Signed Countings of Type B and D Permutations and t,q-Euler numbers

全文

(1)

(2) D EPARTMENT OF M ATHEMATICS N ATIONAL TAIWAN N ORMAL U NIVERSITY. A thesis submitted in partial fulfilment of the requirements for the degree of. M ASTER OF S CIENCE in. M ATHEMATICS. Signed Countings of Type B and D Permutations and t, q-Euler numbers. Supervisor Dr. Sen-Peng E U. Author Hsin-Chieh L IAO. June 27, 2018.

(3) Acknowledgements 兩年時光飛逝, 回想大學畢業至今，經歷過不少打擊卻也讓視野更開闊，對數學看法和態度也改變甚多，這之中首先必須感謝我的指導教授游森棚教官，沒有教官聽我訴苦和適時的鼓勵及建議我恐怕早就在自我懷疑中黯然離開學界。這兩年間教官讓我看見作為一個數學家和一個老師的榜樣，在教官門下也見識了組合學深奧又美麗的一面，不管是上教官的課還是每個禮拜的 meeting 總是令我獲益良多，既開了眼界也讓我了解討論數學的樂趣，我非常榮幸能成為教官的學生。感謝祥峻學長自從兩年前我入教官門下就陪我討論各種組合問題，有時甚至幫我釐清自己的想法，學長對問題快、狠、準的看法和各種創意每每讓我驚嘆不已，也感謝學長隨時提供程式諮詢服務，使我投向 python 的懷抱。謝謝維良在我在師大這兩年跟我討論課業上的問題、分享一些新學到的知識或交換一些數學概念的看法。感謝師大的劉家新老師和台大的康明昌老師，在他們的課堂中我學到很多。感謝教官、祥峻學長、傅東山老師在這篇論文研究的主題上的各種討論促成了這篇論文的誕生，也感謝口試委員林延輯老師和丁建太老師對論文提出的建議，使這篇論文更加完整，沒有他們的幫助這篇論文不可能完稿。最後，我要感謝我的父母，他們營造了一個沒有壓力的環境，讓我可以毫無顧忌追求自己的興趣，對於我任性地做自己想做的事情也總是給予無條件的支持。. 謹以此論文獻給我在天上的母親感謝她 25 年來無怨無悔的付出 2018.6.26 廖信傑.

(4) Abstract A classical result states that the parity balance of number of excedances of all permutations (derangements, respectivly) of length n is the Euler number. In 2010, Josuat-Vergès gives a q-analogue with q representing the number of crossings. We extend this result to the permutations (derangements, respectively) of type B and D. It turns out that the signed counting are related to the derivative polynomials of tan and sec. Springer numbers defined by Springer can be regarded as an analogue of Euler numbers defined on every Coxeter group. In 1992 Arnol’d showed that the Springer numbers of classical types A, B, D count various combinatorial objects, called snakes. In 1999 Hoffman found that derivative polynomials of sec x and tan x and their subtraction evaluated at certain values count exactly the number of snakes of certain types. Then Josuat-Vergès studied the (t, q)-analogs of derivative polynomials Qn (t, q), Rn (t, q) and showed that as setting q = 1 the polynomials are enumerators of snakes with respect to the number of sign changing. Our second result is to find a combinatorial interpretations of Qn (t, q) and Rn (t, q) as enumerator of the snakes, although the outcome is somewhat messy. Key words: Signed permutations, Euler numbers, Springer numbers, qanalogue, continued fractions, weighted bicolored Motzkin paths.

(5) Contents 1. Motivation of the problems 1.1 Signed countings on Permutations and Derangements . . . . 1.2 q-analogue of the signed counting identities . . . . . . . . . .. 2. Signed Permutations and Snakes 2.1 Signed Permutations . . . . . . . . . . . . . . . . 2.2 Crossing of type B . . . . . . . . . . . . . . . . . . 2.3 Refined Enumeration on Singed Permutations . 2.4 Gerneralized Euler numbers: Springer numbers 2.5 Snakes of type B . . . . . . . . . . . . . . . . . . .. 3 3 4. . . . . .. 8 8 9 10 11 12. 3. Weighted Motzkin paths and Signed Permutations 3.1 Continued fractions and weighted Motzkin paths . . . . . . 3.2 Linking signed permutations to bicolored Motzkin paths . . 3.3 Weight Schemes for Qn (t, q) and Rn (t, q) . . . . . . . . . . . .. 16 16 17 18. 4. Signed Countings on type B and D 4.1 Signed Countings on type B and D . . . . . . . . . . . . . . . 4.2 The cases of Bn and Dn . . . . . . . . . . . . . . . . . . . . . . 4.3 The cases of Bn∗ and Dn∗ . . . . . . . . . . . . . . . . . . . . . .. 20 20 22 28. 5. Snakes and (t, q)-analogue of derivative polynomials 5.1 cs-vectors and blocks . . . . . . . . . . . . . . . . . . . . . . . 5.2 The enumerator Qn (t, q) of Sn0 . . . . . . . . . . . . . . . . . . 5.3 The enumerator Rn (t, q) of Sn00+1 . . . . . . . . . . . . . . . . .. 32 32 34 38. 6. Discussions. 42. Bibliography. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. 45. 2.

(6) Chapter 1. Motivation of the problems In this chapter the classical signed counting results of Euler and Roselle on permutations and derangements with respect to Eulerian statistics are presented. Then we introduce Josuat-Vergès’ q-analogues with q representing the number of crossings. These classical signed counting results serve as motivations of our study.. 1.1 Signed countings on Permutations and Derangements Let Sn denote the set of permutations on [n] := {1, 2, . . . , n} and S∗n denote the set of derangements on [n]. A permutation σ ∈ Sn is a bijection on [n] and we may write σ as σ1 σ2 . . . σn (σi ∈ [n]) if σ(i ) = σi for all 1 ≤ i ≤ n. This is called the one-line natation of the permutation σ. Definition 1.1.1. For a permutation σ = σ1 σ2 · · · σn ∈ Sn , an excedance (weak excedance, respectively) is an integer i ∈ [n] such that σi > i (σi ≥ i, respectively). Let exc(σ) and wex(σ) denote the number of excedances and the number of weak excedances of σ, respectively. An elementary result is that the statistics exc and wex have same distribution in Sn and ∑σ∈Sn ywex(σ) = y ∑σ∈Sn yexc(σ) . The polynomial An (y) = ∑σ∈Sn ywex(σ) is called the Eulerian polynomial and An,k = #{σ ∈ Sn |wex(σ) = k} is called the Eulerian number.. 3.

(7) Definition 1.1.2. The classical Euler numbers En are defined by xn. ∑ En n!. = tan x + sec x = 1 + x +. n ≥0. x2 x3 x4 x5 + 2 + 5 + 16 + . . . . 2! 3! 4! 5!. The numbers E2n are called the secant numbers and the numbers E2n+1 are called the tangent numbers. It is well-known that En counts the number of alternating permutations in Sn , i.e., σ ∈ Sn such that σ1 > σ2 < σ3 > . . . σn . For example, when n = 3, there are 2 alternating permutations 213, 312; when n = 4 there are 5 alternating permutations 2143, 3142, 3241, 4132, 4231. A permutation σ satisfying σ1 < σ2 > σ3 < . . . σn is called a reverse alternating permutation. An interesting result states that when we evaluate the Eulerian polynomial An (y) at y = −1 depending on the parity of n we either obtain 0 or tangent numbers. Theorem 1.1.3 (Euler[9]; Foata, Schützenberger[12]). 0, if n is even, exc( σ) = n −1 ∑ (−1) (−1) 2 En , if n is odd. σ ∈Sn. (1.1). This identity was first discovered by Euler[9] in a different form when he introduced Eulerian polynomials, the presenting form of the identity was obtained by Foata and Schützenberger[12]. Interestingly, the other half of the result shows up while we restrict our attention on the derangements in Sn . Theorem 1.1.4 (Roselle[20]).. ∑ σ∈S∗n. (−1). exc( σ). =. . n. (−1) 2 En , if n is even, 0, if n is odd.. (1.2). This was first obtained by Roselle[20] using a slightly different combinatorial interpretation.. 1.2. q-analogue of the signed counting identities. As we can see in (1.1), (1.2), both sides of the identities occur in Sn , it is natural to seek q-analogues of (1.1), (1.2). In fact there are three different q-analogues that have been discovered by Foata and Han [11], JosuatVergès[18], Shin and Zeng [21] respectively. In this section we will introduce the one obtained by Josuat-Vergès.. 4.

(8) 1. 2. 3. 5. 4. 6. 7. Figure 1.1: The diagram of σ = 6453172 To begin with, we need to introduce the corresponding q-analogous of Eulerian polynomials and Euler numbers. Definition 1.2.1. A crossing of a permutation σ = σ1 σ2 . . . σn is a pair of (i, j) (1 ≤ i < j ≤ n) such that i < j ≤ σi < σj or σi < σj < i < j. We denote by cro(σ) the number of crossings in σ. Crossings of a permutation can be visualized via permutation diagram, see Figure 1.1. Let σ = 6453172 then the crossing of σ are (2, 3), (1, 6), (5, 7), hence cro(σ) = 3. Then we have a q-analogue of Eulerian numbers qcro(σ). ∑. An,k (q) =. (1.3). σ ∈Sn wex( σ)= k. and the corresponding q-Eulerian polynomials n. A(y, q) =. ∑ An,k (q)yk = ∑. ywex(σ) qcro(σ) .. σ ∈Sn. k=1. The notion of crossings of a permutation was first introduced by Williams [25] along with another notion called alignments in the study of totally positivity Grassmann cells. In [25], Williams also define An,k (q) though in terms of alignments. But a simple relation between the number of aligenments and the number of crossings shown later by Corteel [4] gives the equivalent definition in (1.3). The following q-analogue of Euler number was introduced by Han, Randrianarivony, Zeng [16].. 5.

(9) Definition 1.2.2. The q-tangent numbers E2n+1 (q) are defined by ∞. 1 [ 1] q [ 2] q z. ∑ E2n+1(q)zn = n =0. 1−. (1.4). [ 2] q [ 3] q z [ 3] q [ 4] q z 1− .. .. 1−. and the q-secant numbers E2n (q) are defined by ∞. 1 [1]2q z. ∑ E2n (q)zn = n =0. 1−. 1−. (1.5). [2]2q z [3]2q z 1− .. .. The first few polynomials are E0 (q) = E1 (q) = E2 (q) = 1, E3 (q) = 1 + q, E4 (q) = 2 + 2q + q2 , E5 (q) = 2 + 5q + 5q2 + 3q3 + q4 . The polynomial En (q) has a combinatorial interpretation [3, 18]:. ∑. En ( q ) =. q31-2(σ). σ∈Altn. where Altn is the set of alternating permutations of length n and 31-2(σ) = #{(i, j) : i + 1 < j, σi+1 < σj < σi }. Using the above En (q) and the number of crossings cro, Josuate-Vergès [18] derived q-analogs of Eqs (1.1) and (1.2). Theorem 1.2.3 (Josuate-Vergès [18]).. ∑. (−1). π ∈Sn. wex( π ) cro( π ). q. =. . and 1 ∑ ∗ (− q )wex(π ) qcro(π ) = π ∈S n. 0 if n is even, n +1 (−1) 2 En (q) if n is odd;. (. 6. (1.6). n. (− 1q ) 2 En (q) if n is even, 0 if n is odd.. (1.7).

(10) Example 1.2.4. When n = 4, from table 1.1 we have 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. ∑ ∗ (− q )wex(π ) qcro(π ) = q2 − q + q + q + 1 + q + q − q + q + q2. π ∈Sn. 2 2 + +1 q2 q 1 =(− )2 (2 + 2q + q2 ) q 1 2 =(− ) E4 (q) q. =. S4∗ 2143 2341 2413 3142 3412 4123 4132 4312 4321. wex 2 3 2 2 2 1 2 2 2. cro 0 2 1 1 2 0 1 1 0. Alt4 2143 3142 3241 4132 4231. 31-2 0 1 0 2 1. Table 1.1 Note that the symmetric group Sn is just the finite irreducible Coxeter group of type A. In type B and type D, there are combinatorial models similar to permutations. Fortunately, the notions we have mentioned, for instance wex, cro, also have type B analogues. One of our purpose in this work is to extend the results of (1.1),(1.2) to type B and D.. 7.

(11) Chapter 2. Signed Permutations and Snakes In this chapter we introduce the type B and D analogues of several notions including signed and even signed permutations, flag weak excedence, crossings of type B, and the corresponding enumerators. Then we briefly describe the type-free analogue of Euler number-Springer numbers. We focus on the Springer number of type B. The combinatorial model of Sn , which is called the snakes of type B, are introduced along with other types of snakes. At last we present the connection between snakes and the derivative polynomials of tan and sec.. 2.1 Signed Permutations The type B and type D analogs of permutations are signed and even sigend permutations respectively. Definition 2.1.1. (i) A signed permutation of [n] is a bijection σ of the set [±n] := {−n, −n + 1, . . . , −1, 1, 2, . . . , n} onto itself such that σ(−i ) = −σ(i ) for all i ∈ [±n]. For convenience, we write −i as i.¯ Sometimes we denote σ as σ1 , σ2 , . . . , σn , which is called the window notation of σ, where σi = σ(i ) for 1 ≤ i ≤ n. (ii) An even signed permutation is a signed permutation with even number of negative entries in its window notation.. 8.

(12) Denote Bn and Dn the set of signed permutations and even signed permutations of [n], and Bn∗ (Dn∗ respectively) the subset of Bn (Dn respectively) without fixed points. ¯ 12, ¯ 21, ¯ 1¯ 2, ¯ 21, 21, ¯ 2¯ 1¯ }, D2 = {12, 1¯ 2, ¯ 21, 2¯ 1¯ }, B∗ = For example, B2 = {12, 12, 2 ∗ ¯ 21, ¯ 21, 21, ¯ 2¯ 1¯ }, D = {1¯ 2, ¯ 21, 2¯ 1¯ }. {1¯ 2, 2 The type B analogous of weak excedance we need is the flag weak excedance of signed permutations, which is defined as following. Definition 2.1.2 (Flag weak excedance). For σ ∈ Bn , we define wex(σ) = #{i ∈ [n] : σi ≥ i } and neg(σ) = #{σi : i ∈ [n], σi < 0}. Then the flag weak excedance number is defined as fwex(σ) = 2wex(σ) + neg(σ).. 2.2 Crossing of type B Notion of crossings of signed permutations were given by Corteel, JosuatVergès and Williams in [6] and was studied further in the later work of Corteel, Josuat-Vergès and Kim [5]. The defintion is defined as following. Definition 2.2.1 (Crossings of type B). For σ = σ1 σ2 · · · σn ∈ Bn , a crossing of σ is a pair (i, j) with i, j ≥ 1 such that • i < j ≤ σi < σj or • −i < j ≤ −σi < σj or • i > j > σi > σj . Similar to the case in type A, we may represent a signed permutation in Bn by diagram which makes the number of crossings easilier to count. Corteel et al. [5] offer several ways to do it. One of them is through the full ¯ 7¯ 2. ¯ pignose diagram, see Figure 2.1 for an example of σ = 63¯ 514 Construct the diagram: We assign two vertices for each i ∈ [±n] and arrange the vertices in a line as in the figure. • For each i > 0 if σi > 0 (σi < 0, resp.), then we connect the first vertex of i to the second vertex of σi (first vertex of σi , resp.) with an arc in the following way: draw the arc above the horizontal line if i ≤ σi , and below the horizontal line if i > σi .. 9.

(13) • for each i < 0 if σi < 0 (σi > 0, resp.), then we connect the second vertex of i to the first vertex of σi (second vertex of σi , resp.) with an arc in the following way: draw the arc below the horizontal line if i ≥ σi , and above the horizontal line if i < σi . We can see that the configuration of upper arcs and that of lower arcs are symmetric, and the number of crossings of σ is exactly the number of crossings between the arcs above the horizontal line. ¯ 7¯ 2, ¯ then the crossings are (7, 1), (3, 1), (2, 1) Example 2.2.2. Let σ = 63¯ 514 (−i < j ≤ −σi < σj ) and (4, 2),(4, 3),(7, 2),(7, 3),(7, 6) (i > j > σi > σj ) so croB (σ) = 8, see figure 2.1 again.. −7 −6 −5 −4 −3 −2 −1. 1. 2. 3. 4. 5. 6. 7. ¯ 7¯ 2¯ Figure 2.1: Full pignose diagram for σ = 63¯ 514. 2.3 Refined Enumeration on Singed Permutations Let Bn (y, t, q) = ∑σ∈ Bn yfwex(σ) tneg (σ) qcroB (σ) . The first few values are B0 (y, t, q) =1, B1 (y, t, q) =y2 + yt, B2 (y, t, q) =y4 + (2t + tq)y3 + (t2 q + t2 + 1)y2 + ty. In particular, when we take t = 0, we have Bn (y, 0, q) = An (y, q). Corteel, Josuat-Vergès and Williams [6] showed that Bn (y, t, q) is also a generating function of permutation tableaux of type B. Using permutation 10.

(14) tableaux, Corteel et al.[6] proved that Bn (y, t, q) satisfied a Matrix Ansatz. (They only consider Bn (1, t, q), but their proofs work for Bn (y, t, q).) Theorem 2.3.1 ([6]). Let D and E be matrices, hW | a row vector, |V i a column vector, satisfying D |V i = |V i ,. hW |E = ythW | D,. DE = qED + D + E.. Then Bn (y, t, q) = hW |(y2 D + E)|V i. By finding a solution to this Matrix Ansatz, Corteel, Josuat-Vergès and Kim [6] obtained the generating fucntion of Bn (y, t, q) in the form of Jfractions (Jacobi continued fractions). Definition 2.3.2. For any two sequences {µh }h≥0 and {λh }h≥1 , let F(µh , λh ) denote the continued fraction F( µ h , λ h ) =. λ1 x 2 λ2 x 2 λ3 x 2 1 ··· 1 − µ0 x − 1 − µ1 x − 1 − µ2 x − 1 − µ3 x −. Theorem 2.3.3. [5] The continued fraction expansion for the generating fucntion of Bn (y, t, q) is (2.1) ∑ Bn (y, t, q) xn = F(µh , λh ) n ≥0. where µh = + 1]q + [h]q + ytqh ([h]q + [h + 1]q ) for h ≥ 0 and λh = [h]2q (y2 + ytqh−1 )(1 + ytqh ) for h ≥ 1 y2 [ h. 2.4 Gerneralized Euler numbers: Springer numbers In 1971, Springer [23] defined an integer K (W ) ,which is usually called Springer number now, for any Coxeter group W. He also computed the quantity for all finite irreducible Coxeter systems. In particular, Sn is the irreducible Coxeter group of type An−1 and K ( An−1 ) = K (Sn ) = En . Definition 2.4.1. Let (W, S) be a Coxeter system, for any w ∈ W the (right) descent set of w is defined to be Des(w) = {s ∈ S : ℓ(ws) < ℓ(w)}. Let J ⊂ S and D J = {w ∈ W : Des(w) = J }, then the Springer number of W is defined to be the cardinality of largest descent class K (W ) := max | D J | J ⊂S. 11.

(15) n. 0. 1. 2. 3. 4. 5. 6. .... En Sn SnD. 1 1 1. 1 1 1. 1 3 1. 2 11 5. 5 57 23. 16 361 151. 61 2763 1141. ... ... .... Table 2.1: Springer number of type A, B and D Example 2.4.2. Let W = S3 and S = {s1 = (12), s2 = (23)}, then when J = {s1 } or {s2 }, D J = {213, 312} or {132, 231} attains its maximun size. In this case, K (S3 ) = 2 = E3 . Des(w) w ∈ A2 123 = id ∅ s2 132 = s2 s1 213 = s1 231 = s1 s2 s2 s1 312 = s2 s1 321 = s1 s2 s1 s1 , s2 For general Sn , we can see that the descent class D J attain its maximun size either when J = {s1 , s3 , . . .} or J = {s2 , s4 . . .}. And D J is exactly RAltn and Altn , where RAltn is the set of reverse alternating permutations in Sn . For more values of K (W ) of classical types, see table 2.1, in which we denote Sn = K ( Bn ) and SnD = K ( Dn ).. 2.5 Snakes of type B By describing Springer number geometrically in terms of Weyl chambers, Arnol’d [1] showed that for irreducible Coxeter systems of type A, B, D the Springer number counts various types of snakes (up-down permutations and up-down signed permutations). Our study mainly focuses on the snakes of type B. Definition 2.5.1. Let σ = σ1 . . . σn ∈ Bn . • The signed permutation σ is a snake if σ1 > σ2 < σ3 > . . . σn . Let Sn ⊂ Bn be the set of snakes of size n. • Let Sn0 ⊂ Sn be the subset consisting of the snakes σ with σ1 > 0. 12.

(16) • Let Sn00 ⊂ Sn0 be the subset consisting of the snakes σ with σ1 > 0 and (−1)n σn < 0. Example 2.5.2. For example, as n = 2 then ¯ 1¯ 2, ¯ 21, 21¯ }, S2 = {12,. ¯ 21, 21¯ }, S20 = {12,. ¯ 21¯ }. S200 = {12,. Note that Sn0 is the subset of snakes of type B so |Sn0 | = Sn . The snakes in Sn are introduced by Arnol’d [1] under the name β-snakes to study snakes of type B and D, with |Sn | = 2n En . The subset Sn00 is a variant introduced by Josuat-Vergès [19] with |Sn00 | = 2n−1 En . There is another surprising link between snakes and derivative polynomials of trigonometric fuctions. Hoffman [17] and Josuat Vergès [19] studied the polynomials Pn (t), Qn (t) and Rn (t) , which are defined as following dn dn dn tan x = P ( tan x ) , sec x = Q ( tan x ) sec x, sec2 x = Rn (tan x) sec2 x. n n dxn dxn dxn Hoffman [17] showed that Pn (1) = 2n En , Qn (1) = Sn and Pn (1) − Qn (1) = SnD . Then Josuat-Vergè [19] defined the polynomail Rn (t) and proved that Rn (1) = 2n En+1 ; morveover, he gave combinatorial interpretations to Pn (t), Qn (t) and Rn (t) in terms of the distributions of number of changes of sign cs on Sn , Sn0 and Sn00 . Definition 2.5.3. For a snake σ = σ1 σ2 · · · σn ∈ Bn , let cs(σ) denote the number of changes of sign through the entries σ1 , σ2 , . . . , σn , i.e., cs(σ) := #{i : σi σi+1 < 0, 0 ≤ i ≤ n} with the following convention for the entries σ0 and σn+1 : • σ0 = −(n + 1) and σn+1 = (−1)n (n + 1) if σ ∈ Sn ; • σ0 = 0 and σn+1 = (−1)n (n + 1) if σ ∈ Sn0 ; • σ0 = 0 and σn+1 = 0 if σ ∈ Sn00 . Theorem 2.5.4 (Josuat-Vergès [19]). For all n ≥ 0, we have Pn (t) =. ∑ σ∈S n. tcs(σ) ,. Q n ( t) =. ∑ σ∈S n0. tcs(σ) ,. R n ( t) =. ∑. tcs(σ) .. σ∈S n00+1. Let D be the ordinary differentiate operator, and U be the operator of multiplying t on the vector space of polynomials C [t]. From the recurrence. 13.

(17) when differentiating tan x and sec x, we can easily see that the definition of polynomials Pn , Qn and Rn can be rephrased as Pn (t) = ( D + UUD )n t Qn (t) = ( D + UDU )n 1 Rn (t) = ( D + DUU )n 1 for n ≥ 0. Josuat-Vergès [19] defined the q-analogs of the derivative polynomials Qn and Rn via the q-derivative. Let D be the q-analog of the differential operator acting on polynomials f (t) by. ( D f )(t) :=. f (qt) − f (t) ( q − 1) t. (2.2). and U be the operator acting on f (t) by multiplication by t. Notice that the q-derivative D (tn ) = [n]q tn−1 and the communication relation DU − qUD = 1 hold. Note that for n ≥ 1 Pn = (( I + UU ) D )n−1 ( I + UU )1 = ( I + UU )( D + DUU )n−11. = ( I + UU ) Rn−1 = (1 + t2 ) Rn−1. Hence we have Pn (t) = (1 + t2 ) Rn−1 (t) for n ≥ 1. This relation also holds for the q-analogs of Pn and Rn , therefore we only focus on q-analogs of Qn and Rn . Definition 2.5.5. The q-analogs of Qn and Rn are defined algebraically by Qn (t, q) := ( D + UDU )n 1,. Rn (t, q) := ( D + DUU )n 1.. (2.3). Several of the initial polynomials are listed below: Q0 (t, q) = 1 Q1 (t, q) = t Q2 (t, q) = 1 + (1 + q)t2 Q3 (t, q) = (2 + 2q + q2 )t + (1 + 2q + 2q2 + q3 )t3 , R0 (t, q) = 1 R1 (t, q) = (1 + q)t R2 (t, q) = (1 + q) + (1 + 2q + 2q2 + q3 )t2 R3 (t, q) = (2 + 5q + 5q2 + 3q3 + q4 )t + (1 + 3q + 5q2 + 6q3 + 5q4 + 3q5 + q6 )t3 . 14.

(18) Applying the Matrix Ansatz approach, Josuat-Vergès obtain the generating function of Qn (t, q) and Rn (t, q). Theorem 2.5.6. (Josuat-Vergès) Q. Q. ∑ Qn (t, q) xn = F(µh , λh ),. where (. ∑ Rn (t, q) xn = F(µhR , λhR ),. n ≥0. n ≥0. µhQ = tqh ([h]q + [h + 1]q ). (. λhQ = (1 + t2 q2h−1 )[h]2q ,. (2.4). µhR = tqh (1 + q)[h + 1]q λhR = (1 + t2 q2h )[h]q [h + 1]q .. Notice that Q2n (0, q) = E2n (q) and R2n+1 (0, q) = E2n+1 (q), the q-secant and q-tangent numbers defined in Eqs. (1.5) and (1.4).. 15.

(19) Chapter 3. Weighted Motzkin paths and Signed Permutations In this chapter we first review Flajolet’s fundamental lemma on continued fractions. It helps us associate combinatorial objects whose generating functions possessing continued fraction expansion with certain weighted Motzkin paths. Then we present the weight schemes of Motzkin paths associated with the enumerator of signed permutations and the (t, q)-derivative polynomials respectively.. 3.1 Continued fractions and weighted Motzkin paths A Motzkin path of length n is a lattice path from the origin to the point (n, 0) staying weakly above the x-axis, using the up step (1, 1), down step (1, −1), and level step (1, 0). Let U, D and L denote an up step, a down step and a level step, accordingly. We consider a Motzkin path µ = w1 w2 · · · wn with a weight function ρ on the steps. The weight of µ, denoted by ρ(µ), is defined to be the product of the weight ρ(w j ) of each step w j for j = 1, 2, . . . , n. The height of a step w j is the y-coordinate of the starting point of w j . Making use of Flajolet’s formula [10, Proposition 7A], the generating function for the weight count of the Motzkin paths can be expressed as a continued fraction. Theorem 3.1.1. (Flajolet) For h ≥ 0, let ah , bh and ch be polynomials such that each monomial has coefficient 1. Let Mn be the set of weighted Motzkin paths of length n such that the weight of an up step (down step or level step, respectively) at height h is one of the monomials appearing in ah (bh or ch , respectively). Then 16.

(20) the generating function for ρ( Mn ) = ∑µ∈ Mn ρ(µ) has the expansion a0 b1 x2. 1. a1 b2 x2. ∑ ρ ( Mn ) x n = 1 − c0 x − 1 − c1 x − 1 − c2 x − · · ·. (3.1). n ≥0. 3.2 Linking signed permutations to bicolored Motzkin paths A bicolored Motzkin path (also known as 2-Motzkin path) is a Motzkin path with two kinds of level steps, say straight and wavy, denoted by L and W, respectively. For a nonnegative integer h, let z(h) denote a step z at height h in a bicolored Motzkin path for z ∈ {U, L, W, D}. With theorem 3.1.1, we may observe that (2.1) and (2.4) both are the generating functions of the weight of some weighted bicolored Motzkin paths. By theorem 2.3.3, the initial part of the expansion of generating function of Bn (y, t, q) is. ∑ Bn (y, t, q) xn = n ≥0. 1−. ( y2. (1 + ytq2 )[2]q ( y2 + ytq )[2]q x2 (1 + ytq )[1]q ( y2 + yt)[1]q x2 1 2 − − + yt)[1]q x 1 − ( y + ytq )[2]q + (1 + ytq )[1]q x 1 − ( y2 + ytq2 )[3]q + (1 + ytq2 )[2]q x . . .. Therefore, by theorem 3.1.1 the following set Mn of weighted bicolored Motzkin paths has the generating function of weights equal to Bn (y, t, q). Definition 3.2.1. Let Mn be the set of weighted bicolored Motzkin paths of length n containing no wavy level steps on the x-axis, with a weight function ρ such that for h ≥ 0, • ρ(U(h) ) ∈ {y2 , y2 q, . . . , y2 qh } ∪ {ytqh , ytqh+1 , . . . , ytq2h }, • ρ(L(h) ) ∈ {y2 , y2 q, . . . , y2 qh } ∪ {ytqh , ytqh+1 , . . . , ytq2h }, • ρ(W(h) ) ∈ {1, q, . . . , qh−1 } ∪ {ytqh , ytqh+1 , . . . , ytq2h−1 } for h ≥ 1, • ρ(D(h+1) ) ∈ {1, q, . . . , qh } ∪ {ytqh+1 , ytqh+2 , . . . , ytq2h+1 }. In fact, extended from Foata–Zeilberger bijection [14], Corteel, JosuatVergès and Kim established a bijection between the set Bn of signed permutations and the set Mn of weighted bicolored Motzkin paths [5, Subsection 7.1]. 17.

(21) Remarks 3.2.2. Rather than the original weight function given in [5], we have interchanged the possible weights of the up steps and the down steps, in the sense of traversing the paths backward. This unifies the possible weights for the initial step (either U(0) or L(0) ), for the purpose of restructuring the weighted bicolored Motzkin paths (Proposition 4.2.2). Theorem 3.2.3. (Corteel, Josuat-Vergès, Kim) There is a bijection Γ between Bn and Mn such that yfwex(σ) tneg (σ) qcro B (σ) = ρ(Mn ).. ∑. (3.2). σ ∈ Bn. 3.3 Weight Schemes for Qn (t, q) and Rn (t, q) By Theorem 2.5.6, the initial part of the expansion of the generating functions for Rn (t, q) and Qn (t, q) are shown below. Rn (t, q) xn =. ∑ n ≥0. ∑. Qn (t, q) xn =. n ≥0. (1 + t2 q2 )[1]q [2]q x2 (1 + t2 q4 )[2]q [3]q x2 1 ··· , 1 − t(1 + q)[1]q x − 1 − tq(1 + q)[2]q x − 1 − tq2 (1 + q)[3]q x (1 + t2 q)[1]2q x2 (1 + t2 q3 )[2]2q x2 1 ··· . 1 − t[1]q x − 1 − tq([1]q + [2]q ) x − 1 − tq2 ([2]q + [3]q ) x. Therefore, by Theorem 3.1.1 the following observations provide feasible weight schemes that realize the polynomials Rn (t, q) and Qn (t, q) in terms of the bicolored Motzkin paths. Proposition 3.3.1. Let Tn be the set of weighted bicolored Motzkin paths of length n with a weight function ρ such that for h ≥ 0, • ρ(U(h) ) ∈ {1, q, . . . , qh } ∪ {t2 q2h+2 , t2 q2h+3 , . . . , t2 q3h+2 }, • ρ(L(h) ) ∈ {tqh+1 , tqh+2 , . . . , tq2h+1 }, • ρ(W(h) ) ∈ {tqh , tqh+1 , . . . , tq2h }, • ρ(D(h+1) ) ∈ {1, q, . . . , qh+1 }. Then we have. ∑ ρ(Tn ) xn = ∑ Rn (t, q) xn . n ≥0. n ≥0. 18.

(22) Proposition 3.3.2. Let Tn∗ be the set of weighted bicolored Motzkin paths of length n containing no wavy level steps on the x-axis, with a weight function ρ such that for h ≥ 0, • ρ(U(h) ) ∈ {1, q, . . . , qh } ∪ {t2 q2h+1 , t2 q2h+2 , . . . , t2 q3h+1 }, • ρ(L(h) ) ∈ {tqh , tqh+1 , . . . , tq2h }, • ρ(W(h) ) ∈ {tqh , tqh+1 , . . . , tq2h−1 } for h ≥ 1, • ρ(D(h+1) ) ∈ {1, q, . . . , qh }. Then we have. ∑ ρ(Tn∗ ) xn = ∑ Qn (t, q) xn . n ≥0. n ≥0. Our second main result is, as Corteel et al. connecting Mn with Bn , to use these set Tn and Tn∗ of paths to encode the snakes in Sn00+1 and Sn0 (Theorem 5.3.3 and Theorem 5.2.5). 19.

(23) Chapter 4. Signed Countings on type B and D The results in chapter 4 and 5 are the joint works with Sen-Peng Eu, TungShan Fu and Hsiang-Chun Hsu, part of them had appeared in [8]. In this chapter we present our type B and D extension of signed counting results. To prove the result, we first apply Corteel et al.’s bijection between Bn and Mn mentioned in chapter 3 with some adjusts. Then we construct an involution on weighted Motzkin paths which encode the signed counting process combinatorially. Then we compared the set of fixed points of the involutions with the set of paths associated with Qn (t, q) and Rn (t, q).. 4.1 Signed Countings on type B and D Our first main result is the type B and type D analogs of Eqs. (1.6) and (1.7), with the sign of σ ∈ Bn depending on the parity of one half of the statistic fwex(σ). Amazingly, the signed counting turns out to be related to the derivative polynomials Qn (t, q) and Rn (t, q). Theorem 4.1.1. For n ≥ 1, we have (i). ∑. (−1). ⌊ fwex2(σ) ⌋ neg ( σ) cro B ( σ). (−1). ⌈ fwex2(σ) ⌉ neg ( σ) cro B ( σ). t. q. =. (. (−1) 2 (t + 1) Rn−1 (t, q) , if n is odd; n −1 2 (−1) (t − 1) Rn−1 (t, q) , if n is even.. =. (. (−1) 2 (t − 1) Rn−1 (t, q) n +1 (−1) 2 (t + 1) Rn−1 (t, q). σ ∈ Bn. (ii). ∑ σ ∈ Bn. t. q. 20. n. n. if n is even; . if n is odd..

(24) Corollary 4.1.2. For n ≥ 1, we have. (−1)⌊. ∑ σ∈ Dn. =. (. fwex( σ ) ⌋ 2. tneg (σ) qcro B (σ) =. ∑. (−1)⌈. fwex( σ ) ⌉ 2. tneg (σ) qcro B (σ). σ∈ Dn. n 2. (−1) tRn−1 (t, q) n +1 (−1) 2 Rn−1 (t, q). if n is even, if n is odd.. Theorem 4.1.3. For n ≥ 1, we have (i). ∑ σ∈ Bn∗. (ii). ∑ σ∈ Bn∗. . 1 − q. ⌊ fwex2(σ) ⌋. . 1 − q. ⌈ fwex2(σ) ⌉. t. neg ( σ) cro B ( σ). t. neg ( σ) cro B ( σ). q. q. =. . 1 − q. ⌊ n2 ⌋. Qn (t, q).. =. . 1 − q. ⌈ n2 ⌉. Qn (t, q).. . ⌈ fwex2(σ) ⌉. Corollary 4.1.4. For n ≥ 1, we have fwex(σ) 1 ⌊ 2 ⌋ neg (σ) cro B (σ) t q = ∑ −q σ∈ Dn∗ ( n − 1q 2 Qn (t, q) if n is even, = 0 if n is odd. . ∑∗. σ∈ Dn. 1 − q. tneg (σ) qcro B (σ). Setting t = 1 and q = 1, we obtain types B and D extensions of the results in Eqs. (1.1) and (1.2). Corollary 4.1.5. For n ≥ 1, we have n (−1) 2 2n En ⌊ fwex2(σ) ⌋ = (i) ∑ (−1) 0 σ∈ B n. (ii). ∑ (−1). ⌈ fwex2(σ) ⌉. =. fwex( σ ) ⌋ 2. =. σ ∈ Bn. (iii). ∑. (−1)⌊. σ∈ Dn. (. 0 n +1 (−1) 2 2n En. ∑. (−1)⌈. fwex( σ ) ⌉ 2. σ∈ Dn. Corollary 4.1.6. For n ≥ 1, we have (i). ∑∗ (−1)⌊. fwex( σ ) ⌋ 2. n. = (−1)⌊ 2 ⌋ Sn .. σ ∈ Bn. 21. if n is even, if n is odd. if n is even; if n is odd.. = (−1)⌊. n +1 2 ⌋. 2n − 1 En ..

(25) (ii). ∑∗ (−1)⌈. fwex( σ ) ⌉ 2. = (−1)⌈ 2 ⌉ Sn .. (−1)⌊. fwex( σ ) ⌋ 2. =. n. σ ∈ Bn. ∑∗. (iii). ∑∗. (−1)⌈. fwex( σ ) ⌉ 2. =. σ∈ Dn. σ∈ Dn. . n. (−1) 2 Sn 0. if n is even, if n is odd.. Subtracting Corollary 4.1.5(i) with 4.1.6i and Corollary 4.1.5(ii) with Corollary 4.1.6(ii), and applying Hoffman’s result Pn (1) − Qn (1) = SnD , we obtain the following identities of Springer numbers of type D. Corollary 4.1.7. For n ≥ 1, we have ( n fwex( σ ) (−1) 2 SnD ⌊ 2 ⌋ (i) = n +1 ∑ (−1) (−1) 2 Sn σ ∈ Bn − B ∗ n. ∑. (ii). (−1). ⌈ fwex2 (σ) ⌉. σ∈ Bn − Bn∗. =. (. n. (−1) 2 +1 Sn n +1 (−1) 2 SnD. if n is even, if n is odd. if n is even, if n is odd.. 4.2 The cases of Bn and Dn In this section we present a combinatorial proof of Theorem 4.1.1 and Corollary 4.1.2, via a sign-reversing involution on corresponding set of paths. √ First, notice that plugging in y = −1 in Bn (y, t, q), we obtain √ √ √ Bn ( −1, t, q) = ∑ ( −1)fwex(σ) tneg (σ) qcro B (σ) + ∑ ( −1)fwex(σ) tneg (σ) qcro B (σ) σ ∈ Bn 2|fwex( σ). ∑. =. σ ∈ Bn 2∤fwex( σ). (−1). fwex( σ ) 2. tneg (σ) qcro B (σ) +. σ ∈ Bn 2|fwex( σ). √. −1. ∑. (−1). fwex( σ )−1 2. tneg (σ) qcro B (σ) .. σ ∈ Bn 2∤fwex( σ). Then it is easy to see that. ∑ (−1)⌊. fwex( σ ) ⌋ 2. σ ∈ Bn. √ √ tneg (σ) qcroB (σ) = Re( Bn ( −1, t, q)) + Im( Bn ( −1, t, q)). and. ∑ (−1)⌈ σ ∈ Bn. fwex( σ ) ⌉ 2. √ √ tneg (σ) qcroB (σ) = Re( Bn ( −1, t, q)) − Im( Bn ( −1, t, q)).. Since Bn (y, t, q) is the generating function of weights of paths in Mn , we want to do the sign counting combinatorially via these paths. In order to 22.

(26) do so, we restructure the weighted bicolored Motzkin paths in Mn . Recall the weight scheme of Mn is the following: • ρ(U(h) ) ∈ {y2 , y2 q, . . . , y2 qh } ∪ {ytqh , ytqh+1 , . . . , ytq2h }, • ρ(L(h) ) ∈ {y2 , y2 q, . . . , y2 qh } ∪ {ytqh , ytqh+1 , . . . , ytq2h }, • ρ(W(h) ) ∈ {1, q, . . . , qh−1 } ∪ {ytqh , ytqh+1 , . . . , ytq2h−1 } for h ≥ 1, • ρ(D(h+1) ) ∈ {1, q, . . . , qh } ∪ {ytqh+1 , ytqh+2 , . . . , ytq2h+1 }. Definition 4.2.1. Let Hn be the set of weighted bicolored Motzkin paths of length n with a weight function ρ such that for h ≥ 0, • ρ(U(h) ) ∈ {y2 , y2 q, . . . , y2 qh+1 } ∪ {ytqh+1 , ytqh+2 , . . . , ytq2h+2 }, • ρ(L(h) ) ∈ {1, q, . . . , qh } ∪ {ytqh+1 , ytqh+2 , . . . , ytq2h+1 }, • ρ(W(h) ) ∈ {y2 , y2 q, . . . , y2 qh } ∪ {ytqh , ytqh+1 , . . . , ytq2h }, • ρ(D(h+1) ) ∈ {1, q, . . . , qh } ∪ {ytqh+1 , ytqh+2 , . . . , ytq2h+1 }. Proposition 4.2.2. There is a two-to-one bijection Φ between Mn and Hn−1 such that ρ(Mn ) = (y2 + yt)ρ(Hn−1 ). Proof. Given a path µ = p1 p2 · · · pn ∈ Mn , we create a weight-preserving path z1 z2 · · · z2n of length 2n from µ as the intermediate stage, where z2i−1 z2i is determined from pi by  UU if pi = U,    UD if pi = L, z2i−1 z2i = DU if pi = W,    DD if pi = D,. with weight ρ(z2i−1 ) = ρ( pi ) and ρ(z2i ) = 1 for 1 ≤ i ≤ n. Note that ρ(z1 ) = ρ( p1 ) ∈ {y2 , yt} since p1 = U(0) or L(0) . Then we construct the corresponding path Φ(µ) = p1′ p2′ · · · p′n−1 from z2 · · · z2n−1 (with z1 and z2n excluded), where p′j is determined by z2j z2j+1 with weight ρ( p′j ) = ρ(z2j )ρ(z2j+1 ) according to the following cases  U if z2j z2j+1 = UU,    L if z2j z2j+1 = UD, p′j = W if z2j z2j+1 = DU,    D if z2j z2j+1 = DD, 23.

(27) for 1 ≤ j ≤ n − 1. Notice that ρ(µ) = ρ( p1 )ρ(Φ(µ)) since ρ( p′j ) = ρ( p j+1 ) for 1 ≤ j ≤ n − 1. Hence ρ(µ) = y2 ρ(Φ(µ)) or ytρ(Φ(µ)). Moreover, the possible weights of p′j can be determined from the steps of µ since  (h) U    (h) L p′j =  W(h)   ( h + 1) D. if if if if. = U(h+1) or L(h+1) , = W(h+1) or D(h+1) , = U(h) or L(h) , = D(h+1) or W(h+1) ,. p j+1 p j+1 p j+1 p j+1. for some h ≥ 0. That Φ(µ) ∈ Hn−1 follows from the weight function of the paths in Mn . It is straightforward to obtain the map Φ−1 by the reverse procedure. Example 4.2.3. See the figure below. Let µ = p1 p2 . . . p8 ∈ M8 be the path shown on the left-hand side in the upper row, where w j = ρ( p j ) for 1 ≤ j ≤ 8. The corresponding bicolored Motzkin path Φ(µ) = p1′ p2′ · · · p7′ ∈ H7 is shown on the right-hand side, where the intermediate stage z1 z2 · · · z16 is shown in the lower row. w2. w3. w4. w7 w5. w1. Φ. w8. w6. w3 w4 w5 w. w2. 6. w7 w8. w4 w2 1. 1. 1 w3. 1 w5. w1. w7 1. 1 w6. 1. w8 1. By a matching pair (U(h) , D(h+1) ) we mean an up step U(h) and a down step D(h+1) that face each other, in the sense that the horizontal line segment from the midpoint of U(h) to the midpoint of D(h+1) stays under the path. Let ρ(U(h) , D(h+1) ) denote the ordered pair (ρ(U(h) ), ρ(D(h+1) )) of weights. We shall establish an involution Ψ1 : Hn → Hn that changes the weight of a path by the factor y2 , with the following set of restricted paths as the fixed points. Definition 4.2.4. Let Fn ⊂ Hn be the subset consisting of the weighted paths satisfying the following conditions. For h ≥ 0, 24.

(28) • ρ(U(h) , D(h+1) ) = (y2 q a , qb ) or (ytqh+1+ a , ytqh+1+b ) for some a ∈ {0, 1, . . . , h + 1} and b ∈ {0, 1, . . . , h}, for any matching pair (U(h) , D(h+1) ), • ρ(L(h) ) ∈ {ytqh+1 , ytqh+2 , . . . , ytq2h+1 }, • ρ(W(h) ) ∈ {ytqh , ytqh+1 , . . . , ytq2h }. Notice that for any matching pair (U(h) , D(h+1) ) with weight ( a, b), it is equivalent to the reassignment ρ(U(h) , D(h+1) ) = ( a′ , b′ ) such that a′ b′ = ab regarding the total weight of a path. Comparing the weight functions of the paths in Fn and in the set Tn given in Proposition 3.3.1, we have the following result. Lemma 4.2.5. We have ρ(Fn ) = yn Rn (t, q). Proof. For any path µ ∈ Fn , notice that the weight of µ contains the factor yn since every matching pair of up step and down step contributes the parameter y2 , while every level step (either straight or wavy) contributes the parameter y. By Theorem 3.1.1, we observe that ρ(Fn ) and yn · ρ(Tn ) have same generating function. By Proposition 3.3.1, the assertion follows. The ‘sign-reversing’ map Ψ1 : Hn → Hn is constructed as follows.. Algorithm A. Given a path µ ∈ Hn , the corresponding path Ψ1 (µ) is constructed by the following procedure. (A1) If µ contains no straight level steps L(h) with weight q a or wavy level steps W(h) with weight y2 q a for any a ∈ {0, 1, . . . , h} then go to (A2). Otherwise, among such level steps find the first step z, say z = L(h) (W(h) , respectively) with weight ρ(z) = q a (y2 q a , respectively), then the corresponding path Ψ1 (µ) is obtained by replacing z by W(h) (L(h) , respectively) with weight y2 q a (q a , respectively). (A2) If µ contains no matching pairs (U(h) , D(h+1) ) with ρ(U(h) , D(h+1) ) = (y2 q a , ytqh+1+b ) or (ytqh+1+ a , qb ) for any a ∈ {0, 1, . . . , h + 1} or b ∈ {0, 1, . . . h} then go to (A3). Otherwise, among such pairs find the first pair with weight, say (y2 q a , ytqh+1+b ) ((ytqh+1+ a , qb ), respectively), then the corresponding path Ψ1 (µ) is obtained by replacing the weight of the pair by (ytqh+1+ a , qb ) ((y2 q a , ytqh+1+b ), respectively). (A3) Then we have µ ∈ Fn . Let Ψ1 (µ) = µ. 25.

(29) Regarding the possibilities of the weighted steps of the paths in Hn , we have the following immediate result. Proposition 4.2.6. The map Ψ1 established by Algorithm A is an involution on the set Hn such that for any path µ ∈ Hn , Ψ1 (µ) = µ if µ ∈ Fn , and ρ(Ψ1 (µ)) = y2 ρ(µ) or y−2 ρ(µ) otherwise. Now we are ready to prove Theorem 4.1.1. Proof of Theorem 4.1.1. For (i), by Theorem 3.2.3 and Proposition 4.2.2, we have ∑ yfwex(σ) tneg(σ) qcroB (σ) = (y2 + yt)ρ(Hn−1 ). σ ∈ Bn. fwex( σ ). Then the expression ∑σ∈ Bn (−1)⌊ 2 ⌋ tneg (σ) qcro B (σ) equals the sum of the real part and √ the imaginary part of the polynomial (y2 + yt)ρ(Hn−1 ) evaluated at y = −1. By Proposition 4.2.6 and Lemma 4.2.5, we have

(30)

(31) (y2 + yt)ρ(Hn−1 )

(32) y=√−1 = (y2 + yt)ρ(Fn−1 )

(33) y=√−1

(34) = (yn+1 + yn t) Rn−1 (t, q)

(35) y=√−1 h i √  (−1) n2 −1 + (−1) n2 t Rn (t, q) , if n is even; i = h √  (−1) n+2 1 + (−1) n−2 1 t −1 Rn−1 (t, q) , if n is odd. Taking the real part and the imaginary part of the above evaluation leads to ( n fwex( σ ) (−1) 2 (t + 1) Rn−1 (t, q) , if n is odd; ⌊ 2 ⌋ neg ( σ) cro B ( σ) t q = n −1 ∑ (−1) (−1) 2 (t − 1) Rn−1 (t, q) , if n is even. σ ∈ Bn as required. fwex( σ ). For (ii), similarly the expression ∑σ∈ Bn (−1)⌈ 2 ⌉ tneg (σ) qcro B (σ) equals the 2 real part subtracting √ the imaginary part of the polynomial (y + yt)ρ(Hn−1 ) evaluated at y = −1. Therefore, we have ( n fwex( σ ) (−1) 2 tRn−1 (t, q) if n is even, ⌊ 2 ⌋ neg ( σ) cro B ( σ) t q = n +1 ∑ (−1) (−1) 2 Rn−1 (t, q) if n is odd. σ∈ Dn In the following we shall prove Corollary 4.1.2. Recall that the set Dn of even-signed permutations consists of the signed permutations with even number of negative entries. 26.

(36) Definition 4.2.7. Let M′n ⊂ Mn be the subset consisting of the paths whose ( 1) ( 2) weights contain even powers of t. Let Hn (Hn , respectively) be the subset of Hn consisting of the paths whose weights contain odd (even, respectively) powers of t. Notice that the bijection Γ : Bn → Mn in Theorem 3.2.3 induces a bijection between Dn and M′n such that. ∑. yfwex(σ) tneg (σ) qcro B (σ) = ρ(M′n ).. (4.1). σ∈ Dn. Moreover, the involution Ψ1 : Hn → Hn and the set Fn of fixed points have the following properties. Lemma 4.2.8. The map Ψ1 established by Algorithm A induces an involution on ( 1) ( 2) Hn and Hn , respectively. Moreover, for any path µ ∈ Fn , the power of t of ρ(µ) has the same parity of n. Proof. By Proposition 4.2.6, we observe that the map Ψ1 : Hn → Hn preserves the powers of t of the weight of the paths. By the weight conditions of µ ∈ Fn given in Definition 4.2.4, we observe that every matching pair (U(h) , D(h+1) ) contributes the parameter t0 or t2 to ρ(µ), while every level step contributes the parameter t to ρ(µ). The assertions follow. Now we are ready to prove Corollary 4.1.2. Proof of Corollary 4.1.2. By Proposition 4.2.2 and Eq. (4.1), taking the terms with even powers of t yields. ∑ σ∈ Dn. ( 2). ( 1). yfwex(σ) tneg (σ) qcro B (σ) = y2 ρ(Hn−1 ) + ytρ(Hn−1 ).. √ ( 1) ( 2) Consider the polynomial y2 ρ(Hn−1 ) + ytρ(Hn−1 ) evaluated at y = −1. ( 1)

(37) By Proposition 4.2.6 and Lemma 4.2.8, for n odd, we have ρ(H )

(38) √ = n −1. 0 and.

(39) ( 2)

(40) y2 ρ(Hn−1 )

(41) y=√−1 = y2 ρ(Fn−1 )

(42) y=√−1

(43) = yn+1 Rn−1 (t, q)

(44) y=√−1 ,. ( 2)

(45) Moreover, for n even, we have ρ(Hn−1 )

(46) y=√−1 = 0 and.

(47) ( 1)

(48) ytρ(Hn−1 )

(49) y=√−1 = ytρ(Fn−1 )

(50) y=√−1

(51) = yn tRn−1 (t, q)

(52) y=√−1 . 27. y = −1.

(53) Hence we have. ∑. (−1). ⌊ fwex2(σ) ⌋ neg ( σ) cro B ( σ). t. q. =. σ∈ Dn. (. n +1. (−1) 2 Rn−1 (t, q) if n is odd, n (−1) 2 tRn−1 (t, q) if n is even. fwex( σ ). fwex( σ ). Note that we obtain the same result if we replace (−1)⌊ 2 ⌋ by (−1)⌈ 2 ⌉ , since every σ ∈ Dn has even number of negative entries, which leads to ⌊ fwex2(σ) ⌋ = ⌈ fwex2(σ) ⌉ = fwex2(σ) . The proof of Corollary 4.1.2 is completed.. 4.3 The cases of Bn∗ and Dn∗ In this section we prove Theorem 4.1.3 and Corollary 4.1.4 in terms of the weighted paths associated to the set Bn∗ of signed permutations without fixed points. Let Bn∗ (y, t, q) = ∑σ∈ Bn∗ yfwex(σ) tneg (σ) qcroB (σ) . Notice that plugging in y = q −1 ∗ q in Bn ( y, t, q), we obtain s. ∑∗. ! −1 , t, q q s !fwex(σ) −1 tneg (σ) qcro B (σ) + q. ∑∗. . Bn∗. =. σ ∈ Bn 2|fwex( σ). =. σ ∈ Bn 2|fwex( σ). −1 q. fwex2(σ). t. neg ( σ) cro B ( σ). q. +. s. ∑∗. σ ∈ Bn 2∤fwex( σ). −1 q. !fwex(σ). s. −1 q. . −1 q. ∑ σ ∈ Bn 2∤fwex( σ). tneg (σ) qcro B (σ). fwex(2σ)−1. tneg (σ) qcro B (σ) .. It is easy to see that. ∑∗. σ ∈ Bn. . −1 q. ⌊ fwex2(σ) ⌋. . −1 q. ⌈ fwex2(σ) ⌉. tneg (σ) qcroB (σ) = Re. Bn∗. s. −1 , t, q q. !!. Bn∗. s. −1 , t, q q. !!. +. √. q · Im. Bn∗. s. −1 , t, q q. !!. Bn∗. s. −1 , t, q q. !!. and. ∑ σ∈ Bn∗. t. neg ( σ) croB ( σ). q. = Re. −. √. q · Im. We show that Bn∗ (y, t, q) is the generating function of weights of some subset M∗n of Mn which is easily described, so we can do the sign counting combinatorially via M∗n . 28.

(54) By the definition of the crossings of signed permutations, for any σ = σ1 σ2 · · · σn ∈ Bn we observe that if (i, j) is a crossing of σ then σi 6= i and σj 6= j, i.e., the fixed points of σ are not involved in any crossing of σ. The following fact is a property of the bijection Γ : Bn → Mn given in Theorem 3.2.3. Lemma 4.3.1. For a σ = σ1 σ2 · · · σn ∈ Bn , let Γ(σ) = z1 z2 · · · zn ∈ Mn be the corresponding weighted bicolored Motzkin path. Then for j ∈ [n], σj = j if and only if the step z j is a straight level step with weight y2 . Let M∗n ⊂ Mn be the subset consisting of the paths containing no straight level steps with weight y2 . It follows from Lemma 4.3.1 that the bijection Γ : Bn → Mn induces a bijection between Bn∗ and M∗n such that ρ(M∗n ) =. ∑∗ yfwex(σ) tneg(σ) qcro. B ( σ). .. (4.2). σ ∈ Bn. and the weight scheme for M∗n is the following:. • ρ(U(h) ) ∈ {y2 , y2 q, . . . , y2 qh } ∪ {ytqh , ytqh+1 , . . . , ytq2h }, • ρ(L(h) ) ∈ {y2 , y2 q, . . . , y2 qh } ∪ {ytqh , ytqh+1 , . . . , ytq2h } for h ≥ 1 and ρ(L(0) ) ∈ {yt} for h = 0. • ρ(W(h) ) ∈ {1, q, . . . , qh−1 } ∪ {ytqh , ytqh+1 , . . . , ytq2h−1 } for h ≥ 1, • ρ(D(h+1) ) ∈ {1, q, . . . , qh } ∪ {ytqh+1 , ytqh+2 , . . . , ytq2h+1 }.. We shall establish an involution Ψ2 : M∗n → M∗n that changes the weight of a path by the factor y2 q, with the following set of restricted paths as the fixed points. Definition 4.3.2. Let Gn ⊂ M∗n be the subset consisting of the paths satisfying the following conditions. For h ≥ 0,. • ρ(U(h) , D(h+1) ) = (y2 q a , qb ) or (ytqh+ a , ytqh+1+b ) for some a, b ∈ {0, 1, . . . , h}, for any matching pair (U(h) , D(h+1) ), • ρ(L(h) ) ∈ {ytqh , ytqh+1 , . . . , ytq2h }, • ρ(W(h) ) ∈ {ytqh , ytqh+1 , . . . , ytq2h−1 } for h ≥ 1.. Comparing the weight functions of the paths in Gn and in the set Tn∗ given in Proposition 3.3.2, the following result can be proved by the same argument as in the proof of Lemma 4.2.5. Lemma 4.3.3. We have ρ(Gn ) = yn Qn (t, q). 29.

(55) We describe the construction of the map Ψ2 : M∗n → M∗n . Algorithm B Given a path µ ∈ M∗n , the corresponding path Ψ2 (µ) is constructed by the following procedure. (B1) If µ contains no straight level steps L(h) with weight y2 q a or wavy level steps W(h) with weight q a−1 for any a ∈ {1, 2, . . . , h} then go to (B2). Otherwise, among such level steps find the first step z, say z = L(h) (W(h) , respectively) with weight ρ(z) = y2 q a (q a−1 , respectively), then the path Ψ2 (µ) is obtained by replacing z by W(h) (L(h) , respectively) with weight q a−1 (y2 q a , respectively). (B2) If µ contains no matching pairs (U(h) , D(h+1) ) with ρ(U(h) , D(h+1) ) = (y2 q a , ytqh+1+b ) or (ytqh+ a , qb ) for any a, b ∈ {0, 1, . . . , h} then go to (B3). Otherwise, among such pairs find the first pair with weight, say (y2 q a , ytqh+1+b ) ((ytqh+ a , qb ), respectively) then the corresponding path is obtained by replacing the weight of the pair by (ytqh+ a , qb ) ((y2 q a , ytqh+1+b ), respectively). (B3) Then we have µ ∈ Gn . Let Ψ2 (µ) = µ. By the possible weighted steps of the paths in M∗n , we have the following result. Proposition 4.3.4. The map Ψ2 established by Algorithm B is an involution on the set M∗n such that for any path µ ∈ M∗n , Ψ2 (µ) = µ if µ ∈ Gn , and ρ(Ψ2 (µ)) = y2 qρ(µ) or y−2 q−1 ρ(µ) otherwise. Now we are ready to prove Theorem 4.1.3. ⌊ fwex2(σ) ⌋ neg (σ) cro (σ) t q B , Proof of Theorem 4.1.3. For (i), the expression ∑σ∈ Bn∗ − 1q by Eq. (4.2), equals the sum of the real part and the imaginary part multiq √ − 1 plying q of the polynomial ρ(M∗n ) evaluated at y = q . By Proposition 4.3.4 and Lemma 4.3.3, we have

(56)

(57) ρ(M∗n )

(58) q −1 = ρ(Gn )

(59) q −1 y=. y=. q. q. n.

(60) = y Qn (t, q)

(61) y=q −1 q  n   −1 2 Qn (t, q) q = q n −1   −1 −1 2 Qn (t, q) q q 30. , if n is even; , if n is odd..

(62) Hence. ∑ σ∈ Bn∗. . 1 − q. ⌊ fwex2(σ) ⌋. t. neg ( σ) cro B ( σ). q. =. . 1 − q. ⌊ 2n ⌋. Qn (t, q).. (4.3). fwex(σ) 1 ⌈ 2 ⌉ neg ( σ) cro B ( σ) t q can be obtained by q q −1 ρ(M∗n ) evaluated at y = q with the imagi-. For (ii), the expression ∑σ∈ Bn∗ −. subtracting the real part of √ nary part multplying q. Therefore, we have. ∑ σ∈ Bn∗. . 1 − q. ⌈ fwex2(σ) ⌉. t. neg ( σ) cro B ( σ). q. =. . 1 − q. ⌈ 2n ⌉. Qn (t, q).. (4.4). Proof of Corollary 4.1.4 Recall that Dn∗ ⊂ Bn∗ is the subset consisting of the fixed point-free signed permutations with even number of negative entries. ′ Let M∗n ⊂ M∗n be the subset consisting of the paths whose weights contain even powers of t. Notice that the bijection Γ : Bn → Mn also induces a ′ bijection between Dn∗ and M∗n such that. ∑ ∗ yfwex(σ) tneg(σ) qcro. B ( σ). ′. = ρ(M∗n ).. (4.5). σ∈ Dn. q ′ Consider the polynomial ρ(M∗n ) evaluated at y = −q1 . By Proposition 4.3.4, we observe that the involution Ψ2 : M∗n → M∗n preserves the powers of t of the weight of the paths. Moreover, for the fixed points µ ∈ Gn of the map Ψ2 , we observe that the power of t of ρ(µ) has the same parity of n. By the expression in Eq. (4.3) and Eq. (4.4), we have. ∑∗. σ∈ Dn. . 1 − q. ⌊ fwex2(σ) ⌋. t. neg ( σ) cro B ( σ). q. =. (. Since ⌊ fwex2(σ) ⌋ = ⌈ fwex2(σ) ⌉ =. when we replace ⌊ is completed.. fwex( σ) ⌋ 2. fwex( σ) for σ 2 fwex( σ) with ⌈ 2 ⌉.. 31. 0. − 1q. n2. Qn (t, q) if n is even, if n is odd.. ∈ Dn∗ . The same result is obtained Hence the proof of Corollary 4.1.4.

(63) Chapter 5. Snakes and ( t, q )-analogue of derivative polynomials In this chapter we shall give a combinatorial interpretation for the polynomials Qn (t, q) and Rn (t, q) as the enumerators for variants of the snakes in signed permutations. We make use of a non-recursive approach, based on Flajolet’s combinatorics of continued fractions, to establish a bijection between the snakes of Sn0 (Sn00+1 , respectively) and the weighted bicolored Motzkin paths of Tn∗ given in Proposition 3.3.2 (Tn given in Proposition 3.3.1, respectively). The bijections are constructed in the spirit of the classical Françon–Viennot bijection [13] by encoding the sign changes and consecutive up-down patterns of snakes into weighted steps of the Motzkin paths.. 5.1 cs-vectors and blocks Given a permutation π = π1 π2 · · · πn ∈ Sn , Arnol’d [1] devised the following rules to determine signs ǫ = ǫ1 ǫ2 · · · ǫn ∈ {±}n such that (ǫ1 π1 , ǫ2 π2 , . . . , ǫn πn ) is a snake; see also [19]. For 2 ≤ i ≤ n − 1, (R1) if πi−1 < πi < πi+1 then ǫi 6= ǫi+1 , (R2) if πi−1 > πi > πi+1 then ǫi−1 6= ǫi , (R3) if πi−1 > πi < πi+1 then ǫi−1 = ǫi+1 . For a snake σ = σ1 σ2 . . . σn ∈ Sn0 (Sn00 , respectively), we make the convention that σ0 = 0 and σn+1 = (−1)n (n + 1) (σ0 = σn+1 = 0, respectively). 32.

(64) Let |σ| denote the permutation obtained from σ by removing the negative signs of σ, i.e., |σ|i = |σi | for 0 ≤ i ≤ n + 1. Recall that the number of sign changes of σ is given by cs(σ) := #{i : σi σi+1 < 0, 0 ≤ i ≤ n}. For 1 ≤ j ≤ n, let j = |σ|i for some i ∈ [n], and define cs(σ, j), the number of sign changes recorded by the element j, by  0 if |σ|i−1 > j < |σ|i+1 and σi−1 σi > 0, σi σi+1 > 0,    2 if |σ|i−1 > j < |σ|i+1 and σi−1 σi < 0, σi σi+1 < 0, cs(σ, j) = 1 if |σ|i−1 < j < |σ|i+1 or |σ|i−1 > j > |σ|i+1 ,    0 if |σ|i−1 < j > |σ|i+1 . We call the sequence (cs(σ, 1), . . . , cs(σ, n)) the cs-vector of σ. Following the rules (R1)-(R3) and the condition σ1 > 0, we observe that the signs of the entries σ1 , . . . , σn of σ can be recovered from left to right by |σ| and the cs-vector of σ.. Lemma 5.1.1. For any snake σ ∈ Sn0 (Sn00 , respectively), the sign of each entry of σ is uniquely determined by |σ| and the vector (cs(σ, 1), . . . , cs(σ, n)). Moreover, we have n. cs(σ) =. ∑ cs(σ, j). j=1. Proof. For the initial condition, we have σ0 = 0 and σ1 > 0. For i ≥ 2, we determine the sign of σi according to the following cases: Case 1. |σ|i−1 > |σ|i . There are two cases. If |σ|i > |σ|i+1 then by (R2) σi has the opposite sign of σi−1 . Otherwise, |σ|i < |σ|i+1 , and by (R3) σi has the opposite (same, respectively) sign of σi−1 if cs(|σ|, |σ|i ) = 2 (0, respectively). Hence the sign change between σi−1 and σi is recorded by cs(|σ|, |σ|i ). Case 2. |σ|i−1 < |σ|i . There are two cases. If |σ|i−2 < |σ|i−1 then by (R1) σi has the opposite sign of σi−1 . Otherwise, |σ|i−2 > |σ|i−1 , and by (R3) σi has the opposite (same, respectively) sign of σi−1 if cs(|σ|, |σ|i−1 ) = 2 (0, respectively). Hence the sign change between σi−1 and σi is recorded by cs(|σ|, |σ|i−1 ). The assertions follow. Example 5.1.2. Given a snake σ = ((0), 5, −2, 4, −7, −1, −8, 10, −9, 6, 3, (11)) ∈ 0 , note that cs( σ) = 6 and the cs-vector of σ is (0, 2, 0, 1, 0, 1, 0, 1, 1, 0). S10 On the other hand, given the permutation |σ| = (5, 2, 4, 7, 1, 8, 10, 9, 6, 3) ∈ S10 and the cs-vector (0, 2, 0, 1, 0, 1, 0, 1, 1, 0), we observe that the snake σ can be recovered, following the rules (R1)-(R3). 33.

(65) Let S0n and S00 n denote two ‘copies’ of Sn with the following convention • π0 = 0 and πn+1 = n + 1 if π ∈ S0n , • π0 = πn+1 = 0 if π ∈ S00 n . Given a permutation π = π1 π2 · · · πn ∈ S0n or S00 n , by a block of π restricted to {0, 1, . . . , k} we mean a maximal sequence of consecutive entries πi πi+1 · · · π j ⊆ {0, 1, . . . , k} for some i ≤ j. For 0 ≤ k ≤ n, let α(π, k) be the number of blocks of π restricted to {0, 1 . . . , k}, and let β(π, k) be the number of such blocks on the right-hand side of the block containing the element k. By the convention on σ0 and σn+1 , for k = 0 we have α(π, 0) = 1 if π ∈ S0n , while α(π, 0) = 2 if π ∈ S00 n . Example 5.1.3. Let π = ((0), 5, 2, 4, 7, 1, 8, 10, 9, 6, 3, (11)) ∈ S010 . Notice that α(π, 6) = 3 and β(π, 6) = 0. The three blocks of π restricted to {0, 1, . . . , 6} are underlined as shown below. (0). 5. 2. 4. 7. 1. 8. 10. 9. 6. 3. (11). For 0 ≤ k ≤ 10, the sequences of α(π, k) and β(π, k) of π are shown in Table 5.1. Table 5.1: The sequences α, β of π = ((0), 5, 2, 4, 7, 1, 8, 10, 9, 6, 3, (11)). k α(π, k) β(π, k). 0 1 0. 1 2 0. 2 3 1. 3 4 0. 4 4 2. 5 3 2. 6 3 0. 7 2 1. 8 2 1. 9 2 0. 10 1 0. 5.2 The enumerator Qn (t, q) of Sn0 . In this section we shall encode the permutation |σ| by a weighted bicolored Motzkin path. We construct a bijection between Sn0 and the set of weighted bicolored Motzkin paths Tn∗ whose generating fucntion of weight is equal Qn (t, q). We shall establish a map Λ1 : Sn0 → Tn∗ by the following procedure.. Algorithm C.. 34.

(66) Given a snake σ = σ1 σ2 · · · σn ∈ Sn0 , we associate σ with a weighted path Λ1 (σ) = z1 z2 · · · zn . For 1 ≤ j ≤ n, let j = |σ|i for some i ∈ [n] and define the step z j according to the following cases: (i) if |σ|i−1 > j < |σ|i+1 then z j = U with weight β(|σ|,j) q if σi−1 σi > 0 and σi σi+1 > 0, ρ( z j ) = 2 β (| σ | ,j )+ 2α (| σ | ,j )− 3 t q if σi−1 σi < 0 and σi σi+1 < 0, (ii) if |σ|i−1 < j < |σ|i+1 then z j = L with weight tq β(|σ|,j)+α(|σ|,j)−1 , (iii) if |σ|i−1 > j > |σ|i+1 then z j = W with weight tq β(|σ|,j)+α(|σ|,j)−1 , (iv) if |σ|i−1 < j > |σ|i+1 then z j = D with weight q β(|σ|,j) . Notice that the value cs(σ, j) is encoded as the power of t in ρ(z j ). For convenience, the element j in (i) is called a valley, in (ii) a double ascent, in (iii) a double descent, and in (iv) a peak of σ. Example 5.2.1. Take the snake σ = ((0), 5, −2, 4, −7, −1, −8, 10, −9, 6, 3, (11)) ∈ 0 . The cs-vector of σ is given in Example 5.1.2 and the sequences α, β S10 of |σ| are given in Example 5.1.3. We observe that z1 = U with weight q β(|σ|,1) = 1 since the element 1 is a valley without sign-changes, and that z2 = U with weight t2 q β(|σ|,2)+2α(|σ|,2)−3 = t2 q4 since the element 2 is a valley with sign-changes. The path Λ1 (σ) is shown in Figure 5.1. tq5 1. q2. tq2 q. t2 q4. tq2. tq 1. 1. Figure 5.1: The corresponding path of the snake in Example 5.2.1. The constraints for the parameters α(|σ|, k) and β(|σ|, k) of |σ| are encoded in the height of the step zk ∈ Λ1 (σ).. Lemma 5.2.2. For a snake σ = σ1 σ2 · · · σn ∈ Sn0 , let Λ1 (σ) = z1 z2 · · · zn be the path constructed by Algorithm C. For 1 ≤ j ≤ n, let h j be the height of the step z j . Then the following properties hold. (i) h j = α(|σ|, j − 1) − 1. 35.

(67) (ii) If z j = W or D then h j ≥ 1 and 0 ≤ β(|σ|, j) ≤ h j − 1. (iii) If z j = U or L then 0 ≤ β(|σ|, j) ≤ h j . Proof. For the initial condition, we have α(|σ|, 0) = 1 and h1 = 0. The first step is either U or L since the element 1 is either a valley or a double ascent. For j ≥ 1, let j = |σ|i for some i ∈ [n]. By induction, we determine the height of z j+1 according to the following cases: • If |σ|i−1 > j < |σ|i+1 then z j = U and the element j itself creates a block of |σ| restricted to {0, 1, . . . , j}. Hence h j+1 = h j + 1 = α(|σ|, j − 1) = α(|σ|, j) − 1. • If |σ|i−1 < j < |σ|i+1 or |σ|i−1 > j > |σ|i+1 then z j = L (W, respectively) and the element j is added to the block with |σ|i−1 (|σ|i+1 , respectively). Hence h j+1 = h j = α(|σ|, j − 1) − 1 = α(|σ|, j) − 1. • If |σ|i−1 < j > |σ|i+1 then z j = D and the element j connects the adjacent two blocks. Hence h j+1 = h j − 1 = α(|σ|, j − 1) − 2 = α(|σ|, j) − 1. The assertion (i) follows. (ii) If z j = W or D then j > |σ|i+1 . The element j is added to the block with |σ|i+1 , which is different from the block containing (0). Then α(|σ|, j − 1) ≥ 2 and hence h j ≥ 1. Moreover, there are at most α(|σ|, j − 1) − 2 blocks on the right-hand side of the block containing |σ|i+1 . Hence β(|σ|, j) ≤ h j − 1. (iii) If z j = U or L then j < |σ|i+1 and there are at most α(|σ|, j − 1) − 1 blocks on the right-hand side of the block containing j. Hence β(|σ|, j) ≤ hj . Comparing the weight function of the paths in Tn∗ in Proposition 3.3.2 and the properties of Λ1 (σ) in Lemma 5.2.2, it follows that the path Λ1 (σ) constructed by Algorithm C is a member of Tn∗ .. Next, we shall construct the map Λ1−1 : Tn∗ → Sn0 by the following procedure. Algorithm D. Given a path µ = z1 z2 · · · zn ∈ Tn∗ , we associate µ with a snake σ′ = Λ1−1 (µ). For 1 ≤ j ≤ n, let cs(|σ′ |, j) (d j , respectively) be the power of t (q, respectively) of ρ(z j ), and let h j be the height of z j . To find |σ′ |, we construct a sequence ω0 , ω1 , . . . , ωn = |σ′ | of words, where ω j is the subword consisting of the blocks of |σ′ | restricted to {0, 1, . . . , j}. The initial word ω0 36.

(68) is a singleton (σ0 ). For j ≥ 1, the word ω j is constructed from ω j−1 and ρ(z j ) according to the following cases: (i) z j = U. There are two cases. If cs(|σ′ |, j) = 0, let ℓ = d j . Otherwise cs(|σ′ |, j) = 2 and let ℓ = d j − 2h j − 1. Then the word ω j is obtained from ω j−1 by inserting j between the ℓth and the (ℓ + 1)st block from right as a new block. (ii) z j = L or W. Then let ℓ = d j − h j . The word ω j is obtained from ω j−1 by appending j to the right end (left end, respectively) of the (ℓ + 1)st block from right as a new member of the block if z j = L (W, respectively). (iii) z j = D. Then let ℓ = d j . The word ω j is obtained from ω j−1 by inserting j between the (ℓ + 1)st and the (ℓ + 2)nd block from right and getting the two blocks combined. Then following the rules (R1)-(R3), we determine the signs of the elements of ωn by the sequence cs(|σ′ |, j) for j = 1, 2, . . . , n. Hence the requested snake σ′ is established. In the following we give an interpretation of the sequences α, β in terms of three-term patterns of the permutation |σ|.. Definition 5.2.3. Let π = π1 π2 · · · πn ∈ S0n or S00 n . For 1 ≤ i ≤ n, we define 13-2(π, i ) = #{ j : 0 ≤ j < i − 1 and π j < πi < π j+1 },. 2-31(π, i ) = #{ j : i < j ≤ n and π j > πi > π j+1 }.. Let also 2-31(π ) = ∑ni=1 2-31(π, i ). For any snake σ = σ1 σ2 · · · σn ∈ Sn0 or Sn00 , we distinguish the following classes X, Y and Z of elements of σ: (i) the valleys with sign changes, (ii) the double ascents or double descents and (iii) the peaks, namely X (σ) = {|σ|i : |σ|i−1 > |σ|i < |σ|i+1 , σi−1 σi < 0 and σi σi+1 < 0}, Y (σ) = {|σ|i : |σ|i−1 < |σ|i < |σ|i+1 or |σ|i−1 > |σ|i > |σ|i+1 }, Z (σ) = {|σ|i : |σ|i−1 < |σ|i > |σ|i+1 }.. The parameters α(|σ|, k) and β(|σ|, k) of the snake σ ∈ Sn0 or Sn00 have the following properties. Lemma 5.2.4. For 0 ≤ k ≤ n, we have (i) β(|σ|, k) = 2-31(|σ|, k), 37.

(69) (ii) α(|σ|, k) = 13-2(|σ|, k) + 2-31(|σ|, k) + 1.. Proof. Suppose there are ℓ (ℓ′ , respectively) blocks on the right-hand (lefthand, respectively) side of the block containing k when |σ| is restricted to {0, 1, . . . , k}. Then along with the element k, the two adjacent entries of |σ| at the left (right, respectively) boundary of each block constitute a 2-31pattern (13-2-pattern, respectively). Hence 2-31(|σ|, k) = ℓ and 13-2(|σ|, k) = ℓ′ . The assertions follow. For example, let π = ((0), 5, 2, 4, 7, 1, 8, 10, 9, 6, 3, (11)) ∈ S010 . As shown in Example 5.1.3, α(π, 6) = 3 and β(π, 6) = 0. We have 13-2(π, 6) = 2, where the two requested 13-2-patterns are (4, 7, 6) and (1, 8, 6). Following the weighting scheme given in Algorithm C, we define the statistic patQ of a snake σ ∈ Sn0 by patQ (σ) = ∑ 2 13-2(|σ|, j) + 2-31(|σ|, j) − #X (σ) j∈ X ( σ). +. ∑. 13-2(|σ|, j) + 2-31(|σ|, j) .. j ∈Y ( σ ). (5.1). By Lemmas 5.2.2 and 5.2.4 and Proposition 3.3.2, we have the following result. Theorem 5.2.5. The map Λ1 established by Algorithm C is a bijection between Sn0 and Tn∗ such that. ∑. tcs(σ) q2-31(|σ|)+patQ (σ) = Qn (t, q).. σ∈S n0. 5.3 The enumerator Rn (t, q) of Sn00+1 . In this section we shall apply a similar procedure to Rn (t, q) and Sn00+1 . We establish a map Λ2 : Sn00+1 → Tn by the same method as in Algorithm C with a modification on the weighting scheme. Given a snake σ = σ1 σ2 · · · σn+1 ∈ Sn00+1 , recall that the cs-vector of σ and the parameters α(|σ|, k), β(|σ|, k) for k = 1, 2, . . . , n are computed under the convention σ0 = σn+2 = 0. 00 . Example 5.3.1. Let σ = ((0), 5, −2, 4, −7, −1, −8, 11, −9, 6, 3, 10, (0)) ∈ S11 Notice that α(|σ|, 6) = 4 and β(|σ|, 6) = 1 as shown below.. (0). 5. 2. 4. 7. 1. 8. 38. 11. 9. 6. 3. 10. (0).

(70) For 0 ≤ k ≤ 10, the sequences α(|σ|, k) and β(|σ|, k) of |σ| are shown in Table 5.2. Table 5.2: The α and β vectors of |σ|. k α(|σ|, k) β(|σ|, k). 0 2. 1 3 1. 2 4 2. 3 5 1. 4 5 3. 5 4 3. 6 4 1. 7 3 2. 8 3 2. 9 3 1. 10 2 0. 11. We associate the snake σ with a weighted path Λ2 (σ) = z1 z2 · · · zn by the following procedure. Algorithm C’. For 1 ≤ j ≤ n, let j = |σ|i for some i ∈ [n + 1] and define the step z j according to the following cases: (i) if |σ|i−1 > j < |σ|i+1 then z j = U with weight ρ( z j ) =. . q β(|σ|,j)−1 t2 q β(|σ|,j)+2α(|σ|,j)−5. if σi−1 σi > 0 and σi σi+1 > 0, if σi−1 σi < 0 and σi σi+1 < 0,. (ii) if |σ|i−1 < j < |σ|i+1 then z j = L with weight tq β(|σ|,j)+α(|σ|,j)−2 , (iii) if |σ|i−1 > j > |σ|i+1 then z j = W with weight tq β(|σ|,j)+α(|σ|,j)−2 , (iv) if |σ|i−1 < j > |σ|i+1 then z j = D with weight q β(|σ|,j) . For example, take the snake σ = ((0), 5, −2, 4, −7, −1, −8, 11, −9, 6, 3, 10, (0)) ∈ 00 . From the parameters α (|σ|, k), β(|σ|, k) of σ given in Example 5.3.1, the S11 corresponding path Λ2 (σ) is shown in Figure 5.2. tq6 1. q3. tq3 q2. t2 q5. 2 tq3 tq. 1. 1. 00 . Figure 5.2: The corresponding path of the snake σ ∈ S11. 39.