Combinatorial Reciprocity Theorems Matthias Beck San Francisco State University math.sfsu.edu/beck/
Joint work with... Thomas Zaslavsky, Binghamton University (SUNY) math.CO/0309330 math.CO/0309331 math.CO/0506315 Combinatorial Reciprocity Theorems Matthias Beck 2
Chromatic polynomials of graphs Γ = ( V, E ) – graph (without loops) k -coloring of Γ : mapping x : V → { 1 , 2 , . . . , k } Combinatorial Reciprocity Theorems Matthias Beck 3
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 x i � = x j if there is an edge ij Combinatorial Reciprocity Theorems Matthias Beck 3
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 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 | . Combinatorial Reciprocity Theorems Matthias Beck 3
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 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 | . Proof: Choose your favorite edge e of Γ and use χ Γ ( k ) = χ (Γ \ e ) ( k ) − χ (Γ · e ) ( k ) and induction. . . Combinatorial Reciprocity Theorems Matthias Beck 3
Stanley’s Acyclic-Orientation Theorem 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.) Combinatorial Reciprocity Theorems Matthias Beck 4
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 Combinatorial Reciprocity Theorems Matthias Beck 5
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 } Combinatorial Reciprocity Theorems Matthias Beck 5
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 . Combinatorial Reciprocity Theorems Matthias Beck 5
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 . What about reciprocity? Combinatorial Reciprocity Theorems Matthias Beck 5
(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 Combinatorial Reciprocity Theorems Matthias Beck 6
(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 . Combinatorial Reciprocity Theorems Matthias Beck 6
(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 which satisfies 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 ) . Combinatorial Reciprocity Theorems Matthias Beck 6
(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 which satisfies 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? Combinatorial Reciprocity Theorems 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 Combinatorial Reciprocity Theorems Matthias Beck 7
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 ) Combinatorial Reciprocity Theorems Matthias Beck 7
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) . Combinatorial Reciprocity Theorems Matthias Beck 7
Ehrhart (quasi-)polynomials P ⊂ R d – convex rational polytope P ∩ 1 � t Z d � For t ∈ Z > 0 let Ehr P ( t ) := # Combinatorial Reciprocity Theorems Matthias Beck 8
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 1 . (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 .) Combinatorial Reciprocity Theorems Matthias Beck 8
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 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 .) Combinatorial Reciprocity Theorems Matthias Beck 8
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 Combinatorial Reciprocity Theorems Matthias Beck 9
Graph coloring a la Ehrhart χ K 2 ( k ) = k ( k − 1) ... k + 1 K 2 k + 1 x 1 = x 2 Combinatorial Reciprocity Theorems Matthias Beck 10
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) ∩ Combinatorial Reciprocity Theorems Matthias Beck 10
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 Combinatorial Reciprocity Theorems Matthias Beck 11
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