Journal of Combinatorial Theory Series A最新文献

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.

We introduce a new family of operators as multi-parameter deformation of the one-row Macdonald polynomials. The matrix coefficients of these operators acting on the space of symmetric functions with rational coefficients in two parameters $q,t$ (denoted by $\Lambda (q,t)$) are computed by assigning some values to skew Macdonald polynomials in λ-ring notation. The new rule is utilized to provide new iterative formulas and also recover various existing formulas in a unified manner. Specifically the following applications are discussed: (i) A $(q,t)$-Murnaghan-Nakayama rule for Macdonald functions is given as a generalization of the q-Murnaghan-Nakayama rule; (ii) An iterative formula for the $(q,t)$-Green polynomial is deduced; (iii) A simple proof of the Murnaghan-Nakayama rule for the Hecke algebra and the Hecke-Clifford algebra is offered; (iv) A combinatorial inversion of the Pieri rule for Hall-Littlewood functions is derived with the help of the vertex operator realization of the Hall-Littlewood functions; (v) Two iterative formulae for the $(q,t)$-Kostka polynomials $K_{\lambda \mu }(q,t)$ are obtained from the dual version of our multiparametric Murnaghan-Nakayama rule, one of which yields an explicit formula for arbitrary λ and μ in terms of the generalized $(q,t)$-binomial coefficient introduced independently by Lassalle and Okounkov.

We revisit the topic of power-free morphisms, focusing on the properties of the class of complementary morphisms. Such morphisms are defined over a 2-letter alphabet, and map the letters 0 and 1 to complementary words. We prove that every prefix of the famous Thue–Morse word t gives a complementary morphism that is $3^{+}$-free and hence α-free for every real number $\alpha >3$. We also describe, using a 4-state binary finite automaton, the lengths of all prefixes of t that give cubefree complementary morphisms. Next, we show that 3-free (cubefree) complementary morphisms of length k exist for all $k\notin \{3,6\}$. Moreover, if k is not of the form $3\cdot 2^n$, then the images of letters can be chosen to be factors of t. Finally, we observe that each cubefree complementary morphism is also α-free for some $\alpha <3$; in contrast, no binary morphism that maps each letter to a word of length 3 (resp., a word of length 6) is α-free for any $\alpha <3$.

In addition to more traditional techniques of combinatorics on words, we also rely on the Walnut theorem-prover. Its use and limitations are discussed.

The k-power hypergraph $G^{\left(k\right)}$ is the k-uniform hypergraph that is obtained by adding $k-2$ new vertices to each edge of a graph G, for $k\ge 3$. A parity-closed walk in G is a closed walk that uses each edge an even number of times. In an earlier paper, we determined the eigenvalues of the adjacency tensor of $G^{\left(k\right)}$ using the eigenvalues of signed subgraphs of G. Here, we express the entire spectrum (that is, we determine all multiplicities and the characteristic polynomial) of $G^{\left(k\right)}$ in terms of parity-closed walks of G. Moreover, we give an explicit expression for the multiplicity of the spectral radius of $G^{\left(k\right)}$. As a side result, we show that the number of parity-closed walks of given length is the corresponding spectral moment averaged over all signed graphs with underlying graph G. By extrapolating the characteristic polynomial of $G^{\left(k\right)}$ to $k=2$, we introduce a pseudo-characteristic function which is shown to be the geometric mean of the characteristic polynomials of all signed graphs on G. This supplements a result by Godsil and Gutman that the arithmetic mean of the characteristic polynomials of all signed graphs on G equals the matching polynomial of G.