Journal of Combinatorial Theory Series A最新文献

For a positive integer m, a group G is said to admit a tournament m-semiregular representation (TmSR for short) if there exists a tournament Γ such that the automorphism group of Γ is isomorphic to G and acts semiregularly on the vertex set of Γ with m orbits. It is easy to see that every finite group of even order does not admit a TmSR for any positive integer m. The T1SR is the well-known tournament regular representation (TRR for short). In 1970s, Babai and Imrich proved that every finite group of odd order admits a TRR except for $Z_3^2$, and every group (finite or infinite) without element of order 2 having an independent generating set admits a T2SR in (1979) [3]. Later, Godsil correct the result by showing that the only finite groups of odd order without a TRR are $Z_3^2$ and $Z_3^3$ by a probabilistic approach in (1986) [11]. In this note, it is shown that every finite group of odd order has a TmSR for every $m\ge 2$.

The exact value of the separating Noether number of an arbitrary finite abelian group of rank two is determined. This is done by a detailed study of the monoid of zero-sum sequences over the group.

The tableau reconstruction problem, posed by Monks (2009), asks the following. Starting with a standard Young tableau T, a 1-minor of T is a tableau obtained by first deleting any cell of T, and then performing jeu de taquin slides to fill the resulting gap. This can be iterated to arrive at the set of k-minors of T. The problem is this: given k, what are the values of n such that every tableau of size n can be reconstructed from its set of k-minors? For $k=1$, the problem was recently solved by Cain and Lehtonen. In this paper, we solve the problem for $k=2$, proving the sharp lower bound $n\ge 8$. In the case of multisets of k-minors, we also give a lower bound for arbitrary k, as a first step toward a sharp bound in the general multiset case.

In this paper, we establish some general expansion formulas for q-series. Three of Liu's identities motivate us to search and find such type of formulas. These expansion formulas include as special cases or limiting cases many q-identities including the q-Gauss summation formula, the q-Pfaff-Saalschütz summation formula, three of Jackson's transformation formulas and Sears' terminating ${}_4\varphi _3$ transformation formula. As applications, we provide a new proof of the orthogonality relation for continuous dual q-Hahn polynomials, establish some generating functions for special values of the Dirichlet L-functions and the Hurwitz zeta functions, give extensions of three of Liu's identities, establish an extension of Dilcher's identity, and deduce various double Rogers-Ramanujan type identities.

Let $r\text{des}$ and $r\text{exc}$ denote the permutation statistics r-descent number and r-excedance number, respectively. We prove that the pairs of permutation statistics $(r\text{des},r\text{maj})$ and $(r\text{exc},r\text{den})$ are equidistributed, where $r\text{maj}$ denotes the r-major index defined by Don Rawlings and $r\text{den}$ denotes the r-Denert's statistic defined by Guo-Niu Han. When $r=1$, this result reduces to the equidistribution of $(\text{des},\text{maj})$ and $(\text{exc},\text{den})$, which was conjectured by Denert in 1990 and proved that same year by Foata and Zeilberger. We call a pair of permutation statistics that is equidistributed with $(r\text{des},r\text{maj})$ and $(r\text{exc},r\text{den})$ an r-Euler-Mahonian statistic, which reduces to the classical Euler-Mahonian statistic when $r=1$.

We then introduce the notions of r-level descent number, r-level excedance number, r-level major index, and r-level Denert's statistic, denoted by ${\text{des}}_r,{\text{exc}}_r,{\text{maj}}_r$, and ${\text{den}}_r$, respectively. We prove that $({\text{des}}_r,{\text{maj}}_r)$ is r-Euler-Mahonian and conjecture that $({\text{exc}}_r,{\text{den}}_r)$ is r-Euler-Mahonian. Furthermore, we give an extension of the above result and conjecture.

The q-Onsager algebra, denoted $O_q$, is defined by two generators $W_0,W_1$ and two relations called the q-Dolan-Grady relations. Recently, Terwilliger introduced some elements of $O_q$, said to be alternating. These elements are denoted${\left\{W_{-k}\right\}}_{k=0}^{\infty },{\left\{W_{k+1}\right\}}_{k=0}^{\infty },{\left\{G_{k+1}\right\}}_{k=0}^{\infty },{\left\{{\tilde{G}}_{k+1}\right\}}_{k=0}^{\infty }.$

The alternating elements of $O_q$ are defined recursively. By construction, they are polynomials in $W_0$ and $W_1$. It is currently unknown how to express these polynomials in closed form.

In this paper, we consider an algebra $T_q$, called the quantum torus. We present a basis for $T_q$ and define an algebra homomorphism $p:O_q\mapsto T_q$. In our main result, we express the p-images of the alternating elements of $O_q$ in the basis for $T_q$. These expressions are in a closed form that we find attractive.

We construct an infinite family of hyperovals on the Klein quadric $Q^{+}(5,q)$, q even. The construction makes use of ovoids of the symplectic generalized quadrangle $W\left(q\right)$ that is associated with an elliptic quadric which arises as solid intersection with $Q^{+}(5,q)$. We also solve the isomorphism problem: we determine necessary and sufficient conditions for two hyperovals arising from the construction to be isomorphic.

Ferrers diagram rank-metric codes were introduced by Etzion and Silberstein in 2009. In their work, they proposed a conjecture on the largest dimension of a space of matrices over a finite field whose nonzero elements are supported on a given Ferrers diagram and all have rank lower bounded by a fixed positive integer d. Since stated, the Etzion-Silberstein conjecture has been verified in a number of cases, often requiring additional constraints on the field size or on the minimum rank d in dependence of the corresponding Ferrers diagram. As of today, this conjecture still remains widely open. Using modular methods, we give a constructive proof of the Etzion-Silberstein conjecture for the class of strictly monotone Ferrers diagrams, which does not depend on the minimum rank d and holds over every finite field. In addition, we leverage on the last result to also prove the conjecture for the class of MDS-constructible Ferrers diagrams, without requiring any restriction on the field size.

A string attractor is a set of positions in a word such that each distinct factor has an occurrence crossing a position from the set. This definition comes from the data compression field, where the size ${\gamma }^{⁎}$ of a smallest string attractor represents a lower bound for the output size of a large family of string compressors exploiting repetitions in words, including BWT-based and LZ-based compressors. For finite words, the combinatorial properties of string attractors have been studied in 2021 by Mantaci et al.. Later, Schaeffer and Shallit introduced the string attractor profile function, a complexity function that evaluates for each $n>0$ the size ${\gamma }^{⁎}$ of the length-n prefix of a one-sided infinite word.

A natural development of the research on the topic is to link string attractors with other classical notions of repetitiveness in combinatorics on words. Our contribution in this sense is threefold. First, we explore the relation between the string attractor profile function and other well-known combinatorial complexity functions in the context of infinite words, such as the factor complexity and the property of recurrence. Moreover, we study its asymptotic growth in the case of purely morphic words and obtain a complete description in the binary case. Second, we introduce two new string attractor-based complexity functions, in which the structure and the distribution of positions in a string attractor are taken into account, and we study their combinatorial properties. We also show that these measures provide a finer classification of some infinite families of words, namely the Sturmian and quasi-Sturmian words. Third, we explicitly give the three complexities for some specific morphic words called k-bonacci words.

A preliminary version of some results presented in this paper can be found in [Restivo, Romana, Sciortino, String Attractors and Infinite Words, LATIN 2022].

Cluster exchange groupoids are introduced by King-Qiu as an enhancement of cluster exchange graphs to study stability conditions and quadratic differentials. In this paper, we introduce the cluster exchange groupoid for any finite Coxeter-Dynkin diagram Δ and show that its fundamental group is isomorphic to the corresponding braid group associated with Δ.