[PPT] - Combinatorial Reciprocity Theorems Matthias Beck San Francisco PowerPoint Presentation

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Combinatorial Reciprocity Theorems

Matthias Beck San Francisco State University math.sfsu.edu/beck/

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Joint work with...

Thomas Zaslavsky, Binghamton University (SUNY) math.CO/0309330 math.CO/0309331 math.CO/0506315

Combinatorial Reciprocity Theorems Matthias Beck 2

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Chromatic polynomials of graphs

Γ = (V, E) – graph (without loops) k-coloring of Γ : mapping x : V → {1, 2, . . . , k}

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Chromatic polynomials of graphs

Γ = (V, E) – graph (without loops) k-coloring of Γ : mapping x : V → {1, 2, . . . , k} Proper k-coloring of Γ : mapping x : V → {1, 2, . . . , k} such that xi = xj if there is an edge ij

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Chromatic polynomials of graphs

Γ = (V, E) – graph (without loops) k-coloring of Γ : mapping x : V → {1, 2, . . . , k} Proper k-coloring of Γ : mapping x : V → {1, 2, . . . , k} such that xi = xj if there is an edge ij Theorem (Birkhoff 1912, Whitney 1932) χΓ(k) := # (proper k-colorings of Γ) is a monic polynomial in k of degree |V |.

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Chromatic polynomials of graphs

Γ = (V, E) – graph (without loops) k-coloring of Γ : mapping x : V → {1, 2, . . . , k} Proper k-coloring of Γ : mapping x : V → {1, 2, . . . , k} such that xi = xj if there is an edge ij Theorem (Birkhoff 1912, Whitney 1932) χΓ(k) := # (proper k-colorings of Γ) is a monic polynomial in k of degree |V |. Proof: Choose your favorite edge e of Γ and use χΓ(k) = χ(Γ\e)(k) − χ(Γ·e)(k) and induction. . .

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Stanley’s Acyclic-Orientation Theorem

Theorem (Stanley 1973) (−1)|V |χΓ(−k) equals the number of pairs (α, x) consisting of an acyclic orientation α of Γ and a compatible k-coloring. In particular, (−1)|V |χΓ(−1) equals the number of acyclic orientations of Γ. (An orientation α of Γ and a k -coloring x are compatible if xj ≥ xi whenever there is an edge oriented from i to j. An orientation is acyclic if it has no directed cycles.)

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Flow polynomials

Nowhere-zero A-flow on a graph Γ = (V, E) : mapping x : E → A \ {0} (A an abelian group) such that for every node v ∈ V

h(e)=v

x(e) =

t(e)=v

x(e) h(e) := head t(e) := tail

f the edge e in a (fixed) orientation of Γ

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Flow polynomials

Nowhere-zero A-flow on a graph Γ = (V, E) : mapping x : E → A \ {0} (A an abelian group) such that for every node v ∈ V

h(e)=v

x(e) =

t(e)=v

x(e) h(e) := head t(e) := tail

f the edge e in a (fixed) orientation of Γ

Nowhere-zero k-flow : Z-flow with values in {1, 2, . . . , k − 1}

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Flow polynomials

Nowhere-zero A-flow on a graph Γ = (V, E) : mapping x : E → A \ {0} (A an abelian group) such that for every node v ∈ V

h(e)=v

x(e) =

t(e)=v

x(e) h(e) := head t(e) := tail

f the edge e in a (fixed) orientation of Γ

Nowhere-zero k-flow : Z-flow with values in {1, 2, . . . , k − 1} Theorem (Tutte 1954) ϕΓ(|A|) := # (nowhere-zero A-flows) is a polynomial in |A|. (Kochol 2002) ϕΓ(k) := # (nowhere-zero k-flows) is a polynomial in k.

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Flow polynomials

Nowhere-zero A-flow on a graph Γ = (V, E) : mapping x : E → A \ {0} (A an abelian group) such that for every node v ∈ V

h(e)=v

x(e) =

t(e)=v

x(e) h(e) := head t(e) := tail

f the edge e in a (fixed) orientation of Γ

Nowhere-zero k-flow : Z-flow with values in {1, 2, . . . , k − 1} Theorem (Tutte 1954) ϕΓ(|A|) := # (nowhere-zero A-flows) is a polynomial in |A|. (Kochol 2002) ϕΓ(k) := # (nowhere-zero k-flows) is a polynomial in k. What about reciprocity?

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(Weak) semimagic squares

Hn(t) – number of nonnegative integral n × n-matrices in which every row and column sums to t 1 1 2 2 1 1 1 2 1

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(Weak) semimagic squares

Hn(t) – number of nonnegative integral n × n-matrices in which every row and column sums to t 1 1 2 2 1 1 1 2 1 Theorem (Ehrhart, Stanley 1973, conjectured by Anand-Dumir-Gupta 1966) Hn(t) is a polynomial in t of degree (n − 1)2.

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(Weak) semimagic squares

Hn(t) – number of nonnegative integral n × n-matrices in which every row and column sums to t 1 1 2 2 1 1 1 2 1 Theorem (Ehrhart, Stanley 1973, conjectured by Anand-Dumir-Gupta 1966) Hn(t) is a polynomial in t of degree (n − 1)2 which satisfies Hn(0) = 1, Hn(−1) = Hn(−2) = · · · = Hn(−n + 1) = 0, and Hn(−n − t) = (−1)n−1Hn(t) .

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(Weak) semimagic squares

Hn(t) – number of nonnegative integral n × n-matrices in which every row and column sums to t 1 1 2 2 1 1 1 2 1 Theorem (Ehrhart, Stanley 1973, conjectured by Anand-Dumir-Gupta 1966) Hn(t) is a polynomial in t of degree (n − 1)2 which satisfies Hn(0) = 1, Hn(−1) = Hn(−2) = · · · = Hn(−n + 1) = 0, and Hn(−n − t) = (−1)n−1Hn(t) . What about “classical” magic squares?

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Characteristic polynomials of hyperplane arrangements

H ⊂ Rd – arrangement of affine hyperplanes L(H) := S : S ⊆ H and S = ∅

, ordered by reverse inclusion

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Characteristic polynomials of hyperplane arrangements

H ⊂ Rd – arrangement of affine hyperplanes L(H) := S : S ⊆ H and S = ∅

, ordered by reverse inclusion

M¨

bius function µ(r, s) :=

         if r ≤ s, 1 if r = s, −

r≤u<s

µ(r, u) if r < s. Characteristic polynomial pH(λ) :=

s∈L(H)

µ

Rd, s
λdim s

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Characteristic polynomials of hyperplane arrangements

H ⊂ Rd – arrangement of affine hyperplanes L(H) := S : S ⊆ H and S = ∅

, ordered by reverse inclusion

M¨

bius function µ(r, s) :=

         if r ≤ s, 1 if r = s, −

r≤u<s

µ(r, u) if r < s. Characteristic polynomial pH(λ) :=

s∈L(H)

µ

Rd, s
λdim s

Theorem (Zaslavsky 1975) If Rd ∈ H then the number of regions into which a hyperplane arrangement H divides Rd is (−1)dpH(−1).

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Ehrhart (quasi-)polynomials

P ⊂ Rd – convex rational polytope For t ∈ Z>0 let EhrP(t) := #

P ∩ 1

tZd

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Ehrhart (quasi-)polynomials

P ⊂ Rd – convex rational polytope For t ∈ Z>0 let EhrP(t) := #

P ∩ 1

tZd

Theorem (Ehrhart 1962) EhrP(t) is a quasipolynomial in t of degree dim P with leading term vol P (normalized to aff P ∩ Zd) and constant term 1. (A quasipolynomial is an expression cd(t) td + · · · + c1(t) t + c0(t) where c0, . . . , cd are periodic functions in t.)

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Ehrhart (quasi-)polynomials

P ⊂ Rd – convex rational polytope For t ∈ Z>0 let EhrP(t) := #

P ∩ 1

tZd

Theorem (Ehrhart 1962) EhrP(t) is a quasipolynomial in t of degree dim P with leading term vol P (normalized to aff P ∩ Zd) and constant term 1. (Macdonald 1971) (−1)dim P EhrP(−t) enumerates the interior lattice points in tP. (A quasipolynomial is an expression cd(t) td + · · · + c1(t) t + c0(t) where c0, . . . , cd are periodic functions in t.)

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Shameless plug

M. Beck & S. Robins

Computing the continuous discretely Integer-point enumeration in polyhedra To appear (late 2006) in Springer Undergraduate Texts in Mathematics Preprint available at math.sfsu.edu/beck

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Graph coloring a la Ehrhart

χK2(k) = k(k − 1) ...

1 k + 1 k +

1 = x 2

x

2

K

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Graph coloring a la Ehrhart

χK2(k) = k(k − 1) ...

1 k + 1 k +

1 = x 2

x

2

K

χΓ(k) = #

(0, 1)V \
H(Γ)
∩

1 k + 1ZV

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Stanley’s Theorem a la Ehrhart

1 k + 1 k +

1 = x 2

x

2

K

χΓ(k) = #

(0, 1)V \ H(Γ)
∩

1 k+1ZV

Write (0, 1)V \

H(Γ) =
j

P◦

j , then by Ehrhart-Macdonald reciprocity

(−1)|V |χΓ(−k) =

j

EhrPj(k − 1)

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Stanley’s Theorem a la Ehrhart

1 k + 1 k +

1 = x 2

x

2

K

χΓ(k) = #

(0, 1)V \ H(Γ)
∩

1 k+1ZV

Write (0, 1)V \

H(Γ) =
j

P◦

j , then by Ehrhart-Macdonald reciprocity

(−1)|V |χΓ(−k) =

j

EhrPj(k − 1) Greene’s observation region of H(Γ) ⇐ ⇒ acyclic orientation of Γ xi < xj ⇐ ⇒ i − → j

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Chromatic polynomials of signed graphs

Σ – signed graph (without loops): each edge is labelled + or − Proper k-coloring of Σ : mapping x : V → {−k, −k + 1, . . . , k} such that, if edge ij has sign ǫ then xi = ǫxj

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Chromatic polynomials of signed graphs

Σ – signed graph (without loops): each edge is labelled + or − Proper k-coloring of Σ : mapping x : V → {−k, −k + 1, . . . , k} such that, if edge ij has sign ǫ then xi = ǫxj Theorem (Zaslavsky 1982) χΣ(2k + 1) := # (proper k-colorings of Σ) and χ∗

Σ(2k) := # (proper zero-free k-colorings of Σ) are monic polynomials of

degree |V |. The number of compatible pairs (α, x) consisting of an acyclic

rientation α and a k-coloring x of Σ is equal to (−1)|V |χΣ(−(2k + 1)).

The number in which x is zero-free equals (−1)|V |χ∗

Σ(−2k). In particular,

(−1)|V |χΣ(−1) equals the number of acyclic orientations of Σ.

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Signed-graph coloring a la Ehrhart

x1 = 1/2 x 2 = 1/2 x1 + x 2 = 1 x1 = x 2 x1 = 1/2 x 2 = 1/2 x1 = x 2 (1,0) (0,1) (0,0) (1,1) (1,0) (0,0) (1,1) (0,1) + x 2 = 1 x1 − + − − pmk2o

Theorem χΣ(2k + 1) and χ∗

Σ(2k) are two halves of one inside-out

quasipolynomial.

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Signed-graph coloring a la Ehrhart

x1 = 1/2 x 2 = 1/2 x1 + x 2 = 1 x1 = x 2 x1 = 1/2 x 2 = 1/2 x1 = x 2 (1,0) (0,1) (0,0) (1,1) (1,0) (0,0) (1,1) (0,1) + x 2 = 1 x1 − + − − pmk2o

Theorem χΣ(2k + 1) and χ∗

Σ(2k) are two halves of one inside-out

quasipolynomial. Open problem Is there a combinatorial interpretation of χ∗

Σ(−1)?

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Flow polynomials revisited

ϕΓ(k) := # (nowhere-zero k-flows) ϕΓ(|A|) := # (nowhere-zero A-flows) Theorem (−1)|E|−|V |+c(Γ)ϕΓ(−k) equals the number of pairs (τ, x) consisting of a totally cyclic orientation τ and a compatible (k + 1) - flow x. In particular, the constant term ϕΓ(0) equals the number of totally cyclic orientations of Γ. (An orientation of Γ is totally cyclic if every edge lies in a coherent circle, that is, where the edges are oriented in a consistent direction around the

circle. A totally cyclic orientation τ and a flow x are compatible if x ≥ 0

when it is expressed in terms of τ.)

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Flow polynomials revisited

ϕΓ(k) := # (nowhere-zero k-flows) ϕΓ(|A|) := # (nowhere-zero A-flows) Theorem (−1)|E|−|V |+c(Γ)ϕΓ(−k) equals the number of pairs (τ, x) consisting of a totally cyclic orientation τ and a compatible (k + 1) - flow x. In particular, the constant term ϕΓ(0) equals the number of totally cyclic orientations of Γ. (An orientation of Γ is totally cyclic if every edge lies in a coherent circle, that is, where the edges are oriented in a consistent direction around the

circle. A totally cyclic orientation τ and a flow x are compatible if x ≥ 0

when it is expressed in terms of τ.) Corollary ϕΓ(0) = (−1)|E|−|V |+c(Γ)ϕΓ(−1) ∃ analogous theorems for signed graphs

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Open problems

Find a formula for, or a combinatorial interpretation of, the leading coefficient of ϕΓ.

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Open problems

Find a formula for, or a combinatorial interpretation of, the leading coefficient of ϕΓ. Is there a combinatorial interpretation of ϕΓ(−k) for k ≥ 2?

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Open problems

Find a formula for, or a combinatorial interpretation of, the leading coefficient of ϕΓ. Is there a combinatorial interpretation of ϕΓ(−k) for k ≥ 2? For some graphs, both ϕΓ and ϕΓ have integral coefficients and ϕΓ is a multiple of ϕΓ. Is there a general reason for these facts?

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Inside-out counting functions

Inside-out polytope : (P, H) Multiplicity of x ∈ Rd : mP,H(x) :=

# closed regions of H in P that contain x

if x ∈ P, if x / ∈ P Closed Ehrhart quasipolynomial EP,H(t) :=

x∈1

tZd

mP,H(x) Open Ehrhart quasipolynomial E◦

P,H(t) := #

1

tZd ∩ [P \ H]

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Basic inside-out results

Theorem If (P, H) is a closed, full-dimensional, rational inside-out polytope, then EP,H(t) and E◦

P◦,H(t) are quasipolynomials in t of degree dim P with

leading term vol P , and with constant term EP,H(0) equal to the number

f regions of (P, H). Furthermore,

E◦

P◦,H(t) = (−1)dEP,H(−t).

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Basic inside-out results

Theorem If (P, H) is a closed, full-dimensional, rational inside-out polytope, then EP,H(t) and E◦

P◦,H(t) are quasipolynomials in t of degree dim P with

leading term vol P , and with constant term EP,H(0) equal to the number

f regions of (P, H). Furthermore,

E◦

P◦,H(t) = (−1)dEP,H(−t).

Theorem (P, H) is a closed, full-dimensional, rational inside-out polytope, then E◦

P,H(t) =

u∈L(H)

µ(Rd, u) EhrP∩u(t), and if H is transverse to P EP,H(t) =

u∈L(H)

|µ(Rd, u)| EhrP∩u(t). (H is transverse to P if every flat u ∈ L(H) that intersects P also intersects P ◦, and P does not lie in any of the hyperplanes of H.)

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(Strong) magic squares

Magn(t) – number of nonnegative integral n × n-matrices with distinct entries in which every row and column sums to t 4 3 8 9 5 1 2 7 6

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(Strong) magic squares

Magn(t) – number of nonnegative integral n × n-matrices with distinct entries in which every row and column sums to t 4 3 8 9 5 1 2 7 6 Corollary Magn(t) is a quasipolynomial in t of degree n − 2n − 1. Open problem Can anything be said about the period of Magn? Even in the weak case, do we ever get a polynomial?

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Enumeration of integer points with distinct entries

P ⊂ Rd – rational convex polytope, transverse to H := H[Kd]aff P – arrangement corresponding to Kd, induced on aff P

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Enumeration of integer points with distinct entries

P ⊂ Rd – rational convex polytope, transverse to H := H[Kd]aff P – arrangement corresponding to Kd, induced on aff P Theorem The number E◦

P◦,H(t) of integer points in tP◦ with distinct entries

is a quasipolynomial with constant term equal to the number of permutations

f [d] that are realizable in P .

Furthermore, (−1)dim sE◦

P◦,H(−t) =

EP,H(t) := the number of pairs (x, σ) consisting of an integer point x ∈ tP and a compatible P-realizable permutation σ of [d]. (The point x ∈ Rd and the permutation τ are compatible if xτ1 < xτ2 < · · · < xτd. τ is realizable in X if there exists a compatible x ∈ X.)

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Enumeration of integer points with distinct entries

P ⊂ Rd – rational convex polytope, transverse to H := H[Kd]aff P – arrangement corresponding to Kd, induced on aff P Theorem The number E◦

P◦,H(t) of integer points in tP◦ with distinct entries

is a quasipolynomial with constant term equal to the number of permutations

f [d] that are realizable in P .

Furthermore, (−1)dim sE◦

P◦,H(−t) =

EP,H(t) := the number of pairs (x, σ) consisting of an integer point x ∈ tP and a compatible P-realizable permutation σ of [d]. (The point x ∈ Rd and the permutation τ are compatible if xτ1 < xτ2 < · · · < xτd. τ is realizable in X if there exists a compatible x ∈ X.) Applications (strong) magic squares, rectangles, cubes, graphs, ...

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