Combinatorics of colored factorizations, flow polytopes and of matrices over finite fields by Alejandro Henry Morales B.M., University of Waterloo (2007) Submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2012 c Massachusetts Institute of Technology 2012. All rights reserved. Author.............................................................. Department of Mathematics May 9, 2012 Certified by. Alexander Postnikov Associate Professor Thesis Supervisor Accepted by . Michel X. Goemans Chairman, Department Committee on Graduate Theses 2 Combinatorics of colored factorizations, flow polytopes and of matrices over finite fields by Alejandro Henry Morales Submitted to the Department of Mathematics on May 9, 2012, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics Abstract In the first part of this thesis we study factorizations of the permutation (1; 2; : : : ; n) into k factors of given cycle type. Using representation theory, Jackson obtained for each k an elegant formula for counting these factorizations according to the number of cycles of each factor. For the case k = 2, Bernardi gave a bijection between these factorizations and tree-rooted maps; certain graphs embedded on surfaces with a distinguished spanning tree. This type of bijection also applies to all k and we use it to show a symmetry property of a refinement of Jackson's formula first exhibited in the case k = 2; 3 by Morales and Vassilieva. We then give applications of this symmetry property. First, we study the mixing properties of permutations obtained as a product of two uniformly random permu- tations of fixed cycle types. For instance, we give an exact formula for the proba- bility that elements 1; 2; : : : ; k are in distinct cycles of the random permutation of 1; 2; : : : ; n obtained as product of two uniformly random n-cycles. Second, we use thef symmetryg to give a short bijective proof of the number of planar trees and cacti with given vertex degree distribution calculated by Goulden and Jackson. In the second part we establish the relationship between volumes of flow polytopes associated to signed graphs and the Kostant partition function. A special case of this relationship, namely, when the graphs are signless, has been studied combinatorially by Postnikov and Stanley and by Baldoni and Vergne using residues. As a special family of flow polytopes, we study the Chan-Robbins-Yuen polytope whose volume is the product of the consecutive Catalan numbers. We introduce generalizations of this polytope and give intriguing conjectures about their volume. In the third part we consider the problem of finding the number of matrices over a finite field with a certain rank and with support that avoids a subset of the entries. These matrices are a q-analogue of permutations with restricted positions (i.e., rook placements). Extending a result of Haglund, we show that when the set of entries is a skew Young diagram, the numbers, up to a power of q 1, are polynomials with nonnegative coefficients. We apply this result to the case− when the set of entries is the Rothe diagram of a permutation. We end by giving conjectures connecting invertible matrices whose support avoids a Rothe diagram and Poincar´epolynomials of the strong Bruhat order. 3 Thesis Supervisor: Alexander Postnikov Title: Associate Professor 4 Acknowledgments I am very grateful to my advisor Alexander Postnikov for his support, guidance and generosity of sharing ideas and insights. I would also like to thank the other members of my thesis committee: Richard Stanley for several interesting discussions and suggesting beautiful problems, and Olivier Bernardi for his mentorship, fruitful discussions and collaborations. I would also like to thank David Jackson for his regular research visits to Boston and for his continuous support and encouragement during my undergraduate and doctoral degrees. I am also indebted to Gilles Schaeffer for invitations to visit Ecole´ Polytechnique in the Summer of 2008 and September 2010 to work on colored fac- torizations, and to Ekaterina Vassilieva for being a mentor and collaborator during those visits and through the years. Thanks to my collaborators: Olivier Bernardi, Rosena Du, Richard Stanley; Karola M´esz´aros; Aaron Klein, and Joel Lewis for allowing me to include our results in this thesis. I also want to express my gratitude to Federico Ardila, Mark Skandera, Sergi Elizalde, Jim Haglund, Fran¸coisBergeron, Sylvie Corteel, Daniel Freedman, Pavel Etingof, Haynes Miller, Ian Goulden, Achim Kempf, Henriette Elvang, Peter Tingley, Gregg Musiker, Sami Assaf, and Guillaume Chapuy. I would like to thank my mother, father and sister, the rest of my family, and Alexia Guuinic for their love and tender support: muchas gracias. Thanks also to Luis Serrano for his friendship and advice since my first day of undergraduate, and to Joel Lewis for sharing every day as graduate students his excitement for combi- natorics. Thanks also to friends Nick Sheridan, Martina Balagovic, Michael Hopp- mann, James Saunderson, Been Kim, Nan Li, Steven Sam, Nikola Kamburov, Lydia Kamps, Benjamin Iriarte, Ang´elicaOsorno, Natalia Duque, Carlos Mendivil, Mario Chamorro, Santiago Alfaro, Pedro Valencia, and Arturo Dominguez for enriching the last five years. This thesis is dedicated to the memory of Manuel Caro and Nicolas E. Del Castillo. 5 6 Contents 1 Introduction 15 1.1 Colored factorizations of permutations and their symmetries . 16 1.2 Flow polytopes and the Kostant partition function . 17 1.3 Counting matrices over finite fields with restricted positions . 19 2 Colored factorizations of permutations 21 2.1 Background on Jackson's formula . 21 2.1.1 Partitions, compositions and permutations . 21 2.1.2 Factorizations of a long cycle and Jackson's formula . 21 2.2 Algebraic proof Jackson's formula . 24 2.2.1 Symmetric functions and irreducible characters of Sn ..... 24 2.2.2 Generating functions for (colored) factorizations . 28 2.2.3 Changing basis from power sums to monomial sums . 28 2.2.4 Coloring some of the permutations . 31 2.3 Symmetry colored factorizations all k .................. 34 2.3.1 Background on maps . 34 2.3.2 From cacti to tree-rooted constellations . 36 2.3.3 Symmetries for tree-rooted constellations . 39 2.4 Applications of symmetry I: enumerating colored factorizations of two and three factors . 42 2.5 Applications of symmetry II: enumerating planar cacti . 47 2.5.1 Introduction . 47 2.5.2 Enumeration of planar rooted k-cacti using symmetry . 47 2.6 Applications of symmetry III: separation probabilities . 55 2.6.1 Background on separation . 55 2.6.2 How to go from separation probabilities to colored factorizations 56 2.6.3 Results on separation probabilities . 60 3 Flow polytopes and the Kostant partition function 67 3.1 Introduction . 67 3.2 Signed graphs, Kostant partition functions, and flows . 70 3.2.1 The Ehrhart function of the flow polytope (a)....... 74 FG 3.3 The vertices of the flow polytope (a)................. 74 FG 3.3.1 Vertices of (a)......................... 74 FG 7 3.3.2 Vertices of the type Cn+1 and type Dn+1 Chan-Robbins-Yuen polytope . 78 3.4 Reduction rules of the flow polytope G(a)............... 80 3.4.1 Reduction rules for signed graphsF . 80 3.5 Subdivision of flow polytopes . 81 3.5.1 Noncrossing trees . 82 3.5.2 Removing vertex i from a signed graph G ........... 83 3.5.3 Subdivision Lemma . 85 3.6 Volume of flow polytopes . 87 3.6.1 A correspondence between integer flows and simplices in a tri- angulation of H (1; 0;:::; 0; 1), where H only has negative edges . .F . .− . 87 3.6.2 A correspondence between dynamic integer flows and simplices in a triangulation of G(2; 0;:::; 0), where G is a signed graph 90 3.7 The volumes of the (signed)F Chan-Robbins-Yuen polytopes . 94 3.7.1 Chan-Robbins-Yuen polytope of type An ............ 94 3.7.2 Volumes of Chan-Robbins-Yuen polytopes of type Cn and type Dn.................................. 97 4 Counting matrices over finite fields with restricted support 99 4.1 Introduction . 99 4.2 Definitions . 102 4.3 Polynomial formula for the rank-one case matq(n; S; 1) . 103 4.4 Formula for matq(n; B; r) when B has NE Property . 105 4.5 Studying matq(n; S; r) when S is a Rothe diagram . 107 4.5.1 Properties of matq(n; S; r) when S is a Rothe diagram . 108 4.5.2 Skew-vexillary permutations . 109 4.6 Poincar´epolynomials, matq(n; Rw; n) and q-rook numbers . 112 4.6.1 matq(n; Rw; n) for skew-vexillary permutations is a Poincar´e polynomial . 113 4.6.2 Further relationships between matq(n; Rw; n) and Poincar´epoly- nomials . 115 A Computations to prove Corollary 2.4.2 119 8 List of Figures 1-1 Examples of (a) factorizations of the cycle (1; 2; 3) viewed as graphs embedded in surfaces, (b) a projection of a type A3 Chan-Robbins- Yuen polytope, a flow polytope that is a convex hull of four 3 3 permutation matrices, and (c) using those four permutation matrices× to count invertible 3 3 matrices over Fq with the entry (3; 1) forced to be zero. .× . 15 2-1 Examples of (a) a Young diagram of shape λ = 4331, (b) a skew Young diagram of shape λ/µ = 4331=211, and (c) a skew Young diagram of shape λ/µ = 5322=211 that is also a rim hook. 24 2-2 Examples, in English notation, of (a) a semistandard Young tableau (SSYT) of shape 4331 and type (2; 2; 3; 2; 2), (b) a standard Young tableau (SYT) of shape 4331, (c) a semistandard skew Young tableau of shape 5433=211 and type (2; 2; 1; 1; 2), and (d) a rim-hook tableau of shape 6553 of type (4; 6; 2; 3; 4) and height 3.