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In this paper we consider mixed volumes of combinations of hypersimplices. These numbers, called "mixed Eulerian numbers", were first considered by A. Postnikov and were shown to satisfy many properties related to Eulerian numbers, Catalan…
In this dissertation, we explore the structure of inversion graphs of permutations--a class of graphs that naturally arises by representing each permutation as a graph, where vertices correspond to entries and edges encode inversions.…
We study the combinatorial properties of vexillary signed permutations, which are signed analogues of the vexillary permutations first considered by Lascoux and Sch\"utzenberger. We give several equivalent characterizations of vexillary…
The purpose of this article is to motivate the study of invariant, and especially conformally invariant, differential pairings. Since a general theory is lacking, this work merely presents some interesting examples of these pairings,…
We introduce a guessing game, permutation Wordle, in which a guesser attempts to recover a hidden permutation in $S_n$. In each round, the guesser guesses a permutation (using information from previous rounds) and is told which entries of…
We give several families of polynomials which are related by Eulerian summation operators. They satisfy interesting combinatorial properties like being integer-valued at integral points. This involves nearby-symmetries and a recursion for…
We give an account on what is known on the subject of permutation matchings, which are bijections of a finite regular semigroup that map each element to one of its inverses. This includes partial solutions to some open questions, including…
The Eulerian numbers form a triangular array with many interesting properties. The numbers arise from various combinatorial and probabilistic interpretations, and have been studied in a variety of mathematical contexts. In this article we…
We derive explicit formulas for the matroidal mixed Eulerian numbers. We resolve a question posed by Berget, Spink, and Tseng, demonstrating that the invariant defined by matroidal mixed Eulerian numbers is precisely equivalent to Derksen's…
An equivalence relation in the symmetric group, where is a positive integer has been considered. An algorithm for calculation of the number of the equivalence classes by this relation for arbitrary integer has been described.
The paper focuses on some versions of connected dominating set problems: basic problems and multicriteria problems. A literature survey on basic problem formulations and solving approaches is presented. The basic connected dominating set…
In this work we propose a combinatorial model that generalizes the standard definition of permutation. Our model generalizes the degenerate Eulerian polynomials and numbers of Carlitz from 1979 and provides missing combinatorial proofs for…
We describe the role of algebraic extensions in the theory of commutative, unital normed algebras, with special attention to uniform algebras. We shall also compare these constructions and show how they are related to each other.
Combinatorial enumeration leads to counting generating functions presenting a wide variety of analytic types. Properties of generating functions at singularities encode valuable information regarding asymptotic counting and limit…
We survey a research program on the strong convergence of unitary and permutation representations of discrete groups. We also take the opportunity to flesh out details that have not appeared elsewhere.
In this paper we introduce mixed coloured permutation, permutations with certain coloured cycles, and study the enumerative properties of these combinatorial objects. We derive the generating function, closed forms, recursions and…
We present a few combinatorial identities which were encountered in our work on the spectral theory of quantum graphs. They establish a new connection between the theory of random matrix ensembles and combinatorics.
The object of this paper is to give a systematic treatment of excedance-type polynomials. We first give a sufficient condition for a sequence of polynomials to have alternatingly increasing property, and then we present a systematic study…
Andr\'e proved that the number of alternating permutations on $\{1, 2, \dots, n\}$ is equal to the Euler number $E_n$. A refinement of Andr\'e's result was given by Entringer, who proved that counting alternating permutations according to…
Goulden and Jackson introduced a very powerful method to study the distributions of certain consecutive patterns in permutations, words, and other combinatorial objects which is now called the cluster method. There are a number of natural…