Combinatorial Reciprocity Theorems
|
|
- Alberta Norton
- 5 years ago
- Views:
Transcription
1 Combinatorial Reciprocity Theorems Matthias Beck San Francisco State University math.sfsu.edu/beck Based on joint work with Thomas Zaslavsky Binghamton University (SUNY)
2 In mathematics you don t understand things. You just get used to them. John von Neumann ( ) Combinatorial Reciprocity Theorems Matthias Beck 2
3 The Theme Combinatorics counting function depending on k Z >0 What is p(0)? p( 1)? p( 2)? polynomial p(k) Combinatorial Reciprocity Theorems Matthias Beck 3
4 The Theme Combinatorics counting function depending on k Z >0 What is p(0)? p( 1)? p( 2)? polynomial p(k) Two-for-one charm of combinatorial reciprocity theorems Big picture motivation: understand/classify these polynomials Combinatorial Reciprocity Theorems Matthias Beck 3
5 Chromatic Polynomials of Graphs G = (V, E) graph (without loops) k-coloring of G mapping x {1, 2,..., k} V Combinatorial Reciprocity Theorems Matthias Beck 4
6 Chromatic Polynomials of Graphs G = (V, E) graph (without loops) Proper k-coloring of G x {1, 2,..., k} V such that x i x j if ij E χ G (k) := # (proper k-colorings of G) Example: Combinatorial Reciprocity Theorems Matthias Beck 4
7 Chromatic Polynomials of Graphs G = (V, E) graph (without loops) Proper k-coloring of G x {1, 2,..., k} V such that x i x j if ij E χ G (k) := # (proper k-colorings of G) Example: χ K3 (k) = k Combinatorial Reciprocity Theorems Matthias Beck 4
8 Chromatic Polynomials of Graphs G = (V, E) graph (without loops) Proper k-coloring of G x {1, 2,..., k} V such that x i x j if ij E χ G (k) := # (proper k-colorings of G) Example: χ K3 (k) = k(k 1) Combinatorial Reciprocity Theorems Matthias Beck 4
9 Chromatic Polynomials of Graphs G = (V, E) graph (without loops) Proper k-coloring of G x {1, 2,..., k} V such that x i x j if ij E χ G (k) := # (proper k-colorings of G) Example: χ K3 (k) = k(k 1)(k 2) Combinatorial Reciprocity Theorems Matthias Beck 4
10 Chromatic Polynomials of Graphs χ K3 (k) = k(k 1)(k 2) Theorem (Birkhoff 1912, Whitney 1932) χ G (k) is a polynomial in k. Combinatorial Reciprocity Theorems Matthias Beck 5
11 Chromatic Polynomials of Graphs χ K3 (k) = k(k 1)(k 2) Theorem (Birkhoff 1912, Whitney 1932) χ G (k) is a polynomial in k. χ K3 ( 1) = 6 counts the number of acyclic orientations of K 3. Combinatorial Reciprocity Theorems Matthias Beck 5
12 Chromatic Polynomials of Graphs χ K3 (k) = k(k 1)(k 2) Theorem (Birkhoff 1912, Whitney 1932) χ G (k) is a polynomial in k. χ K3 ( 1) = 6 counts the number of acyclic orientations of K 3. Theorem (Stanley 1973) ( 1) V χ G ( k) equals the number of pairs (α, x) consisting of an acyclic orientation α of G and a compatible k-coloring x. In particular, ( 1) V χ G ( 1) equals the number of acyclic orientations of G. Combinatorial Reciprocity Theorems Matthias Beck 5
13 If you get bored... Show that the coefficients of χ G alternate in sign. [old news] Show that the absolute values of the coefficients form a unimodal sequence. [J. Huh, arxiv: ] Show that χ G (4) > 0 for any planar graph G. [impressive with or without a computer] Show that χ G has no real root 4. [open] Classify chromatic polynomials. [wide open] Combinatorial Reciprocity Theorems Matthias Beck 6
14 Hyperplane Arrangements H R d arrangement of affine hyperplanes L(H) all nonempty intersections of hyperplanes in H Möbius function µ(f ) := 1 if F = R d µ(g) otherwise GF Characteristic polynomial p H (k) := F L(H) µ(f ) k dim F R 2 Combinatorial Reciprocity Theorems Matthias Beck 7
15 Hyperplane Arrangements H R d arrangement of affine hyperplanes L(H) all nonempty intersections of hyperplanes in H Möbius function µ(f ) := 1 if F = R d µ(f ) otherwise GF Characteristic polynomial p H (k) := F L(H) µ(f ) k dim F 1 R 2 Combinatorial Reciprocity Theorems Matthias Beck 7
16 Hyperplane Arrangements H R d arrangement of affine hyperplanes L(H) all nonempty intersections of hyperplanes in H Möbius function µ(f ) := 1 if F = R d µ(f ) otherwise GF Characteristic polynomial p H (k) := F L(H) µ(f ) k dim F 1 1 R 2 Combinatorial Reciprocity Theorems Matthias Beck 7
17 Hyperplane Arrangements H R d arrangement of affine hyperplanes L(H) all nonempty intersections of hyperplanes in H Möbius function µ(f ) := 1 if F = R d µ(f ) otherwise GF Characteristic polynomial p H (k) := F L(H) µ(f ) k dim F R 2 Combinatorial Reciprocity Theorems Matthias Beck 7
18 Hyperplane Arrangements H R d arrangement of affine hyperplanes L(H) all nonempty intersections of hyperplanes in H Möbius function µ(f ) := 1 if F = R d µ(f ) otherwise GF Characteristic polynomial p H (k) := F L(H) µ(f ) k dim F R 2 Combinatorial Reciprocity Theorems Matthias Beck 7
19 Hyperplane Arrangements H R d arrangement of affine hyperplanes L(H) all nonempty intersections of hyperplanes in H Möbius function µ(f ) := 1 if F = R d µ(f ) otherwise GF Characteristic polynomial p H (k) := F L(H) µ(f ) k dim F = k 2 3k R 2 Combinatorial Reciprocity Theorems Matthias Beck 7
20 Hyperplane Arrangements R 2 p H (k) = F L(H) µ(f ) k dim F = k 2 3k + 2 Note that H divides R 2 into p H ( 1) = 6 regions... Combinatorial Reciprocity Theorems Matthias Beck 8
21 Hyperplane Arrangements R 2 p H (k) = F L(H) µ(f ) k dim F = k 2 3k + 2 Note that H divides R 2 into p H ( 1) = 6 regions... Theorem (Zaslavsky 1975) ( 1) d p H ( 1) equals the number of regions into which a hyperplane arrangement H divides R d. Combinatorial Reciprocity Theorems Matthias Beck 8
22 If you get bored... Compute p H (k) for [old news] the Boolean arrangement H = {x j = 0 : 1 j d} the braid arrangement H = {x j = x k : 1 j < k d} an arrangement H in R d of n hyperplanes in general position. Show that the coefficients of p H (k) alternate in sign. [old news] Show that the absolute values of the coefficients form a unimodal sequence. [J. Huh, arxiv: ] Classify characteristic polynomials. [wide open] Combinatorial Reciprocity Theorems Matthias Beck 9
23 Ehrhart Polynomials Lattice polytope P R d convex hull of finitely points in Z d For k Z >0 let L P (k) := # ( kp Z d) Combinatorial Reciprocity Theorems Matthias Beck 10
24 Ehrhart Polynomials Lattice polytope P R d convex hull of finitely points in Z d For k Z >0 let L P (k) := # ( kp Z d) Example: = conv {(0, 0), (1, 0), (0, 1)} = { (x, y) R 2 : x, y 0, x + y 1 } L (k) =... Combinatorial Reciprocity Theorems Matthias Beck 10
25 Ehrhart Polynomials Lattice polytope P R d convex hull of finitely points in Z d For k Z >0 let L P (k) := # ( kp Z d) Example: = conv {(0, 0), (1, 0), (0, 1)} = { (x, y) R 2 : x, y 0, x + y 1 } L (k) = ( k+2 2 ) = 1(k + 1)(k + 2) 2 Combinatorial Reciprocity Theorems Matthias Beck 10
26 Ehrhart Polynomials Lattice polytope P R d convex hull of finitely points in Z d For k Z >0 let L P (k) := # ( kp Z d) Example: = conv {(0, 0), (1, 0), (0, 1)} = { (x, y) R 2 : x, y 0, x + y 1 } L (k) = ( k+2 2 L ( k) = ( ) k 1 2 ) = 1(k + 1)(k + 2) 2 Combinatorial Reciprocity Theorems Matthias Beck 10
27 Ehrhart Polynomials Lattice polytope P R d convex hull of finitely points in Z d For k Z >0 let L P (k) := # ( kp Z d) Example: = conv {(0, 0), (1, 0), (0, 1)} = { (x, y) R 2 : x, y 0, x + y 1 } L (k) = ( k+2 2 ) = 1(k + 1)(k + 2) L ( k) = ( ) k 1 2 = L (k) 2 For example, the evaluations L ( 1) = L ( 2) = 0 point to the fact that neither nor 2 contain any interior lattice points. Combinatorial Reciprocity Theorems Matthias Beck 10
28 Ehrhart Polynomials Lattice polytope P R d convex hull of finitely points in Z d For k Z >0 let L P (k) := # ( kp Z d) Theorem (Ehrhart 1962) L P (k) is a polynomial in k. Combinatorial Reciprocity Theorems Matthias Beck 11
29 Ehrhart Polynomials Lattice polytope P R d convex hull of finitely points in Z d For k Z >0 let L P (k) := # ( kp Z d) Theorem (Ehrhart 1962) L P (k) is a polynomial in k. Theorem (Macdonald 1971) ( 1) dim P L P ( k) enumerates the interior lattice points in kp. Combinatorial Reciprocity Theorems Matthias Beck 11
30 If you get bored... Show how the previous page for d = 2 follows from Pick s Theorem. Compute the Ehrhart polynomial of your favorite lattice polytope. Here are two of my favorites: the cross polytope, the convex hull of the unit vectors in R d and their negatives [old news] the Birkhoff von Neumann polytope of all doubly-stochastic n n matrices. [open for n 10] Classify Ehrhart polynomials of lattice polygons. [Scott 1976] Classify Ehrhart polynomials of lattice 3-polytopes. [open] Combinatorial Reciprocity Theorems Matthias Beck 12
31 Combinatorial Reciprocity Common theme: a combinatorial function, which is a priori defined on the positive integers, (1) can be algebraically extended beyond the positive integers (e.g., because it is a polynomial), and (2) has (possibly quite different) meaning when evaluated at negative integers. Combinatorial Reciprocity Theorems Matthias Beck 13
32 The Mother of All Combinatorial Reciprocity Theorems Polyhedron P intersection of finitely many halfspaces f P (k) := F face of P k dim F = k 3 + 6k k + 8 Note that f P ( 1) = 1... Combinatorial Reciprocity Theorems Matthias Beck 14
33 The Mother of All Combinatorial Reciprocity Theorems Polyhedron P intersection of finitely many halfspaces f P (k) := k dim F = k 3 + 5k 2 + 8k + 4 F face of P Note that f P ( 1) = 0... Combinatorial Reciprocity Theorems Matthias Beck 14
34 The Mother of All Combinatorial Reciprocity Theorems Polyhedron P intersection of finitely many halfspaces f P (k) := k dim F = k 3 + 5k 2 + 8k + 4 F face of P Note that f P ( 1) = 0... Theorem (Euler Poincaré) For any polyhedron, f P ( 1) = 0 or ±1. Combinatorial Reciprocity Theorems Matthias Beck 14
35 Graph Coloring a la Ehrhart χ K2 (k) = k(k 1)... k + 1 K 2 k + 1 x 1 = x 2 (Blass Sagan) Combinatorial Reciprocity Theorems Matthias Beck 15
36 Graph Coloring a la Ehrhart χ K2 (k) = k(k 1)... k + 1 K 2 k + 1 x 1 = x 2 (Blass Sagan) ( ( χ G (k) = # {1, 2,..., k} V \ H ) ) Z V Combinatorial Reciprocity Theorems Matthias Beck 15
37 Graph Coloring a la Ehrhart χ K2 (k) = k(k 1)... k + 1 K 2 k + 1 x 1 = x 2 (Blass Sagan) ( ( χ G (k) = # {1, 2,..., k} V \ H ) ) Z V = # ( (k + 1) ( \ H) Z V ) where is the unit cube in R V. Combinatorial Reciprocity Theorems Matthias Beck 15
38 Stanley s Theorem a la Ehrhart χ G (k) = # ( (k + 1) ( \ H) Z V ) Write \ H = j P j, then by Ehrhart Macdonald reciprocity ( 1) V χ G ( k) = j L Pj (k 1) k + 1 K 2 k + 1 x 1 = x 2 So ( 1) V χ G ( k) counts lattice points in k with multiplicity #regions. Combinatorial Reciprocity Theorems Matthias Beck 16
39 Stanley s Theorem a la Ehrhart χ G (k) = # ( (k + 1) ( \ H) Z V ) Write \ H = j P j, then by Ehrhart Macdonald reciprocity ( 1) V χ G ( k) = j Greene s observation L Pj (k 1) x 2 = x 3 region of H(G) acyclic orientation of G x i < x j i j x 1 = x 3 x 1 = x 2 Stanley s Theorem ( 1) V χ G ( k) equals the number of pairs (α, x) consisting of an acyclic orientation α of G and a compatible k-coloring x. Combinatorial Reciprocity Theorems Matthias Beck 16
40 Inside-Out Polytopes Underlying setup of our proof of Stanley s theorem: P (rational) polytope in R d H (rational) hyperplane arrangement P \ H = j P j Say we re interested in the counting function f(k) := # ( k (P \ H) Z d) = j L P j (k). Combinatorial Reciprocity Theorems Matthias Beck 17
41 Inside-Out Polytopes Underlying setup of our proof of Stanley s theorem: P (rational) polytope in R d H (rational) hyperplane arrangement P \ H = j P j Say we re interested in the counting function f(k) := # ( k (P \ H) Z d) = j L P j (k). Ehrhart says that this is a (quasi-)polynomial, and by Ehrhart Macdonald reciprocity, f( k) = ( 1) d L Pj (k) j i.e., ( 1) d f( k) counts lattice points in kp with multiplicity #regions. Combinatorial Reciprocity Theorems Matthias Beck 17
42 Make-Your-Own Combinatorial Reciprocity Theorem counting function given inside-out function guaranteed reciprocal counting function? reciprocal inside-out function your world my world Combinatorial Reciprocity Theorems Matthias Beck 18
43 Applications Nowhere-zero flow polynomials (MB Zaslavsky 2006, Breuer Dall 2011, Breuer Sanyal 2011) Magic & Latin squares (MB Zaslavsky 2006, MB Van Herick 2011) Antimagic graphs (MB Zaslavsky 2006, MB Jackanich 201?) Nowhere-harmonic & bivariable graph colorings (MB Braun 2011, MB Chavin Hardin 201?) Golomb rulers (MB Bogart Pham 201?)... with lots of open questions. Combinatorial Reciprocity Theorems Matthias Beck 19
44 Golomb Rulers Sequences of n distinct integers whose pairwise differences are distinct Combinatorial Reciprocity Theorems Matthias Beck 20
45 Golomb Rulers Sequences of n distinct integers whose pairwise differences are distinct Natural applications to error-correcting codes and phased array radio antennas Classical studies in additive number theory, more recent studies on existence problems (e.g., optimal Golomb rulers) Combinatorial Reciprocity Theorems Matthias Beck 20
46 Golomb Rulers Sequences of n distinct integers whose pairwise differences are distinct Natural applications to error-correcting codes and phased array radio antennas Classical studies in additive number theory, more recent studies on existence problems (e.g., optimal Golomb rulers) g m (t) := # { x Z m+1 : 0 = x 0 < x 1 < < x m 1 < x m = t all x j x k distinct } Combinatorial Reciprocity Theorems Matthias Beck 20
47 Enumeration of Golomb Rulers Goal Study/compute g m (t) := # = # { x Z m+1 : { z Z m >0 : 0 = x 0 < x 1 < < x m 1 < x m = t all x j x k distinct z 1 + z z m = t j U z j j V z j for all dpcs U, V [m] } } where dpcs means disjoint proper consecutive subset. Combinatorial Reciprocity Theorems Matthias Beck 21
48 Enumeration of Golomb Rulers Goal Study/compute { g m (t) := # x Z m+1 : = # { z Z m >0 : 0 = x 0 < x 1 < < x m 1 < x m = t all x j x k distinct z 1 + z z m = t j U z j j V z j for all dpcs U, V [m] } } where dpcs means disjoint proper consecutive subset. Combinatorial Reciprocity Theorems Matthias Beck 21
49 Golomb Ruler Reciprocity Real Golomb ruler z R m 0 satisfying z 1 + z z m = t and j U z j j V z j for all dpcs U, V [m] Combinatorial Reciprocity Theorems Matthias Beck 22
50 Golomb Ruler Reciprocity Real Golomb ruler z R m 0 satisfying z 1 + z z m = t and j U z j j V z j for all dpcs U, V [m] z, w R m 0 are combinatorially equivalent if for any dpcs U, V [m] z j < j V z j w j j U j U w j < j V Golomb multiplicty of z Z m 0 number of combinatorially different real Golomb rulers in an ɛ-neighborhood of z Combinatorial Reciprocity Theorems Matthias Beck 22
51 Golomb Ruler Reciprocity Real Golomb ruler z R m 0 satisfying z 1 + z z m = t and j U z j j V z j for all dpcs U, V [m] z, w R m 0 are combinatorially equivalent if for any dpcs U, V [m] z j < j V z j w j j U j U w j < j V Golomb multiplicty of z Z m 0 number of combinatorially different real Golomb rulers in an ɛ-neighborhood of z Theorem g m (t) is a quasipolynomial in t whose evaluation ( 1) m g m ( t) equals the number of rulers in Z m 0 of length t, each counted with its Golomb multiplicity. Furthermore, ( 1) m g m (0) equals the number of combinatorially different Golomb rulers. Combinatorial Reciprocity Theorems Matthias Beck 22
52 More Golomb Counting Natural correspondence to certain mixed graphs Regions of the Golomb inside-out polytope correspond to acyclic orientations General reciprocity theorem for mixed graphs Combinatorial Reciprocity Theorems Matthias Beck 23
53 For much more... math.sfsu.edu/beck/crt.html Combinatorial Reciprocity Theorems Matthias Beck 24
Combinatorial Reciprocity Theorems
Combinatorial Reciprocity Theorems Matthias Beck San Francisco State University math.sfsu.edu/beck based on joint work with Raman Sanyal Universität Frankfurt JCCA 2018 Sendai Thomas Zaslavsky Binghamton
More informationand Other Combinatorial Reciprocity Instances
and Other Combinatorial Reciprocity Instances Matthias Beck San Francisco State University math.sfsu.edu/beck [Courtney Gibbons] Act 1: Binomial Coefficients Not everything that can be counted counts,
More information10 Years BADGeometry: Progress and Open Problems in Ehrhart Theory
10 Years BADGeometry: Progress and Open Problems in Ehrhart Theory Matthias Beck (San Francisco State University) math.sfsu.edu/beck Thanks To... 10 Years BADGeometry: Progress and Open Problems in Ehrhart
More informationEnumerating integer points in polytopes: applications to number theory. Matthias Beck San Francisco State University math.sfsu.
Enumerating integer points in polytopes: applications to number theory Matthias Beck San Francisco State University math.sfsu.edu/beck It takes a village to count integer points. Alexander Barvinok Outline
More informationThe Arithmetic of Graph Polynomials. Maryam Farahmand. A dissertation submitted in partial satisfaction of the. requirements for the degree of
The Arithmetic of Graph Polynomials by Maryam Farahmand A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division
More informationComputing the continuous discretely: The magic quest for a volume
Computing the continuous discretely: The magic quest for a volume Matthias Beck San Francisco State University math.sfsu.edu/beck Joint work with... Dennis Pixton (Birkhoff volume) Ricardo Diaz and Sinai
More informationMAXIMAL PERIODS OF (EHRHART) QUASI-POLYNOMIALS
MAXIMAL PERIODS OF (EHRHART QUASI-POLYNOMIALS MATTHIAS BECK, STEVEN V. SAM, AND KEVIN M. WOODS Abstract. A quasi-polynomial is a function defined of the form q(k = c d (k k d + c d 1 (k k d 1 + + c 0(k,
More informationThe partial-fractions method for counting solutions to integral linear systems
The partial-fractions method for counting solutions to integral linear systems Matthias Beck, MSRI www.msri.org/people/members/matthias/ arxiv: math.co/0309332 Vector partition functions A an (m d)-integral
More informationLattice polygons. P : lattice polygon in R 2 (vertices Z 2, no self-intersections)
Lattice polygons P : lattice polygon in R 2 (vertices Z 2, no self-intersections) A, I, B A = area of P I = # interior points of P (= 4) B = #boundary points of P (= 10) Pick s theorem Georg Alexander
More informationA thesis presented to the faculty of San Francisco State University In partial fulfilment of The Requirements for The Degree
ON THE POLYHEDRAL GEOMETRY OF t DESIGNS A thesis presented to the faculty of San Francisco State University In partial fulfilment of The Requirements for The Degree Master of Arts In Mathematics by Steven
More informationThe Hopf monoid of generalized permutahedra. SIAM Discrete Mathematics Meeting Austin, TX, June 2010
The Hopf monoid of generalized permutahedra Marcelo Aguiar Texas A+M University Federico Ardila San Francisco State University SIAM Discrete Mathematics Meeting Austin, TX, June 2010 The plan. 1. Species.
More informationENUMERATING COLORINGS, TENSIONS AND FLOWS IN CELL COMPLEXES
ENUMERATING COLORINGS, TENSIONS AND FLOWS IN CELL COMPLEXES MATTHIAS BECK, FELIX BREUER, LOGAN GODKIN, AND JEREMY L. MARTIN Abstract. We study quasipolynomials enumerating proper colorings, nowherezero
More informationON WEAK CHROMATIC POLYNOMIALS OF MIXED GRAPHS
ON WEAK CHROMATIC POLYNOMIALS OF MIXED GRAPHS MATTHIAS BECK, DANIEL BLADO, JOSEPH CRAWFORD, TAÏNA JEAN-LOUIS, AND MICHAEL YOUNG Abstract. A mixed graph is a graph with directed edges, called arcs, and
More informationA MINIMAL-DISTANCE CHROMATIC POLYNOMIAL FOR SIGNED GRAPHS
A MINIMAL-DISTANCE CHROMATIC POLYNOMIAL FOR SIGNED GRAPHS A thesis presented to the faculty of San Francisco State University In partial fulfilment of The Requirements for The Degree Master of Arts In
More informationEhrhart Polynomials of Zonotopes
Matthias Beck San Francisco State University Katharina Jochemko Kungliga Tekniska Högskolan Emily McCullough University of San Francisco AMS Session Ehrhart Theory an its Applications Hunter College 2017
More informationA Finite Calculus Approach to Ehrhart Polynomials. Kevin Woods, Oberlin College (joint work with Steven Sam, MIT)
A Finite Calculus Approach to Ehrhart Polynomials Kevin Woods, Oberlin College (joint work with Steven Sam, MIT) Ehrhart Theory Let P R d be a rational polytope L P (t) = #tp Z d Ehrhart s Theorem: L p
More informationTop Ehrhart coefficients of integer partition problems
Top Ehrhart coefficients of integer partition problems Jesús A. De Loera Department of Mathematics University of California, Davis Joint Math Meetings San Diego January 2013 Goal: Count the solutions
More informationSix Little Squares and How Their Numbers Grow
3 47 6 3 Journal of Integer Sequences, Vol. 3 (00), Article 0.3.8 Six Little Squares and How Their Numbers Grow Matthias Beck Department of Mathematics San Francisco State University 600 Holloway Avenue
More informationOutline. Some Reflection Group Numerology. Root Systems and Reflection Groups. Example: Symmetries of a triangle. Paul Renteln
Outline 1 California State University San Bernardino and Caltech 2 Queen Mary University of London June 13, 2014 3 Root Systems and Reflection Groups Example: Symmetries of a triangle V an n dimensional
More informationASPHERIC ORIENTATIONS OF SIMPLICIAL COMPLEXES
ASPHERIC ORIENTATIONS OF SIMPLICIAL COMPLEXES A thesis presented to the faculty of San Francisco State University In partial fulfillment of The requirements for The degree Master of Arts In Mathematics
More informationBIVARIATE ORDER POLYNOMIALS. Sandra Zuniga Ruiz
BIVARIATE ORDER POLYNOMIALS Sandra Zuniga Ruiz Contents 1 Introduction 3 2 Background 4 2.1 Graph Theory................................... 4 2.1.1 Basic Properties.............................. 4 2.1.2
More informationHodge theory for combinatorial geometries
Hodge theory for combinatorial geometries June Huh with Karim Adiprasito and Eric Katz June Huh 1 / 48 Three fundamental ideas: June Huh 2 / 48 Three fundamental ideas: The idea of Bernd Sturmfels that
More informationCombining the cycle index and the Tutte polynomial?
Combining the cycle index and the Tutte polynomial? Peter J. Cameron University of St Andrews Combinatorics Seminar University of Vienna 23 March 2017 Selections Students often meet the following table
More informationNon-Attacking Chess Pieces: The Bishop
Non-Attacking Chess Pieces: The Bishop Thomas Zaslavsky Binghamton University (State University of New York) C R Rao Advanced Institute of Mathematics, Statistics and Computer Science 9 July 010 Joint
More informationHigher Spin Alternating Sign Matrices
Higher Spin Alternating Sign Matrices Roger E. Behrend and Vincent A. Knight School of Mathematics, Cardiff University, Cardiff, CF24 4AG, UK behrendr@cardiff.ac.uk, knightva@cardiff.ac.uk Submitted: Aug
More informationarxiv: v2 [math.co] 2 Dec 2015
SCHEDULING PROBLEMS FELIX BREUER AND CAROLINE J. KLIVANS arxiv:1401.2978v2 [math.co] 2 Dec 2015 Abstract. We introduce the notion of a scheduling problem which is a boolean function S over atomic formulas
More informationThe Ehrhart polynomial of the Birkhoff polytope 1
The Ehrhart polynomial of the Birkhoff polytope Matthias Beck and Dennis Pixton All means (even continuous sanctify the discrete end. Doron Zeilberger 2 Abstract: The n th Birkhoff polytope is the set
More informationReciprocal domains and Cohen Macaulay d-complexes in R d
Reciprocal domains and Cohen Macaulay d-complexes in R d Ezra Miller and Victor Reiner School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA ezra@math.umn.edu, reiner@math.umn.edu
More informationThe degree of lattice polytopes
The degree of lattice polytopes Benjamin Nill - FU Berlin Graduiertenkolleg MDS - December 10, 2007 Lattice polytopes having h -polynomials with given degree and linear coefficient. arxiv:0705.1082, to
More informationThe Triangle Closure is a Polyhedron
The Triangle Closure is a Polyhedron Amitabh Basu Robert Hildebrand Matthias Köppe January 8, 23 Abstract Recently, cutting planes derived from maximal lattice-free convex sets have been studied intensively
More informationFlows on Simplicial Complexes
FPSAC 2012, Nagoya, Japan DMTCS proc. AR, 2012, 827 836 Flows on Simplicial Complexes Matthias Beck 1 and Yvonne Kemper 2 1 Department of Mathematics, San Francisco State University, San Francisco, CA
More informationGeneralized Ehrhart polynomials
FPSAC 2010, San Francisco, USA DMTCS proc. (subm.), by the authors, 1 8 Generalized Ehrhart polynomials Sheng Chen 1 and Nan Li 2 and Steven V Sam 2 1 Department of Mathematics, Harbin Institute of Technology,
More informationMAGIC COUNTING WITH INSIDE-OUT POLYTOPES. Louis Ng
MAGIC COUNTING WITH INSIDE-OUT POLYTOPES Louis Ng Version of May 13, 018 Contents 1 Introduction 3 Background 5 3 Methodology 6 4 Strong 4 4 Pandiagonal Magic Squares 11 4.1 Structure.....................................
More informationTORIC WEAK FANO VARIETIES ASSOCIATED TO BUILDING SETS
TORIC WEAK FANO VARIETIES ASSOCIATED TO BUILDING SETS YUSUKE SUYAMA Abstract. We give a necessary and sufficient condition for the nonsingular projective toric variety associated to a building set to be
More informationarxiv: v1 [math.co] 10 Aug 2016
POLYTOPES OF STOCHASTIC TENSORS HAIXIA CHANG 1, VEHBI E. PAKSOY 2 AND FUZHEN ZHANG 2 arxiv:1608.03203v1 [math.co] 10 Aug 2016 Abstract. Considering n n n stochastic tensors (a ijk ) (i.e., nonnegative
More informationFinding parking when not commuting. Jon McCammond U.C. Santa Barbara
Finding parking when not commuting PSfrag replacements 7 8 1 2 6 3 5 {{1, 4, 5}, {2, 3}, {6, 8}, {7}} 4 Jon McCammond U.C. Santa Barbara 1 A common structure The goal of this talk will be to introduce
More informationOrder of Operations. Real numbers
Order of Operations When simplifying algebraic expressions we use the following order: 1. Perform operations within a parenthesis. 2. Evaluate exponents. 3. Multiply and divide from left to right. 4. Add
More informationInteger points enumerator of hypergraphic polytopes
arxiv:1812.09770v1 [math.co] 23 Dec 2018 Integer points enumerator of hypergraphic polytopes Marko Pešović Faculty of Civil Engineering, University of Belgrade mpesovic@grf.bg.ac.rs Mathematics Subject
More informationDiscrete Geometry. Problem 1. Austin Mohr. April 26, 2012
Discrete Geometry Austin Mohr April 26, 2012 Problem 1 Theorem 1 (Linear Programming Duality). Suppose x, y, b, c R n and A R n n, Ax b, x 0, A T y c, and y 0. If x maximizes c T x and y minimizes b T
More informationBoolean Product Polynomials and the Resonance Arrangement
Boolean Product Polynomials and the Resonance Arrangement Sara Billey University of Washington Based on joint work with: Lou Billera and Vasu Tewari FPSAC July 17, 2018 Outline Symmetric Polynomials Schur
More informationOn Weak Chromatic Polynomials of Mixed Graphs
On Weak Chromatic Polynomials of Mixed Graphs Daniel Blado California Institute of Technology Joseph Crawford Morehouse College July 27, 2012 Taïna Jean-Louis Amherst College Abstract Modeling of metabolic
More informationReal-rooted h -polynomials
Real-rooted h -polynomials Katharina Jochemko TU Wien Einstein Workshop on Lattice Polytopes, December 15, 2016 Unimodality and real-rootedness Let a 0,..., a d 0 be real numbers. Unimodality a 0 a i a
More informationRESEARCH ARTICLE. An extension of the polytope of doubly stochastic matrices
Linear and Multilinear Algebra Vol. 00, No. 00, Month 200x, 1 15 RESEARCH ARTICLE An extension of the polytope of doubly stochastic matrices Richard A. Brualdi a and Geir Dahl b a Department of Mathematics,
More informationHamiltonian Tournaments and Gorenstein Rings
Europ. J. Combinatorics (2002) 23, 463 470 doi:10.1006/eujc.2002.0572 Available online at http://www.idealibrary.com on Hamiltonian Tournaments and Gorenstein Rings HIDEFUMI OHSUGI AND TAKAYUKI HIBI Let
More informationSolving the MWT. Recall the ILP for the MWT. We can obtain a solution to the MWT problem by solving the following ILP:
Solving the MWT Recall the ILP for the MWT. We can obtain a solution to the MWT problem by solving the following ILP: max subject to e i E ω i x i e i C E x i {0, 1} x i C E 1 for all critical mixed cycles
More informationFinding parking when not commuting. Jon McCammond U.C. Santa Barbara
Finding parking when not commuting 7 6 8 1 2 3 5 4 {{1,4,5}, {2,3}, {6,8}, {7}} Jon McCammond U.C. Santa Barbara 1 A common structure The main goal of this talk will be to introduce you to a mathematical
More informationAn Introduction to Hyperplane Arrangements. Richard P. Stanley
An Introduction to Hyperplane Arrangements Richard P. Stanley Contents An Introduction to Hyperplane Arrangements 1 Lecture 1. Basic definitions, the intersection poset and the characteristic polynomial
More informationPatterns of Counting: From One to Zero and to Infinity
Patterns of Counting: From One to Zero and to Infinity Beifang Chen Department of Mathematics Hong Kong University of Science and Technology Clear Water Bay, Kowloon, Hong Kong mabfchen@ust.hk, International
More informationA notion of Total Dual Integrality for Convex, Semidefinite and Extended Formulations
A notion of for Convex, Semidefinite and Extended Formulations Marcel de Carli Silva Levent Tunçel April 26, 2018 A vector in R n is integral if each of its components is an integer, A vector in R n is
More informationarxiv:math/ v1 [math.co] 2 Jan 2004
arxiv:math/0401006v1 [math.co] 2 Jan 2004 GEOMETRICALLY CONSTRUCTED BASES FOR HOMOLOGY OF PARTITION LATTICES OF TYPES A, B AND D ANDERS BJÖRNER1 AND MICHELLE WACHS 2 Dedicated to Richard Stanley on the
More informationEhrhart Positivity. Federico Castillo. December 15, University of California, Davis. Joint work with Fu Liu. Ehrhart Positivity
University of California, Davis Joint work with Fu Liu December 15, 2016 Lattice points of a polytope A (convex) polytope is a bounded solution set of a finite system of linear inequalities, or is the
More informationSurprising Examples of Manifolds in Toric Topology!
Surprising Examples of Manifolds in Toric Topology! Djordje Baralić Mathematical Institute SASA Belgrade Serbia Lazar Milenković Union University Faculty of Computer Science Belgrade Serbia arxiv:17005932v1
More informationThe Triangle Closure is a Polyhedron
The Triangle Closure is a Polyhedron Amitabh Basu Robert Hildebrand Matthias Köppe November 7, 21 Abstract Recently, cutting planes derived from maximal lattice-free convex sets have been studied intensively
More informationClassification of Ehrhart polynomials of integral simplices
FPSAC 2012, Nagoya, Japan DMTCS proc. AR, 2012, 591 598 Classification of Ehrhart polynomials of integral simplices Akihiro Higashitani Department of Pure and Applied Mathematics, Graduate School of Information
More informationAlgebraic Methods in Combinatorics
Algebraic Methods in Combinatorics Po-Shen Loh 27 June 2008 1 Warm-up 1. (A result of Bourbaki on finite geometries, from Răzvan) Let X be a finite set, and let F be a family of distinct proper subsets
More informationChapter 9: Relations Relations
Chapter 9: Relations 9.1 - Relations Definition 1 (Relation). Let A and B be sets. A binary relation from A to B is a subset R A B, i.e., R is a set of ordered pairs where the first element from each pair
More informationarxiv:math/ v1 [math.co] 3 Sep 2000
arxiv:math/0009026v1 [math.co] 3 Sep 2000 Max Min Representation of Piecewise Linear Functions Sergei Ovchinnikov Mathematics Department San Francisco State University San Francisco, CA 94132 sergei@sfsu.edu
More informationAlgebraic and Geometric ideas in the theory of Discrete Optimization
Algebraic and Geometric ideas in the theory of Discrete Optimization Jesús A. De Loera, UC Davis Three Lectures based on the book: Algebraic & Geometric Ideas in the Theory of Discrete Optimization (SIAM-MOS
More informationACYCLIC ORIENTATIONS AND CHROMATIC GENERATING FUNCTIONS. Ira M. Gessel 1
ACYCLIC ORIENTATIONS AND CHROMATIC GENERATING FUNCTIONS Ira M. Gessel 1 Department of Mathematics Brandeis University Waltham, MA 02454-9110 gessel@brandeis.edu www.cs.brandeis.edu/ ~ ira June 2, 1999
More informationA Survey of Parking Functions
A Survey of Parking Functions Richard P. Stanley M.I.T. Parking functions... n 2 1... a a a 1 2 n Car C i prefers space a i. If a i is occupied, then C i takes the next available space. We call (a 1,...,a
More informationCOUNTING INTEGER POINTS IN POLYHEDRA. Alexander Barvinok
COUNTING INTEGER POINTS IN POLYHEDRA Alexander Barvinok Papers are available at http://www.math.lsa.umich.edu/ barvinok/papers.html Let P R d be a polytope. We want to compute (exactly or approximately)
More informationA quasisymmetric function generalization of the chromatic symmetric function
A quasisymmetric function generalization of the chromatic symmetric function Brandon Humpert University of Kansas Lawrence, KS bhumpert@math.ku.edu Submitted: May 5, 2010; Accepted: Feb 3, 2011; Published:
More informationSinks in Acyclic Orientations of Graphs
Sinks in Acyclic Orientations of Graphs David D. Gebhard Department of Mathematics, Wisconsin Lutheran College, 8800 W. Bluemound Rd., Milwaukee, WI 53226 and Bruce E. Sagan Department of Mathematics,
More informationThe Complexity of Computing the Sign of the Tutte Polynomial
The Complexity of Computing the Sign of the Tutte Polynomial Leslie Ann Goldberg (based on joint work with Mark Jerrum) Oxford Algorithms Workshop, October 2012 The Tutte polynomial of a graph G = (V,
More informationThe cycle polynomial of a permutation group
The cycle polynomial of a permutation group Peter J. Cameron School of Mathematics and Statistics University of St Andrews North Haugh St Andrews, Fife, U.K. pjc0@st-andrews.ac.uk Jason Semeraro Department
More informationCharacterization of Fixed Points in Sequential Dynamical Systems
Characterization of Fixed Points in Sequential Dynamical Systems James M. W. Duvall Virginia Polytechnic Institute and State University Department of Mathematics Abstract Graph dynamical systems are central
More informationh -polynomials of dilated lattice polytopes
h -polynomials of dilated lattice polytopes Katharina Jochemko KTH Stockholm Einstein Workshop Discrete Geometry and Topology, March 13, 2018 Lattice polytopes A set P R d is a lattice polytope if there
More informationarxiv: v1 [math.co] 30 Sep 2017
LAPLACIAN SIMPLICES ASSOCIATED TO DIGRAPHS GABRIELE BALLETTI, TAKAYUKI HIBI, MARIE MEYER, AND AKIYOSHI TSUCHIYA arxiv:70.005v [math.co] 0 Sep 07 Abstract. We associate to a finite digraph D a lattice polytope
More informationAlgebraic Methods in Combinatorics
Algebraic Methods in Combinatorics Po-Shen Loh June 2009 1 Linear independence These problems both appeared in a course of Benny Sudakov at Princeton, but the links to Olympiad problems are due to Yufei
More informationA Geometric Approach to Graph Isomorphism
A Geometric Approach to Graph Isomorphism Pawan Aurora and Shashank K Mehta Indian Institute of Technology, Kanpur - 208016, India {paurora,skmehta}@cse.iitk.ac.in Abstract. We present an integer linear
More informationThe problematic art of counting
The problematic art of counting Ragni Piene SMC Stockholm November 16, 2011 Mikael Passare A (very) short history of counting: tokens and so on... Partitions Let n be a positive integer. In how many ways
More informationLinear Algebra Review: Linear Independence. IE418 Integer Programming. Linear Algebra Review: Subspaces. Linear Algebra Review: Affine Independence
Linear Algebra Review: Linear Independence IE418: Integer Programming Department of Industrial and Systems Engineering Lehigh University 21st March 2005 A finite collection of vectors x 1,..., x k R n
More informationarxiv: v3 [math.co] 1 Oct 2018
NON-SPANNING LATTICE 3-POLYTOPES arxiv:7.07603v3 [math.co] Oct 208 Abstract. We completely classify non-spanning 3-polytopes, by which we mean lattice 3-polytopes whose lattice points do not affinely span
More informationPARTITIONS WITH FIXED DIFFERENCES BETWEEN LARGEST AND SMALLEST PARTS
PARTITIONS WITH FIXED DIFFERENCES BETWEEN LARGEST AND SMALLEST PARTS GEORGE E. ANDREWS, MATTHIAS BECK, AND NEVILLE ROBBINS Abstract. We study the number p(n, t) of partitions of n with difference t between
More informationA q-queens PROBLEM IV. ATTACKING CONFIGURATIONS AND THEIR DENOMINATORS
A q-queens PROBLEM IV. ATTACKING CONFIGURATIONS AND THEIR DENOMINATORS SETH CHAIKEN, CHRISTOPHER R. H. HANUSA, AND THOMAS ZASLAVSKY Abstract. In Parts I III we showed that the number of ways to place q
More informationA Characterization of (3+1)-Free Posets
Journal of Combinatorial Theory, Series A 93, 231241 (2001) doi:10.1006jcta.2000.3075, available online at http:www.idealibrary.com on A Characterization of (3+1)-Free Posets Mark Skandera Department of
More informationPreliminaries and Complexity Theory
Preliminaries and Complexity Theory Oleksandr Romanko CAS 746 - Advanced Topics in Combinatorial Optimization McMaster University, January 16, 2006 Introduction Book structure: 2 Part I Linear Algebra
More informationBinomial Exercises A = 1 1 and 1
Lecture I. Toric ideals. Exhibit a point configuration A whose affine semigroup NA does not consist of the intersection of the lattice ZA spanned by the columns of A with the real cone generated by A.
More informationEhrhart polynome: how to compute the highest degree coefficients and the knapsack problem.
Ehrhart polynome: how to compute the highest degree coefficients and the knapsack problem. Velleda Baldoni Università di Roma Tor Vergata Optimization, Moment Problems and Geometry I, IMS at NUS, Singapore-
More informationOrdering Events in Minkowski Space
Ordering Events in Minkowski Space Richard P. Stanley 1 Department of Mathematics Massachusetts Institute of Technology Cambridge, MA 02139 e-mail: rstan@math.mit.edu version of 26 May 2005 Abstract Let
More informationPARTITIONS WITH FIXED DIFFERENCES BETWEEN LARGEST AND SMALLEST PARTS
PARTITIONS WITH FIXED DIFFERENCES BETWEEN LARGEST AND SMALLEST PARTS GEORGE E. ANDREWS, MATTHIAS BECK, AND NEVILLE ROBBINS Abstract. We study the number p(n, t) of partitions of n with difference t between
More informationApplications of the Lopsided Lovász Local Lemma Regarding Hypergraphs
Regarding Hypergraphs Ph.D. Dissertation Defense April 15, 2013 Overview The Local Lemmata 2-Coloring Hypergraphs with the Original Local Lemma Counting Derangements with the Lopsided Local Lemma Lopsided
More informationPolytopes and Algebraic Geometry. Jesús A. De Loera University of California, Davis
Polytopes and Algebraic Geometry Jesús A. De Loera University of California, Davis Outline of the talk 1. Four classic results relating polytopes and algebraic geometry: (A) Toric Geometry (B) Viro s Theorem
More informationA Course in Combinatorics
A Course in Combinatorics J. H. van Lint Technical Universüy of Eindhoven and R. M. Wilson California Institute of Technology H CAMBRIDGE UNIVERSITY PRESS CONTENTS Preface xi 1. Graphs 1 Terminology of
More informationRational Catalan Combinatorics: Intro
Rational Catalan Combinatorics: Intro Vic Reiner Univ. of Minnesota reiner@math.umn.edu AIM workshop Dec. 17-21, 2012 Goals of the workshop 1 Reinforce existing connections and forge new connections between
More informationZero sum games Proving the vn theorem. Zero sum games. Roberto Lucchetti. Politecnico di Milano
Politecnico di Milano General form Definition A two player zero sum game in strategic form is the triplet (X, Y, f : X Y R) f (x, y) is what Pl1 gets from Pl2, when they play x, y respectively. Thus g
More informationInteger Programming ISE 418. Lecture 12. Dr. Ted Ralphs
Integer Programming ISE 418 Lecture 12 Dr. Ted Ralphs ISE 418 Lecture 12 1 Reading for This Lecture Nemhauser and Wolsey Sections II.2.1 Wolsey Chapter 9 ISE 418 Lecture 12 2 Generating Stronger Valid
More informationThe Lopsided Lovász Local Lemma
Joint work with Linyuan Lu and László Székely Georgia Southern University April 27, 2013 The lopsided Lovász local lemma can establish the existence of objects satisfying several weakly correlated conditions
More informationPARKING FUNCTIONS. Richard P. Stanley Department of Mathematics M.I.T Cambridge, MA
PARKING FUNCTIONS Richard P. Stanley Department of Mathematics M.I.T. -75 Cambridge, MA 09 rstan@math.mit.edu http://www-math.mit.edu/~rstan Transparencies available at: http://www-math.mit.edu/~rstan/trans.html
More informationCOEFFICIENTS AND ROOTS OF EHRHART POLYNOMIALS
COEFFICIENTS AND ROOTS OF EHRHART POLYNOMIALS M. BECK, J. A. DE LOERA, M. DEVELIN, J. PFEIFLE, AND R. P. STANLEY Abstract. The Ehrhart polynomial of a convex lattice polytope counts integer points in integral
More informationMAT-INF4110/MAT-INF9110 Mathematical optimization
MAT-INF4110/MAT-INF9110 Mathematical optimization Geir Dahl August 20, 2013 Convexity Part IV Chapter 4 Representation of convex sets different representations of convex sets, boundary polyhedra and polytopes:
More informationTRISTRAM BOGART AND REKHA R. THOMAS
SMALL CHVÁTAL RANK TRISTRAM BOGART AND REKHA R. THOMAS Abstract. We introduce a new measure of complexity of integer hulls of rational polyhedra called the small Chvátal rank (SCR). The SCR of an integer
More informationCOUNTING INTEGER POINTS IN POLYTOPES ASSOCIATED WITH DIRECTED GRAPHS. Ilse Fischer
COUNTING INTEGER POINTS IN POLYTOPES ASSOCIATED WITH DIRECTED GRAPHS Ilse Fischer Fakultät für Mathematik, Universität Wien Oskar-Morgenstern-Platz 1, 1090 Wien, Austria ilse.fischer@univie.ac.at Tel:
More informationInteger Programming Methods LNMB
Integer Programming Methods LNMB 2017 2018 Dion Gijswijt homepage.tudelft.nl/64a8q/intpm/ Dion Gijswijt Intro IntPM 2017-2018 1 / 24 Organisation Webpage: homepage.tudelft.nl/64a8q/intpm/ Book: Integer
More informationarxiv: v1 [math.ac] 6 Jan 2019
GORENSTEIN T-SPREAD VERONESE ALGEBRAS RODICA DINU arxiv:1901.01561v1 [math.ac] 6 Jan 2019 Abstract. In this paper we characterize the Gorenstein t-spread Veronese algebras. Introduction Let K be a field
More informationOn the cells in a stationary Poisson hyperplane mosaic
On the cells in a stationary Poisson hyperplane mosaic Matthias Reitzner and Rolf Schneider Abstract Let X be the mosaic generated by a stationary Poisson hyperplane process X in R d. Under some mild conditions
More informationIncreasing spanning forests in graphs and simplicial complexes
Increasing spanning forests in graphs and simplicial complexes Joshua Hallam Jeremy L. Martin Bruce E. Sagan October 11, 016 Key Words: chromatic polynomial, graph, increasing forest, perfect elimination
More informationAlgebraic and geometric structures in combinatorics
Algebraic and geometric structures in combinatorics Federico Ardila San Francisco State University Mathematical Sciences Research Institute Universidad de Los Andes AMS Invited Address Joint Math Meetings
More informationAcyclic orientations of graphs
Discrete Mathematics 306 2006) 905 909 www.elsevier.com/locate/disc Acyclic orientations of graphs Richard P. Stanley Department of Mathematics, University of California, Berkeley, Calif. 94720, USA Abstract
More informationThe Lopsided Lovász Local Lemma
Department of Mathematics Nebraska Wesleyan University With Linyuan Lu and László Székely, University of South Carolina Note on Probability Spaces For this talk, every a probability space Ω is assumed
More information