Combinatorial Reciprocity Theorems Matthias Beck Based on joint - - PowerPoint PPT Presentation

Combinatorial Reciprocity Theorems

Matthias Beck San Francisco State University math.sfsu.edu/beck Based on joint work with Thomas Zaslavsky Binghamton University (SUNY)

“In mathematics you don’t understand things. You just get used to them.” John von Neumann (1903–1957)

Combinatorial Reciprocity Theorems Matthias Beck 2

The Theme

Combinatorics What is p(0)? p(−1)? p(−2)? polynomial p(k) counting function depending on k ∈ Z>0

Combinatorial Reciprocity Theorems Matthias Beck 3

The Theme

Combinatorics What is p(0)? p(−1)? p(−2)? polynomial p(k) counting function depending on k ∈ Z>0 ◮ Two-for-one charm of combinatorial reciprocity theorems ◮ “Big picture” motivation: understand/classify these polynomials

Combinatorial Reciprocity Theorems Matthias Beck 3

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

Chromatic Polynomials of Graphs

G = (V, E) — graph (without loops) Proper k-coloring of G — x ∈ {1, 2, . . . , k}V such that xi = xj if ij ∈ E χG(k) := # (proper k-colorings of G) Example:

✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏

• Combinatorial Reciprocity Theorems

Matthias Beck 4

Chromatic Polynomials of Graphs

G = (V, E) — graph (without loops) Proper k-coloring of G — x ∈ {1, 2, . . . , k}V such that xi = xj if ij ∈ E χG(k) := # (proper k-colorings of G) Example:

✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏

• χK3(k) = k · · ·

Combinatorial Reciprocity Theorems Matthias Beck 4

Chromatic Polynomials of Graphs

G = (V, E) — graph (without loops) Proper k-coloring of G — x ∈ {1, 2, . . . , k}V such that xi = xj if ij ∈ E χG(k) := # (proper k-colorings of G) Example:

✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏

• χK3(k) = k(k − 1) · · ·

Combinatorial Reciprocity Theorems Matthias Beck 4

Chromatic Polynomials of Graphs

G = (V, E) — graph (without loops) Proper k-coloring of G — x ∈ {1, 2, . . . , k}V such that xi = xj if ij ∈ E χG(k) := # (proper k-colorings of G) Example:

✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏

• χK3(k) = k(k − 1)(k − 2)

Combinatorial Reciprocity Theorems Matthias Beck 4

Chromatic Polynomials of Graphs

✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏

• χK3(k) = k(k − 1)(k − 2)

Theorem (Birkhoff 1912, Whitney 1932) χG(k) is a polynomial in k.

Combinatorial Reciprocity Theorems Matthias Beck 5

Chromatic Polynomials of Graphs

✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏

• χK3(k) = k(k − 1)(k − 2)

Theorem (Birkhoff 1912, Whitney 1932) χG(k) is a polynomial in k. |χK3(−1)| = 6 counts the number

• f acyclic orientations of K3.

Combinatorial Reciprocity Theorems Matthias Beck 5

Chromatic Polynomials of Graphs

✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏

• χK3(k) = k(k − 1)(k − 2)

Theorem (Birkhoff 1912, Whitney 1932) χG(k) is a polynomial in k. |χK3(−1)| = 6 counts the number

• f acyclic orientations of K3.

Theorem (Stanley 1973) (−1)|V |χG(−k) equals the number of pairs (α, x) consisting of an acyclic

• rientation α of G and a compatible k-coloring x.

In particular, (−1)|V |χG(−1) equals the number

• f acyclic orientations of G.

Combinatorial Reciprocity Theorems Matthias Beck 5

If you get bored. . .

◮ Show that the coefficients of χG alternate in sign. [old news] ◮ Show that the absolute values of the coefficients form a unimodal

• sequence. [J. Huh, arXiv:1008.4749]

◮ Show that χG(4) > 0 for any planar graph G . [impressive with or without a computer] ◮ Show that χG has no real root ≥ 4. [open] ◮ Classify chromatic polynomials. [wide open]

Combinatorial Reciprocity Theorems Matthias Beck 6

Hyperplane Arrangements

H ⊂ Rd — arrangement of affine hyperplanes L(H) — all nonempty intersections of hyperplanes in H M¨

• bius function µ(F) :=

     1 if F = Rd −

• GF

µ(G)

• therwise

Characteristic polynomial pH(k) :=

• F ∈L(H)

µ(F) kdim F

✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏ ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅

• ❅

❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅

• R2
• Combinatorial Reciprocity Theorems

Matthias Beck 7

Hyperplane Arrangements

H ⊂ Rd — arrangement of affine hyperplanes L(H) — all nonempty intersections of hyperplanes in H M¨

• bius function µ(F) :=

     1 if F = Rd −

• GF

µ(F)

• therwise

Characteristic polynomial pH(k) :=

• F ∈L(H)

µ(F) kdim F

✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏ ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅

• ❅

❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅

• R2

1

• Combinatorial Reciprocity Theorems

Matthias Beck 7

Hyperplane Arrangements

H ⊂ Rd — arrangement of affine hyperplanes L(H) — all nonempty intersections of hyperplanes in H M¨

• bius function µ(F) :=

     1 if F = Rd −

• GF

µ(F)

• therwise

Characteristic polynomial pH(k) :=

• F ∈L(H)

µ(F) kdim F

✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏ ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅

• ❅

❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅

• R2

1

• −1
• Combinatorial Reciprocity Theorems

Matthias Beck 7

Hyperplane Arrangements

H ⊂ Rd — arrangement of affine hyperplanes L(H) — all nonempty intersections of hyperplanes in H M¨

• bius function µ(F) :=

     1 if F = Rd −

• GF

µ(F)

• therwise

Characteristic polynomial pH(k) :=

• F ∈L(H)

µ(F) kdim F

✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏ ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅

• ❅

❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅

• R2

1

• −1
• −1
• −1
• Combinatorial Reciprocity Theorems

Matthias Beck 7

Hyperplane Arrangements

H ⊂ Rd — arrangement of affine hyperplanes L(H) — all nonempty intersections of hyperplanes in H M¨

• bius function µ(F) :=

     1 if F = Rd −

• GF

µ(F)

• therwise

Characteristic polynomial pH(k) :=

• F ∈L(H)

µ(F) kdim F

✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏ ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅

• ❅

❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅

• R2

1

• −1
• −1
• −1
• 2

Combinatorial Reciprocity Theorems Matthias Beck 7

Hyperplane Arrangements

H ⊂ Rd — arrangement of affine hyperplanes L(H) — all nonempty intersections of hyperplanes in H M¨

• bius function µ(F) :=

     1 if F = Rd −

• GF

µ(F)

• therwise

Characteristic polynomial pH(k) :=

• F ∈L(H)

µ(F) kdim F = k2 − 3k + 2

✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏ ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅

• ❅

❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅

• R2

1

• −1
• −1
• −1
• 2

Combinatorial Reciprocity Theorems Matthias Beck 7

Hyperplane Arrangements

✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏ ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅

• ❅

❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅

• R2

1

• −1
• −1
• −1
• 2

pH(k) =

• F ∈L(H)

µ(F) kdim F = k2 − 3k + 2 Note that H divides R2 into pH(−1) = 6 regions...

Combinatorial Reciprocity Theorems Matthias Beck 8

Hyperplane Arrangements

✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏ ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅

• ❅

❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅

• R2

1

• −1
• −1
• −1
• 2

pH(k) =

• F ∈L(H)

µ(F) kdim F = k2 − 3k + 2 Note that H divides R2 into pH(−1) = 6 regions... Theorem (Zaslavsky 1975) (−1)d pH(−1) equals the number

• f regions into which a hyperplane arrangement H divides Rd.

Combinatorial Reciprocity Theorems Matthias Beck 8

If you get bored. . .

◮ Compute pH(k) for [old news]

• the Boolean arrangement H = {xj = 0 : 1 ≤ j ≤ d}
• the braid arrangement H = {xj = xk : 1 ≤ j < k ≤ d}
• an arrangement H in Rd of n hyperplanes in general position.

◮ Show that the coefficients of pH(k) alternate in sign. [old news] ◮ Show that the absolute values of the coefficients form a unimodal

• sequence. [J. Huh, arXiv:1008.4749]

◮ Classify characteristic polynomials. [wide open]

Combinatorial Reciprocity Theorems Matthias Beck 9

Ehrhart Polynomials

Lattice polytope P ⊂ Rd – convex hull of finitely points in Zd For k ∈ Z>0 let LP(k) := #

• kP ∩ Zd

Combinatorial Reciprocity Theorems Matthias Beck 10

Ehrhart Polynomials

Lattice polytope P ⊂ Rd – convex hull of finitely points in Zd For k ∈ Z>0 let LP(k) := #

• kP ∩ Zd

Example: ∆ = conv {(0, 0), (1, 0), (0, 1)} =

• (x, y) ∈ R2 : x, y ≥ 0, x + y ≤ 1
• L∆(k) = . . .

Combinatorial Reciprocity Theorems Matthias Beck 10

Ehrhart Polynomials

Lattice polytope P ⊂ Rd – convex hull of finitely points in Zd For k ∈ Z>0 let LP(k) := #

• kP ∩ Zd

Example: ∆ = conv {(0, 0), (1, 0), (0, 1)} =

• (x, y) ∈ R2 : x, y ≥ 0, x + y ≤ 1
• L∆(k) =

k+2

2

• = 1

2(k + 1)(k + 2)

Combinatorial Reciprocity Theorems Matthias Beck 10

Ehrhart Polynomials

Lattice polytope P ⊂ Rd – convex hull of finitely points in Zd For k ∈ Z>0 let LP(k) := #

• kP ∩ Zd

Example: ∆ = conv {(0, 0), (1, 0), (0, 1)} =

• (x, y) ∈ R2 : x, y ≥ 0, x + y ≤ 1
• L∆(k) =

k+2

2

• = 1

2(k + 1)(k + 2)

L∆(−k) = k−1

2

• Combinatorial Reciprocity Theorems

Matthias Beck 10

Ehrhart Polynomials

Lattice polytope P ⊂ Rd – convex hull of finitely points in Zd For k ∈ Z>0 let LP(k) := #

• kP ∩ Zd

Example: ∆ = conv {(0, 0), (1, 0), (0, 1)} =

• (x, y) ∈ R2 : x, y ≥ 0, x + y ≤ 1
• L∆(k) =

k+2

2

• = 1

2(k + 1)(k + 2)

L∆(−k) = k−1

2

• = L∆◦(k)

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

Ehrhart Polynomials

Lattice polytope P ⊂ Rd – convex hull of finitely points in Zd For k ∈ Z>0 let LP(k) := #

• kP ∩ Zd

Theorem (Ehrhart 1962) LP(k) is a polynomial in k.

Combinatorial Reciprocity Theorems Matthias Beck 11

Ehrhart Polynomials

Lattice polytope P ⊂ Rd – convex hull of finitely points in Zd For k ∈ Z>0 let LP(k) := #

• kP ∩ Zd

Theorem (Ehrhart 1962) LP(k) is a polynomial in k. Theorem (Macdonald 1971) (−1)dim PLP(−k) enumerates the interior lattice points in kP.

Combinatorial Reciprocity Theorems Matthias Beck 11

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 Rd 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

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

The Mother of All Combinatorial Reciprocity Theorems

✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇

Polyhedron P – intersection of finitely many halfspaces fP(k) :=

• F face of P

kdim F = k3 + 6k2 + 12k + 8 Note that fP(−1) = 1 . . .

Combinatorial Reciprocity Theorems Matthias Beck 14

The Mother of All Combinatorial Reciprocity Theorems

✇ ✇ ✇ ✇

Polyhedron P – intersection of finitely many halfspaces fP(k) :=

• F face of P

kdim F = k3 + 5k2 + 8k + 4 Note that fP(−1) = 0 . . .

Combinatorial Reciprocity Theorems Matthias Beck 14

The Mother of All Combinatorial Reciprocity Theorems

✇ ✇ ✇ ✇

Polyhedron P – intersection of finitely many halfspaces fP(k) :=

• F face of P

kdim F = k3 + 5k2 + 8k + 4 Note that fP(−1) = 0 . . . Theorem (Euler–Poincar´ e) For any polyhedron, fP(−1) = 0 or ±1.

Combinatorial Reciprocity Theorems Matthias Beck 14

Graph Coloring a la Ehrhart

χK2(k) = k(k − 1) ...

1 k + 1 k +

1 = x 2

x

2

K

(Blass–Sagan)

Combinatorial Reciprocity Theorems Matthias Beck 15

Graph Coloring a la Ehrhart

χK2(k) = k(k − 1) ...

1 k + 1 k +

1 = x 2

x

2

K

(Blass–Sagan) χG(k) = #

• {1, 2, . . . , k}V \ H
• ∩ ZV

Combinatorial Reciprocity Theorems Matthias Beck 15

Graph Coloring a la Ehrhart

χK2(k) = k(k − 1) ...

1 k + 1 k +

1 = x 2

x

2

K

(Blass–Sagan) χG(k) = #

• {1, 2, . . . , k}V \ H
• ∩ ZV

= #

• (k + 1) (✷◦ \ H) ∩ ZV

where ✷ is the unit cube in RV .

Combinatorial Reciprocity Theorems Matthias Beck 15

Stanley’s Theorem a la Ehrhart

χG(k) = #

• (k + 1) (✷◦ \ H) ∩ ZV

Write ✷◦ \ H =

• j

P◦

j , then by Ehrhart–Macdonald reciprocity

(−1)|V |χG(−k) =

• j

LPj(k − 1)

1 k + 1 k +

1 = x 2

x

2

K

So (−1)|V |χG(−k) counts lattice points in k ✷ with multiplicity #regions.

Combinatorial Reciprocity Theorems Matthias Beck 16

Stanley’s Theorem a la Ehrhart

✉ ✉ ✉

x1 = x2 x1 = x3 x2 = x3

✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉

3 1 2

χG(k) = #

• (k + 1) (✷◦ \ H) ∩ ZV

Write ✷◦ \ H =

• j

P◦

j , then by Ehrhart–Macdonald reciprocity

(−1)|V |χG(−k) =

• j

LPj(k − 1) Greene’s observation region of H(G) ⇐ ⇒ acyclic orientation of G xi < xj ⇐ ⇒ i − → j Stanley’s Theorem (−1)|V |χG(−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

Inside-Out Polytopes

Underlying setup of our proof of Stanley’s theorem: P — (rational) polytope in Rd H — (rational) hyperplane arrangement P◦ \ H =

• j

P◦

j

Say we’re interested in the counting function f(k) := #

• k (P◦ \ H) ∩ Zd

=

• j

LP ◦

j (k) . Combinatorial Reciprocity Theorems Matthias Beck 17

Inside-Out Polytopes

Underlying setup of our proof of Stanley’s theorem: P — (rational) polytope in Rd H — (rational) hyperplane arrangement P◦ \ H =

• j

P◦

j

Say we’re interested in the counting function f(k) := #

• k (P◦ \ H) ∩ Zd

=

• j

LP ◦

j (k) .

Ehrhart says that this is a (quasi-)polynomial, and by Ehrhart–Macdonald reciprocity, f(−k) = (−1)d

j

LPj(k) i.e., (−1)df(−k) counts lattice points in kP with multiplicity #regions.

Combinatorial Reciprocity Theorems Matthias Beck 17

Make-Your-Own Combinatorial Reciprocity Theorem

counting function inside-out function reciprocal inside-out function reciprocal counting function your world my world ? given guaranteed

Combinatorial Reciprocity Theorems Matthias Beck 18

Applications

◮ Nowhere-zero flow polynomials (M B–Zaslavsky 2006, Breuer–Dall 2011, Breuer–Sanyal 2011) ◮ Magic & Latin squares (M B–Zaslavsky 2006, M B–Van Herick 2011) ◮ Antimagic graphs (M B–Zaslavsky 2006, M B–Jackanich 201?) ◮ Nowhere-harmonic & bivariable graph colorings (M B–Braun 2011, M B–Chavin–Hardin 201?) ◮ Golomb rulers (M B–Bogart–Pham 201?) . . . with lots of open questions.

Combinatorial Reciprocity Theorems Matthias Beck 19

Golomb Rulers

Sequences of n distinct integers whose pairwise differences are distinct

Combinatorial Reciprocity Theorems Matthias Beck 20

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

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) gm(t) := #

• x ∈ Zm+1 : 0 = x0 < x1 < · · · < xm−1 < xm = t

all xj − xk distinct

• Combinatorial Reciprocity Theorems

Matthias Beck 20

Enumeration of Golomb Rulers

Goal Study/compute gm(t) := #

• x ∈ Zm+1 : 0 = x0 < x1 < · · · < xm−1 < xm = t

all xj − xk distinct

• =

#

• z ∈ Zm

>0 :

z1 + z2 + · · · + zm = t

• j∈U zj =

j∈V zj 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 gm(t) := #

• x ∈ Zm+1 : 0 = x0 < x1 < · · · < xm−1 < xm = t

all xj − xk distinct

• =

#

• z ∈ Zm

>0 :

z1 + z2 + · · · + zm = t

• j∈U zj =

j∈V zj for all dpcs U, V ⊂ [m]

• where dpcs means disjoint

proper consecutive subset.

Combinatorial Reciprocity Theorems Matthias Beck 21

Golomb Ruler Reciprocity

Real Golomb ruler — z ∈ Rm

≥0 satisfying z1 + z2 + · · · + zm = t and

• j∈U

zj =

• j∈V

zj for all dpcs U, V ⊂ [m]

Combinatorial Reciprocity Theorems Matthias Beck 22

Golomb Ruler Reciprocity

Real Golomb ruler — z ∈ Rm

≥0 satisfying z1 + z2 + · · · + zm = t and

• j∈U

zj =

• j∈V

zj for all dpcs U, V ⊂ [m] z, w ∈ Rm

≥0 are combinatorially equivalent if for any dpcs U, V ⊂ [m]

• j∈U

zj <

• j∈V

zj ⇐ ⇒

• j∈U

wj <

• j∈V

wj Golomb multiplicty of z ∈ Zm

≥0 — number of combinatorially different real

Golomb rulers in an ǫ-neighborhood of z

Combinatorial Reciprocity Theorems Matthias Beck 22

Golomb Ruler Reciprocity

Real Golomb ruler — z ∈ Rm

≥0 satisfying z1 + z2 + · · · + zm = t and

• j∈U

zj =

• j∈V

zj for all dpcs U, V ⊂ [m] z, w ∈ Rm

≥0 are combinatorially equivalent if for any dpcs U, V ⊂ [m]

• j∈U

zj <

• j∈V

zj ⇐ ⇒

• j∈U

wj <

• j∈V

wj Golomb multiplicty of z ∈ Zm

≥0 — number of combinatorially different real

Golomb rulers in an ǫ-neighborhood of z Theorem gm(t) is a quasipolynomial in t whose evaluation (−1)mgm(−t) equals the number of rulers in Zm

≥0 of length t , each counted with its

Golomb multiplicity. Furthermore, (−1)mgm(0) equals the number of combinatorially different Golomb rulers.

Combinatorial Reciprocity Theorems Matthias Beck 22

More Golomb Counting

◮ Natural correspondence to certain mixed graphs ◮ Regions of the Golomb inside-out polytope correspond to acyclic

• rientations

◮ General reciprocity theorem for mixed graphs

Combinatorial Reciprocity Theorems Matthias Beck 23

For much more. . .

math.sfsu.edu/beck/crt.html

Combinatorial Reciprocity Theorems Matthias Beck 24