Combinatorial Reciprocity Theorems Matthias Beck based on joint - PowerPoint PPT Presentation

Combinatorial Reciprocity Theorems Matthias Beck based on joint work with San Francisco State University Raman Sanyal math.sfsu.edu/beck Universit¨ at Frankfurt Thomas Zaslavsky JCCA 2018 Sendai Binghamton University

The Theme Combinatorics counting function depending on k ∈ Z > 0 polynomial p ( k ) What is p (0) ? p ( − 1) ? p ( − 2) ? Combinatorial Reciprocity Theorems Matthias Beck 2

The Theme Combinatorics counting function depending on k ∈ Z > 0 polynomial p ( k ) What is p (0) ? p ( − 1) ? p ( − 2) ? Two-for-one charm of combinatorial reciprocity theorems ◮ “Big picture” motivation: understand/classify these polynomials ◮ Combinatorial Reciprocity Theorems Matthias Beck 2

Chromatic Polynomials G = ( V, E ) — graph (without loops) Proper k -coloring of G — x ∈ [ k ] V such that x i � = x j if ij ∈ E χ G ( k ) := # ( proper k -colorings of G ) Example: • ✡ ❏ ✡ ❏ ✡ ❏ χ K 3 ( k ) = k ( k − 1)( k − 2) ✡ ❏ ✡ ❏ ✡ ❏ ✡ ❏ • • ✡ ❏ Combinatorial Reciprocity Theorems Matthias Beck 3

Chromatic Polynomials • ✡ ❏ ✡ ❏ ✡ ❏ χ K 3 ( k ) = k ( k − 1)( k − 2) ✡ ❏ ✡ ❏ ✡ ❏ ✡ ❏ • • ✡ ❏ Theorem (Birkhoff 1912, Whitney 1932) χ G ( k ) is a polynomial in k . | χ K 3 ( − 1) | = 6 counts the number of acyclic orientations of K 3 . Theorem (Stanley 1973) ( − 1) | V | χ G ( − k ) equals the number of pairs ( α, x ) consisting of an acyclic orientation α of G and a compatible k -coloring x . In particular, ( − 1) | V | χ G ( − 1) equals the number of acyclic orientations of G . Combinatorial Reciprocity Theorems Matthias Beck 3

Order Polynomials (Π , � ) — finite poset φ ∈ [ k ] Π : a � b = � � Ω Π ( k ) := # ⇒ φ ( a ) ≤ φ ( b ) φ ∈ [ k ] Π : a ≺ b = Ω ◦ � � Π ( k ) := # ⇒ φ ( a ) < φ ( b ) � − k = ( − 1) d � k + d − 1 � k → Ω ◦ � � � Example: Π = [ d ] − Π ( k ) = and d d d Combinatorial Reciprocity Theorems Matthias Beck 4

Order Polynomials (Π , � ) — finite poset φ ∈ [ k ] Π : a � b = � � Ω Π ( k ) := # ⇒ φ ( a ) ≤ φ ( b ) φ ∈ [ k ] Π : a ≺ b = Ω ◦ � � Π ( k ) := # ⇒ φ ( a ) < φ ( b ) � − k = ( − 1) d � k + d − 1 � k → Ω ◦ � � � Example: Π = [ d ] − Π ( k ) = and d d d Theorem (Stanley 1970) Ω Π ( k ) and Ω ◦ Π ( k ) are Π ( − k ) = ( − 1) | Π | Ω Π ( k ) . polynomials related via Ω ◦ Combinatorial Reciprocity Theorems Matthias Beck 4

Order Polynomials h ( ◦ ) Π ( z ) Π ( k ) z k = Ω ( ◦ ) � (1 − z ) | Π | +1 (Π , � ) — finite poset k ≥ 0 φ ∈ [ k ] Π : a � b = � � Ω Π ( k ) := # ⇒ φ ( a ) ≤ φ ( b ) φ ∈ [ k ] Π : a ≺ b = Ω ◦ � � Π ( k ) := # ⇒ φ ( a ) < φ ( b ) � − k = ( − 1) d � k + d − 1 � k → Ω ◦ � � � Example: Π = [ d ] − Π ( k ) = and d d d Theorem (Stanley 1970) Ω Π ( k ) and Ω ◦ Π ( k ) are Π ( − k ) = ( − 1) | Π | Ω Π ( k ) . polynomials related via Ω ◦ Equivalently, z | Π | +1 h ◦ Π ( 1 z ) = h Π ( z ) . Combinatorial Reciprocity Theorems Matthias Beck 4

Eulerian Simplicial Complexes Γ — simplicial complex (collection of subsets of a finite set, closed under taking subsets) Γ is Eulerian if it is pure and every interval has as many elements of even rank as of odd rank f j := # ( j + 1) -subsets = # faces of dimension j d +1 f j − 1 z j (1 − z ) d +1 − j � h ( z ) := j =0 Theorem (Everyone 19xy) If Γ is Eulerian then z d +1 h ( 1 z ) = h ( z ) . Key example (Dehn–Sommerville): Γ = boundary complex of a simplicial polytope Combinatorial Reciprocity Theorems Matthias Beck 5

Ehrhart Polynomials Lattice polytope P ⊂ R d – convex hull of finitely points in Z d k P ∩ Z d � � For k ∈ Z > 0 let ehr P ( k ) := # x 1 = x 2 x 2 Example: x 2 = 6 ∆ = conv { (0 , 0) , (0 , 1) , (1 , 1) } ( x, y ) ∈ R 2 : 0 ≤ x 1 ≤ x 2 ≤ 1 � � = � k +2 = 1 � ehr ∆ ( k ) = 2 ( k + 1)( k + 2) 2 x 1 � k − 1 � ehr ∆ ( − k ) = = ehr ∆ ◦ ( k ) 2 Combinatorial Reciprocity Theorems Matthias Beck 6

Ehrhart Polynomials Lattice polytope P ⊂ R d – convex hull of finitely points in Z d k P ∩ Z d � � For k ∈ Z > 0 let ehr P ( k ) := # x 1 = x 2 x 2 Example: x 2 = 6 ∆ = conv { (0 , 0) , (0 , 1) , (1 , 1) } ( x, y ) ∈ R 2 : 0 ≤ x 1 ≤ x 2 ≤ 1 � � = � k +2 = 1 � ehr ∆ ( k ) = 2 ( k + 1)( k + 2) 2 x 1 � k − 1 � ehr ∆ ( − k ) = = ehr ∆ ◦ ( k ) 2 For example, the evaluations ehr ∆ ( − 1) = ehr ∆ ( − 2) = 0 point to the fact that neither ∆ nor 2∆ contain any interior lattice points. Combinatorial Reciprocity Theorems Matthias Beck 6

Ehrhart Polynomials Lattice polytope P ⊂ R d – convex hull of finitely points in 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 . Theorem (Macdonald 1971) ( − 1) dim P ehr P ( − k ) enumerates the interior lattice points in k P . Combinatorial Reciprocity Theorems Matthias Beck 7

Ehrhart Polynomials Lattice polytope P ⊂ R d – convex hull of finitely points in 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 . h ∗ P ( z ) ehr P ( k ) z k = � Ehr P ( z ) := 1 + (1 − z ) dim( P )+1 k> 0 Theorem (Macdonald 1971) ( − 1) dim P ehr P ( − k ) enumerates the interior lattice points in k P . z dim( P ) h ∗ P ( 1 z ) = h ∗ P ◦ ( z ) Combinatorial Reciprocity Theorems Matthias Beck 7

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. Generating-function version: evaluate at reciprocals. Combinatorial Reciprocity Theorems Matthias Beck 8

Ehrhart − → Order Polynomials (Π , � ) — finite poset φ ∈ [0 , 1] Π : a � b = � � Order polytope O Π := ⇒ φ ( a ) ≤ φ ( b ) φ ∈ [ k ] Π : a � b = � � Ω Π ( k ) = # ⇒ φ ( a ) ≤ φ ( b ) = ehr O Π ( k − 1) φ ∈ [ k ] Π : a ≺ b = Ω ◦ � � Π ( k ) = # ⇒ φ ( a ) < φ ( b ) = ehr O ◦ Π ( k + 1) P ( − k ) = ( − 1) dim P ehr P ( k ) implies Ω ◦ Π ( − k ) = ( − 1) | Π | Ω Π ( k ) and so ehr ◦ Combinatorial Reciprocity Theorems Matthias Beck 9

Order − → Chromatic Polynomials � Ω ◦ χ G ( k ) = # ( proper k -colorings of G ) = Π ( k ) Π acyclic ( − 1) | V | χ G ( − k ) = ( − 1) | Π | Ω ◦ � � Π ( − k ) = Ω Π ( k ) Π acyclic Π acyclic k + 1 K 2 k + 1 x 1 = x 2 Combinatorial Reciprocity Theorems Matthias Beck 10

Chain Partitions (Π , � ) — finite graded poset with ˆ 0 and ˆ 1 φ : Π \ { ˆ 0 , ˆ 1 } → Z > 0 order preserving (Π , φ ) -chain partition of n ∈ Z > 0 : n = φ ( c m ) + φ ( c m − 1 ) + · · · + φ ( c 1 ) for some multichain ˆ 1 ≻ c m � c m − 1 � · · · � c 1 ≻ ˆ 0 � cp Π ,φ ( k ) z k cp Π ,φ ( k ) := # (chain partitions of k ) CP Π ,φ ( z ) := 1+ k> 0 Combinatorial Reciprocity Theorems Matthias Beck 11

Chain Partitions (Π , � ) — finite graded poset with ˆ 0 and ˆ 1 φ : Π \ { ˆ 0 , ˆ 1 } → Z > 0 order preserving (Π , φ ) -chain partition of n ∈ Z > 0 : n = φ ( c m ) + φ ( c m − 1 ) + · · · + φ ( c 1 ) for some multichain ˆ 1 ≻ c m � c m − 1 � · · · � c 1 ≻ ˆ 0 � cp Π ,φ ( k ) z k cp Π ,φ ( k ) := # (chain partitions of k ) CP Π ,φ ( z ) := 1+ k> 0 Example: A = { a 1 < a 2 < · · · < a d } ⊂ Z > 0 Π = [ d ] φ ( j ) := a j − → cp Π ,φ ( k ) is the restricted partition function with parts in A Combinatorial Reciprocity Theorems Matthias Beck 11

Chain Partitions (Π , � ) — finite graded poset with ˆ 0 and ˆ 1 φ : Π \ { ˆ 0 , ˆ 1 } → Z > 0 order preserving (Π , φ ) -chain partition of n ∈ Z > 0 : n = φ ( c m ) + φ ( c m − 1 ) + · · · + φ ( c 1 ) for some multichain ˆ 1 ≻ c m � c m − 1 � · · · � c 1 ≻ ˆ 0 � cp Π ,φ ( k ) z k cp Π ,φ ( k ) := # (chain partitions of k ) CP Π ,φ ( z ) := 1+ k> 0 φ is ranked if rank( a ) = rank( b ) = ⇒ φ ( a ) = φ ( b ) Theorem If Π is Eulerian then � distinct ranks CP Π ,φ ( z ) � 1 � ( − 1) rank(Π) CP Π ,φ = z z Combinatorial Reciprocity Theorems Matthias Beck 11

Chain Partitions for Simplicial Complexes Π = Γ ∪ { ˆ 1 } for a d -simplicial complex Γ with ground set V φ ( σ ) = rank( σ ) = | σ | (Π , φ ) -chain partition of n ∈ Z > 0 : n = φ ( c m ) + φ ( c m − 1 ) + · · · + φ ( c 1 ) for some multichain ˆ 1 ≻ c m � c m − 1 � · · · � c 1 ≻ ˆ 0 d +1 � k � � cp Π ,φ ( k ) := # (chain partitions of k ) = f j − 1 j j =0 Canonical geometric realization of Γ in R V : � � R [Γ] := conv { e v : v ∈ σ } : σ ∈ Γ Combinatorial Reciprocity Theorems Matthias Beck 12