Inversion Formulae on Permutations Avoiding 321

Transcription

1 Inversion Formulae on Permutations Avoiding 31 Pingge Chen College of Mathematics and Econometrics Hunan University Changsha, P. R. China. Suijie Wang College of Mathematics and Econometrics Hunan University Changsha, P. R. China. Zhousheng Mei College of Mathematics and Econometrics Hunan University Changsha, P. R. China. Submitted: July 9, 015; Accepted: Nov 4, 015; Published: Nov 13, 015 Abstract We will study the inversion statistic of 31-avoiding permutations, and obtain that the number of 31-avoiding permutations on [n] with m inversions is given by S n,m (31) = b m (b) ). l(b) where the sum runs over all compositions b = (b 1,b,...,b k ) of m, i.e., m = b 1 +b + +b k and b i 1, l(b) = k is the length of b, and (b) := b 1 + b b b k b k 1 + b k. We obtain a new bijection from 31-avoiding permutations to Dyck paths which establishes a relation on inversion number of 31-avoiding permutations and valley height of Dyck paths. Keywords: pattern avoidance; Catalan number; Dyck path; generating function Supported by NSFC & the electronic journal of combinatorics (4) (015), #P4.8 1

2 1 Introduction Let S n denote the permutation group on [n] = {1,,...,n}. Write σ S n in the form σ = σ 1 σ σ n. For m n, if σ S n and π = π 1 π m S m, we say that σ contains the pattern π if there is an index subsequence 1 i 1 < i < < i m n such that σ ij < σ ik iff π j < π k for 1 j,k m, that is, σ has a subsequence which is order isomorphic to π. Otherwise, σ avoids the pattern π, or say, σ is π-avoiding. We denote by S n (π) the set of all permutations σ S n that are π-avoiding, i.e., S n (π) = {σ S n σ avoids the pattern π}. For example, the permutation 4153 avoids the pattern 31, but contains the pattern 13 since its subsequence 153 is order isomorphic to 13, hence 4153 S 5 (31) and 4153 / S 5 (13). In 1970 s, Knuth [1, 13] obtained a well known result on permutations avoiding patterns, that is for any π S 3, S n (π) = C n = 1 ( ) n, n+1 n where C n is the n-th Catalan number which counts the number of Dyck paths of length n. In past decades, various articles considered the bijections between 31-avoiding permutations and Dyck paths, see [4, 7, 10, 11, 14, 15, 17, 18, 19, 1, ]. In this paper, we will study the inversion distribution of 31-avoiding permutations. For σ = σ 1 σ σ n S n (π), we define the inversion set Inv(σ) to be Inv(σ) = {(σ i,σ j ) i < j and σ i > σ j }, and denote by inv(σ) = #Inv(σ), called the inversion number of σ, where the hash sign denotes cardinality. The generating function I n (π,q) of the inversion numbers is I n (π,q) = σ S n(π) q inv(σ). for σ S n (π). This generating function was first introduced and explored in [8, 0] and some recurrence formulae have been obtained for π S 3 and π 31. Conjecture 3. of [8] states that, for all n 1, n I n (31,q) = I n 1 (31,q)+ q i+1 I i (31,q)I n i 1 (31,q). (1) i=0 Soon afterwards a bijective proof of the recursive formula (1) was obtained by Szu-En Cheng et al. [6]. There are some other works on inversions of restricted permutations, see [1, 3, 5, 9, 15, 16]. In 014, M. Barnabei, F. Bonetti, S. Elizalde and M. Silimbani [] studied the distribution of descents and major indexes of 31-avoiding involutions. the electronic journal of combinatorics (4) (015), #P4.8

3 Motivated by [, 6], in this paper we will study the inversion distribution of 31- avoiding permutations. As the main result, we give an explicit formula counting the number of 31-avoiding permutations with the fixed inversion number. We also find a bijection between 31-avoiding permutations and Dyck paths, which is new to the best of our knowledge. From this bijection, we show that the inversion number of 31-avoiding permutations and the valley-sum of Dyck paths are equally distributed. Inversions of Permutations Avoiding 31 For 1 k n, let Sn k (31) be the collection of 31-avoiding permutations of [n] and containing 1 k as a subsequence, S k n(31) = {σ S n (31) σ 1 (1) < σ 1 () < < σ 1 (k)}. More precisely, if σ = σ 1 σ σ n S k n(31) and σ i1 = 1,σ i =,...,σ ik = k, then i 1 < i < < i k. Obviously, we have S n (31) = Sn 1 (31) S n (31) Sn n (31) = {id}. For 1 k n, let In k (31,q) be the generating function defined by In k (31,q) = q inv(σ). σ Sn k(31) Then we have I n (31,q) = In 1(31,q) and In n (31,q) = 1 for all n 1. Lemma 1. For 1 k n, we have I k n(31,q) = I k+1 n (31,q)+ q i In+i k 1(31,q). i Proof. Given σ Sn k(31) with 1 k n 1, consider the position of σ 1 (k + 1). Assuming σ 1 (0) = 0, we have either σ 1 (k) < σ 1 (k + 1), or σ 1 (i) < σ 1 (k + 1) < σ 1 (i+1) for some i k 1. (i): If σ Sn k(31) and σ 1 (k) < σ 1 (k+1), it follows that σ Sn k+1 (31). So this case contributes a term In k+1 (31, q) to the generating function In(31,q). k (ii): If σ Sn(31) k and σ 1 (i) < σ 1 (k +1) < σ 1 (i+1) for some i k 1, since σ avoids the pattern 31, it forces that σ 1 (j) > σ 1 (k + 1) for all j k +. Otherwise, we have σ 1 (j) < σ 1 (k+1) < σ 1 (i+1) which is obviously a contradiction. It implies that σ = σ 1 σ σ n satisfies σ 1 = 1,σ =,...,σ i = i,σ i+1 = k + 1. Denote by σ = σ i+ σ i+3 σ n. Then σ is a permutation of {i+1,...,k,k +,...,n} satisfying σ 1 (i+1) < σ 1 (i+) < < σ 1 (k) and inv(σ) = k i+inv( σ). It implies that case (ii)contributes a term q k i I k i n i 1 (31,q) to Ik n (31,q) for 0 i k 1. Changing the index i to k i, the proof will be complete by combining (i)and (ii). the electronic journal of combinatorics (4) (015), #P4.8 3

4 In the sequel, we always denote by δ : R {0,1} a function such that δ(u,v) = { 0, u = v; 1, otherwise. In order to characterize the generating function I n (31,q) as a counting function of lattice points in a lattice polytope, we introduce the following lemma. Lemma. Assuming x 0 = 0, for all 1 t n, we have 1 t t In+1 1 (31,q) = I x 1 =0 x =x 1 t+1 xt n+1 x t (31,q) x t=x t 1 q δ(x i,x i 1 )(i+1 x i ) Proof. The statement is true for t = 1 by Lemma 1. To use induction on t, suppose the above equality holds for t. From Lemma 1, we have I t+1 xt n+1 x t (31,q) = t+1 I t+ xt+1 n+1 x t+1 (31,q) q δ(x t+1,x t)(t+ x t+1 ). x t+1 =x t Using above formula to substitute the term I t+1 xt n+1 x t (31,q) in the formula of this Lemma, we can easily conclude that the equality holds for the case t+1. Let Ω n be a convex lattice polytope defined by Ω n = {(x 1...,x n ) Z n 0 x 1 x i i for all 1 i n}. Recall that I n+1 (31,q) = In+1 1 (31,q) and In+1 xn n (31,q) = 1. From above lemma by taking t = n we can easily obtain Proposition 3. Assuming x 0 = 0, we have I n+1 (31,q) = x Ω n n q δ(x i,x i 1 )(i+1 x i ). In the following we will give a more explicit interpretation about this formula. Let inv k (σ) be the number of inversions of σ whose first element is k, i.e, inv k (σ) = #{i (k,i) Inv(σ)} It is obvious that inv k (σ) k 1. From the definition of I n+1 (31,q) and Proposition 3, we have q inv(σ) = q n δ(x i,x i 1 )(i+1 x i ) x Ω n σ S n+1 (31) Below we recursively define a map ϕ : S n+1 (31) Ω n, ϕ(σ) = (x 1,...,x n ) = x, () the electronic journal of combinatorics (4) (015), #P4.8 4

5 Figure 1: ϕ : σ = (0,0,,0,1,0,,0,3) x = (0,1,1,4,4,5,5,6) such that x 1 = inv (σ) and for k, { x x k = k 1, if inv k+1 (σ) = 0; k +1 inv k+1 (σ), otherwise. Figure.1 shows an example, where the second vector is ( inv 1 (σ),...,inv 9 (σ) ). Theorem 4. The map ϕ defined above is a bijection. Moreover, if ϕ(σ) = (x 1,...,x n ), then n inv(σ) = δ(x i,x i 1 )(i+1 x i ). Proof. We first show that ϕ is well defined in the sense that if x = ϕ(σ) then x Ω n = {(x 1,...,x n ) Z n 0 x 1 x i i for all 1 i n}. We use induction on i. For i = 1, it is obvious x 1 = inv (σ) 1. Suppose 0 x 1 x i 1 i 1. If inv i+1 (σ) = 0, then x i = x i 1 i by the induction hypothesis. If inv i+1 (σ) 0, then x i = i + 1 inv i+1 (σ) i. It remains to show that if inv i+1 (σ) 0, then x i 1 x i. Let k i be maximal such that inv k (σ) 0, i.e., inv k+1 (σ) = = inv i (σ) = 0. It follows that there exists an inversion (k,l) Inv(σ). Since σ is 31-avoiding, we have σ 1 (k) < σ 1 (i+1), otherwise (i+1,k,l) is a subsequence of σ and of type 31. Hence we obtain inv i+1 (σ) inv k (σ)+i k, and x i = i+1 inv i+1 (σ) k +1 inv k (σ) = x k 1 +1 > x k 1. Note that inv k+1 (σ) = = inv i (σ) = 0. By definitions, we have x k 1 = = x i 1 which provesthatϕiswelldefined. Toprovethemapϕisabijection, notethateachpermutation σ can be uniquely recovered from its inversion vector ( inv 1 (σ),...,inv n+1 (σ) ). Now we construct an inverse map ψ : Ω n S n+1 (31) of ϕ recursively as follows. Given x = (x 1,...,x n ), define ψ(x) = σ = σ 1 σ n+1 such that inv 1 (σ) = 0 and for k n+1, { 0 if x inv k (σ) = k 1 = x k ; k x k 1 otherwise. It is not difficult to see that both ψ ϕ and ϕ ψ are identity map, i.e., ϕ is a bijection. This completes the proof. the electronic journal of combinatorics (4) (015), #P4.8 5

6 Figure : x = (0,1,1,4,4,5,5,6) D = uuduuddduuduududdd When n = 3, the inversion polynomial of S n (31) is I 3 (31,q) = q 4 +4q 3 +5q +3q+1. Below is the list of the bijection ϕ, q 0 : {134} ϕ {(0,0,0)}; q 1 : {143,134,134} ϕ {(0,0,3),(1,1,1),(0,,)}; q : {134,143,143,314,314} ϕ {(0,,3),(0,0,),(1,1,3),(1,,),(0,1,1)}; q 3 : {341,413,314,413} ϕ {(1,,3),(1,1,),(0,1,3),(0,0,1)}; q 4 : {341} ϕ {(0,1,)}. A Dyck path D is a lattice path from (0,0) to (n,0) in the (x,y)-plane with up-steps (1,1) (abbreviated as u ) and down-steps (1, 1) (abbreviated as d ), such that D never falls below the x-axis. A valley du of the Dyck path D is a down-step followed by an up-step. The height of a valley is defined to be the y-coordinate of its bottom. Denote by D n the set of all Dyck paths of length n. Several bijections between S n (31) and D n have been established in the literature, see [4, 7, 10, 11, 14, 15, 17, 18, 19, 1, ]. Here we will give a new bijection obtained easily from the above theorem. Morever, this bijection will allow to read the inversion number of a permutation as the sum of all valley heights and the number of valleys in the corresponding Dyck path. Indeed, for x = (x 1,...,x n ) Ω n, assuming x 0 = 0 and x n+1 = n+1, we construct a Dyck path D x as follows. By reading i from 1 to n+1, for each i we add an up-step and x i x i 1 down-steps from left to right. Figure. presents an example. It is obvious that this construction gives an bijection from Ω n+1 to D n+1. If all valleys of a Dyck path D have heights a 1,...,a k, denote by v(d) = (a i +1). Combining with Theorem 4, we can easily obtain our first main result. Theorem 5. The map σ D ϕ(σ) is a bijection from S n+1 (31) to D n+1 such that where ϕ is defined in (). inv(σ) = v(d ϕ(σ) ), As an application of Theorem 5, we will give a counting formula on the number of 31-avoiding permutations with a fixed inversion number. For any D D n, we define a the electronic journal of combinatorics (4) (015), #P4.8 6

7 tunnel of D to be a horizontal segment between two lattice points of D that intersects D only in these two points, and stays always below D. From Theorem 5, for m 0, there is a bijection S n,m (31) := {σ S n (31) : inv(σ) = m} D n,m := {D D n : v(d) = m}. Theorem 6. For every m 0, S n,m (31) = b m (b) ). l(b) where the sum runs over all compositions b = (b 1,b,...,b k ) of m, denoted b m, i.e., m = b 1 +b + +b k and b i 1, l(b) = k is the length of b, and (b) := b 1 + b b b k b k 1 + b k. Proof. It is sufficient to consider D n,m. For any D D n,m, suppose that D has k valleys with heights a 1,a,,a k, then m = v(d) = k (a i + 1). Let l i be the length of the path D located between the i-th and (i+1)-th valley, for i = 0,1,,,k. Then we have l i = a i+1 a i +t i, l i = n, t i 1. i=0 Where t i is the number of tunnels between the i-th and (i+1)-th valley. Let a 0 = 0 and a k+1 = 0 be the heights of the starting point and the terminal point of the Dyck path D, respectively. Write (a) = a i+1 a i. Then i=0 #{D D n all valleys of D have heights a 1,a,...,a k } = #{(l 0,l 1,,l k ) l i = a i+1 a i +t i, l i = n,t i 1}. = #{(t 0,t 1,,t k ) t 0 +t 1 + +t k = n (a),t i 1}. (a) ) 1 = k i=0 So we have D n,m = (a 1 +1)+(a +1)+ +(a k +1)=m a i +1 1 (a) ) 1 k the electronic journal of combinatorics (4) (015), #P4.8 7

8 Let b i = a i +1 for 1 i k, b 0 = b k+1 = 0, obviously (b) = k i=0 b i+1 b i = (a)+. Hence (b) ) D n,m =. k b 1 +b + +b k =m b i 1 Acknowledgements We would like to express our gratitude to the anonymous referee for many useful comments. References [1] J. Bandlow, Eric S. Egge, and K. Killpatrick. A weight-preserving bijection between Schröer paths and Schröer permutations. Ann. Comb., 6(3-4): 35 48, 00. [] M. Barnabei, F. Bonetti, S. Elizalde, and M. Silimbani. Descent sets on 31-avoiding involutions and hook decompositions of partitions. J. Combin. Theory Ser. A, 18:13 148, 014. [3] Andrew M. Baxter. Refining enumeration schemes to count according to the inversion number. Pure Math. Appl. (PU. M. A.), 1(): , 010. [4] S. C. Billey, W. Jockusch, and R. P. Stanley. Some combinatorial properties of Schubert polynomials. J. Algebraic Combin., (4): , [5] William Y. C. Chen, Yu-Ping Deng, and Laura L.M. Yang. Motzkin paths and reduced decompositions for permutations with forbidden patterns. Electron. J. Combin., 9(), #R15, 003. [6] S. E. Cheng, S. Elizalde, A. Kasraouic, and B. E. Sagan. Inversion polynomials for 31-avoiding permutations. Discrete Math., 313():55 565, 013 [7] A. Claesson and S. Kitaev. Classification of bijections between 31- and 13-avoiding permutations. Sém. Lothar. Combin. 60, Art. B60d, 30 pp, 008. [8] T. Dokos, T. Dwyer, B. P. Johnson, B. E. Sagan, and K. Selsor. Permutation patterns and statistics. Discrete Math., 31(18): , 01. [9] Eric S. Egge. Restricted 341-avoiding involutions, continued fractions, and Chebyshev polynomials. Adv. in Appl. Math., 33(3): , 004. [10] S. Elizalde and E. Deutsch. A Simple and Unusual Bijection for Dyck Paths and its Consequences. Ann. Comb., 7(3):81 97, 003. [11] S. Elizalde and I. Pak. Bijections for refined restricted permutations. J. Combin. Theory Ser. A, 105():07 19, 004. [1] D. E. Knuth. The Art of Computer Programming I: Fundamental Algorithms, Addison-Wesley, Publishing Co., Reading, Mass.-London-Don Mills, Ont, the electronic journal of combinatorics (4) (015), #P4.8 8

9 [13] D. E. Knuth. The Art of Computer Programming III: Sorting and Searching, Addison-Wesley, Reading, MA, [14] C. Krattenthaler. Permutations with restricted patterns and Dyck paths. Adv. in Appl. Math., 7(-3): , 001. [15] T. Mansour, Eva Y. D. Deng, and Rosena R. X. Du. Dyck paths and restricted permutations. Discrete Appl. Math., 154(11): , 006. [16] S. Min and S. Park. The maximal-inversion statistic and pattern-avoiding permutations. Discrete Math., 309(9): , 009. [17] A. Reifegerste. A generalization of Simion-Schmidt s bijection for restricted permutations. Electron. J. Combin., 9(), #R14, 003. [18] D. Richards. Ballot sequences and restricted permutations. Ars Combin., 5:83 86, [19] D. Rotem. On a correspondence between binary trees and a certain type of permutations. 4(3):58 61, [0] B. E. Sagan and C. D. Savage. Mahonian pairs. J. Combin. Theory Ser. A, 119(3):56 545, 01. [1] R. Simion and F. W. Schmidt. Restricted Permutations. European J. Combin., 6(4): , [] J. West. Generating trees and the Catalan and Schröder numbers. Discrete Math., 146(1-3):47 6, the electronic journal of combinatorics (4) (015), #P4.8 9