Inside-Out Polytopes Matthias Beck, San Francisco State University Thomas Zaslavsky, Binghamton University (SUNY) math.sfsu.edu/beck/ arXiv: math.CO/0309330 & math.CO/0309331 & math.CO/0506315
Chromatic Polynomials of Graphs Γ = ( V, E ) – graph (without loops) k -coloring of Γ : mapping x : V → { 1 , 2 , . . . , k } Inside-Out Polytopes Matthias Beck 2
Chromatic Polynomials of Graphs Γ = ( V, E ) – graph (without loops) Proper k -coloring of Γ : mapping x : V → { 1 , 2 , . . . , k } such that x i � = x j if there is an edge ij ∈ E Inside-Out Polytopes Matthias Beck 2
Chromatic Polynomials of Graphs Γ = ( V, E ) – graph (without loops) Proper k -coloring of Γ : mapping x : V → { 1 , 2 , . . . , k } such that x i � = x j if there is an edge ij ∈ E Theorem (Birkhoff 1912, Whitney 1932) χ Γ ( k ) := # ( proper k -colorings of Γ) is a monic polynomial in k of degree | V | . Inside-Out Polytopes Matthias Beck 2
Chromatic Polynomials of Graphs Γ = ( V, E ) – graph (without loops) Proper k -coloring of Γ : mapping x : V → { 1 , 2 , . . . , k } such that x i � = x j if there is an edge ij ∈ E Theorem (Birkhoff 1912, Whitney 1932) χ Γ ( k ) := # ( proper k -colorings of Γ) is a monic polynomial in k of degree | V | . Theorem (Stanley 1973) ( − 1) | V | χ Γ ( − 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 | χ Γ ( − 1) equals the number of acyclic orientations of Γ . (An orientation α of Γ and a k -coloring x are compatible if x j ≥ x i whenever there is an edge oriented from i to j . An orientation is acyclic if it has no directed cycles.) Inside-Out Polytopes Matthias Beck 2
Graphical Hyperplane Arrangements We associate with Γ = ( V, E ) the hyperplane arrangement H Γ := { x i = x j : ij ∈ E } ✡ ✡❏ ❏ x 2 � � � � � • • � � � ◗ ◗ ✑ ✑ � K 2 x 1 � � � � � � � Inside-Out Polytopes Matthias Beck 3
Graphical Hyperplane Arrangements We associate with Γ = ( V, E ) the hyperplane arrangement H Γ := { x i = x j : ij ∈ E } ✡ ✡❏ ❏ x 2 � � � � � • • � � � ◗ ◗ ✑ ✑ � K 2 x 1 � � � � � � � Greene’s observation region of H Γ ← → acyclic orientation of Γ x i < x j ← → orient ij ∈ E from i to j Inside-Out Polytopes Matthias Beck 3
Ehrhart Polynomials P ⊂ R d – lattice polytope, i.e., the convex hull of finitely points in Z d P ∩ 1 � k Z d � � k P ∩ Z d � For k ∈ Z > 0 let Ehr P ( k ) := # = # Inside-Out Polytopes Matthias Beck 4
Ehrhart Polynomials P ⊂ R d – lattice polytope, i.e., the convex hull of finitely points in Z d P ∩ 1 � k Z d � � k P ∩ Z d � For k ∈ Z > 0 let Ehr P ( k ) := # = # Theorem (Ehrhart 1962) Ehr P ( k ) is a polynomial in k of degree dim P with leading term vol P (normalized to aff P ∩ Z d ) and constant term Ehr P (0) = 1 . (Macdonald 1971) ( − 1) dim P Ehr P ( − k ) = Ehr P ◦ ( k ) , where P ◦ denotes the interior of P . Inside-Out Polytopes Matthias Beck 4
Ehrhart Polynomials P ⊂ R d – lattice polytope, i.e., the convex hull of finitely points in Z d P ∩ 1 � k Z d � � k P ∩ Z d � For k ∈ Z > 0 let Ehr P ( k ) := # = # Theorem (Ehrhart 1962) Ehr P ( k ) is a polynomial in k of degree dim P with leading term vol P (normalized to aff P ∩ Z d ) and constant term Ehr P (0) = 1 . (Macdonald 1971) ( − 1) dim P Ehr P ( − k ) = Ehr P ◦ ( k ) , where P ◦ denotes the interior of P . Idea A k -coloring of Γ is an interior lattice point of ( k + 1) P , where P = [0 , 1] V . Inside-Out Polytopes Matthias Beck 4
Graph Coloring a la Ehrhart χ K 2 ( k ) = k ( k − 1) ... k + 1 K 2 k + 1 x 1 = x 2 Inside-Out Polytopes Matthias Beck 5
Graph Coloring a la Ehrhart χ K 2 ( k ) = k ( k − 1) ... k + 1 K 2 k + 1 x 1 = x 2 �� 1 � (0 , 1) V \ � � k + 1 Z V χ Γ ( k ) = # H Γ ∩ Inside-Out Polytopes Matthias Beck 5
Stanley’s Theorem a la Ehrhart k + 1 K 2 k + 1 (0 , 1) V \ � H Γ �� k +1 Z V � 1 � χ Γ ( k ) = # ∩ x 1 = x 2 Write (0 , 1) V \ � � P ◦ H Γ = j , then by Ehrhart-Macdonald reciprocity j � ( − 1) | V | χ Γ ( − k ) = Ehr P j ( k − 1) j Inside-Out Polytopes Matthias Beck 6
Stanley’s Theorem a la Ehrhart k + 1 K 2 k + 1 (0 , 1) V \ � H Γ �� k +1 Z V � 1 � χ Γ ( k ) = # ∩ x 1 = x 2 Write (0 , 1) V \ � � P ◦ H Γ = j , then by Ehrhart-Macdonald reciprocity j � ( − 1) | V | χ Γ ( − k ) = Ehr P j ( k − 1) j Greene’s observation region of H Γ ← → acyclic orientation of Γ Inside-Out Polytopes Matthias Beck 6
Inside-Out Counting Functions Inside-out polytope : ( P , H ) Multiplicity of x ∈ R d : � # closed regions of H in P that contain x if x ∈ P , m P , H ( x ) := 0 if x / ∈ P � Closed Ehrhart quasipolynomial E P, H ( k ) := m P , H ( x ) x ∈ 1 k Z d k Z d ∩ [ P \ � H ] Open Ehrhart quasipolynomial E ◦ � 1 � P , H ( k ) := # Inside-Out Polytopes Matthias Beck 7
Inside-Out Philosophy Theorem If ( P , H ) is a closed, full-dimensional, rational inside-out polytope, then E P , H ( k ) and E ◦ P ◦ , H ( k ) are quasipolynomials in k of degree dim P with leading term vol P , and with constant term E P , H (0) equal to the number of regions of ( P , H ) . Furthermore, E ◦ P ◦ , H ( k ) = ( − 1) d E P , H ( − k ) . Inside-Out Polytopes Matthias Beck 8
Inside-Out Philosophy Theorem If ( P , H ) is a closed, full-dimensional, rational inside-out polytope, then E P , H ( k ) and E ◦ P ◦ , H ( k ) are quasipolynomials in k of degree dim P with leading term vol P , and with constant term E P , H (0) equal to the number of regions of ( P , H ) . Furthermore, E ◦ P ◦ , H ( k ) = ( − 1) d E P , H ( − k ) . Philosophy If you have an enumeration problem that can be encoded as k Z d ∩ [ P ◦ \ � H ] E ◦ � 1 � P ◦ , H ( k ) = # for some inside-out polytope ( P , H ) and you have a combinatorial interpretation for the multiplicities m P , H ( x ) , then you’ll have a combinatorial reciprocity theorem for E ◦ P ◦ , H ( k ) . Inside-Out Polytopes Matthias Beck 8
Inside-Out Philosophy Theorem If ( P , H ) is a closed, full-dimensional, rational inside-out polytope, then E P , H ( k ) and E ◦ P ◦ , H ( k ) are quasipolynomials in k of degree dim P with leading term vol P , and with constant term E P , H (0) equal to the number of regions of ( P , H ) . Furthermore, E ◦ P ◦ , H ( k ) = ( − 1) d E P , H ( − k ) . Theorem ( P , H ) is a closed, full-dimensional, rational inside-out polytope, then � E ◦ µ ( R d , u ) Ehr P∩ u ( k ) , P , H ( k ) = u ∈L ( H ) and if H is transverse to P � | µ ( R d , u ) | Ehr P∩ u ( k ) . E P , H ( k ) = u ∈L ( H ) ( 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 .) Inside-Out Polytopes Matthias Beck 8
Flow Polynomials A nowhere-zero k -flow on a graph Γ = ( V, E ) is a mapping x : E → {− k + 1 , − k + 2 , . . . , − 2 , − 1 , 1 , 2 , . . . , k − 2 , k − 1 } such that for every node v ∈ V � � x ( e ) = x ( e ) h ( e )= v t ( e )= v h ( e ) := head of the edge e in a (fixed) orientation of Γ t ( e ) := tail Inside-Out Polytopes Matthias Beck 9
Flow Polynomials A nowhere-zero k -flow on a graph Γ = ( V, E ) is a mapping x : E → {− k + 1 , − k + 2 , . . . , − 2 , − 1 , 1 , 2 , . . . , k − 2 , k − 1 } such that for every node v ∈ V � � x ( e ) = x ( e ) h ( e )= v t ( e )= v h ( e ) := head of the edge e in a (fixed) orientation of Γ t ( e ) := tail Theorem (Kochol 2002) ϕ Γ ( k ) := # ( nowhere-zero k -flows ) is a polynomial in k . Inside-Out Polytopes Matthias Beck 9
Flow Polynomial Reciprocity Let C denote the subspace of R E determined by the flow-conservation equations, P := [ − 1 , 1] E ∩ C , and H the arrangement of all coordinate hyperplanes in R E . Then ϕ Γ ( k ) = E ◦ P ◦ , H ( k ) . Inside-Out Polytopes Matthias Beck 10
Flow Polynomial Reciprocity Let C denote the subspace of R E determined by the flow-conservation equations, P := [ − 1 , 1] E ∩ C , and H the arrangement of all coordinate hyperplanes in R E . Then ϕ Γ ( k ) = E ◦ P ◦ , H ( k ) . Greene–Zaslavsky’s Observation Every region of the hyperplane arrangement induced by H in C corresponds to a totally cyclic orientation. (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.) Inside-Out Polytopes Matthias Beck 10
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