Related papers: Multigraph Hyperplane Arrangements and Parking Fun…

Church-Ellenberg-Farb used the language of FI-modules to prove that the cohomology of certain sequences of hyperplane arrangements with S_n-actions satisfies representation stability. Here we lift their results to the level of the…

Let p(k) denote the partition function of k. For each k >= 2, we describe a list of p(k)-1 quasirandom properties that a k-uniform hypergraph can have. Our work connects previous notions on linear hypergraph quasirandomness of…

In a recent paper [J. Combin. Theory Ser. B}, 113 (2015), pp. 1-17], the authors have extended the concept of quadrangulation of a surface to higher dimension, and showed that every quadrangulation of the $n$-dimensional projective space…

We propose a new representation of $k$-partite, $k$-uniform hypergraphs, that is, a hypergraph with a partition of vertices into $k$ parts such that each hyperedge contains exactly one vertex of each type; we call them $k$-hypergraphs for…

In recent years, there has been extensive work on inequalities among partition functions. In particular, Nicolas, and independently DeSalvo--Pak, proved that the partition function $p(n)$ is eventually log-concave. Inspired by this and…

The modified Macdonald polynomials introduced by Garsia and Haiman (1996) have many remarkable combinatorial properties. One such class of properties involves applying the $\nabla$ operator of Bergeron and Garsia (1999) to basic symmetric…

We prove here that the polynomial <nabla(C_p(1)), e_a h_b h_c> q, t-enumerates, by the statistics dinv and area, the parking functions whose supporting Dyck path touches the main diagonal according to the composition p of size a + b + c and…

We give a very short proof of the fact that the number of $(a,b)$-parking functions of length $n$ equals $a(a+bn)^{n-1}$. This was first proved in 2003 by Kung and Yan, via a very long and torturous route, as a corollary of a more general…

A half-square of a bipartite graph $B=(X,Y,E_B)$ has one color class of $B$ as vertex set, say $X$; two vertices are adjacent whenever they have a common neighbor in $Y$. If $G=(V,E_G)$ is the half-square of a planar bipartite graph…

We introduce the $k$-Plane Insertion into Plane drawing ($k$-PIP) problem: given a plane drawing of a planar graph $G$ and a set $F$ of edges, insert the edges in $F$ into the drawing such that the resulting drawing is $k$-plane. In this…

We present an algorithm for determining whether a bipartite graph $G$ is 2-chordal (formerly doubly chordal bipartite). At its core this algorithm is an extension of the existing efficient algorithm for determining whether a graph is…

We established a hyperplane restriction theorem for the local holomorphic mappings between projective spaces, which is inspired by the corresponding theorem of Green for homogeneous ideals in polynomial rings. Our theorem allows us to give…

We present a functorial computation of the equivariant intersection cohomology of a hypertoric variety, and endow it with a natural ring structure. When the hyperplane arrangement associated with the hypertoric variety is unimodular, we…

Chung and Graham began the systematic study of k-uniform hypergraph quasirandom properties soon after the foundational results of Thomason and Chung-Graham-Wilson on quasirandom graphs. One feature that became apparent in the early work on…

Given an arrangement of hyperplanes in $\P^n$, possibly with non-normal crossings, we give a vanishing lemma for the cohomology of the sheaf of $q$-forms with logarithmic poles along our arrangement. We give a basis for the ideal $\cal J$…

We introduce the 2-sorted counting logic $GC^k$ that expresses properties of hypergraphs. This logic has available k variables to address hyperedges, an unbounded number of variables to address vertices, and atomic formulas E(e,v) to…

Let $k$ and $l$ be integers, both at least 2. A $(k,l)$-bipartite graph is an $l$-regular bipartite multigraph with coloured bipartite sets of size $k$. Define $\chi(k,l)$ and $\mu(k,l)$ to be the minimum and maximum order of automorphism…

In this paper, we view parking functions viewed as labeled Dyck paths in order to study a notion of pattern avoidance first introduced by Remmel and Qiu. In particular we enumerate the parking functions avoiding any set of two or more…

In this paper we present new results on the enumeration of parking functions and labeled forests. We introduce new statistics on parking functions, which are then extended to labeled forests via bijective correspondences. We determine the…

We introduce a statistic on overpartitions called the $\overline{k}$-rank. When there are no overlined parts, this coincides with the $k$-rank of a partition introduced by Garvan. Moreover, it reduces to the D-rank of an overpartition when…