Journal of Algebraic Combinatorics最新文献

Given a finite permutation group G with domain (Omega ), we associate two subsets of natural numbers to G, namely ({mathcal {I}}(G,Omega )) and ({mathcal {M}}(G,Omega )), which are the sets of cardinalities of all the irredundant and minimal bases of G, respectively. We prove that ({mathcal {I}}(G,Omega )) is an interval of natural numbers, whereas ({mathcal {M}}(G,Omega )) may not necessarily form an interval. Moreover, for a given subset of natural numbers (X subseteq {mathbb {N}}), we provide some conditions on X that ensure the existence of both intransitive and transitive groups G such that ({mathcal {I}}(G,Omega ) = X) and ({mathcal {M}}(G,Omega ) = X).

In (Striker in Discret Math Theor Comput Sci 20, 2018), Striker generalized Cameron and Fon-Der-Flaass’s notion of a toggle group. In this paper, we begin the study of transitive generalized toggle groups that contain a cycle. We first show that if such a group has degree n and contains a transposition or a 3-cycle, then the group contains (A_n). Using the result about transpositions, we then prove that a transitive generalized toggle group that contains a short cycle must be primitive. Employing a result of Jones (Bull Aust Math Soc 89(1):159-165, 2014), which relies on the classification of the finite simple groups, we conclude that any transitive generalized toggle group of degree n that contains a cycle with at least 3 fixed points must also contain (A_n). Finally, we look at imprimitive generalized toggle groups containing a long cycle and show that they decompose into a direct product of primitive generalized toggle groups each containing a long cycle.

The problem of determining the largest possible number of distinct Hamming weights in several classes of codes over finite fields was studied recently in several papers (Shi et al. in Des Codes Cryptogr 87(1):87–95, 2019, in IEEE Trans Inf Theory 66(11):6855–6862, 2020; Chen et al. in IEEE Trans Inf Theory 69(2):995–1004, 2022). A further problem is to find the minimum length of codes meeting those bounds with equality. These two questions are extended here to linear codes over chain rings for the homogeneous weight. An explicit upper bound is given for codes of given type and arbitrary length as a function of the residue field size. This bound is then shown to be tight by an argument based on Hjemslev geometries. The second question is studied for chain rings with residue field of order two.

Let (I(G,textbf{w})) be the edge ideal of an edge-weighted graph ((G,textbf{w})). We prove that (I(G,textbf{w})) is sequentially Cohen–Macaulay for all weight functions (textbf{w}) if and only if G is a Woodroofe graph.

The (epsilon )-BBS is the family of solitonic cellular automata obtained via the ultradiscretization of the elementary Toda orbits, which is a parametrized family of integrable systems unifying the Toda equation and the relativistic Toda equation. In this paper, we derive the (epsilon )-BBS with many kinds of balls and give its conserved quantities by the Schensted insertion algorithm which is introduced in combinatorics. To prove this, we extend birational transformations of the continuous elementary Toda orbits to the discrete hungry elementary Toda orbits.

The aim of this work is to construct families of weighing matrices via their automorphism group action. The matrices can be reconstructed from the 0, 1, 2-cohomology groups of the underlying automorphism group. We use this mechanism to (re)construct the matrices out of abstract group datum. As a consequence, some old and new families of weighing matrices are constructed. These include the Paley conference, the projective space, the Grassmannian, and the flag variety weighing matrices. We develop a general theory relying on low-dimensional group cohomology for constructing automorphism group actions and in turn obtain structured matrices that we call cohomology-developed matrices. This ‘cohomology development’ generalizes the cocyclic and group developments. The algebraic structure of modules of cohomology-developed matrices is discussed, and an orthogonality result is deduced. We also use this algebraic structure to define the notion of quasiproducts, which is a generalization of the Kronecker product.

In this short note we prove a local version of Philip Hall’s theorem on the nilpotency of the stability group of a chain of subgroups by only using elementary commutator calculus (Hall’s theorem is a direct consequence of our result). This provides a new way of dealing with stability groups.

We determine the minimal degree of a faithful permutation representation for each group of order (p^6) where p is an odd prime. We also record how to obtain such a representation.

For a subgraph G of a complete graph (K_n), the (K_n)-complement of G, denoted by (K_n-G), is the graph obtained from (K_n-G) by removing all the edges of G. In this paper, we express the number of spanning trees of the (K_n)-complement (K_n-G) of a bipartite graph G in terms of the determinant of the biadjcency matrices of all induced balanced bipartite subgraphs of G, which are nonsingular, and we derive formulas of the number of spanning trees of (K_n-G) for various important classes of bipartite graphs G, some of which generalize some previous results.