Enumeration of minimal acyclic automata via generalized parking functions

We give an exact enumerative formula for the minimal acyclic deterministic finite automata. This formula is obtained from a bijection between a family of generalized parking functions and the transitions functions of acyclic automata

GG-Parking Functions, Acyclic Orientations and Spanning Trees

Given an undirected graph $G=(V,E)$, and a designated vertex $q\in V$, the notion of a $G$-parking function (with respect to $q$) was independently developed and studied by various authors, and has recently gained renewed attention. This notion generalizes the classical notion of a parking function associated with the complete graph. In this work, we study properties of {\em maximum} $G$-parking functions and provide a new bijection between them and the set of spanning trees of $G$ with no broken circuit. As a case study, we specialize some of our results to the graph corresponding to the discrete $n$-cube $Q_n$. We present the article in an expository self-contained form, since we found the combinatorial aspects of $G$-parking functions somewhat scattered in the literature, typically treated in conjunction with sandpile models and closely related chip-firing games.Comment: Added coauthor, extension of v2 with additional results and references. 28 pages, 2 figure

Bigraphical Arrangements

We define the bigraphical arrangement of a graph and show that the Pak-Stanley labels of its regions are the parking functions of a closely related graph, thus proving conjectures of Duval, Klivans, and Martin and of Hopkins and Perkinson. A consequence is a new proof of a bijection between labeled graphs and regions of the Shi arrangement first given by Stanley. We also give bounds on the number of regions of a bigraphical arrangement.Comment: Added Remark 19 addressing arbitrary G-parking functions; minor revision