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Abelian Combinatorics on Words: a Survey [PDF]

open access: yes, 2022
We survey known results and open problems in abelian combinatorics on words. Abelian combinatorics on words is the extension to the commutative setting of the classical theory of combinatorics on words. The extension is based on \emph{abelian equivalence}, which is the equivalence relation defined in the set of words by having the same Parikh vector ...
arxiv   +1 more source

The combinatorics of Jeff Remmel [PDF]

open access: yesarXiv, 2021
We give a brief overview of the life and combinatorics of Jeff Remmel, a mathematician with successful careers in both logic and combinatorics.
arxiv  

A determinacy approach to Borel combinatorics [PDF]

open access: yesJ. Amer. Math. Soc. 29 (2016), 579-600, 2013
We introduce a new method, involving infinite games and Borel determinacy, which we use to answer several well-known questions in Borel combinatorics.
arxiv   +1 more source

Enumerative and Algebraic Combinatorics in the 1960's and 1970's [PDF]

open access: yesarXiv, 2021
A partly autobiographical survey of the development of enumerative and algebraic combinatorics in the 1960's and 1970's.
arxiv  

Borel chain conditions of Borel posets [PDF]

open access: yesarXiv, 2021
We study the coarse classification of partial orderings using chain conditions in the context of descriptive combinatorics. We show that (unlike the Borel counterpart of many other combinatorics), we have a distinct hierarchy of different chain conditions, similar to the classical case.
arxiv  

Lecture notes on algebraic methods in combinatorics [PDF]

open access: yesarXiv, 2023
These are lecture notes of a course taken in Leipzig 2023, spring semester. It deals with extremal combinatorics, algebraic methods and combinatorial geometry. These are not meant to be exhaustive, and do not contain many proofs that were presented in the course.
arxiv  

Combinatorics and N-Koszul algebras [PDF]

open access: yesarXiv, 2009
The numerical Hilbert series combinatorics and the comodule Hilbert series combinatorics are introduced, and some applications are presented, including the MacMahon Master Theorem.
arxiv  

Refuting conjectures in extremal combinatorics via linear programming [PDF]

open access: yesarXiv, 2019
We apply simple linear programming methods and an LP solver to refute a number of open conjectures in extremal combinatorics.
arxiv  

Pythagorean Hyperplane Arrangements: Combinatorics of Gain Genericity [PDF]

open access: yesarXiv, 2023
We study Pythagorean hyperplane arrangements, originally defined by Zaslavsky. In this first part of a series on such arrangements, we introduce a new notion of genericity for such arrangements. Using this notion we construct an auxiliary hyperplane arrangement whose combinatorics determines the combinatorics of all possible Pythagorean arrangements ...
arxiv  

A structural approach to subset-sum problems [PDF]

open access: yesarXiv, 2008
We discuss a structural approach to subset-sum problems in additive combinatorics. The core of this approach are Freiman-type structural theorems, many of which will be presented through the paper. These results have applications in various areas, such as number theory, combinatorics and mathematical physics.
arxiv  

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