On tiling the integers with 4-sets of the same gap sequence

Partitioning a set into similar, if not, identical, parts is a fundamental research topic in combinatorics. The question of partitioning the integers in various ways has been considered throughout history. Given a set $\{x_1,\dots ,x_n\}$ of integers where $x_1<?<x_n$, let the gap sequence of this set be the unordered multiset $\{d_1,\dots ,d_{n?1}\}=\{x_{i+1}?x_i:i\in \{1,\dots ,n?1\}\}$. This paper addresses the following question, which was explicitly asked by Nakamigawa: can the set of integers be partitioned into sets with the same gap sequence? The question is known to be true for any set where the gap sequence has length at most two. This paper provides evidence that the question is true when the gap sequence has length three. Namely, we prove that given positive integers $p$ and $q$, there is a positive integer $r_0$ such that for all $r\ge r_0$, the set of integers can be partitioned into 4-sets with gap sequence $p,q$, $r$.     

Minimal solutions of the isometry equation

For a finite vector space $W$ over ${\mathbb{F}}_q$, there are described all the pairs of multisets $\{V_1,\dots ,V_{q+1}\}$ and $\{U_1,\dots ,U_{q+1}\}$ of subspaces in $W$ such that for all $w\in W$ the equality $\left|\{i\mid w\in V_i\}\right|=\left|\{i\mid w\in U_i\}\right|$ holds.     

On parabolic Kazhdan–Lusztig R-polynomials for the symmetric group

Parabolic R-polynomials were introduced by Deodhar as parabolic analogues of ordinary R-polynomials defined by Kazhdan and Lusztig. In this paper, we are concerned with the computation of parabolic R-polynomials for the symmetric group. Let $S_n$ be the symmetric group on $\{1,2,\dots ,n\}$, and let $S=\{s_i|1\le i\le n?1\}$ be the generating set of $S_n$, where for $1\le i\le n?1$, $s_i$ is the adjacent transposition. For a subset $J?S$, let ${(S_n)}_J$ be the parabolic subgroup generated by J, and let ${(S_n)}^J$ be the set of minimal coset representatives for $S_n/{(S_n)}_J$. For $u\le v\in {(S_n)}^J$ in the Bruhat order and $x\in \{q,?1\}$, let $R_{u,v}^{J,x}(q)$ denote the parabolic R-polynomial indexed by u and v. Brenti found a formula for $R_{u,v}^{J,x}(q)$ when $J=S?\{s_i\}$, and obtained an expression for $R_{u,v}^{J,x}(q)$ when $J=S?\{s_{i?1},s_i\}$. In this paper, we provide a formula for $R_{u,v}^{J,x}(q)$, where $J=S?\{s_{i?2},s_{i?1},s_i\}$ and i appears after $i?1$ in v. It should be noted that the condition that i appears after $i?1$ in v is equivalent to that v is a permutation in ${(S_n)}^{S?\{s_{i?2},s_i\}}$. We also pose a conjecture for $R_{u,v}^{J,x}(q)$, where $J=S?\{s_k,s_{k+1},\dots ,s_i\}$ with $1\le k\le i\le n?1$ and v is a permutation in ${(S_n)}^{S?\{s_k,s_i\}}$.     

Some multicolor bipartite Ramsey numbers involving cycles and a small number of colors

For bipartite graphs $G_1,G_2,\dots ,G_k$, the bipartite Ramsey number $b(G_1,G_2,\dots ,G_k)$ is the least positive integer $b$ so that any coloring of the edges of $K_{b,b}$ with $k$ colors will result in a copy of $G_i$ in the $i$th color for some $i$. In this paper, our main focus will be to bound the following numbers: $b(C_{2t_1},C_{2t_2})$ and $b(C_{2t_1},C_{2t_2},C_{2t_3})$ for all $t_i\ge 3,$$b(C_{2t_1},C_{2t_2},C_{2t_3},C_{2t_4})$ for $3\le t_i\le 9,$ and $b(C_{2t_1},C_{2t_2},C_{2t_3},C_{2t_4},C_{2t_5})$ for $3\le t_i\le 5.$ Furthermore, we will also show that these mentioned bounds are generally better than the bounds obtained by using the best known Zarankiewicz-type result.     

Pruned discrete random samples

Let $X_i,i\in \mathbb{N}$, be independent and identically distributed random variables with values in ${\mathbb{N}}_0$. We transform (‘prune’) the sequence $\{X_1,\dots ,X_n\},n\in \mathbb{N}$, of discrete random samples into a sequence $\{0,1,2,\dots ,Y_n\},n\in \mathbb{N}$, of contiguous random sets by replacing $X_{n+1}$ with $Y_n+1$ if $X_{n+1}>Y_n$. We consider the asymptotic behaviour of $Y_n$ as $n\to \infty$. Applications include path growth in digital search trees and the number of tables in Pitmanʼs Chinese restaurant process if the latter is conditioned on its limit value.     

On the sum of reciprocals of least common multiples

Let ${\{a_i\}}_{i=1}^{\infty }$ be a strictly increasing sequence of positive integers ($a_i<a_j$ if $i<j$). In 1978, Borwein showed that for any positive integer n, we have ${\sum }_{i=1}^n\frac{1}{\mathrm{lcm}(a_i,a_{i+1})}\le 1?\frac{1}{2^n}$, with equality occurring if and only if $a_i=2^{i?1}$ for $1\le i\le n+1$. Let $3\le r\le 7$ be an integer. In this paper, we investigate the sum ${\sum }_{i=1}^n\frac{1}{\mathrm{lcm}(a_i,...,a_{i+r?1})}$ and show that ${\sum }_{i=1}^n\frac{1}{\mathrm{lcm}(a_i,...,a_{i+r?1})}\le U_r(n)$ for any positive integer n, where $U_r(n)$ is a constant depending on r and n. Further, for any integer $n\ge 2$, we also give a characterization of the sequence ${\{a_i\}}_{i=1}^{\infty }$ such that the equality ${\sum }_{i=1}^n\frac{1}{\mathrm{lcm}(a_i,...,a_{i+r?1})}=U_r(n)$ holds.     

A note on conservative galaxies,Skolem systems,cyclic cycle decompositions,and Heffter arrays

The conservative number of a graph $G$ is the minimum positive integer $M$, such that $G$ admits an orientation and a labeling of its edges by distinct integers in $\{1,2,\dots ,M\}$, such that at each vertex of degree at least three, the sum of the labels on the in-coming edges is equal to the sum of the labels on the out-going edges. A graph is conservative if $M=\left|E\left(G\right)\right|$. It is worth noting that determining whether certain biregular graphs are conservative is equivalent to find integer Heffter arrays.In this work we show that the conservative number of a galaxy (a disjoint union of stars) of size $M$ is $M$ for $M\equiv 0$, $3(mod4)$, and $M+1$ otherwise. Consequently, given positive integers $m_1$, $m_2$, …, $m_n$ with $m_i\ge 3$ for $1\le i\le n$, we construct a cyclic $(m_1,m_2,\dots ,m_n)$-cycle system of infinitely many circulant graphs, generalizing a result of Bryant, Gavlas and Ling (2003). In particular, it allows us to construct a cyclic $(m_1,m_2,\dots ,m_n)$-cycle system of the complete graph $K_{2M+1}$, where $M={\sum }_{i=1}^n{m}_i$. Also, we prove necessary and sufficient conditions for the existence of a cyclic $(m_1,m_2,\dots ,m_n)$-cycle system of $K_{2M+2}?F$, where $F$ is a 1-factor. Furthermore, we give a sufficient condition for a subset of ${\mathbb{Z}}_v?\left\{0\right\}$ to be sequenceable.     

Degree and neighborhood conditions for hamiltonicity of claw-free graphs

For a graph $H$, let ${\sigma }_t\left(H\right)=min\left\{{\Sigma }_{i=1}^t{d}_H\left(v_i\right)\right|\{v_1,v_2,\dots ,v_t\}\text{is an independent set in }H\}$ and let $U_t\left(H\right)=min\left\{\right|{?}_{i=1}^t{N}_H\left(v_i\right)\left|\right|\{v_1,v_2,?,v_t\}\text{is an independent set in }H\}$. We show that for a given number $?$ and given integers $p\ge t>0$, $k\in \{2,3\}$ and $N=N(p,?)$, if $H$ is a $k$-connected claw-free graph of order $n>N$ with $\delta \left(H\right)\ge 3$ and its Ryjác?ek’s closure $cl\left(H\right)=L\left(G\right)$, and if $d_t\left(H\right)\ge t(n+?)∕p$ where $d_t\left(H\right)\in \{{\sigma }_t\left(H\right),U_t\left(H\right)\}$, then either $H$ is Hamiltonian or $G$, the preimage of $L\left(G\right)$, can be contracted to a $k$-edge-connected $K_3$-free graph of order at most $max\{4p?5,2p+1\}$ and without spanning closed trails. As applications, we prove the following for such graphs $H$ of order $n$ with $n$ sufficiently large:(i) If $k=2$, $\delta \left(H\right)\ge 3$, and for a given $t$ ($1\le t\le 4$) $d_t\left(H\right)\ge \frac{tn}{4}$, then either $H$ is Hamiltonian or $cl\left(H\right)=L\left(G\right)$ where $G$ is a graph obtained from $K_{2,3}$ by replacing each of the degree 2 vertices by a $K_{1,s}$ ($s\ge 1$). When $t=4$ and $d_t\left(H\right)={\sigma }_4\left(H\right)$, this proves a conjecture in Frydrych (2001).(ii) If $k=3$, $\delta \left(H\right)\ge 24$, and for a given $t$ ($1\le t\le 10$) $d_t\left(H\right)>\frac{t(n+5)}{10}$, then $H$ is Hamiltonian. These bounds on $d_t\left(H\right)$ in (i) and (ii) are sharp. It unifies and improves several prior results on conditions involved ${\sigma }_t$ and $U_t$ for the hamiltonicity of claw-free graphs. Since the number of graphs of orders at most $max\{4p?5,2p+1\}$ are fixed for given $p$, improvements to (i) or (ii) by increasing the value of $p$ are possible with the help of a computer.