Patterns of Counting: From One to Zero and to Infinity

Transcription

1 Patterns of Counting: From One to Zero and to Infinity Beifang Chen Department of Mathematics Hong Kong University of Science and Technology Clear Water Bay, Kowloon, Hong Kong International Conference on Combinatorics Institute of Math, Academia Sinica, Taipei, Taiwan May 19-22, 2017 (Math Dept, HKUST) Patterns of Counting May 19-22, / 40

2 Outline 1 Counting and Euler Numbers 2 Hilbert s Third Problem 3 Finitely Additive Measures 4 Subspace Arrangement 5 Chromatic Polynomial 6 Counting over Q 7 Group Arrangement 8 Grassmannians 9 Counting Points of Algebraic Varieties 10 Conclusion (Math Dept, HKUST) Patterns of Counting May 19-22, / 40

3 A Prologue It was the best time, it was the worst time;... It was the season of Light, it was the season of Darkness;... We had everything before us, we had nothing before us;... Charles Dickens A Tale of Two Cities A Dipicture of Math It is the simplest subject, it is the most complicated subject;... Why we think of this way? why we think of the other way?... To be and not to be? This is the problem. (Math Dept, HKUST) Patterns of Counting May 19-22, / 40

4 Counting and Euler Numbers (Math Dept, HKUST) Patterns of Counting May 19-22, / 40

5 Cardinal Numbers and Counting Principles Cantor s Idea of Cardinal Numbers: 0, 1, 2, 3,..., ω, ω + 1, ω + 2,..., 2ω, 2ω + 1, 2ω + 2,..., 3ω, 3ω + 1, 3ω + 2,..., ω 2, ω 2 + 1, ω 2 + 2,..., ω 2 +ω, ω 2 +ω + 1, ω 2 +ω + 2,..., ω 2 + 2ω,... Addition Rule: Given two disjoint sets A and B, we should have A B = A + B. Product Rule: Given two sets A and B, we have A B = A B. (Math Dept, HKUST) Patterns of Counting May 19-22, / 40

6 First Example Given two subsets A,B of R 2, see the figure below. Do we still have A B = A + B A B? A B Naive Counting: #(A) = 2, #(B) = 3, #(A B) = 2, #(A B) = 4. If the addition rule is true, then 4 = #(A B) = #(A)+#(B) #(A B) = = 3. Correct Counting: If we count the ring shape zero as its shape looks, i.e., χ(a B) = 0, then the addition rule is still valid. 3 = χ(a B) = χ(a)+χ(b) χ(a B) = = 3. (Math Dept, HKUST) Patterns of Counting May 19-22, / 40

7 Hilbert s Third Problem (Math Dept, HKUST) Patterns of Counting May 19-22, / 40

8 Volume Theory A dissection of an n-polytope P is a collection of polytopes P 1,...,P m whose union is P and intersections P i P j for all i j have lower dimensions, written P = P 1 P m. Two n-polytopes P,Q are said to be scissors congruent if there are dissections P = P 1 P m, Q = Q 1 Q m such that Q i = g i (P i ) for some rigid motions g i of R n. Hilbert asked in 1900: Are two n-polytopes P,Q of R n necessarily scissors congruent when they have the same volume? The problem was solved negatively by his student Max Dehn by certain valuations (finitely additive measures), called Dehn invariants. (Math Dept, HKUST) Patterns of Counting May 19-22, / 40

9 Volume Theory Cont d Consider the set function D : P n R Z (R/Qπ) defined by D(P) = F vol(f) θ(f), where the sum is extended over all 1-faces F of P, θ(f) is the dihedral angle between the two facets meeting at F, and R is viewed as a vector space over Q. Valuation Property (finitely additive measure): If P admits a dissection P = P 1 P m, then D(P) = m D(P i ). i=1 For any rectangular parallelotope P and simplex T, we have D(P) = 0, D(T) 0. (Math Dept, HKUST) Patterns of Counting May 19-22, / 40

10 Finitely Additive Measures (Math Dept, HKUST) Patterns of Counting May 19-22, / 40

11 Valuations (=Finitely Additive Measures) Let S be an intersectional collection of subsets of a set S. Let B(S) be the relative boolean algebra generated by the members of S by taking unions, intersections, and relative complement finitely many times. A valuation on S with values in an abelian group A is a function µ : S A such that µ( ) = 0, µ(s 1 S 2 ) = µ(s 1 )+µ(s 2 ) µ(s 1 S 2 ), whenever S 1,S 2,S 1 S 2 S. Groemer s Extension Theorem: A set function µ : S A can be extended to a valuation on B(S) if and only if it satisfies the the Inclusion-Exclusion Principle: µ(s 1 S n ) = ( ( 1) J 1 µ S j ). J [n] j J (Math Dept, HKUST) Patterns of Counting May 19-22, / 40

12 Elementary Mixed Volumes Let K n be the class of all compact convex sets of R n. A valuation µ : K n R is said to be continuous if µ(k m ) µ(k) as K m K under the Hausdorff distance { } d(k 1,K 2 ) = max sup inf d(x,y), sup inf d(x,y). y K 2 x K 1 y K 1 x K 2 Minkowski Sum: K 1 + K 2 = {x + y : x K 1,y K 2 }. Steiner s Formula: Let B denote the unit ball of R n. Given any convex body K of R n, vol n (K + rb) = n µ i (K)ω n i r n i, i=0 where µ i : K n R are continuous valuations, called the elementary mixed volumes, and ω n i is the volume of the (n i)-dim unit ball. (Math Dept, HKUST) Patterns of Counting May 19-22, / 40

13 Hadwiger s Characterization Every valuation on K n can be uniquely extended to a valuation on the relative Boolean algebra B(K n ) generated by K n. Let P = n i=1 [0,a i], the rectangular box of side length a i. Then µ k (P) = e k (a 1,...,a n ), where e k is the kth elementary symmetric polynomial. Let Q k = k i=1 [0,1) be the half-closed and half-open rectangular unit boxes; Q 0 := {0}. Then µ k (Q k ) = 1 for all 0 k n. Hadwiger Theorem: The elementary mixed volumes µ 0,µ 1,...,µ n form a basis of the vector space of all continuous rigid motion invariant valuations on K n. (Math Dept, HKUST) Patterns of Counting May 19-22, / 40

14 Polytope Algebra Let R be the vector space of sequences (a 1,a 2,...) of real numbers having a i = 0 for large enough i. A polytope P in R is a convex hull of some finite number of points. The polytope algebra Π is an abelian group generated by [P], where P are polytopes of R, subject to Translation Invariance: [P] = [a+p]; Valuation Property: [ ] = 0; where P,Q,P Q,P Q P n. [P]+[Q] = [P Q]+[P Q], (Math Dept, HKUST) Patterns of Counting May 19-22, / 40

15 Polytope Algebra Cont ed Minkowski Multiplication: [P] [Q] = [P + Q]. Canonical Basis I: Let t i denote the class of the simplex whose vertices are the origin 0 = e 0 and the coordinate unit vectors e 1,...,e i, where 0 i n. Then t 0, t 1,..., t n form a basis of the invariant lattice polytope algebra Π n. Canonical Basis II: Let u denote the class of the half-closed and half-open unit interval [0,1). Then u 0,u 1,u 2,... form a basis of the invariant lattice polytope algebra. Moreover, if P is a d-polytope then [P] = d a i (P)u i i=0 is a polynomial of the variable u. The invariant lattice polytope algebra is isomorphic to the polynomial algebra Q[u] of one indeterminant u. (Math Dept, HKUST) Patterns of Counting May 19-22, / 40

16 Ehrhart Polynomial Let P be a lattice polytope. It is well known that the number of lattice points of np := {np : p P} is a polynomial function L(P,n) = dim P i=0 a i (P)n i of n, called the Ehrhart polynomial of P. Betke and Kneser s Theorem: The set functions a i are valuations on lattice polytopes and form a basis for all valuations. Since L(mP,n) = dim P i=0 a i (P)m i n i, it is suggested by Danilov that L(P,t) may be written as L(P,t) = dim P i=0 σ F(P) dim σ=i f(cone(p,σ))vol i (σ)t i to interpret the coefficients of the Ehrhart polynomials. The function f (defined on lattice convex cones) are still not constructive in general. (Math Dept, HKUST) Patterns of Counting May 19-22, / 40

17 Subspace Arrangement (Math Dept, HKUST) Patterns of Counting May 19-22, / 40

18 Subspace Arrangement A subspace arrangement is a finite collection of affine subspaces of a finite-dimensional vector space V. Associated with a subspace arrangement A is the characteristic polynomial χ(a,t) = µ(v,x)t dim X, X L(A) where L(A) is the intersection semilattice of A, µ is the Möbius function on the poset L(A) under the set inclusion. Ehrenberg and Readdy s Theorem: Let V be a finite-dimensional vector space over an infinite field K, L(V) the semilattice of all affine subspace of V, and B(V) the Boolean algebra generated by L(V). Then there exists a unique valuation ν : B(V) Z[t] such that for any affine subspace W V, ν(w) = t dim W. (Math Dept, HKUST) Patterns of Counting May 19-22, / 40

19 Subspace Arrangement For any affine subspace arrangement A, the complement M(A) of A is a member of B(V) and ( χ(a,t) = ν V ) H. H A This says that the characteristic polynomial χ(a,t) is the measure of the characteristic function of the complement of the arrangement. This explanation was known to me and Gian-Carlo Rota was told by me in MIT in his office in 1992, and paper was published later by Ehrenberg and Readdy in Zaslavsky Formula: χ(a, 1) = #{regions of the complement}, χ(a,1) = #{relatively bonded regions of the complement}. (Math Dept, HKUST) Patterns of Counting May 19-22, / 40

20 Chromatic Polynomial (Math Dept, HKUST) Patterns of Counting May 19-22, / 40

21 Chromatic Polynomial The number χ(g,n) of proper colorings of a graph G = (V,E) with n colors such that no two adjacent vertices receiving a same color is a polynomial function of n, known as chromatic polynomial of G. The number of proper colorings of graph a b c by n colors is χ(g,n) = χ{a,b,c} χ{a = b,c} χ{a,b = c} +χ{a = b = c} = n 3 n 2 n 2 + n = n 3 2n 2 + n. If the color set is Z and consider the the number" of colors in Z is t, we still have the same pattern χ(g,t) = t 3 t 2 t 2 + t = t 3 2t + t. For rational matroid, similar chromatic polynomial" was introduced by Welsh by mimic the pattern of chromatic polynomial without t factor, better called tension polynomial. (Math Dept, HKUST) Patterns of Counting May 19-22, / 40

22 Counting over Q (Math Dept, HKUST) Patterns of Counting May 19-22, / 40

23 Counting over Q Counting the number of elements of a finite set of Q gives an integer. Standard: number of elements of Z is t. Then The number of elements of 2Z (set of even integers) should be 1 2 t. The number of integers having remainders 3 or 4 modulo 5 is 2 5 t. The number of elements of 1 3Z is 3t. The number of elements of 2Z 1 3 Z is 3t 1 2 t = 5 2 t. Observation: There exists a unique translation invariant valuation (finitely additive measure) ν : B(Q) {a+bt a,b Q} such that ν({0}) = 1, ν(z) = t, where B(Q) is the Boolean algebra generated by the cosets of lattices (subgroups generated by one element) of Q. (Math Dept, HKUST) Patterns of Counting May 19-22, / 40

24 Counting over Q Q = {(a 1,a 2,...) : a i Q}, where a i = 0 for i large enough. All Q n are identified as the subsets Q n 0. A lattice of rank r of Q is a subgroup generated by r linearly independent vectors. L(Q ): set of all translates of lattices of Q. B(Q ): Boolean algebra generated by L(Q ), whose objects are obtained by taking unions, intersections, and complements finitely many times of objects in L(Q ). A valuation on B(Q ) with values in a commutative ring R with unity 1 is a function ν : B(Q ) R such that ν( ) = 0, ν(a B) = ν(a)+ν(b) ν(a B). A valuation ν is said to be translation invariant if ν(a+a) = ν(a), a Q. (Math Dept, HKUST) Patterns of Counting May 19-22, / 40

25 Unique Valuation A valuation ν is said to satisfy the multiplicativity if A+B is a direct sum of lattices A,B, and A+B is a direct summand of Z then ν(a+b) = ν(a)ν(b). Let L be a lattice with a basis a 1,...,a r in Q ; A = r -matrix with rows a i. The determinant of L is det L = det AA T. Let V be a finite-dimensional rational subspace of Q, i.e., the lattice V Z has rank dim V, set det V = det(v Z ). Theorem. There exists a unique translation invariant valuation λ : B(Q) Q[t] such that λ({0}) = 1, λ(z) = t. Moreover, if L is a lattice of rank r, then λ(l) = det(l R) t r. det L (Math Dept, HKUST) Patterns of Counting May 19-22, / 40

26 Group Arrangement (Math Dept, HKUST) Patterns of Counting May 19-22, / 40

27 Group Arrangement A group arrangement of M is a finite collection A of flats of M. Let L(A) be the intersection semilattice of A. Associated with A is the characteristic polynomial χ(a,t) := µ(o,x)t r(l), L L(A) where µ is the Möbius function on L(A), r(l) is the rank of the lattice L. Theorem: For an arrangement A of a lattice M of finite rank in Q, ( det M χ(a,t) = det(m R) λ M ) L. L A (Math Dept, HKUST) Patterns of Counting May 19-22, / 40

28 Case of Real Numbers L(R): set of translates of all lattices of R generated by finite linear independent vectors chosen from R. Given a lattice L of R. The rational core of L is a maximal sublattice K L such that the rank of (K R) Z is dim(k R), denoted K = K(L). L(R) is a semilattice (closed under intersection). The theorem can be modified as follows: There exists a unique translation invariant valuation λ : B(Q) R[t] such that for any lattice L, det K(L) ν(l) = det L trank(l). Moreover, if L is rational, then the coefficient of t rank(l) is rational. (Math Dept, HKUST) Patterns of Counting May 19-22, / 40

29 Case of Real Numbers Let L,M be lattices of the same rank and L M. Then M = F. F M/L Any translation invariant valuation on B(R) must satisfy the relation ν(m) = M/L ν(l). Theorem. A translation invariant set function ν : B(R) R is a valuation if and only if ν(m) = M/L ν(l). For each vector space V, assign a real number a(v). Then the set function ν : L 0 (R) R[t] given by ν(l) = a(l R) det(l) trank(l) can be uniquely extended to a translation invariant valuation on B(R). (Math Dept, HKUST) Patterns of Counting May 19-22, / 40

30 Grassmannians (Math Dept, HKUST) Patterns of Counting May 19-22, / 40

31 Grassmannians Gr(k,K n ): Grassmannian of k-subspaces of K n. Gauss polynomial (q-analog of binomial coefficient): ( ) n : = #Gr(k,F n k q) q = (qn 1)(q n q) (q n q k 1 ) (q k 1)(q k q) (q k q k 1 ). Gr(k,K ): Grassmannian of k-subspaces of K. Infinite type of q-analog of binomial coefficient: ( ) : = #Gr(k,F q k ) q = 1 (1 q)(1 q 2 ) (1 q k ). (Math Dept, HKUST) Patterns of Counting May 19-22, / 40

32 Grassmannian of -subspaces Identification: Gr(0,K ) Gr(1,K ) Gr(2,K ) V K V. Gr(,K ): direct limt of {Gr(k,K )} k=0. Each element of Gr(,K ) can be viewed as an equivalence class of finite-dimensional subspaces V, K V, K 2 V, K 3 V,. #Gr(,F q ) = i=1 1 1 q i. (Math Dept, HKUST) Patterns of Counting May 19-22, / 40

33 Partition Variety and q-multinomial Coefficients M n,d (K): space of all n-by-d matrices of rank n over K, having only finite number of non-zero entries. Given a partition λ = (λ 1,...,λ n ) of positive integer or with n parts, where 0 < λ 1 λ n. Ferrers board F λ of λ: having the parts λ 1,..., λ n arranged from the top to the bottom and justified on the left side. Consider the subspace where d = λ n. M λ (K) = { [a ij ] M n,d (K) : a ij = 0 for (i,j) F λ }, (Math Dept, HKUST) Patterns of Counting May 19-22, / 40

34 Partition Variety and q-multinomial Coefficients Cont d Ferrers board F λ for partition λ = (2,3,5,, ) has the shape and the matrices of M λ have the form (Math Dept, HKUST) Patterns of Counting May 19-22, / 40

35 Partition Variety and q-multinomial Coefficients Cont d Given a composition γ = (γ 1,...,γ m ) of the positive integer n, i.e., γ i > 0 and m i=1 γ i = n. Set ν 0 = 0 and ν j = j i=1 γ i for j = 1,...,m. We say that λ and γ are compatible if λ i = µ j for ν j i ν j, i.e., λ = ( ) µ 1,...,µ }{{ 1, µ } 2,...,µ 2,..., µ }{{} m,...,µ m. }{{} γ 1 γ m P γ (K): parabolic subgroup of GL n (K) of γ-block lower triangular invertible n-by-n matrices over K. Partition variety: M λ (K)/P γ (K). S d : symmetric group of Z +, 1 d. Inversion set Inv(σ) = {a 1,a 2,...} for each σ S d, where γ 2 a k = # { i Z + : σ(i) > σ(j) = k, i < j }. (Math Dept, HKUST) Patterns of Counting May 19-22, / 40

36 Partition Variety and q-multinomial Coefficients Cont d Introduce the index sets: S λ := { σ S d : d = λ n,σ(i) λ i,i [n] }, S d (γ) := { σ S d : Inv(σ) {ν 0,ν 1,...,ν m } }, S λ (γ) := S λ (γ 1,...,γ m ) = S λ S d (γ). Flag space of type (λ,γ): subspace of m i=1 Gr(ν i,k µ i) defined by F(γ,K λ ) := {(V 1,...,V m ) V 1 V m,v i Gr(ν i,k µ i )}. Ding s Theorem: M λ /P γ F(γ,K λ ), [M] (V 1,...,V m ), where V i is the span of rows of M from 1 to µ i. Ding s Formula: For K = F q, we have m ( ) q inv(σ) µi ν i 1 =. σ S λ (γ) i=1 γ i q (Math Dept, HKUST) Patterns of Counting May 19-22, / 40

37 Partition Variety and q-multinomial Coefficients Cont d If λ j = n for all 1 j n, then µ i ν i 1 = n ν i 1. Write γ i = a i and recall that ( ) ( ) ( ) ( ) n n n a1 n a1 a m 1 =, a 1,...,a m q a 1 q a 2 a m one sees σ S n(a 1,...,a m) For any positive integers a 1,...,a m, q ( ) q inv(σ) n =. a 1,...,a m q q q inv(σ) = σ S (a 1,...,a m) = m 1 (1 q)(1 q 2 ) (1 q a i ) i=1 ( ) a 1,...,a m. q (Math Dept, HKUST) Patterns of Counting May 19-22, / 40

38 Counting Points of Algebraic Varieties (Math Dept, HKUST) Patterns of Counting May 19-22, / 40

39 Last Example: Algebraic Variety over Z (with Chai Chin-Li) Let X be an algebraic variety whose defining equations have integer coefficients. Then X/F is well-defined over F, where F {F q,r,c,h}. Let f(x,q) = #(X/F q ). If f(x,q) is a polynomial functions of q, then f(x,1) = χ(x/c) = χ(x/h). where χ means the Euler number of the algebraic varieties. f(x, 1) = χ c (X/R), where χ c means the Euler number with compact support. This is a joint work with Chai Ching-Li (UPenn). (Math Dept, HKUST) Patterns of Counting May 19-22, / 40

40 Conclusion from Patterns God s Universe is beautiful Exact, perfect, and powerful No little less No little more Thank you! (Math Dept, HKUST) Patterns of Counting May 19-22, / 40