Transitive cycle factorizations and prime parking functions

Transcription

1 Transitive cycle factorizations and prime parking functions Dongsu Kim Department of Mathematics KAIST, Daejeon 0-0, Korea and Seunghyun Seo Department of Mathematics KAIST, Daejeon 0-0, Korea June, 00 Abstract Minimal transitive cycle factorizations and parking functions are related very closely. Using the correspondence between them, we find a bijection between minimal transitive factorizations of permutations of type (,n ) and prime parking functions of length n. Keywords: Minimal transitive factorization, parking function, noncrossing partition Introduction For any natural number n, lets n denote the group of all permutations of [n] ={,,...,n}. A cycle of length n can be expressed as a product of n transpositions. The total number of different such expressions is n n. In general, we may count different ways to form a given permutation as a product of transpositions with some interesting conditions. For example, we may require that the transpositions in the product generate S n and the minimal number of transpositions are used. This case has been considered in various literature [,,, ]. Clearly the count depends only on the cycle type of the permutation. An integer partition of n is a weakly increasing sequence of positive integers, which sums to n. Given an integer partition λ =(λ,...,λ l )ofn, letλ denote the permutation ( λ )(λ + λ + λ ) (n λ l + n) of[n] in the cycle notation. The cycle type of λ is λ. LetF λ be the set of all m-tuples of transpositions (σ,...,σ m ) such that (f) σ σ m = λ, (f) {σ,...,σ m } generates S n, Running head: Transitive cycle factorizations Corresponding author

2 (f) m = n + l, i.e., m is minimal subject to (f), (f). Elements of F λ are called minimal transitive factorizations of λ (or simply λ). These factorizations are connected with branched covers of the sphere, that is treated in [,,, ]. Goulden and Jackson [] proved that F λ =(n + l )! n l l i= λ λi i (λ i )!, () which was originally suggested by Hurwitz []. They proved it for arbitrary λ, but the proof was not combinatorial. Recently, Bousquet-Mélou and Schaeffer [] enumerated the transitive factorizations by interpreting them as planar constellations and Eulerian trees, and obtained the formula () by using the principle of Inclusion Exclusion. In case λ =(n), there have been several combinatorial proofs [,, 9, 0]. For any other λ, there has been no simple combinatorial proof yet. When λ =(,n ), the expression () reduces to F (,n ) =(n ) n. () The aim of this paper is to give a combinatorial proof of (), using prime parking functions. Preliminaries Let π be a permutation and σ =(wz), with w<z, a transposition in S n. If w and z appear in the same cycle in π, then this cycle is cut into two cycles in the product σπ; otherwise, the cycles containing w and z are joined into one cycle in σπ. We call σ a cut of π in the former case and a join in the latter. For an element (σ,...,σ m )inf λ, where λ =(λ,...,λ l ) is an integer partition of n, σ i is called a cut (resp. a join), if σ i is a cut (resp. join) of σ i+ σ m. Note that, from Goulden and Jackson [, Proposition.], there are exactly l cuts and n joins in these factorizations. When λ =(n), a minimal transitive factorization has n transpositions which generate S n. A circle chord diagram, first introduced in [0], is a circle drawn on the plane with n points on it, labelled,...,n clockwise, and with n chords on these n points, numbered,...,n distinctly. There is a natural injection Γ from factorizations to circle chord diagrams: Γ maps a factorization (σ,...,σ n ) F (n), where σ i =(w i z i ), to the circle chord diagram in which the chords joining points w i and z i are numbered i, fori =,...,n. The circle chord diagrams corresponding to the factorizations are characterized below. Proposition The following three properties (c), (c), (c) completely characterize the image set of Γ, Γ(F (n) ). (c) The chords form a tree on [n]. (c) The chords meet only at the end points. (c) The numbers on the chords of the boundary of any region increase clockwise, starting immediately after the unique arc. Proof. See Theorem. in [0].

3 9 Figure : A circle chord diagram with n =9 Let C n be the set of all circle chord diagrams which satisfy the above conditions (c), (c) and (c). Clearly, Γ is a bijection from F (n) onto C n. For example, the circle chord diagram illustrated in Figure is Γ(F ), with F = ( ( ), ( ), ( ), ( ), ( ), ( ), ( 9), ( ) ). Note that there are 9 regions and each region satisfies (c) in the proposition. Using the planarity of circle chord diagrams, Goulden and Yong [0] found a bijection between C n and the set T n of all labelled trees on [n]. A partition of [n] is a family of nonempty, pairwise disjoint sets B,...,B k, called blocks, whose union is [n]. The collection of all partitions of [n] will be denoted by Π n. A partition of [n] is called noncrossing when it has the property that for any four elements a<b<c<d,ifa and c are in the same block and b is in another, then b and d are not in the same block. The collection of all noncrossing partitions of [n] will be denoted by NC n. The collection Π n can be viewed as a partially ordered set (poset) with the following order (called refinement order): for μ, ν Π n,wehaveμ ν if each block of μ is contained in a block of ν. The set NC n is also regarded as a poset with the refinement order. The Hasse diagram of NC is shown in Figure. In fact, the poset NC n has many nice properties. See [] for extensive studies of noncrossing partitions. A parking function of length n is a sequence (x,...,x n ) of positive integers such that the increasing rearrangement y y n of x,...,x n satisfies y i i for all i. LetP n denote the set of all parking functions of length n. It is well known that P n = n n. For example, the set P is as follows: (An array abc means (a, b, c).) P = {,,,,,,,,,,,,,,, } Parking functions are related to labelled trees, lattice paths, noncrossing partitions and hyperplane arrangements. For further information see [] and [, pp. 9]. Combining the results of Goulden and Yong [0] and Stanley [, Theorem.], we can define a combinatorial bijection Φ : F (n) P n by Φ( (w z ),...,(w n z n ))=(w,...,w n ). () We will use Φ to establish our main result later. Since the mapping Φ is not shown explicitly in [0, ], we present an explicit description of Φ in the next section.

4 ˆ = / / / / / / // // // // // // ˆ0 =/// Bijection Φ Figure : The Hasse diagram of NC Let M n be the set of all maximal chains in NC n. We define a mapping Σ : C n M n through the following process. Let C be a circle chord diagram in C n. Then C =Γ((w z ),...,(w n z n )) for some factorization ( (w z ),...,(w n z n )) F (n).letm =[μ 0 <μ < <μ n ] be a maximal chain in M n such that μ 0 = ˆ0, μ n = ˆ, and while i changes from n to,μ i is refined to μ i by splitting the blocks in μ i with the chord (w i z i ): To be more specific, B is a block in μ i if and only if for some block B in μ i, B = B [w i +,z i ] or B = B ([n] \ [w i +,z i ]), where [a, b] denotes the set {a, a +,...,b}. Since C satisfies (c), the chords (w z ),...,(w n z n ) do not cross in C, and so each μ i is an element of NC n with bk(μ i ) bk(μ i ) +, where bk(μ) is the number of blocks in μ. Moreover,w i and z i are in the same block in μ i, which implies bk(μ i ) bk(μ i )+; if they are not in the same block, then for some j>i,thechord(w i z i ) crosses or coincides with (w j z j ), contradicting to the condition (c). Thus μ i covers μ i for each i and m is really in M n. Set Σ(C) =m. Now let us find the inverse of Σ. Suppose that μ, ν NC n, where ν covers μ. Then there exist A ν and A,A μ satisfying A = A A, with min A > min A. We define the base, middle, left, right and left-right sets of (μ, ν), denoted by B, M, L, R and LR respectively, as follows (Figure ): B(μ, ν) =A, M(μ, ν) =A, L(μ, ν) ={i A : i<min A }, R(μ, ν) ={i A : i>max A }, LR(μ, ν) =L(μ, ν) R(μ, ν) =A. Note that L(μ, ν) andm(μ, ν) arenot empty. For μ, ν NC n such that ν covers μ, define Δ (μ, ν) and Δ (μ, ν) by Δ (μ, ν) = max L(μ, ν), Δ (μ, ν) = max M(μ, ν),

5 B(μ, ν) A = B(μ, ν) ν splitting M(μ, ν) A = M(μ, ν) μ L(μ, ν) R(μ, ν) A = LR(μ, ν) μ Figure : Base, middle, left, right and left-right set of (μ, ν). and define Δ(μ, ν) to be the transposition (Δ (μ, ν)δ (μ, ν)) in S n. We now define Ω : M n F (n) as follows: Given a maximal chain m =[μ 0 <μ < <μ n ], Ω(m) = (Δ(μ 0,μ ),...,Δ(μ n,μ n )). It easily follows from construction that Ω is well-defined and injective. Proposition The composition map Γ Ω is the inverse of Σ. So the mappings Σ and Ω are all bijections. Proof. Consider the maximal chain Σ(C) =[μ 0 <μ < <μ n ], where C =Γ((w z ),...,(w n z n ) ). Recall that w i and z i belong to the same block A in μ i.the block A is split into two blocks A and A in μ i, where A = A [w i +,z i ], A = A ([,w i ] [z i +,n]). On the other hand, Since w i, z i are in A, Thus Δ(μ i,μ i )=(w i z i )and M(μ i,μ i )=A, L(μ i,μ i )=A [,w i ]. max L(μ i,μ i )=w i, max M(μ i,μ i )=z i. Ω(Σ(C))=((w z ),...,(w n z n )). Applying the both sides of the above with Γ yields that (Γ Ω) Σ is the identity map on C n.thus Γ Ω is surjective. On the other hand, since Ω is injective, Γ Ω is the inverse of Σ.

6 Given a maximal chain m =[μ 0 < <μ n ] M n, define Ω (m) by Ω (m) =(Δ (μ 0,μ ),...,Δ (μ n,μ n )). Stanley [, Theorem.] showed that the map Ω (in his notation, Λ) is a bijection between M n and P n. Thus we have the following consequence. Theorem The map Φ:F (n) P n defined by is a bijection. Φ( (w z ),...,(w n z n ))=(w,...,w n ) Proof. It is easy to show that So Φ is a bijection. Φ=Ω Σ Γ. Remark. It is well known that P n = n n. So Theorem gives a bijective proof of () for the case λ =(n). In fact, the resulting bijection Φ is the same as given by Biane [], who proved it without Stanley s result. In Figure, we give an example which summarizes all the consequences in this section. Note that all maps are bijective and all diagrams commute. Main Result We now proceed to construct a combinatorial proof of (). Let (τ,...,τ m ) be an element of F (,n ). Since λ has two parts, i.e. l =,wehavem = n and there is only one cut among τ,...,τ n (Note that τ n cannot be a cut). Moreover, since τ τ n = ()( n), the unique cut has the form ( r) for some r, r n. LetG r n,k be the set of all minimal factorizations (τ,...,τ n ) F (,n ) which have a cut ( r) atthek-th position, i.e. τ k =(r). Set Clearly, G r n = n k= F (,n ) = G r n,k. n Gn. r We first observe that the cardinality of Gn,k r is independent of r. Lemma Let θ be the permutation ()( n). For any r and s, r s n, the mapping Θ s r : Gn,k r Gs n,k defined by r= Θ s r (τ,...,τ n )=(θ s r τ θ r s,...,θ s r τ n θ r s ) is a bijection. Moreover, Θ s r can be extended to the bijection between G r n and G s n.

7 F : ( ( ),( ),( ),( ),( ),( ),( 9),( ) ) Φ P : (,,,,,,, ) Γ Ω Ω C : m : / / / 9 / / / 9 Σ / / / / 9 9 / / / / / 9 / / / / / / 9 / / / / / / / 9 / / / / / / / / 9 Figure : Bijections between F (n), C n, M n and P n.

8 Proof. Clearly it is a bijection since θ s r τ i θ r s fixes and changes r into s in τ i. A parking function (x,...,x n ) is said to be prime, if the increasing rearrangement y y n of x,...,x n satisfies y i <ifor i =,...,n. Let Q n denote the set of prime parking functions Q =(x,...,x n ) of length n and define Q n,k to be the set of all Q Q n in which x k = is the leftmost occurrence of. Clearly, we have Q n = n k= Q n,k. It is clear that the prime parking functions satisfy the following: Lemma (x,...,x n ) Q n,k if and only if (x,...,x k,x k+,...,x n ) P n with x i for i =,,..., k. We will eventually show that G n,k and Q n,k have the same cardinality. Let A n,k be the set consisting of all (σ,...,σ n ) F (n) such that (a) occurs in σ k,...,σ n. (a) does not occur in σ,...,σ k, Lemma The map Ψ : G n,k A n,k defined by is a bijection. Ψ (τ,...,τ n ) = (( )τ ( ),...,( )τ k ( ),τ k+,...,τ n ), Proof. Set τ i =()τ i ( ). Assume that (τ,...,τ n ) G n,k. Since (τ,...,τ n ) F (,n ), τ τ n = ()( n) andτ k = ( ). Multiplying both sides with ( ) yields τ τ k τ k+ τ n =( n). This implies that ( τ,..., τ k,τ k+,...,τ n ) F (n). In addition, since τ k = ( ) is a cut, τ k+ τ n doesn t fix and so there exists a j greater than k such that τ j contains. Moreover, since τ k is the unique cut in (τ,...,τ n ), doesn t appear in any of τ,...,τ k (otherwise, can not be a fixed point in τ τ n ). Thus τ,..., τ k do not have. Hence ( τ,..., τ k,τ k+,...,τ n )isina n,k. We now show that Ψ has the inverse. Define the map Ψ : A n,k G n,k by Ψ (σ,...,σ n )=( σ,..., σ k, ( ),σ k,...,σ n ). We first prove that Ψ is well-defined. Since (σ,...,σ n ) F (n),wehaveσ σ n =( n). Multiplying both sides with ( ) yields σ σ k ( )σ k σ n = ()(...n). This implies that F =( σ,..., σ k, ( ),σ k,...,σ n ) F (,n ). Since (σ,...,σ n ) F (n) has no cut, none of σ k,..., σ n is a cut in F. We now show that ( ) is a cut in F. Suppose that it is a join in F. The unique cut of F is σ j for some j less than k. So is a fixed point of σ j σ k ( )σ k σ n.but from the condition (a), appears at least twice among σ j+,..., σ k, ( ),σ k,...,σ n and is contained in a cycle of length at least in σ j+ σ k ( )σ k σ n. So σ j is of the form ( m) and appears in σ j, contradicting to the condition (a). Therefore, the k-th transposition ( ) is a cutandwehave( σ,..., σ k, ( ),σ k,...,σ n ) G n,k. Clearly, Ψ Ψ is the identity map and so Ψ is a bijection. We now show that A n,k and Q n,k are in - correspondence.

9 Lemma The map Ψ : A n,k Q n,k defined by Ψ (σ,...,σ n )=(ã,...,ã k,,a k,...,a n ), where σ i =(a i b i ) and ã i is the image of a i under the transposition ( ), is a bijection. Proof. Let (σ,...,σ n ) be an element of A n,k. Then (a,...,a n ) P n with / {a,...,a k } and {a k,...,a n }, which imply (ã,...,ã k,a k,...,a n ) P n with / {ã,...,ã k }.So by Lemma, (ã,...,ã k,,a k,...,a n ) Q n,k. The map Ψ has the inverse. It can be shown easily that the map Ψ : Q n,k A n,k defined by is the inverse of Ψ. Hence Ψ is a bijection. Ψ (x,...,x k,,x k+,...,x n )=Φ ( x,..., x k,w k+,...,x n ) Theorem The mapping Ψ:G n,k Q n,k defined by Ψ( (a b ),...,(a n b n ))=(a,...,a n ), is a bijection. Moreover, Ψ can be extended to a bijection between G n and Q n. Proof. Since (a i b i ) ()fori =,,...,k, we clearly have Ψ = Ψ Ψ, where Ψ and Ψ are the bijections defined in Lemmas and. Since Gn = n k= G n,k and Q n = n k= Q n,k, Ψ can be defined as a mapping from Gn to Q n. Now we are ready to finish a combinatorial proof of (). Corollary The mapping Θ:Q n [n ] F (,n ) defined by is a bijection. Θ(Q, t) =Θ t (Ψ (Q)), Proof. Θ t and Ψ are bijections and Θ t (Gn)=G n t+ for t =,...,n. Since n F (,n ) = Gn, r Θ is a bijection. (See Figure.) Since Q n =(n ) n ([, pp. 9]), Corollary is a combinatorial proof of F (,n ) =(n ) n. r= Remark With a variation of the depth-first-search, we are able to find a bijection between Q n,k and the set R n,k of rooted labelled trees on [n] such that the root k is a local minimum of the tree. Chauve, Dulucq and Guibert [] enumerated the cardinality of R n,k with a bijective proof. Their result is: R n,k =(n k) n n k (n ) k. So we are able to count not only F (,n ) but also Gn,k, k =,,...,n, which are refinements of F (,n ). Moreover, by this refinement, our approach can be extended to handle the case λ =(,n ). 9

10 G n {} Θ 0 G n Q n [n ] Ψ id G n {} Θ G n = F (,n ) G n {n } Θ n G n n Acknowledgment Figure : The map Θ : Q n [n ] F (,n ) We are thankful to an anonymous referee and Mireille Bousquet-Mélou for bringing [] to our attention. This work was partially supported by KOSEF: R References [] V.I. Arnold, Topological classification of trigonometric polynomials and combinatorics of graphs with an equal number of vertices and edges, (English. Russian original) Funct. Anal. Appl. 0 no. (99), ; translation from Funkts. Anal. Prilozh. 0 no. (99),. [] P. Biane, Parking functions of types A and B, Electronic J. Combinatorics 9 no. (00), N. [] M. Bousquet-Mélou and G. Schaeffer, Enumeration of planar constellations, Adv. in Appl. Math. no. (000),. [] C. Chauve, S. Dulucq and O. Guibert, Enumeration of some labelled trees, Technical Report RR--99, LaBRI, Université Bordeaux I (999). [] M. Crescimanno and W. Taylor, Large N phases of chiral QCD, Neuclear Phys. B, (99),. [] J. Dénes, The representation of a permutation as the product of a minimal number of transpositions and its connection with the theory of graphs, Publ. Math. Institute Hung. Acad. Sci. (99), 0. [] C.L. Ezell, Branch point structure of covering maps onto nonorientable surfaces, Trans. Amer. Math. Soc. (9),. [] I. P. Goulden and D. M. Jackson, Transitive factorizations into transpositions and holomorphic mappings on the sphere, Proc. Amer. Math. Soc. (99), 0. [9] I. P. Goulden and S. Pepper, Labelled trees and factorizations of a cycle into transpositions, Discrete Math. (99),. [0] I. P. Goulden and A. Yong, Tree-like properties of cycle factorizations, Journal of Combinatorial Theory, Series A 9 (00), 0. 0

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