Combinatorial Reciprocity Theorems

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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

χ K3 ( 1) = 6 counts the number of acyclic orientations of K 3.

consisting of an acyclic orientation α of G and a compatible k-coloring x.

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

Hyperplane Arrangements 2 1 1 1 1 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.

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

8k + 4 F face of P Note that f P ( 1) = 0.

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

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 2 +

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