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 (denoted by ) 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 -Murnaghan-Nakayama rule for Macdonald functions is given as a generalization of the q-Murnaghan-Nakayama rule; (ii) An iterative formula for the -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 -Kostka polynomials 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 -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 -free and hence α-free for every real number . 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 . Moreover, if k is not of the form , 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 ; 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 .
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 is the k-uniform hypergraph that is obtained by adding new vertices to each edge of a graph G, for . 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 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 in terms of parity-closed walks of G. Moreover, we give an explicit expression for the multiplicity of the spectral radius of . 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 to , 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.
In this paper we extend the study of linear spaces of upper triangular matrices endowed with the flag-rank metric. Such metric spaces are isometric to certain spaces of degenerate flags and have been suggested as suitable framework for network coding. In this setting we provide a Singleton-like bound which relates the parameters of a flag-rank-metric code. This allows us to introduce the family of maximum flag-rank distance codes, that are flag-rank-metric codes meeting the Singleton-like bound with equality. Finally, we provide several constructions of maximum flag-rank distance codes.
We prove several linkage properties of graphs with flows, generalizing some results on linkage of graphs. This translates in properties of connectedness through codimension one of certain posets. For example, the poset of flows and the posets of odd and even tropical spin curves. These posets are, respectively, the posets underlying the moduli space of roots of divisors on tropical curves and the moduli spaces of odd and even tropical spin curves.
A group code is a linear code which can be realized as a two-sided ideal of a group algebra over a finite field. When the characteristic of the field is prime to the order of the group, we will give explicit expressions for primitive central idempotents in the group algebra, which enables us to determine the number of equivalent classes of minimal group codes. Then, we apply our formula to calculate the number of equivalent classes of minimal general dihedral group codes.
Let H be an HD0L-system. We show that there are only finitely many primitive words v with the property that , for all integers k, is an element of the factorial language of H. In particular, this result applies to the set of all factors of a morphic word. We provide a formalized proof in the proof assistant Isabelle/HOL as part of the Combinatorics on Words Formalized project.
Let denote the family of all graph-group pairs where Γ is finite, 4-valent, connected, and G-oriented (G-half-arc-transitive). A subfamily of has recently been identified as ‘basic’ in the sense that all graphs in this family are normal covers of at least one basic member. In this paper we provide a description of such basic pairs which have at least one G-normal quotient which is isomorphic to a cycle graph. In doing so, we produce many new infinite families of examples and solve several problems posed in the recent literature on this topic. This result completes a research project aiming to provide a description of all basic pairs in .
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