Some statistics on Stirling permutations and Stirling derangements

A permutation of the multiset $\{1,1,2,2,\dots ,n,n\}$ is called a Stirling permutation of order $n$ if every entry between the two occurrences of $i$ is greater than $i$ for each $i\in \{1,2,\dots ,n\}$. In this paper, we introduce the definitions of block, even indexed entry, odd indexed entry, Stirling derangement, marked permutation and bicolored increasing binary tree. We first study the joint distribution of ascent plateaux, even indexed entries and left-to-right minima over the set of Stirling permutations of order $n$. We then present an involution on Stirling derangements.     

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 $p,q$, $r$.     

A sufficient condition for DP-4-colorability

DP-coloring of a simple graph is a generalization of list coloring, and also a generalization of signed coloring of signed graphs. It is known that for each $k\in \{3,4,5,6\}$, every planar graph without $C_k$ is 4-choosable. Furthermore, Jin et al. (2016) showed that for each $k\in \{3,4,5,6\}$, every signed planar graph without $C_k$ is signed 4-choosable. In this paper, we show that for each $k\in \{3,4,5,6\}$, every planar graph without $C_k$ is 4-DP-colorable, which is an extension of the above results.     

Computation and theory of Euler sums of generalized hyperharmonic numbers

Recently, Dil and Boyadzhiev [10] proved an explicit formula for the sum of multiple harmonic numbers whose indices are the sequence $({\left\{0\right\}}_r,1)$. In this paper, we show that the sums of multiple harmonic numbers whose indices are the sequence $({\left\{0\right\}}_r,1;{\left\{1\right\}}_{k?1})$ can be expressed in terms of (multiple) zeta values, (multiple) harmonic numbers, and Stirling numbers of the first kind, and give an explicit formula.     

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.     

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.     

On the existence of retransmission permutation arrays

We investigate retransmission permutation arrays (RPAs) that are motivated by applications in overlapping channel transmissions. An RPA is an $n\times n$ array in which each row is a permutation of $\{1,\dots ,n\}$, and for $1?i?n$, all $n$ symbols occur in each $i\times ?\frac{n}{i}?$ rectangle in specified corners of the array. The array has types 1, 2, 3 and 4 if the stated property holds in the top left, top right, bottom left and bottom right corners, respectively. It is called latin if it is a latin square. We show that for all positive integers $n$, there exists a type-1, 2, 3, 4 $\mathrm{RPA}\left(n\right)$ and a type-1, 2 latin $\mathrm{RPA}\left(n\right)$.     

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.