Journal of Combinatorial Theory Series A最新文献

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.

In this paper, we present a study on polyominoes, which are polygons created by connecting unit squares along their edges. Specifically, we focus on a related concept called a bargraph, which is a path on a lattice in $Z_{\ge 0}\times Z_{\ge 0}$ traced along the boundaries of a column convex polyomino where the lower edge is on the x-axis. To explore new variations of bargraphs, we introduce the notion of non-decreasing bargraphs, which incorporate an additional restriction concerning the valleys within the path. We establish intriguing connections between these novel objects and unimodal compositions. To facilitate our analysis, we employ generating functions, including q-series, as well as various closed formulas. These tools enable us to enumerate the different types of bargraphs based on their semi-perimeter, area, and the number of peaks. Furthermore, we provide combinatorial justifications for some of the derived closed formulas.

In this paper, we refine a result of Andrews and Merca on truncated pentagonal number series. Subsequently, we establish some positivity results involving Andrews–Gordon–Bressoud identities and d-regular partitions. In particular, we prove several conjectures of Merca and Krattenthaler–Merca–Radu on truncated pentagonal number series.

The difference graph $D\left(G\right)$ of a finite group G is the difference of the enhanced power graph of G and the power graph of G, where all isolated vertices are removed. In this paper we study the connectedness and perfectness of $D\left(G\right)$ with respect to various properties of the underlying group G. We also find several connections between the difference graph of G and the Gruenberg-Kegel graph of G. We also examine the operation of twin reduction on graphs, a technique which produces smaller graphs which may be easier to analyze. Applying this technique to simple groups can have a number of outcomes, not fully understood, but including some graphs with large girth.

In this note we show that a flag-transitive automorphism group G of a non-trivial 2-$(v,k,\lambda )$ design with $\lambda \ge {(r,\lambda )}^2$ is not of product action type. In conclusion, G is a primitive group of affine or almost simple type.

A subset $F$ of a finite transitive group $G\le \mathrm{Sym}\left(\Omega \right)$ is intersecting if any two elements of $F$ agree on an element of Ω. The intersection density of G is the number$\rho \left(G\right)=\max \{\left|F\right|/|G_{\omega }\left|\right|F\subset G\text{ is intersecting}\},$ where $\omega \in \Omega$ and $G_{\omega }$ is the stabilizer of ω in G. It is known that if $G\le \mathrm{Sym}\left(\Omega \right)$ is an imprimitive group of degree a product of two odd primes $p>q$ admitting a block of size p or two complete block systems, whose blocks are of size q, then $\rho \left(G\right)=1$.

In this paper, we analyze the intersection density of imprimitive groups of degree pq with a unique block system with blocks of size q based on the kernel of the induced action on blocks. For those whose kernels are non-trivial, it is proved that the intersection density is larger than 1 whenever there exists a cyclic code C with parameters ${[p,k]}_q$ such that any codeword of C has weight at most $p-1$, and under some additional conditions on the cyclic code, it is a proper rational number. For those that are quasiprimitive, we reduce the cases to almost simple groups containing $\mathrm{Alt}\left(5\right)$ or a projective special linear group. We give some examples where the latter has intersection density equal to 1, under some restrictions on p and q.