25 years of the Electronic Journal of Combinatorics I
Org: Richard A. Brualdi, Bruce Sagan, Maya Stein et David Wood
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MATTHIAS BECK, San Francisco State University, United States
Classification of combinatorial polynomials (in particular, Ehrhart polynomials of zonotopes)  [PDF]

The Ehrhart polynomial of a lattice polytope $P$ encodes fundamental arithmetic data of $P$, namely, the number of integer lattice points in positive integral dilates of $P$. Mirroring Herb Wilf's much-cherished and still-wide-open question which polynomials are chromatic polynomials?, we give a brief survey of attempts during the last half century to classify Ehrhart polynomials. This classification problem is related to that of a whole family of polynomials in combinatorics. We will also present some new results for Ehrhart polynomials of zonotopes, i.e., projections of (higher dimensional) cubes, based on joint work with Katharina Jochemko and Emily McCullough.

RONALD GRAHAM, University of California, San Diego, United States
A few of my favorite combinatorial problems  [PDF]

In this talk I will discuss a few of my favorite combinatorial problems.

IAN WANLESS, Monash University, Australia
Generalised transversals of Latin squares  [PDF]

A $k$-plex in a Latin square is a selection of entries which has exactly $k$ representatives from each row, column and symbol. The $1$-plexes are transversals and have been studied since Euler. In E-JC Vol.9, I proved the first theorems for general $k$-plexes, and conjectured that for all even orders $n>4$ there is a Latin square that has $3$-plexes but no transversal.

Much more recently, in joint work with Nick Cavenagh published in E-JC Vol.24, we proved this conjecture and that there are very many Latin squares without transversals. I will discuss these two papers and the intervening history.

CATHERINE YAN, Texas A&M University, United States
Vector parking functions with periodic boundaries and rational parking functions  [PDF]

Vector parking functions are sequences of non-negative integers whose order statistics are bounded by a given boundary $(a_0, a_1, ..., a_n)$. We combine the theory of fractional power series, the Newton-Puiseux Theorem, and the theory of Goncarov polynomials to study the enumeration of vector parking functions with a periodic boundary. As an application, we obtained an explicit formula for the exponential generating function of rational parking functions, for which the boundary is linear with a rational slope. This is a joint work with Yue Cai.