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Cardiff University

1. Izanloo, Hassan. The volume and ehrhart polynomial of the alternating sign matrix polytope.

Degree: PhD, 2019, Cardiff University

URL: http://orca.cf.ac.uk/125150/

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.

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APA (6^{th} Edition):

Izanloo, H. (2019). The volume and ehrhart polynomial of the alternating sign matrix polytope. (Doctoral Dissertation). Cardiff University. Retrieved from http://orca.cf.ac.uk/125150/

Chicago Manual of Style (16^{th} Edition):

Izanloo, Hassan. “The volume and ehrhart polynomial of the alternating sign matrix polytope.” 2019. Doctoral Dissertation, Cardiff University. Accessed September 19, 2019. http://orca.cf.ac.uk/125150/.

MLA Handbook (7^{th} Edition):

Izanloo, Hassan. “The volume and ehrhart polynomial of the alternating sign matrix polytope.” 2019. Web. 19 Sep 2019.

Vancouver:

Izanloo H. The volume and ehrhart polynomial of the alternating sign matrix polytope. [Internet] [Doctoral dissertation]. Cardiff University; 2019. [cited 2019 Sep 19]. Available from: http://orca.cf.ac.uk/125150/.

Council of Science Editors:

Izanloo H. The volume and ehrhart polynomial of the alternating sign matrix polytope. [Doctoral Dissertation]. Cardiff University; 2019. Available from: http://orca.cf.ac.uk/125150/