Inside-out polytopes & a tale of seven polynomials Matthias Beck, San Francisco State University Thomas Zaslavsky, Binghamton University (SUNY) math.sfsu.edu/beck/ arXiv: math.CO/0309330 & math.CO/0309331 & . . .
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 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 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
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 � � 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 3
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 � � 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 Nowhere-zero k -flow : Z -flow with values in { 1 , 2 , . . . , k − 1 } Inside-Out Polytopes Matthias Beck 3
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 � � 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 Nowhere-zero k -flow : Z -flow with values in { 1 , 2 , . . . , k − 1 } Theorem (Tutte 1954) ϕ Γ ( | A | ) := # ( nowhere-zero A -flows ) is a polynomial in | A | . (Kochol 2002) ϕ Γ ( k ) := # ( nowhere-zero k -flows ) is a polynomial in k . Inside-Out Polytopes Matthias Beck 3
(Weak) semimagic squares H n ( 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 Inside-Out Polytopes Matthias Beck 4
(Weak) semimagic squares H n ( 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) H n ( t ) is a polynomial in t of degree ( n − 1) 2 satisfying H n (0) = 1 , H n ( − 1) = H n ( − 2) = · · · = H n ( − n + 1) = 0 , and H n ( − n − t ) = ( − 1) n − 1 H n ( t ) . Inside-Out Polytopes Matthias Beck 4
(Weak) semimagic squares H n ( 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) H n ( t ) is a polynomial in t of degree ( n − 1) 2 satisfying H n (0) = 1 , H n ( − 1) = H n ( − 2) = · · · = H n ( − n + 1) = 0 , and H n ( − n − t ) = ( − 1) n − 1 H n ( t ) . What about “classical” magic squares? Inside-Out Polytopes Matthias Beck 4
Ehrhart (quasi-)polynomials P ⊂ R d – convex rational polytope P ∩ 1 � t Z d � For t ∈ Z > 0 let Ehr P ( t ) := # Inside-Out Polytopes Matthias Beck 5
Ehrhart (quasi-)polynomials P ⊂ R d – convex rational polytope P ∩ 1 � t Z d � For t ∈ Z > 0 let Ehr P ( t ) := # Theorem (Ehrhart 1962) Ehr P ( t ) is a quasipolynomial in t of degree dim P with leading term vol P (normalized to aff P ∩ Z d ) and constant term Ehr P (0) = χ ( P ) = 1 . (Macdonald 1971) ( − 1) dim P Ehr P ( − t ) enumerates the interior lattice points in t P . (A quasipolynomial is an expression c d ( t ) t d + · · · + c 1 ( t ) t + c 0 ( t ) where c 0 , . . . , c d are periodic functions in t .) Inside-Out Polytopes Matthias Beck 5
Characteristic polynomials of hyperplane arrangements H ⊂ R d – arrangement of affine hyperplanes �� S : S ⊆ H and � S � = ∅ � L ( H ) := , ordered by reverse inclusion Inside-Out Polytopes Matthias Beck 6
Characteristic polynomials of hyperplane arrangements H ⊂ R d – arrangement of affine hyperplanes �� S : S ⊆ H and � S � = ∅ � L ( H ) := , ordered by reverse inclusion 0 if r �≤ s, 1 if r = s, M¨ obius function µ ( r, s ) := � − µ ( r, u ) if r < s. r ≤ u<s Characteristic polynomial � λ dim s R d , s � � p H ( λ ) := µ s ∈L ( H ) Inside-Out Polytopes Matthias Beck 6
Characteristic polynomials of hyperplane arrangements H ⊂ R d – arrangement of affine hyperplanes �� S : S ⊆ H and � S � = ∅ � L ( H ) := , ordered by reverse inclusion if r �≤ s, 0 1 if r = s, M¨ obius function µ ( r, s ) := � − µ ( r, u ) if r < s. r ≤ u<s Characteristic polynomial � λ dim s R d , s � � p H ( λ ) := µ s ∈L ( H ) Theorem (Zaslavsky 1975) If R d �∈ H then the number of regions into which a hyperplane arrangement H divides R d is ( − 1) d p H ( − 1) . Inside-Out Polytopes Matthias Beck 6
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 7
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 7
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 8
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 x i < x j ⇐ ⇒ i − → j Inside-Out Polytopes Matthias Beck 8
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 x i � = ǫx j Inside-Out Polytopes Matthias Beck 9
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 x i � = ǫx j Theorem (Zaslavsky 1982) χ Σ (2 k + 1) := # ( proper k -colorings of Σ) and χ ∗ Σ (2 k ) := # ( proper zero-free k -colorings of Σ) are monic polynomials of degree | V | . The number of compatible pairs ( α, x ) consisting of an acyclic orientation α and a k -coloring x of Σ is equal to ( − 1) | V | χ Σ ( − (2 k + 1)) . The number in which x is zero-free equals ( − 1) | V | χ ∗ Σ ( − 2 k ) . In particular, ( − 1) | V | χ Σ ( − 1) equals the number of acyclic orientations of Σ . Inside-Out Polytopes Matthias Beck 9
Signed-graph coloring a la Ehrhart x 1 = 1/2 x 1 = 1/2 x 1 = x 2 x 1 = x 2 (0,1) (1,1) (0,1) (1,1) = 1/2 = 1/2 x 2 x 2 (0,0) (1,0) (0,0) (1,0) = 1 = 1 x 1 + x 2 x 1 + x 2 − − − pmk2o + Theorem χ Σ (2 k + 1) and χ ∗ Σ (2 k ) are two halves of one inside-out quasipolynomial. Inside-Out Polytopes Matthias Beck 10
Signed-graph coloring a la Ehrhart x 1 = 1/2 x 1 = 1/2 x 1 = x 2 x 1 = x 2 (0,1) (1,1) (0,1) (1,1) = 1/2 = 1/2 x 2 x 2 (0,0) (1,0) (0,0) (1,0) = 1 = 1 x 1 + x 2 x 1 + x 2 − − − pmk2o + Theorem χ Σ (2 k + 1) and χ ∗ Σ (2 k ) are two halves of one inside-out quasipolynomial. Open problem Is there a combinatorial interpretation of χ ∗ Σ ( − 1) ? Inside-Out Polytopes Matthias Beck 10
Flow polynomials revisited ϕ Γ ( k ) := # ( nowhere-zero k -flows ) ϕ Γ ( | A | ) := # ( nowhere-zero A -flows ) Theorem ( − 1) | E |−| V | + c (Γ) ϕ Γ ( − 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 ϕ Γ (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 .) Inside-Out Polytopes Matthias Beck 11
Recommend
More recommend
