n+b_{2}k+b_3)R_{n-1,k-1}+(c_{1}n+c_{2}k+c_3)R_{n-1,k-2}$ with $R_{0,0}=1$ and $R_{n,k}=0$ unless $0\le k\le n$. In this paper, we first prove that the array ${\left[R_{n,k}\right]}_{n,k\ge 0}$ can be generated by some context-free Grammars, which gives a unified proof of many known results. Furthermore, we present criteria for real rootedness of row-generating functions and asymptotical normality of rows of ${\left[R_{n,k}\right]}_{n,k\ge 0}$. Applying the criteria to some arrays related to tree-like tableaux, interior and left peaks, alternating runs, flag descent numbers of group of type $B$, and so on, we get many results in a unified manner. Additionally, we also obtain the continued fraction expansions for generating functions related to above examples. As results, we prove the strong $q$-log-convexity of some generating functions.     

On three color Ramsey numbers R(C4,C4,K1,n)

For $k$ given graphs $G_1,G_2,\dots ,G_k$, $k\ge 2$, the $k$-color Ramsey number, denoted by $R(G_1,G_2,\dots ,G_k)$, is the smallest integer $N$ such that if we arbitrarily color the edges of a complete graph of order $N$ with $k$ colors, then it always contains a monochromatic copy of $G_i$ colored with $i$, for some $1\le i\le k$. Let $C_m$ be a cycle of length $m$ and $K_{1,n}$ a star of order $n+1$. In this paper, firstly we give a general upper bound of $R(C_4,C_4,\dots ,C_4,K_{1,n})$. In particular, for the 3-color case, we have $R(C_4,C_4,K_{1,n})\le n+?\sqrt{4n+5}?+3$ and this bound is tight in some sense. Furthermore, we prove that $R(C_4,C_4,K_{1,n})\le n+?\sqrt{4n+5}?+2$ for all $n={?}^2??$ and $?\ge 2$, and if $?$ is a prime power, then the equality holds.     

Forcing clique immersions through chromatic number

Building on recent work of Dvořák and Yepremyan, we show that every simple graph of minimum degree $7t+7$ contains $K_t$ as an immersion and that every graph with chromatic number at least $3.54t+4$ contains $K_t$ as an immersion. We also show that every graph on $n$ vertices with no independent set of size three contains $K_{2\lfloor n∕5\rfloor }$ as an immersion.     

Erdős–Gallai stability theorem for linear forests

The Erd?s–Gallai Theorem states that every graph of average degree more than $l?2$ contains a path of order $l$ for $l\ge 2$. In this paper, we obtain a stability version of the Erd?s–Gallai Theorem in terms of minimum degree. Let $G$ be a connected graph of order $n$ and $F=\left({?}_{i=1}^k{P}_{2a_i}\right)?\left({?}_{i=1}^l{P}_{2b_i+1}\right)$ be $k+l$ disjoint paths of order $2a_1,\dots ,2a_k,2b_1+1,\dots ,2b_l+1,$ respectively, where $k\ge 0$, $0\le l\le 2$, and $k+l\ge 2$. If the minimum degree $\delta \left(G\right)\ge {\sum }_{i=1}^k{a}_i+{\sum }_{i=1}^l{b}_i?1$, then $F?G$ except several classes of graphs for sufficiently large $n$, which extends and strengths the results of Ali and Staton for an even path and Yuan and Nikiforov for an odd path.     

The component counts of random functions

Consider functions $f:A\to A\cup C$, where A and C are disjoint finite sets. The weakly connected components of the digraph of such a function are cycles of rooted trees, as in random mappings, and isolated rooted trees. Let $n_1=|A|$ and $n_3=|C|$. When a function is chosen from all ${(n_1+n_3)}^{n_1}$ possibilities uniformly at random, then we find the following limiting behaviour as $n_1\to \infty$. If $n_3=o(n_1)$, then the size of the maximal mapping component goes to infinity almost surely; if $n_3～\gamma n_1$, $\gamma >0$ a constant, then process counting numbers of mapping components of different sizes converges; if $n_1=o(n_3)$, then the number of mapping components converges to 0 in probability. We get estimates on the size of the largest tree component which are of order $\log ?n_3$ when $n_3～\gamma n_1$ and constant when $n_3～n_1^{\alpha }$, $\alpha >1$. These results are similar to ones obtained previously for random injections, for which the weakly connected components are cycles and linear trees.     

On a conjecture about a class of permutation quadrinomials

Very recently, Tu et al. presented a sufficient condition on $(a_1,a_2,a_3)$, see Theorem 1.1, such that $f(x)=x^{3\cdot 2^m}+a_1{x}^{2^{m+1}+1}+a_2{x}^{2^m+2}+a_3{x}^3$ is a class of permutation polynomials over ${\mathbb{F}}_{2^n}$ with $n=2m$ and m odd. In this present paper, we prove that the sufficient condition is also necessary.     

Spanning trees with at most 4 leaves in K1,5-free graphs

In 2009, Kyaw proved that every $n$-vertex connected $K_{1,4}$-free graph $G$ with ${\sigma }_4\left(G\right)\ge n?1$ contains a spanning tree with at most 3 leaves. In this paper, we prove an analogue of Kyaw’s result for connected $K_{1,5}$-free graphs. We show that every $n$-vertex connected $K_{1,5}$-free graph $G$ with ${\sigma }_5\left(G\right)\ge n?1$ contains a spanning tree with at most 4 leaves. Moreover, the degree sum condition “${\sigma }_5\left(G\right)\ge n?1$” is best possible.     

New upper bound for multicolor Ramsey number of odd cycles

Let $r_k\left(C_{2m+1}\right)$ be the $k$-color Ramsey number of an odd cycle $C_{2m+1}$ of length $2m+1$. It is shown that for each fixed $m\ge 2$, $r_k\left(C_{2m+1}\right)<c^k\sqrt{k!}$for all sufficiently large $k$, where $c=c\left(m\right)>0$ is a constant. This improves an old result by Bondy and Erd?s (1973).     

A generalization of Schönemann’s theorem via a graph theoretic method

Recently, Grynkiewicz et al. (2013), using tools from additive combinatorics and group theory, proved necessary and sufficient conditions under which the linear congruence $a_1{x}_1+?+a_k{x}_k\equiv b(modn)$, where $a_1,\dots ,a_k,b,n$ ($n\ge 1$) are arbitrary integers, has a solution $〈x_1,\dots ,x_k〉\in {\mathbb{Z}}_n^k$ with all $x_i$ distinct. So, it would be an interesting problem to give an explicit formula for the number of such solutions. Quite surprisingly, this problem was first considered, in a special case, by Schönemann almost two centuries ago(!) but his result seems to have been forgotten. Schönemann (1839), proved an explicit formula for the number of such solutions when $b=0$, $n=p$ a prime, and ${\sum }_{i=1}^k{a}_i\equiv 0(modp)$ but ${\sum }_{i\in I}{a}_i?\equiv 0(modp)$ for all $0?\ne I??\{1,\dots ,k\}$. In this paper, we generalize Schönemann’s theorem using a result on the number of solutions of linear congruences due to D. N. Lehmer and also a result on graph enumeration. This seems to be a rather uncommon method in the area; besides, our proof technique or its modifications may be useful for dealing with other cases of this problem (or even the general case) or other relevant problems.     

Maximal sets of Hamilton cycles in Knr;λ1,λ2

Let $K\left(n^r;{\lambda }_1,{\lambda }_2\right)$ be the $r$-partite multigraph in which each part has size $n$, where two vertices in the same part or different parts are joined by exactly ${\lambda }_1$ edges or ${\lambda }_2$ edges, respectively. It is proved that there exists a maximal set of $t$ edge-disjoint Hamilton cycles in $K\left(n^r;{\lambda }_1,{\lambda }_2\right)$ for ${\lambda }_{2}n\lfloor \frac{r+3}{4}\rfloor \le t\le min\left\{\lfloor \frac{{\lambda }_2{n}^2(r-1)}{2}\rfloor ,\lfloor \frac{{\lambda }_1(n-1)+{\lambda }_{2}n(r-1)}{2}\rfloor \right\}$, the upper bound being best possible. The results proved make use of the method of amalgamations.     

Gaps in the number of generators of monomial Togliatti systems

Let $I_{d,n}?k[x_0,?,x_n]$ be a minimal monomial Togliatti system of forms of degree d. In [4], Mezzetti and Miró-Roig proved that the minimal number of generators $\mu (I_{d,n})$ of $I_{d,n}$ lies in the interval $[2n+1,\left(\begin{array}{c}n+d?1\\ n?1\end{array}\right)]$. In this paper, we prove that for $n\ge 4$ and $d\ge 3$, the integer values in $[2n+3,3n?1]$ cannot be realized as the number of minimal generators of a minimal monomial Togliatti system. We classify minimal monomial Togliatti systems $I_{d,n}?k[x_0,?,x_n]$ of forms of degree d with $\mu (I_{d,n})=2n+2$ or 3n (i.e. with the minimal number of generators reaching the border of the non-existence interval). Finally, we prove that for $n=4$, $d\ge 3$ and $\mu \in [9,\left(\begin{array}{c}d+3\\ 3\end{array}\right)]?\{11\}$ there exists a minimal monomial Togliatti system $I_{d,n}?k[x_0,?,x_n]$ of forms of degree d with $\mu (I_{n,d})=\mu$.     

The combinatorial properties of the Benoumhani polynomials for the Whitney numbers of Dowling lattices

In the papers (Benoumhani 1996;1997), Benoumhani defined two polynomials $F_{m,n,1}\left(x\right)$ and $F_{m,n,2}\left(x\right)$. Then, he defined $A_m(n,k)$ and $B_m(n,k)$ to be the polynomials satisfying $F_{m,n,1}\left(x\right)={\sum }_{k=0}^n{A}_m(n,k)x^{n?k}{(x+1)}^k$ and $F_{m,n,1}\left(x\right)={\sum }_{k=0}^n{B}_m(n,k)x^{n?k}{(x+1)}^k$. In this paper, we give a combinatorial interpretation of the coefficients of $A_{m+1}(n,k)$ and prove a symmetry of the coefficients, i.e., $\left[m^s\right]A_{m+1}(n,k)=\left[m^{n?s}\right]A_{m+1}(n,n?k)$. We give a combinatorial interpretation of $B_{m+1}(n,k)$ and prove that $B_{m+1}(n,n?1)$ is a polynomial in $m$ with non-negative integer coefficients. We also prove that if $n\ge 6$ then all coefficients of $B_{m+1}(n,n?2)$ except the coefficient of $m^{n?1}$ are non-negative integers. For all $n$, the coefficient of $m^{n?1}$ in $B_{m+1}(n,n?2)$ is $?(n?1)$, and when $n\le 5$ some other coefficients of $B_{m+1}(n,n?2)$ are also negative.     

Permutations with small maximal k-consecutive sums

Let $n$ and $k$ be positive integers with $n>k$. Given a permutation $({\pi }_1,\dots ,{\pi }_n)$ of integers $1,\dots ,n$, we consider $k$-consecutive sums of $\pi$, i.e., $s_i?{\sum }_{j=0}^{k?1}{\pi }_{i+j}$ for $i=1,\dots ,n$, where we let ${\pi }_{n+j}={\pi }_j$. What we want to do in this paper is to know the exact value of $\mathsf{msum}(n,k)?min\left\{max\{s_i:i=1,\dots ,n\}?\frac{k(n+1)}{2}:\pi \in S_n\right\},$ where $S_n$ denotes the set of all permutations of $1,\dots ,n$. In this paper, we determine the exact values of $\mathsf{msum}(n,k)$ for some particular cases of $n$ and $k$. As a corollary of the results, we obtain $\mathsf{msum}(n,3)$, $\mathsf{msum}(n,4)$ and $\mathsf{msum}(n,6)$ for any $n$.