The volume and ehrhart polynomial of the alternating sign matrix polytope

Alternating sign matrices (ASMs), polytopes and partially-ordered sets are fascinating combinatorial objects which form the main themes of this thesis. In Chapter 1, the origins and various aspects of ASMs are discussed briefly. In particular, bijections between ASMs and other objects, including monotone triangles, corner sum matrices, configurations of the six-vertex model with domain-wall boundary conditions, configurations of simple flow grids and height function matrices, are presented. The ASM lattice and ASM partially ordered set are also introduced. In Chapter 2, the ASM polytope and related polytopes, including the Birkhoff polytope, Chan-Robbins-Yuen polytope, ASM order polytope and ASM Chan-Robbins-Yuen polytope, are defined and their properties are summarised. In Chapter 3, new results for the volume and Ehrhart polynomial of the ASM polytope are obtained. In particular, by constructing an explicit bijection between higher spin ASMs and a disjoint union of sets of certain (P, ω)-partitions (where P is a subposet of the ASM poset and ω is a labeling), a formula is derived for the number of higher spin ASMs, or equivalently for the Ehrhart polynomial of the ASM polytope. The relative volume of the ASM polytope is then given by the leading term of its Ehrhart polynomial. Evaluation of the formula involves computing numbers of linear extensions of certain subposets of the ASM poset, and numbers of descents in these linear extensions. Details of this computation are presented for the cases of the ASM polytope of order 4, 5, 6 and 7. In Chapter 4, some directions for further work are outlined. A joint paper with Roger Behrend, based on Chapter 3 of the thesis, is currently in preparation for submission