Combinatorial Reciprocity Theorems

Matthias Beck Based on joint work with San Francisco State University Thomas Zaslavsky math.sfsu.edu/beck 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 counting function depending on k ∈ Z>0

What is p(0)? polynomial p(k) p(−1)? p(−2)?

Combinatorial Reciprocity Theorems Matthias Beck 3 The Theme

Combinatorics counting function depending on k ∈ Z>0

What is p(0)? polynomial p(k) p(−1)? p(−2)?

I Two-for-one charm of combinatorial reciprocity theorems

I “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)

V Proper k-coloring of G — x ∈ {1, 2, . . . , k} such that xi 6= xj if ij ∈ E

χG(k) := # (proper k-colorings of G)

Example:

• JJ J J J J J • J•

Combinatorial Reciprocity Theorems Matthias Beck 4 Chromatic Polynomials of Graphs

G = (V,E) — graph (without loops)

V Proper k-coloring of G — x ∈ {1, 2, . . . , k} such that xi 6= xj if ij ∈ E

χG(k) := # (proper k-colorings of G)

Example:

• JJ J J χK (k) = k ··· J 3 J J • J•

Combinatorial Reciprocity Theorems Matthias Beck 4 Chromatic Polynomials of Graphs

G = (V,E) — graph (without loops)

V Proper k-coloring of G — x ∈ {1, 2, . . . , k} such that xi 6= xj if ij ∈ E

χG(k) := # (proper k-colorings of G)

Example:

• JJ J J χK (k) = k(k − 1) ··· J 3 J J • J•

Combinatorial Reciprocity Theorems Matthias Beck 4 Chromatic Polynomials of Graphs

G = (V,E) — graph (without loops)

V Proper k-coloring of G — x ∈ {1, 2, . . . , k} such that xi 6= xj if ij ∈ E

χG(k) := # (proper k-colorings of G)

Example:

• JJ J J χK (k) = k(k − 1)(k − 2) J 3 J J • J•

Combinatorial Reciprocity Theorems Matthias Beck 4 Chromatic Polynomials of Graphs • JJ J J χK (k) = k(k − 1)(k − 2) J 3 J J • J•

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

Combinatorial Reciprocity Theorems Matthias Beck 5 Chromatic Polynomials of Graphs • JJ J J χK (k) = k(k − 1)(k − 2) J 3 J J • J•

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

|χK3(−1)| = 6 counts the number of acyclic orientations of K3.

Combinatorial Reciprocity Theorems Matthias Beck 5 Chromatic Polynomials of Graphs • JJ J J χK (k) = k(k − 1)(k − 2) J 3 J J • J•

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

|χK3(−1)| = 6 counts the number of acyclic orientations of K3.

|V | Theorem (Stanley 1973) (−1) χG(−k) equals the number of pairs (α, x) consisting of an acyclic orientation α of G and a compatible k-coloring x. |V | In particular, (−1) χG(−1) equals the number of acyclic orientations of G.

Combinatorial Reciprocity Theorems Matthias Beck 5 If you get bored. . .

I Show that the coeﬃcients of χG alternate in sign. [old news]

I Show that the absolute values of the coeﬃcients form a unimodal sequence. [J. Huh, arXiv:1008.4749]

I Show that χG(4) > 0 for any planar graph G . [impressive with or without a computer]

I Show that χG has no real root ≥ 4. [open]

I 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

 d 1 if F = R X M¨obiusfunction µ(F ) := − µ(G) otherwise   G)F

X dim F Characteristic polynomial pH(k) := µ(F ) k F ∈L(H) • @ @ @ @ @ @ @ @ @ @ @ hh @ hhh @ hhhh@ ••• h@hh @ • hhhh @ @ hhh hhhh @ @ @ @ @ @ @ @ @ @• 2 @ R

Combinatorial Reciprocity Theorems Matthias Beck 7 Hyperplane Arrangements

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

 d 1 if F = R X M¨obiusfunction µ(F ) := − µ(F ) otherwise   G)F

X dim F Characteristic polynomial pH(k) := µ(F ) k F ∈L(H) • @ @ @ @ @ @ @ @ @ @ @ hh @ hhh @ hhhh@ ••• h@hh @ • hhhh @ @ hhh hhhh @ @ @ @ @ @ @ @ @ @• 2 @ 1 R

Combinatorial Reciprocity Theorems Matthias Beck 7 Hyperplane Arrangements

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

 d 1 if F = R X M¨obiusfunction µ(F ) := − µ(F ) otherwise   G)F

X dim F Characteristic polynomial pH(k) := µ(F ) k F ∈L(H) • @ @ @ @ @ @ @ @ @ @ @ hh @ hhh @ hhhh@ • •• h@hh @−1 • hhhh @ @ hhh hhhh @ @ @ @ @ @ @ @ @ @• 2 @ 1 R

Combinatorial Reciprocity Theorems Matthias Beck 7 Hyperplane Arrangements

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

 d 1 if F = R X M¨obiusfunction µ(F ) := − µ(F ) otherwise   G)F

X dim F Characteristic polynomial pH(k) := µ(F ) k F ∈L(H) • @ @ @ @ @ @ @ @ @ @ @ hh @ hhh @ hhhh@ • • • h@hh @−1 −1 −1 • hhhh @ @ hhh hhhh @ @ @ @ @ @ @ @ @ @• 2 @ 1 R

Combinatorial Reciprocity Theorems Matthias Beck 7 Hyperplane Arrangements

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

 d 1 if F = R X M¨obiusfunction µ(F ) := − µ(F ) otherwise   G)F

X dim F Characteristic polynomial pH(k) := µ(F ) k F ∈L(H) •2 @ @ @ @ @ @ @ @ @ @ @ hh @ hhh @ hhhh@ • • • h@hh @−1 −1 −1 • hhhh @ @ hhh hhhh @ @ @ @ @ @ @ @ @ @• 2 @ 1 R

Combinatorial Reciprocity Theorems Matthias Beck 7 Hyperplane Arrangements

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

 d 1 if F = R X M¨obiusfunction µ(F ) := − µ(F ) otherwise   G)F

X dim F 2 Characteristic polynomial pH(k) := µ(F ) k = k − 3k + 2 F ∈L(H) •2 @ @ @ @ @ @ @ @ @ @ @ hh @ hhh @ hhhh@ • • • h@hh @−1 −1 −1 • hhhh @ @ hhh hhhh @ @ @ @ @ @ @ @ @ @• 2 @ 1 R

Combinatorial Reciprocity Theorems Matthias Beck 7 Hyperplane Arrangements

•2 @ @ @ @ @ @ @ @ @ @ @ h @ hhh @ hhhh @ hhh •@−1 •−1 •−1 @• hhhh @ hhh @ hhhh @ @ h @ @ @ @ @ @ @ @• 2 @ 1 R

X dim F 2 pH(k) = µ(F ) k = k − 3k + 2 F ∈L(H)

2 Note that H divides R into pH(−1) = 6 regions...

Combinatorial Reciprocity Theorems Matthias Beck 8 Hyperplane Arrangements

•2 @ @ @ @ @ @ @ @ @ @ @ h @ hhh @ hhhh @ hhh •@−1 •−1 •−1 @• hhhh @ hhh @ hhhh @ @ h @ @ @ @ @ @ @ @• 2 @ 1 R

X dim F 2 pH(k) = µ(F ) k = k − 3k + 2 F ∈L(H)

2 Note that H divides R into pH(−1) = 6 regions...

d Theorem (Zaslavsky 1975) (−1) pH(−1) equals the number of regions into which a hyperplane arrangement H divides Rd.

Combinatorial Reciprocity Theorems Matthias Beck 8 If you get bored. . .

I 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.

I Show that the coeﬃcients of pH(k) alternate in sign. [old news]

I Show that the absolute values of the coeﬃcients form a unimodal sequence. [J. Huh, arXiv:1008.4749]

I Classify characteristic polynomials. [wide open]

Combinatorial Reciprocity Theorems Matthias Beck 9 Ehrhart Polynomials

Lattice polytope P ⊂ Rd – convex hull of ﬁnitely points in Zd

d For k ∈ Z>0 let LP(k) := # kP ∩ Z

Combinatorial Reciprocity Theorems Matthias Beck 10 Ehrhart Polynomials

Lattice polytope P ⊂ Rd – convex hull of ﬁnitely points in Zd

d For k ∈ Z>0 let LP(k) := # kP ∩ Z

Example:

∆ = conv {(0, 0), (1, 0), (0, 1)}

2 = (x, y) ∈ R : 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

d For k ∈ Z>0 let LP(k) := # kP ∩ Z

Example:

∆ = conv {(0, 0), (1, 0), (0, 1)}

2 = (x, y) ∈ R : x, y ≥ 0, x + y ≤ 1

k+2 1 L∆(k) = 2 = 2(k + 1)(k + 2)

Combinatorial Reciprocity Theorems Matthias Beck 10 Ehrhart Polynomials

Lattice polytope P ⊂ Rd – convex hull of ﬁnitely points in Zd

d For k ∈ Z>0 let LP(k) := # kP ∩ Z

Example:

∆ = conv {(0, 0), (1, 0), (0, 1)}

2 = (x, y) ∈ R : x, y ≥ 0, x + y ≤ 1

k+2 1 L∆(k) = 2 = 2(k + 1)(k + 2)

k−1 L∆(−k) = 2

Combinatorial Reciprocity Theorems Matthias Beck 10 Ehrhart Polynomials

Lattice polytope P ⊂ Rd – convex hull of ﬁnitely points in Zd

d For k ∈ Z>0 let LP(k) := # kP ∩ Z

Example:

∆ = conv {(0, 0), (1, 0), (0, 1)}

2 = (x, y) ∈ R : x, y ≥ 0, x + y ≤ 1

k+2 1 L∆(k) = 2 = 2(k + 1)(k + 2)

k−1 ◦ L∆(−k) = 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

d For k ∈ Z>0 let LP(k) := # kP ∩ Z

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

d For k ∈ Z>0 let LP(k) := # kP ∩ Z

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

dim P Theorem (Macdonald 1971) (−1) LP(−k) enumerates the interior lattice points in kP.

Combinatorial Reciprocity Theorems Matthias Beck 11 If you get bored. . .

I Show how the previous page for d = 2 follows from Pick’s Theorem.

I 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]

I Classify Ehrhart polynomials of lattice polygons. [Scott 1976]

I 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

X dim F 3 2 fP(k) := k = k + 6k + 12k + 8 w w

F face of P ww

Note that fP(−1) = 1 ... w w

w w

Combinatorial Reciprocity Theorems Matthias Beck 14 The Mother of All Combinatorial Reciprocity Theorems

Polyhedron P – intersection of ﬁnitely many halfspaces

X dim F 3 2 fP(k) := k = k + 5k + 8k + 4

F face of P ww

Note that fP(−1) = 0 ...

w w

Combinatorial Reciprocity Theorems Matthias Beck 14 The Mother of All Combinatorial Reciprocity Theorems

Polyhedron P – intersection of ﬁnitely many halfspaces

X dim F 3 2 fP(k) := k = k + 5k + 8k + 4

F face of P ww

Note that fP(−1) = 0 ...

Theorem (Euler–Poincar´e)For any polyhedron, w w fP(−1) = 0 or ±1.

Combinatorial Reciprocity Theorems Matthias Beck 14 Graph Coloring a la Ehrhart

χK2(k) = k(k − 1) ... k + 1

K2 k + 1

x1 = x 2 (Blass–Sagan)

Combinatorial Reciprocity Theorems Matthias Beck 15 Graph Coloring a la Ehrhart

χK (k) = k(k − 1) ... 2 k + 1

K2 k + 1

x1 = x 2 (Blass–Sagan)

V V χG(k) = # {1, 2, . . . , k} \H ∩ Z

Combinatorial Reciprocity Theorems Matthias Beck 15 Graph Coloring a la Ehrhart

χK2(k) = k(k − 1) ... k + 1

K2 k + 1

x1 = x 2 (Blass–Sagan)

V V χG(k) = # {1, 2, . . . , k} \H ∩ Z ◦ V = # (k + 1) (2 \H) ∩ Z where 2 is the unit cube in RV .

Combinatorial Reciprocity Theorems Matthias Beck 15 Stanley’s Theorem a la Ehrhart

◦ V χG(k) = # (k + 1) (2 \H) ∩ Z

◦ [ ◦ Write 2 \H = Pj , then by Ehrhart–Macdonald reciprocity j X (−1)|V |χ (−k) = L (k − 1) G Pj j

k + 1

K2 k + 1

x1 = x 2

|V | So (−1) χG(−k) counts lattice points in k 2 with multiplicity #regions.

Combinatorial Reciprocity Theorems Matthias Beck 16 Stanley’s Theorem a la Ehrhart

◦ V χG(k) = # (k + 1) (2 \H) ∩ Z

◦ [ ◦ Write 2 \H = Pj , then by Ehrhart–Macdonald reciprocity j X (−1)|V |χ (−k) = L (k − 1) G Pj 3

j u x2 = x3 u Greene’s observation 1 2uu uu u region of H(G) ⇐⇒ acyclic orientation of G uu u x = x xi < xj ⇐⇒ i −→ j 1 3 uu u u uu

uu x1 = x2

|V | Stanley’s Theorem (−1) 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 ◦ [ ◦ P \H = Pj H — (rational) hyperplane arrangement j

Say we’re interested in the counting function

◦ d X f(k) := # k (P \H) ∩ = L ◦(k) . Z Pj j

Combinatorial Reciprocity Theorems Matthias Beck 17 Inside-Out Polytopes

Underlying setup of our proof of Stanley’s theorem:

P — (rational) polytope in Rd ◦ [ ◦ P \H = Pj H — (rational) hyperplane arrangement j

Say we’re interested in the counting function

◦ d X f(k) := # k (P \H) ∩ = L ◦(k) . Z Pj j Ehrhart says that this is a (quasi-)polynomial, and by Ehrhart–Macdonald reciprocity, X f(−k) = (−1)d L (k) Pj j 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

given counting function inside-out function

guaranteed

reciprocal reciprocal counting function inside-out function

your world my world

Combinatorial Reciprocity Theorems Matthias Beck 18 Applications

I Nowhere-zero ﬂow polynomials (MB–Zaslavsky 2006, Breuer–Dall 2011, Breuer–Sanyal 2011)

I Magic & Latin squares (MB–Zaslavsky2006, MB–Van Herick 2011)

I Antimagic graphs (MB–Zaslavsky2006, MB–Jackanich 201?)

I Nowhere-harmonic & bivariable graph colorings (MB–Braun2011, MB–Chavin–Hardin 201?)

I Golomb rulers (MB–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

I Natural applications to error-correcting codes and phased array radio antennas

I 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

I Natural applications to error-correcting codes and phased array radio antennas

I Classical studies in additive number theory, more recent studies on existence problems (e.g., optimal Golomb rulers) m+1 0 = x0 < x1 < ··· < xm−1 < xm = t gm(t) := # x ∈ Z : all xj − xk distinct

Combinatorial Reciprocity Theorems Matthias Beck 20 Enumeration of Golomb Rulers

Goal Study/compute

m+1 0 = x0 < x1 < ··· < xm−1 < xm = t gm(t) := # x ∈ Z : all xj − xk distinct m z1 + z2 + ··· + zm = t = # z ∈ Z>0 : P P j∈U zj 6= 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 m+1 0 = x0 < x1 < ··· < xm−1 < xm = t gm(t) := # x ∈ Z : all xj − xk distinct m z1 + z2 + ··· + zm = t = # z ∈ Z>0 : P P j∈U zj 6= 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

m Real Golomb ruler— z ∈ R≥0 satisfying z1 + z2 + ··· + zm = t and

X X zj 6= zj for all dpcs U, V ⊂ [m] j∈U j∈V

Combinatorial Reciprocity Theorems Matthias Beck 22 Golomb Ruler Reciprocity

m Real Golomb ruler— z ∈ R≥0 satisfying z1 + z2 + ··· + zm = t and

X X zj 6= zj for all dpcs U, V ⊂ [m] j∈U j∈V

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

X X X X zj < zj ⇐⇒ wj < wj j∈U j∈V j∈U j∈V

m Golomb multiplicty of z ∈ Z≥0 — number of combinatorially diﬀerent real Golomb rulers in an -neighborhood of z

Combinatorial Reciprocity Theorems Matthias Beck 22 Golomb Ruler Reciprocity

m Real Golomb ruler— z ∈ R≥0 satisfying z1 + z2 + ··· + zm = t and

X X zj 6= zj for all dpcs U, V ⊂ [m] j∈U j∈V

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

X X X X zj < zj ⇐⇒ wj < wj j∈U j∈V j∈U j∈V

m Golomb multiplicty of z ∈ Z≥0 — number of combinatorially diﬀerent real Golomb rulers in an -neighborhood of z

m Theorem gm(t) is a quasipolynomial in t whose evaluation (−1) gm(−t) m equals the number of rulers in Z≥0 of length t , each counted with its m Golomb multiplicity. Furthermore, (−1) gm(0) equals the number of combinatorially diﬀerent Golomb rulers.

Combinatorial Reciprocity Theorems Matthias Beck 22 More Golomb Counting

I Natural correspondence to certain mixed graphs

I Regions of the Golomb inside-out polytope correspond to acyclic orientations

I 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