On the Combinatorics of Valuations PDF
Preview On the Combinatorics of Valuations
Fachbereich Mathematik und Informatik der Freien Universität Berlin Dissertation On the Combinatorics of Valuations Q P ϕ(P ∪ Q) = ϕ(P) + ϕ(Q) − ϕ(P ∩ Q) presented by Katharina Victoria Jochemko Berlin 2014 Advisor and first reviewer: Professor Dr. Raman Sanyal Second reviewer: Professor Matthias Beck, Ph.D. Date of the defense: December 12, 2014 Für meine Eltern. To my parents. Summary This thesis deals with structural results for translation invariant valuations on polytopes and certain related enumeration problems together with geo- metric approaches to them. The starting point of the first part are two theorems by Richard Stanley. The first one is his famous Nonnegativity Theorem [42] stating that the Ehrhart h∗-vector of every lattice polytope has nonnegative integer entries. In [48] he further proves that the entries satisfy a monotonicity property. In Chapter 2 we consider the h∗-vector for arbitrary translation invariant valuations. Our main theorem states that monotonicity and nonnegativity of the h∗-vector are, in fact, equivalent properties and we give a simple characterization. In Chapter 3 we consider the h∗-vector of zonotopes and show that the entries of their h∗-vector form a unimodal sequence for all translation invariant val- uations that satisfy the nonnegativity condition. Thesecondpartdealswithcertainenumerationproblemsfororder preserving maps. Given a suitable pair of finite posets A ⊆ P and an order preserving mapλfromAto[n]weconsidertheproblemofenumeratingorderpreserving extensions of λ to P. In Chapter 4 we show that their number is given by a piecewise multivariate polynomial. We apply our results to counting exten- sions of graph colorings and generalize a theorem by Herzberg and Murty [21]. We further apply our results to counting monotone triangles, which are closely related to alternating sign matrices, and give a short geometric proof of a reciprocity theorem by Fischer and Riegler [17]. In Chapter 5 we consider counting order preserving maps from P to [n] up to symmetry. We show that their number is given by a polynomial in n, thus, giving an or- der theoretic generalization of Pólya’s enumeration theorem [33]. We further prove a reciprocity theorem and apply our results to counting graph colorings up to symmetry. Chapters 2 and 4 are based on joint work with Raman Sanyal. Chapter 4 appeared in [23]. Chapter 3 is part of a joint project with Matthias Beck and Emily McCullough. The content of Chapter 5 is published in [22]. vii Acknowledgements Starting in chronological order, my deepest thanks go to Günter Ziegler. It was he, who invited me to come to Berlin for my PhD at the very beginning three years ago. Thank you for your confidence and your open ear and advice at every stage of my PhD. The most important person for a PhD student is the advisor; in my case this is Raman Sanyal. Thank you for the numerous spontaneous mini-talks you gavetome, forteachingmehowtowritepapersandhowtogivetalks, andfor your encouragement throughout. I really enjoyed the various mathematical discussionswithyou. TheymademylifeatFUBerlinawonderfulexperience. I am proud to be your first PhD student. I want to say Thank You to the Berlin Mathematical School for the financial and non-material support, especially for funding my unforgettable research stay in San Francisco. Here, I also want to thank my host Matthias Beck from San Francisco State University: Thank you for your great hospitality and a productive time in the Bay Area, and for reviewing this thesis. Further, I thank the RTG Methods for Discrete Structures for funding of conferences and interesting Monday lectures. Thanks go to Elke Pose, Dorothea Kiefer and the BMS One-Stop-Office, especially to Nadja Wisniewski and Chris Seyffer for keeping bureaucracy at an easy level. I am deeply grateful to Emily McCullough and Giulia Battiston for proof- reading parts of this thesis, and to Dagmar Timmreck for teaching me xfig. I further want to thank my mentors Heike Siebert and Carsten Lange, and all my colleagues at the Villa and at FU, most of all Bruno Benedetti, Hao Chen, Francesco Grande, Dirk Frettlöh, Tobias Friedl, Bernd Gonska, Al- bert Haase, Marie-Sophie Litz, Arnau Padrol, Nevena Palic, Moritz Schmitt, Louis Theran, Monica Blanco and Christian Haase. Furthermore, I am grateful to my friends Felizitas Weidner, Thomas Schach- ix x erer, Christian Reiher, Simon Lentner, Darij Grinberg, Linus Mattauch, Olga Heismann, Ariane Papke and Vi Huynh, who were always at my side when I needed them. And last but not least, I want to thank my parents, to whom I dedicate this thesis. Thank you for your unconditional support and your strong belief in me.
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