Bumpless Pipedreams, Reduced Word Tableaux and Stanley Symmetric Functions
aa r X i v : . [ m a t h . C O ] O c t Bumpless Pipedreams, Reduced Word Tableaux andStanley Symmetric Functions
Neil J.Y. Fan , Peter L. Guo , Sophie C.C. Sun Department of MathematicsSichuan University, Chengdu, Sichuan 610064, P.R. China , Center for Combinatorics, LPMC-TJKLCNankai University, Tianjin 300071, P.R. China [email protected], [email protected], [email protected] Abstract
Lam, Lee and Shimozono introduced the structure of bumpless pipedreamsin their study of back stable Schubert calculus. They found that a specific fam-ily of bumpless pipedreams, called EG-pipedreams, can be used to interpret theEdelman-Greene coefficients appearing in the expansion of a Stanley symmetricfunction in the basis of Schur functions. It is well known that the Edelman-Greenecoefficients can also be interpreted in terms of reduced word tableaux for permuta-tions. Lam, Lee and Shimozono proposed the problem of finding a shape preservingbijection between reduced word tableaux for a permutation w and EG-pipedreamsof w . In this paper, we construct such a bijection. The key ingredients are twonew developed isomorphic tree structures associated to w : the modified Lascoux-Sch¨utzenberger tree of w and the Edelman-Greene tree of w . Using the Little map,we show that the leaves in the modified Lascoux-Sch¨utzenberger of w are in bijec-tion with the reduced word tableaux for w . On the other hand, applying the droopoperation on bumpless pipedreams also introduced by Lam, Lee and Shimozono,we show that the leaves in the Edelman-Greene tree of w are in bijection with theEG-pipedreams of w . This allows us to establish a shape preserving one-to-onecorrespondence between reduced word tableaux for w and EG-pipedreams of w . The structure of bumpless pipedreams was recently introduced by Lam, Lee and Shi-mozono [19] in their study of backward stable Schubert calculus. They proved thatbumpless pipedreams can generate back stable Schubert polynomials, which are poly-nomial representatives of the Schubert classes of the infinite flag varieties. Restrictingto finite flag varieties, bumpless pipedreams serve as a new combinatorial model for theSchubert polynomials. Moreover, they found that for a permutation w , the coefficients c wλ in the expansion of the Stanley symmetric function F w ( x ) = X λ c wλ s λ ( x ) (1.1)1n the basis of Schur functions s λ ( x ) can be interpreted by a specific family of bumplesspipedreams, namely, the EG-pipedreams of w .Stanley symmetric functions were invented by Stanley in his seminal paper [28] inorder to enumerate the reduced decompositions of permutations. The coefficients c wλ , nowcalled the Edelman-Greene coefficients or the Stanley coefficients, were first conjecturedby Stanley [28] and then proved by Edelman and Greene [9] to be nonnegative integers.More precisely, by developing the Edelman-Greene algorithm (also called the Coxeter-Knuth algorithm), Edelman and Greene [9] showed that the coefficient c wλ is equal to thenumber of reduced word tableaux for w with shape λ , see also Fomin and Greene [8],Lam [18] or Stanley [29].Lam, Lee and Shimozono [19] proposed the following problem. Problem 1.1 (Lam-Lee-Shimozono [19, Problem 5.19]) . Find a direct shape preservingbijection between EG-pipedreams of w and reduced word tableaux for w . In this paper, we provide a desired bijection as asked in Problem 1.1. The construc-tion of our bijection relies on the Edelman-Greene insertion algorithm [9], the Littlemap [23] as well as two new structures developed in this paper: the modified Lascoux-Sch¨utzenberger tree and the Edelman-Greene tree.Before outlining our strategy, let us recall the classical Lascoux-Sch¨utzenberger tree(LS-tree), which is an alternative approach to explain the Edelman-Greene coefficients.For a permutation w , the LS-tree of w was introduced by Lascoux and Sch¨utzenberger[21] by utilizing the maximal transition formula for Stanley symmetric functions. In theLS-tree of w , the children of a node u are the results of applying maximal transitionsto u . A combinatorial proof of such transition relations was found by Little [23]. Inthe LS-tree of w , each leaf is labeled by a Grassmannian permutation, whose Stanleysymmetric function is a Schur function. Thus the coefficient c wλ is equal to the numberof leaves in the LS-tree of w whose Stanley symmetric function is s λ ( x ).We introduce the modified Lascoux-Sch¨utzenberger tree (modified LS-tree) of a per-mutation w , and show that each leaf in the modified LS-tree of w corresponds to areduced word tableau for w . Unlike the classical LS-tree, the construction of a modifiedLS-tree is based on general transition relations satisfied by Stanley symmetric functions.More precisely, in the modified LS-tree of w , the children of a node u are the results of ap-plying some general (not necessarily maximal) transitions to u . Each leaf in the modifiedLS-tree of w is a dominant permutation. It is known that for a dominant permutation u , the corresponding Stanley symmetric function F u ( x ) equals a Schur function s λ ( u ) ( x ),where λ ( u ) is the Lehmer code of u . Therefore, the modified LS-tree of w also allows usto expand F w ( x ) in terms of Schur functions, that is, the Edelman-Greene coefficient c wλ is equal to the number of leaves in the modified LS-tree of w whose Lehmer codes are λ . Moreover, employing the Little map [23], we can associate each leaf in the modifiedLS-tree of w to a reduced word tableau for w .It is worth mentioning another difference between the modified LS-tree and the clas-sical LS-tree. Let w be a permutation on { , , . . . , n } . Since maximal transitions mayincrease the size of permutations, there may exist nodes in the LS-tree of w which arelabeled with permutations on { , , . . . , m } with m > n . However, the general transi-2ions used in this paper do not increase the size of permutations, that is, each node inthe modified LS-tree of w is also labeled with a permutation on { , , . . . , n } . Therefore,in some sense, the modified LS-tree seems to be more controllable than the LS-tree inthe process of expanding a Stanley symmetric function into Schur functions.On the other hand, we show that the EG-pipedreams of w can also be generated as theleaves of a tree associated to w , which is isomorphic to the modified LS-tree of w . Lam,Lee and Shimozono [19] introduced an operation, called droops, on bumpless pipedreams.They showed that any bumpless pipedream of w can be obtained by applying a sequenceof droops to the Rothe pipedream of w . By applying the droop operations, we constructa tree of bumpless pipedreams of w , called the Edelman-Greene tree (EG-tree) of w . Inthe EG-tree of w , each node is a bumpless pipedream of w , and the children of the node u are obtained by applying some specific droops to u , which correspond to the generaltransitions in the process of constructing the modified LS-tree of w . Thus the EG-tree of w and the modified LS-tree of w are isomorphic. In particular, the leaves in the EG-treeof w are exactly labeled with the EG-pipedreams of w . Since the leaves in the modifiedLS-tree of w are in one-to-one correspondence with the reduced word tableaux for w , weobtain a bijection between reduced word tableaux for w and EG-pipedreams of w .This paper is organized as follows. In Section 2, we give overviews of the Stanley sym-metric function, the Edelman-Greene insertion algorithm, the Lascoux-Sch¨utzenbergertree and the Little map. In Section 3, we describe the structure of bumpless pipedreamsas well as the droop operation introduced by Lam, Lee and Shimozono [19]. In Section4, we introduce the structure of a modified LS-tree. Based on Section 4, we constructthe structure of an EG-tree from a modified LS-tree in Section 5. In Section 6, using thestructures developed in Sections 4 and 5 together with the Edelman-Greene algorithmand the Little map, we establish a shape preserving bijection between reduced wordtableaux and EG-pipedreams. In this section, we collect some notions and structures that we are concerned with inthis paper, including the Stanley symmetric function, the Edelman-Greene insertionalgorithm, the Lascoux-Sch¨utzenberger tree and the Little map. The reader familiarwith these structures could skip this section.
Let w = w w · · · w n ∈ S n be a permutation on { , , . . . , n } . As usual, let s i denotethe simple transposition interchanging the elements i and i + 1. Note that ws i is thepermutation obtained from w by swapping w i and w i +1 . A decomposition of w as aproduct of simple transpositions is called reduced if it consists of a minimum numberof simple transpositions. A sequence ( a , a , . . . , a ℓ ) is called a reduced word of w if s a s a · · · s a ℓ is a reduced decomposition of w . The length of w , denoted ℓ ( w ), is thenumber of simple transpositions in a reduced decomposition of w .3tanley symmetric functions were introduced by Stanley [28] to enumerate the re-duced decompositions of a permutation. For a permutation w , the Stanley symmetricfunction F w ( x ) is defined as F w ( x ) = X ( a ,a ,...,a ℓ ) X ≤ b ≤ b ≤···≤ bℓai 1, then set S w ( x ; y ) = Y i + j ≤ n ( x i − y j ) . If w = w , then choose a simple transposition s i such that ℓ ( ws i ) = ℓ ( w ) + 1, and let S w ( x ; y ) = ∂ i S ws i ( x ; y ) . Here, ∂ i is the divided difference operator applies only to the x variables. That is, ∂ i f = ( f − s i f ) / ( x i − x i +1 ) , where f is a polynomial in x and s i f is obtained byinterchanging x i and x i +1 in f . Setting y i = 0, S w ( x ; y ) reduces to the (single) Schubertpolynomial S w ( x ).There have been several combinatorial rules to generate S w ( x ) and S w ( x ; y ), see, forexample [1–3, 10, 11, 14, 15, 32, 33]. To explain the relation between F w ( x ) and S w ( x ), letus recall the combinatorial construction of S w ( x ) due to Billey, Jockusch and Stanley [3]: S w ( x ) = X ( a ,a ,...,a ℓ ) X ≤ b ≤ b ≤···≤ bℓbi ≤ aiai 1) has been constructed. Letus generate P i +1 by inserting the integer a i +1 into P i . Set x = a i +1 . Let R be the firstrow of P i . Roughly speaking, an element in R may be bumped out and then insertedinto the next row. The process is repeated until no element is bumped out. There aretwo cases.Case 1: The integer x is strictly larger than all the entries in R . Let P i +1 be thetableau by adding x as a new box to the end of R , and the process terminates.Case 2: The integer x is strictly smaller than some element in R . Let y be theleftmost entry in R that is strictly larger than x . If replacing y by x results in anincreasing tableau, then y is bumped out by x and y will be inserted into the nextrow. If replacing y by x does not result in an increasing tableau, then keep the row R unchanged and the element y will be inserted into the next row.Iterating the above procedure, we finally get the tableau P i +1 . Set P ( a ) = P ℓ . Itshould be noted that in the process of inserting an integer into a row of P i , we will notencounter the situation that this integer is equal to the largest element in that row. Thetableau Q ( a ) is the standard Young tableau which records the changes of the shapes of P , P , . . . , P ℓ as the insertion is performed.For example, let a = (2 , , , , , , 2) be a reduced word of the permutation w =3514276. Then P ( a ) and Q ( a ) can be constructed as in Figure 2.1. P i : Q i : 21 2 31 2 12 313 2 12 3 613 2 4 12 36 413 25 4 126 34 4136 25 4 1246 23 41367 25 4 Figure 2.1: Construction of P ( a ) and Q ( a ) for a = (2 , , , , , , P ( a ), we need the notion of the row readingword of a tableau T , that is, the word obtained by reading the entries of T along therows from left to right, bottom to top. For example, the tableau P ( a ) in Figure 2.1 hasrow reading word (6 , , , , , , heorem 2.1 (Edelman-Greene [9]) . The Edelman-Greene correspondence is a bijectionbetween the set of reduced words of w and the set of pairs ( P, Q ) of tableaux of the sameshape, where P is an increasing tableaux whose row reading word is a reduced word of w ,and Q is a standard Young tableau. The following theorem is attributed to Edelman and Greene [9], see also Fomin andGreene [8], Lam [18] or Stanley [29]. Theorem 2.2 (Edelman-Greene [9]) . The coefficient c wλ equals the number of increasingtableaux of shape λ whose row reading words are reduced words of w − . Theorem 2.2 has an equivalent statement in terms of column reading words. Thecolumn reading word of T , denoted column( T ), is obtained by reading the entries of T along the columns from top to bottom, right to left.The following theorem follows from [5, Theorem 1] by restricting a Hecke word to areduced word. Here we give a self-contained proof based on properties of the Edelman-Greene algorithm. Theorem 2.3 (Buch-Kresch-Shimozono-Tamvakis-Yong [5]) . The coefficient c wλ equalsthe number of increasing tableaux of shape λ whose column reading words are reducedwords of w .Proof. Let T be an increasing tableau such that the row reading word of T is a reducedword of w − . We claim that column( T ) is a reduced word of w . Fix a standard Youngtableau Q which has the same shape as T . Let a = ( a , a , . . . , a ℓ ) be the reduced wordof w − corresponding to the pair ( T, Q ). Denote T t by the transpose of T , and write a rev = ( a ℓ , a ℓ − , . . . , a ) (2.4)for the reverse of a . Note that a rev is a reduced word of w . By Edelman and Greene [9](see also Felsner [7]), the insertion tableau of a rev is T t . By Theorem 2.1, the row readingword of T t is a reduced word of w , or equivalently, the column reading word of T is areduced word of w . This verifies the claim.Conversely, we can show that if T is an increasing tableau whose column readingword is a reduced word of w , then its row reading word is a reduced word of w − . Thiscompletes the proof.Throughout this paper, an increasing tableau T is called a reduced word tableau for w if column( T ) is a reduced word of w . Equivalently, by the proof of Theorem 2.3, anincreasing tableau T is a reduced word tableau for w if its row reading word is a reducedword of w − . The LS-tree is built based on maximal transitions on Stanley symmetric functions. Fora permutation w = w w · · · w n , let r = max { i | w i > w i +1 } , = max { i > r | w i < w r } ,I ( w ) = { i < r | w i < w s and ∀ j ∈ ( i, r ) , w j ( w i , w s ) } , where for two integers a and b with a < b , we use ( a, b ) to denote the interval { a + 1 , a +2 , . . . , b − } . In other words, r is the last decent of w , s is the unique position after r such that ℓ ( wt r,s ) = ℓ ( w ) − 1, where wt r,s is the permutation obtained by swapping w r and w s . And I ( w ) is the set of positions i before r such that ℓ ( wt r,s t i,r ) = ℓ ( w ). Withthe above notation, setΦ( w ) = ( { wt r,s t i,r | i ∈ I ( w ) } , if I ( w ) = ∅ ;Φ(1 × w ) , if I ( w ) = ∅ ,where 1 × w = 1( w + 1) · · · ( w n + 1) as defined in Section 2.1.Each permutation in the set Φ( w ) is called a child of w . The LS-tree of w is obtainedby recursively applying the above operation until every leaf is a Grassmanian permuta-tion. A Grassmanian permutation is a permutation with at most one descent. In Figure2.2, we illustrate the LS-tree of w = 231654, where, as used in [23], the entries w r , w s and w i with i ∈ I ( w ) are boxed, barred and underlined, respectively.2 3 1 6 5 ¯42 4 1 6 3 ¯5 2 3 4 6 1 ¯52 5 1 4 ¯3 6 2 4 5 1 ¯3 6 2 3 5 4 ¯1 61 3 4 6 5 ¯2 73 5 1 2 4 6 2 5 3 ¯1 4 61 3 6 4 ¯2 5 7 3 4 2 ¯1 5 61 4 5 3 ¯2 6 7 2 3 4 6 1 5 72 3 6 1 4 5 7 2 4 5 1 3 6 7Figure 2.2: The LS-tree for w = 231654.According to the transition formula for Schubert polynomials due to Lascoux andSch¨utzenberger [21], one has the following transition relation for Stanley symmetricfunctions: F w ( x ) = X w ′ ∈ Φ( w ) F w ′ ( x ) , (2.5)see also [23, Theorem 1] or Garsia [12]. Since each leaf of a LS-tree is labeled with aGrassmannian permutation, F w ( x ) can be eventually written as a sum of Stanley sym-metric functions indexed by Grassmannian permutations. For a Grassmannian permu-tation, the Stanley symmetric function becomes a single Schur function. More generally,7s proved by Stanley [28], the Stanley symmetric function for a vexillary permutationis a single Schur function. Since Grassmannian permutations are vexillary, the LS-treeimplies the Schur positivity of Stanley symmetric functions.Recall that a permutation is called a vexillary (or 2143-avoiding) permutation if itdoes not contain a subsequence order-isormorphic to 2143, that is, there are no indices i < i < i < i such that w i < w i < w i < w i . For a permutation w ∈ S n , let c ( w ) = ( c ( w ) , c ( w ) , . . . , c n ( w ))be the Lehmer code of w , where c i ( w ) = |{ j | j > i, w j < w i }| . Define a partition λ ( w ) byrearranging the Lehmer code of w in weakly decreasing order. For example, the Lehmercode of w = 35412 is (2 , , , , λ ( w ) = (3 , , w , Stanley [28] proved that F w ( x ) = s λ ( w ) ( x ) , which also follows from (2.3)together with a tableau formula for Schubert polynomials of vexillary permutations dueto Wachs [31] or Knutson, Miller and Yong [16].In particular, a permutation w is called a dominant permutation if w is 132-avoiding.It is clear that a dominant permutation is also a vexillary permutation. The Lehmer codeof a dominant permutation w is a weakly decreasing sequence which form a partitionshape λ ( w ), see Stanley [27, Chapter 1]. Hence, for a dominant permutation w , we have F w ( x ) = s λ ( w ) ( x ) , where λ ( w ) = c ( w ) is the partition equal to the Lehmer code of w .Therefore, there is only one reduced word tableau for w . In fact, the only reduced wordtableau T ( w ) for w can be obtained as follows: fill the entry in the box ( i, j ) of λ ( w )with i + j − 1. The tableau T ( w ) is also called a frozen tableau, see [22].There is a more general relation satisfied by Stanley symmetric functions, including(2.5) as a special case. For a permutation u = u u · · · u n and 1 ≤ k ≤ n , let I ( u, k ) = { i < k | ℓ ( ut i,k ) = ℓ ( u ) + 1 } , (2.6)and S ( u, k ) = { j > k | ℓ ( ut k,j ) = ℓ ( u ) + 1 } . (2.7)Define two sets of permutations byΦ( u, k ) = ( { ut i,k | i ∈ I ( u, k ) } , if I ( u, k ) = ∅ ;Φ(1 × u, k + 1) , otherwise , (2.8)and Ψ( u, k ) = ( { ut k,j | j ∈ S ( u, k ) } , if S ( u, k ) = ∅ ;Ψ( u × , k ) , otherwise , (2.9)where u × u u · · · u n ( n + 1).By the Monk’s rule for Schubert polynomials (see for example [1, 26]) together with(2.3), one can easily establish the following relation: X w ∈ Ψ( u,k ) F w ( x ) = X w ∈ Φ( u,k ) F w ( x ) . (2.10)Indeed, (2.10) contains (2.5) as a special case, sinceΦ( wt r,s , r ) = Φ( w ) and Ψ( wt r,s , r ) = { w } . .4 Little map Little [23] developed a bijection, known as the Little map, to give a combinatorial proofof (2.10). Let Red( w ) denote the set of reduced words of a permutation w . The Littlemap is a descent preserving bijection: θ k : [ w ∈ Ψ( u,k ) Red( w ) −→ [ w ′ ∈ Φ( u,k ) Red( w ′ ) . (2.11)Indeed, in view of (2.1), the Little map yields a combinatorial proof of (2.10).The Little map is defined based on a bumping algorithm acting on the line diagramsfor permutations. Assume that a = ( a , a , . . . , a ℓ ) is a word (not necessarily reduced)of a permutation w . The line diagram of a is the array { , , . . . , ℓ } × { , , . . . , n } inthe Cartesian coordinates, which describes the trajectories of the numbers 1 , , . . . , n asthey are arranged into the permutation w by successive simple transpositions. Note that a is reduced if and only if no two lines cross more than once. For example, Figure 2.3 isthe line diagram of the reduced word a = (5 , , , , 5) of w = 231654. Figure 2.3: The line diagram of a = (5 , , , , a = ( a , a , . . . , a m ) and an integer 1 ≤ t ≤ m , let a ( t ) = ( a , a , . . . , a t − , a t +1 , . . . , a m )and define a ↑ t = ( a , a , . . . , a t − , a t − , a t +1 , . . . , a m ) , if a t > a + 1 , , a + 1 , . . . , a t − + 1 , a t , a t +1 + 1 , . . . , a m + 1) , if a t = 1 , where a ↑ t is called the word obtained from a by bumping at time t . Denote the wordobtained after a sequence of bumps by a ↑ t ,t ,...,t i = ((( a ↑ t ) ↑ t ) · · · ) ↑ t i . The bumping algorithm, called the Little bump, transforms a reduced word intoanother reduced word, which can be sketched as follows. For more detailed information,9ee Little [23] or Hamaker and Young [25]. Let a = ( a , a , . . . , a m ) be a reduced word.Assume that t is an index such that a ( t ) is also reduced. Consider the word a ↑ t .If a ↑ t is a reduced word, then the algorithm terminates. Otherwise, it can be shownthat there is a unique index, say t , such that ( a ↑ t ) ( t ) is reduced. Now consider theword a ↑ t ,t . Repeating the above procedure, one is eventually left with a reduced word a ↑ t ,t ,...,t i . The above process is referred to as the Little bump.Using the above bumping algorithm, we can define the Little map θ k . Given apermutation w ∈ Ψ( u, k ), let a be a reduced word of w . By the definition of the setΨ( u, k ), there is an index j > k in S ( u, k ) such that w = u t k,j and ℓ ( w ) = ℓ ( u ) + 1. Letus first define a map θ k,w j as follows. Since k < j , w k > w j and a is a reduced word, thereis exactly one letter, say a t , that interchanges w k and w j . Because ℓ ( w ) = ℓ ( u ) + 1, theword a ( t ) is reduced. Thus we can invoke the Little bump beginning at the position t .Let a ′ be the reduced word after applying the Little bump. Define θ k ( a ) = θ k,w j ( a ) = a ′ . Little [23] showed that a ′ ∈ Red( w ′ ) for some w ′ ∈ Φ( u, k ) . (2.12)We remark that although the subscript w j in θ k,w j is determined once a reduce word of w ∈ Ψ( u, k ) is given, we would like to use the two parameters k and w j since this wouldbe more convenient to describe the inverse of the Little map.For example, let u = 241536 and k = 5. Then w = u t , = 241563 is the uniquepermutation in Ψ( u, k ). Let a = (3 , , , , 2) be a reduced word of w . Then w k = 6 , w j =3 and t = 4. Applying the Little map θ to a , the resulting word is (2 , , , , w ′ = 341526 in Φ( u, k ). The bumping process isillustrated in Figure 2.4, where the dotted crossings indicate the bumped positions. Figure 2.4: Applying the Little map θ to a = (3 , , , , Theorem 2.4 (Little [23]) . The map θ k is a descent preserving bijection. As described by Little [23], the inverse of θ k can be stated as follows. Assume that w ′ is a permutation in Φ( u, k ) and that i < k is the position in I ( u, k ) such that w ′ = wt i,k .Let a ′ = ( a ′ , a ′ , . . . , a ′ ℓ ) be a reduced word of w ′ and denote ( a ′ ) c = ( n − a ′ , . . . , n − a ′ ℓ ).Note that ( a ′ ) c is a reduced word of the permutation v = v v · · · v n where v i = n + 1 − w ′ n +1 − i . Then one has θ − k ( a ′ ) = ( θ n +1 − k,n +1 − w ′ i (( a ′ ) c )) c . (2.13)10e conclude with two properties of the Little bump due to Hamaker and Young [25],which will be used in Section 6. First, the Little bump preserves the recording tableauxof reduced words when applying the Edelman-Greene algorithm, which was conjecturedby Lam [18] and proved by Hamaker and Young [25]. Theorem 2.5 (Hamaker-Young [25, Theorem 2]) . Let a and a ′ be two reduced wordssuch that there exists a sequence of Little bumps changing a to a ′ . Then Q ( a ) = Q ( a ′ ) . The second property was essentially implied in the proof of Lemma 6 in [25]. Theorem 2.6 (Hamaker-Young [25, Lemma 6]) . Let T be a reduced word tableau and a = column( T ) . Assume that b is obtained from a by applying a Little bump, and let T ′ be the insertion tableau of the reverse b rev of b under the Edelman-Greene algorithm.Then column( T ′ ) = b . In this section, we give an overview of the structure of bumpless pipedreams as well as thedroop operation on bumpless pipedreams. The droop operation has a close connectionto the modified LS-tree, as will be seen in Section 4.Lam, Lee and Shimozono [19] introduced several versions of bumpless pipedreamsfor a permutation w ∈ S n . For the purpose of this paper, we shall only be concernedwith the bumpless pipedreams in the region of a square grid, which are called w -squarebumpless pipedreams in [19]. Given an n × n square grid, we use the matrix coordinatesfor unit squares, that is, the row coordinates increase from top to bottom and the columncoordinates increase from left to right. Let us use ( i, j ) to denote the box in row i andcolumn j . A bumpless pipedream for w consists of n pipes labeled 1 , , . . . , n , flowingfrom the south boundary of the n × n square grid to the east boundary, such that(1) the pipe labeled i enters from the south boundary in column i and exits from theeast boundary in row w − ( i );(2) pipes can only go north or east;(3) no two pipes overlap any step or cross more than once;(4) no two pipes change their directions in a box simultaneously. In other words, eachbox looks like one of the first six tiles as shown in Figure 3.5: an empty box, anNW elbow, an SE elbow, a horizontal line, a vertical line, and a crossing.Figure 3.5: Boxes in a bumpless pipedream.11or ease of drawing pictures, an NW elbow and an SE elbow are replaced by the tilesgiven in Figure 3.6. Since two pipes cannot bump at the same box, there is no ambiguityfor this simplification. We illustrate four bumpless pipedreams of w = 2761453 in Figure3.7. −→ −→ Figure 3.6: NW elbow and SE elbow. (a) (b) (c) (d)Figure 3.7: Four Bumpless pipedreams of w = 2761453.Lam, Lee and Shimozono [19] showed that the bumpless pipedreams of w can gener-ate the Schubert polynomial S w ( x ), or more generally the double Schubert polynomial S w ( x ; y ). For a bumpless pipedream P , define the weight wt( P ) of P to be the productof x i − y j over all empty boxes of P in row i and column j . Theorem 3.1 (Lam-Lee-Shimozono [19, Theorem 5.13]) . For any permutation w , S w ( x ; y ) = X P wt( P ) , (3.1) where the sum is over the bumpless pipedreams of w . Lam, Lee and Shimozono [19] also discovered that a specific family of bumplesspipedreams, called EG-pipedreams, can be used to interpret the Edelman-Greene coef-ficients. A bumpless pipedream P is called an EG-pipedream if all the empty boxes of P are at the northwest corner, where they form a Young diagram λ = λ ( P ), called theshape of P . For example, Figure 3.7(d) is an EG-pipedream with shape λ = (5 , , , Theorem 3.2 (Lam-Lee-Shimozono [19, Theorem 5.14]) . The Edelman-Greene coeffi-cient c wλ is equal to the number of EG-pipedreams of w with shape λ . For example, there are three EG-pipedreams for w = 321654, as illustrated in Figure3.8. By Theorem 3.2, we see that F w ( x ) = s (4 , ( x ) + 2 s (3 , , ( x ) . In the remaining of this section, we recall the droop operation on bumpless pipedreamsintroduced by Lam, Lee and Shimozono [19]. As will be seen in Corollary 4.7, the droopoperation is closely related to the modified LS-tree.12 Figure 3.8: The EG-pipedreams of w = 321654.Let P be a bumpless pipedream of w . The pipes in P are determined by the locationsof the NW elbows and the SE elbows. Generally speaking, a droop is a local move whichswaps an SE elbow e with an empty box t , when the SE elbow e lies strictly to thenorthwest of the empty box t . To be more specific, let R be the rectangle with northwestcorner e and southeast corner t , and let L be the pipe passing through the SE elbow e .A droop is allowed only if(1) the pipe L passes through the westmost column and northmost row of R ;(2) the rectangle R contains only one elbow: the SE elbow which is at e ;(3) after the droop we obtain another bumpless pipedream.After a droop, the pipe L travels along the southmost row and eastmost column of R ,and an NW elbow occupies the box that used to be empty while the box that containedan SE elbow becomes an empty box. Pipes in P except the pipe L do not change afterthe droop. Figure 3.9 is an illustration of the local move of the droop operation. Formore examples, Figure 3.7(b) is obtained from Figure 3.7(a) by a droop, and Figure3.7(c) is obtained from Figure 3.7(b) by a droop. droop −−−−→ Figure 3.9: The droop operation.Lam, Lee and Shimozono [19] proved that each bumpless pipedream of w can begenerated from the Rothe pipedream of w by applying a sequence of droops. The Rothepipedream of w , denoted D ( w ), is the unique bumpless pipedream of w that does notcontain any NW elbows, see Figure 3.7(a) for an example. Clearly, for 1 ≤ i ≤ n , pipe i passes through exactly the SE elbow ( i, w i ) in row i and column w i .The Rothe pipedream D ( w ) can also be constructed as follows. From the center ofeach box ( i, w i ), draw a horizonal line to the right and a vertical line to the bottom. Thisforms n hooks with turning points at the center of the boxes ( i, w i ). Thinking of each13ook as a pipe, the pipes together with the n × n grid form the Rothe pipedream D ( w ).The empty boxes of D ( w ) are known as the Rothe diagram of w , denoted Rothe( w ),which encode the positions of inversions of w . That is, there is an empty box of Rothe( w )at ( i, j ) if and only if w i > j and the number j appears in w after the position i . Proposition 3.3 (Lam-Lee-Shimozono [19, Proposition 5.3]) . For a permutation w ,every bumpless pipedream of w can be obtained from the Rothe pipedream D ( w ) by asequence of droops. In this section, we introduce the structure of the modified LS-tree of a permutation. Wealso explain the relation between the modified LS-trees and the droop operations.For a permutation w = w w · · · w n , let p be the largest index in w such that thereare indices i and j satisfying i < p < j and w i < w j < w p , (4.1)namely, p = max { t | ≤ t ≤ n − , ∃ i < t < j, s.t. w i < w j < w t } . (4.2)In other words, p is the largest position such that there is a subsequence w i w p w j whichis order-isomorphic to 132. For example, for the permutation w = 2431 we have p = 2,whereas for the permutation w = 3421 we cannot find a position satisfying the conditionin (4.1).By definition, there does not exist an index p satisfying (4.1) if and only if w is132-avoiding. If a permutation w contains a 132 pattern, we also call w a non-dominantpermutation.Assume that w is a non-dominant permutation. Let q be the largest index after p such that w p > w q and there exists an index i < p such that w i < w q , namely, q = max { j | j > p, w j < w p , ∃ i < p, s.t. w i < w j } . (4.3)For example, for w = 645978321, we have p = 4 and q = 6. We have the followingequivalent description of the index q . Lemma 4.1. Let w be a non-dominant permutation. Then the index q defined in (4.3) is the index of the largest number following w p which is less than w p .Proof. Suppose otherwise that q ′ = q is the index of the largest number following w p which is less than w p . By the definition of q in (4.3), we see that p < q ′ < q . On theother hand, by the definition of p , there is an index i < p such that w i < w q < w p . Thuswe have w i < w q < w q ′ , implying that q ′ satisfies (4.1). This contradicts the choice ofthe index p . So the proof is complete.The following proposition implies that the indices p and q play similar roles to theindices r and s as defined in Section 2.3. 14 roposition 4.2. Let w be a non-dominant permutation. Then we have Ψ( wt p,q , p ) = { w } . (4.4) Proof. Write u = wt p,q = u u · · · u n , where u p = w q , u q = w p , and u i = w i for i = p, q .By Lemma 4.1, we see that ℓ ( wt p,q ) = ℓ ( w ) − , which implies ℓ ( ut p,q ) = ℓ ( w ) = ℓ ( u ) + 1 . So we have w ∈ Ψ( u, p ).Suppose that there is another permutation w ′ ∈ Ψ( u, p ) which is not equal to w . Bythe definition of Ψ( u, p ) in (2.9), there exists an index q ′ = q such that q ′ > p , w ′ = u t p,q ′ and ℓ ( w ′ ) = ℓ ( u ) + 1 . Then we have u q ′ > u p , that is, w q ′ > w q = u p . (4.5)There are two cases.Case 1: p < q ′ < q . By the definition of q , there is an index i < p such that w i < w q < w p . In view of (4.5), we see that w i < w q < w q ′ , which implies that the index q ′ satisfies the condition in (4.1). This contradicts the choice of the index p .Case 2: q ′ > q . In this case, since ℓ ( w ′ ) = ℓ ( u ) + 1, we must have w q ′ < w p , which,together with (4.5), implies that w q < w q ′ < w p . This is contrary to Lemma 4.1. So theproof is complete.Combining (2.10) and Proposition 4.2, we arrive at the following transition relationsatisfied by Stanley symmetric functions. Theorem 4.3. Let w ∈ S n be a non-dominant permutation. Then we have F w ( x ) = X w ′ ∈ Φ( wt p,q ,p ) F w ′ ( x ) . (4.6) Moreover, by the choices of the indices p and q , the set I ( wt p,q , p ) is not empty, andhence each permutation in Φ( wt p,q , p ) is still a permutation on { , , . . . , n } . For example, for w = 645978321, we see that p = 4, q = 6 and so we have wt p,q =645879321 . Moreover, one can check that I ( wt p,q , p ) = { , } . Therefore,Φ( wt p,q , p ) = { , } . The construction of the modified LS-tree of w relies on Theorem 4.3. To be specific,for a permutation w ∈ S n , iterate the relation in (4.6) until each leaf is a dominantpermutation on { , , . . . , n } . The resulting tree is called the modified LS-tree of w .Figure 4.10 illustrates the modified LS-tree of w = 231654. Remark 4.4. The construction of a modified LS-tree is feasible because the processeventually terminates. This can be seen as follows. For two sequences c = ( c , c , . . . , c n ) and c ′ = ( c ′ , c ′ , . . . , c ′ n ) of nonnegative integers of length n , write c ≥ c ′ if for ≤ i ≤ n , c + · · · + c i ≥ c ′ + · · · + c ′ i . 15 3 1 6 5 ¯42 4 1 6 3 ¯5 2 3 4 6 1 ¯52 5 1 4 ¯3 6 2 4 5 1 ¯3 6 2 3 5 ¯4 1 63 5 1 2 4 6 2 5 3 1 ¯4 6 3 4 2 1 5 6 2 4 ¯3 5 1 64 3 1 2 5 6 4 2 3 1 5 6 3 2 4 5 1 6Figure 4.10: The modified LS-tree of w = 231654. This defines a partial order on the sequences of nonnegative integers of length n . Assumethat w ∈ S n is a non-dominant permutation. It is easy to check that for w ′ ∈ Φ( wt p,q , p ) ,the Lehmer code of w is smaller than the Lehmer code of w ′ under the above order.Hence, the nodes in any path from the root to a leaf in the modified LS-tree are labeledby distinct permutations on { , , . . . , n } . So the process terminates. The differences between a modified LS-tree and an ordinary LS-tree are obvious.First, each node in a modified LS-tree is labeled with a permutaton on { , , . . . , n } ,whereas in an ordinary LS-tree, there may exist nodes which are labeled with permuta-tions on { , , . . . , m } with m > n . Second, each leaf in a modified LS-tree is labeledwith a dominant permutation, whereas each leaf in an ordinary LS-tree is labeled witha Grassmannian permutation. To make a comparison, see Figure 2.2 and Figure 4.10.Since each leaf in the modified LS-tree of w is a dominant permutation, according toTheorem 4.3, we obtain the following interpretation for the Edelman-Greene coefficients. Theorem 4.5. The Edelman-Greene coefficient c wλ equals the number of leaves in themodified LS-tree of w whose labels are the dominant permutations with Lehmer code λ . In the modified LS-tree of w , if replacing the label of each node with its correspondingRothe pipedream, then we obtain a tree labeled with Rothe pipedreams. Figure 4.11is the Rothe pipedream version of Figure 4.10. By using Rothe pipedreams to labelthe nodes, we can directly read off the Schur functions in the expansion of a Stanleysymmetric function. Moreover, it will be more convenient to use the Rothe pipedreamversion of a modified LS-tree in the construction of an EG-tree in Section 5.The remaining of this section are devoted to several propositions that will be used inSection 5 to construct the EG-tree of w from the modified LS-tree of w .16igure 4.11: The Rothe pipedream version of Figure 4.10.Propositions 4.6 and 4.7 tell us how to use the droop operation to generate thechildren of a node in a modified LS-tree. To describe these two propositions, we definethe notion of a pivot of an empty box in the Rothe pipedream of a permutation w . Notethat for the specific empty box ( r, w s ), where r and s are defined in Section 2.3, thenotion of pivots has been defined by Knutson and Yong [17]. Let ( i, j ) be an empty boxin the Rothe pipedream D ( w ) of w . We say that an SE elbow e of D ( w ) is a pivot of ( i, j )if e is northwest of ( i, j ) and there are no other elbows contained in the rectangle withnorthwest corner e and southeast corner ( i, j ). In other words, among the SE elbowsthat are northwest of ( i, j ), the pivots are maximally southeast. For example, the Rothepipedream of w = 2761453 is displayed in Figure 3.7(a). The empty box (3 , 1) has nopivots, whereas the empty box (6 , 3) has two pivots: (1 , 2) and (4 , w , it is easily seen that ( p, w q ) is an empty box of D ( w ). The following proposition gives a characterization of the pivots of this specificempty box. Proposition 4.6. Let w be a non-dominant permutation. Then the set of pivots of theempty box ( p, w q ) is { ( i, w i ) | i ∈ I ( wt p,q , p ) } , (4.7) where I ( wt p,q , p ) is the index set as defined in (2.6) .Proof. The assertion follows from the definition of pivots as well as the definition of theindex set I ( wt p,q , p ).Clearly, we can apply a droop to the Rothe pipedream of w with respect to the emptybox ( p, w q ) and a pivot of ( p, w q ). 17 roposition 4.7. Assume that w is a non-dominant permutation. Let P be the bumplesspipedream of w obtained from the Rothe pipedream D ( w ) by applying a droop swappingthe empty box ( p, w q ) and a pivot ( i, w i ) , where i ∈ I ( wt p,q , p ) . Then the Rothe diagramof w ′ = wt p,q t i,p ∈ Φ( wt p,q , p ) equals the set of empty boxes of P .Proof. Let us first explain that the Rothe diagram Rothe( w ′ ) of w ′ can be constructedfrom the Rothe diagram Rothe( w ) of w as follows. First, note that Rothe( wt p,q ) can beobtained from Rothe( w ) by deleting the box ( p, w q ), see Figure 4.12 for an illustration.Let us proceed to determine Rothe( w ′ ) from Rothe( wt p,q ). Locate the rectangle R in the −−−→ Figure 4.12: Construction of Rothe( wt p,q ) from Rothe( w ). n × n square grid, such that the northwest corner is ( i, w i ) and the southeast corner is( p, w q ). Then Rothe( w ′ ) can be obtained from Rothe( wt p,q ) as follows:(1) Move each box of Rothe( wt p,q ) in the bottom row of R to the top row of R ;(2) Move each box of Rothe( wt p,q ) in the rightmost column of R to the leftmost columnof R ;(3) Add a new box to the position ( i, w i ).Figure 4.13 illustrates the above construction of Rothe( w ′ ) from Rothe( wt p,q ). −→ −→ −→ Figure 4.13: Construction of Rothe( w ′ ) from Rothe( wt p,q ).One the other hand, one can apply a droop on D ( w ) swapping ( i, w i ) and ( p, w q ) toobtain a bumpless pipedream P of w , see Figure 4.14 for an illustration.Evidently, the droop has the same effect on the empty boxes of D ( w ) (that is, theRothe diagram of w ) as the operation illustrated in Figures 4.12 and 4.13. This competesthe proof. 18 −−→ Figure 4.14: A droop operation on D ( w ). Remark 4.8. When p is the last descent r of w , Knutson and Yong [17] introducedthe marching operation to generate the Rothe diagram of a child of w from the Rothediagram Rothe( w ) of w . In this specific case, the marching operation has the same effecton Rothe( w ) as the droop operation. Propositions 4.6 and 4.7 give an explicit way to generate the children of a non-dominant permutation w in the modified LS-tree of w . Assume that e , . . . , e m are thepivots of ( p, w q ). For 1 ≤ i ≤ m , let P i be the bumpless pipedream of w obtained from D ( w ) by applying a droop with respect to ( p, w q ) and e i . Denote by Box( P i ) the emptyboxes of P i . Then Box( P ) , . . . , Box( P m ) are the Rothe diagrams of the children of w .Equivalently, Box( P ) , . . . , Box( P m ) are the empty boxes of Rothe pipedreams of thechildren of w .The next two propositions investigate the properties concerning the specific emptybox ( p, w q ). Define a total order on the boxes of the n × n grid by letting ( i, j ) < ( k, ℓ ) ifeither i < k , or i = k and j < ℓ. (4.8)Let pivot( w ) denote the set of empty boxes in the Rothe pipedream of w which has atleast one pivot. Proposition 4.9. Let w be a non-dominant permutation. Then ( p, w q ) is the largest boxin the set pivot( w ) under the total order defined in (4.8) .Proof. Suppose otherwise that ( p ′ , j ) = ( p, w q ) is the largest box in pivot( w ). Then thereis an index q ′ > p ′ such that w q ′ = j . Since ( p, w q ) is the rightmost empty box in the p -th row of D ( w ), we must have p ′ > p . Let ( i ′ , w i ′ ) be a pivot of ( p ′ , w q ′ ), where i ′ < p ′ .By the definition of a pivot, we have w i ′ < j = w q ′ . Moreover, notice that w q ′ = j < w p ′ .So, for the indices i ′ < p ′ < q ′ , we find that w i ′ < w q ′ < w p ′ . This implies that p ′ satisfiesthe condition (4.1), contrary to the choice of p . This concludes the proof.Using Proposition 4.9, we can prove the following assertion. Proposition 4.10. Let w be a non-dominant permutation, and let w ′ = wt p,q t i,p ∈ Φ( wt p,q , p ) be a child of w . Suppose that w ′ is also a non-dominant permutation. Write p ′ and q ′ for the indices as defined in (4.2) and (4.3) for w ′ , respectively. Then ( p ′ , w ′ q ′ ) < ( p, w q ) (4.9) under the total order defined in (4.8) . roof. The proof is best understood by means of the pictures of the Rothe pipedreamsof w and w ′ , as illustrated in Figure 4.15. The dashed rectangle in Figure 4.15 signifies w i w p w q w i w q w p w q w i w p w i w q w p Figure 4.15: Rothe pipedreams of w and w ′ .the rectangle where the droop operation on D ( w ), as described in Proposition 4.7, tookplace. The shaded area of D ( w ′ ) in Figure 4.15 contains the boxes of D ( w ′ ) smaller thanthe SE elbow ( p, w i ) under the total order (4.8). Since w ′ p = w i , there are no emptyboxes in the p -th row of D ( w ′ ) after the box ( p, w i ). So (4.9) is equivalent to saying thatthe empty box ( p ′ , w ′ q ′ ) of D ( w ′ ) lies in the shaded area.Suppose otherwise that ( p ′ , w ′ q ′ ) > ( p, w q ). Then ( p ′ , w ′ q ′ ) lies strictly below row p .Note that ( p ′ , w ′ q ′ ) is also an empty box of D ( w ). There are two cases to discuss.Case 1: w i < w ′ q ′ . In this case, in the Rothe pipedream of w , ( i, w i ) is an SE elbow thatis northwest of ( p ′ , w ′ q ′ ). Hence ( p ′ , w ′ q ′ ) has a pivot in the Rothe pipedream of w , whichis contrary to Proposition 4.9.Case 2: w i > w ′ q ′ . Let e be an SE elbow in the Rothe pipedream of w ′ which is apivot of ( p ′ , w ′ q ′ ). Still, from Figure 4.15, we see that e is also an SE elbow in the Rothepipedream of w . This implies that ( p ′ , w ′ q ′ ) has a pivot in the Rothe pipedream of w ,which again contradicts Proposition 4.9. In this section, we introduce the structure of the Edelman-Greene tree (EG-tree) of apermutation w . Each node in the EG-tree of w is labeled with a bumpless pipedream of w . In particular, we show that the leaves in the EG-tree of w are exactly labeled withthe EG-pipedreams of w .Let us proceed with the construction of the EG-tree of w . The idea behind is that foreach path in the modified LS-tree of w , we construct a corresponding path of bumplesspipedreams of w . Assume that w = w (0) → w (1) → · · · → w ( m ) is a path in the modified LS-tree of w from the root w (0) = w to a leaf w ( m ) . We aim toconstruct a sequence P → P → · · · → P m (5.1)of bumpless pipedreams of w such that 201) P is the Rothe pipedream of w and P m is an EG-pipedream of w ;(2) For 0 ≤ j ≤ m , the empty boxes of P j are the same as the empty boxes of theRothe pipedream D ( w ( j +1) ) of w ( j ) .We now describe the construction of the pipedreams in (5.1). For 0 ≤ j ≤ m − p j and q j be the indices of the permutation w ( j ) as defined in (4.2) and (4.3). By theconstruction of the modified LS-tree of w , there exists an index i j ∈ I ( w ( j ) t p j ,q j , p j ) suchthat w ( j +1) = w ( j ) t p j ,q j t i j ,p j . By Proposition 4.7, we can generate the empty boxes of D ( w ( j +1) ) by applying a droop to D ( w ( j ) ) with respect to the empty box ( p j , w ( j ) q j ) andthe pivot ( i j , w ( j ) i j ). As will be seen in the proof of Proposition 5.1, we can also apply adroop to P j with respect to ( p j , w ( j ) q j ) and ( i j , w ( j ) i j ). Let P j +1 be the bumpless pipedreamof w obtained by applying a droop to P j with respect to ( p j , w ( j ) q j ) and ( i j , w ( j ) i j ). Proposition 5.1. The above construction from P j to P j +1 is feasible.Proof. Let us first consider the case j = 0. Since P = D ( w ), by Proposition 4.7, wecan apply a droop to P by swapping the empty box ( p , w (0) q ) and its pivot ( i , w (0) i ),resulting in the bumpless pipedream P .We proceed to consider the case j = 1. By Proposition 4.7, the empty boxes of D ( w (2) ) can be obtained from D ( w (1) ) by applying a droop with respect to the emptybox ( p , w (1) q ) and its pivot ( i , w (1) i ). By Proposition 4.10, there holds that ( p , w (1) p ) < ( p , w (0) p ). By the construction of a droop operation, it is easy to check that P and D ( w (1) ) have the same tile at each position smaller than ( p , w (0) q ). Thus we can applya droop to P with respect to ( p , w (1) q ) and ( i , w (1) i ), yielding the bumpless pipedream P of w .For the same reason as above, in the general case for j ≥ 2, we can check that P j and D ( w ( j ) ) have the same tile at each position smaller than ( p j − , w ( j − q j − ), and so thatwe can apply a droop to P j with respect to ( p j , w (1) q j ) and ( i j , w ( j ) i j ). This generates thebumpless pipedream P j +1 of w .By the proof of Proposition 5.1, the empty boxes of P j are the same as the emptyboxes of the Rothe pipedream of w ( j ) . Notice that the Rothe diagram of a dominantpermutation is a partition shape at the northwest corner of the n × n grid, see Fulton [13]or Stanley [27, Chapter 1]. Thus the bumpless pipedream P m is an EG-pipedream of w .We apply the above procedure to each path from the root to a leaf in the modifiedLS-tree of w . The resulting tree is called the Edelman-Greene tree (EG-tree) of w . Figure5.16 is the EG-tree of w = 231654.By the construction of an EG-tree of w , we summarize its relation to a modifiedLS-tree of w in the following theorem. Theorem 5.2. For permutation w , the modified LS-tree of w and the EG-tree of w areisomorphic. For any given node labeled with a permutation u in the modified LS-tree of w , let P be the corresponding bumpless pipedream of w in the EG-tree of w . Then theempty boxes of the Rothe pipedream of u are the same as the empty boxes of P . w = 231654.We have obtained a map from the leaves of an EG-tree of w to the EG-pipedreamsof w . In the following theorem, we shall give the reverse procedure. Theorem 5.3. Let w ∈ S n be a permutation and P be an EG-pipedream of w . Thenthere is a leaf in the EG-tree of w whose label is P .Proof. Assume that P has m NW elbows, say,( i m , j m ) < ( i m − , j m − ) < · · · < ( i , j ) , which are listed in the total order as defined in (4.8). We construct a sequence P = P m → P m − → · · · → P (5.2)of bumpless pipedreams of w such that P m = P and P = D ( w ).First, we construct the pipedream P m − from P m = P . The construction is the sameas that in the proof of [19, Proposition 5.3] and is sketched below. Let L be the pipe in P m passing through the NW elbow ( i m , j m ). Then L passes through an SE elbow ( i m , y )(respectively, ( x, j m )) in the same row (respectively, column). Let R be the rectanglewith corners ( i m , y ) and ( x, j m ). It is easy to check that the northwest corner of R is anempty box, and that there are no any other elbows in R . Let P m − be the pipedreamobtained from P m by a “reverse droop”, that is, change the pipe L to travel along thewestmost column and northmost row of R , see Figure 5.17 for an illustration.Using the same procedure as above, we can construct P k − from P k (1 ≤ k ≤ m ) byapplying a “reverse droop” corresponding to the NW elbow ( i k , j k ). Figure 5.18 gives anexample to illustrate the generation of the chain from an EG-pipedream.22 i m y j m −−−→ xi m y j m Figure 5.17: Reverse droop operation. (2 , −−→ (4 , −−→ (4 , −−→ (5 , −−→ (2 , −−→ (4 , −−→ (4 , −−→ (5 , −−→ Figure 5.18: Example of the construction of the sequence in (5.2).We show that the sequence in (5.2) is a path from a leaf to the root in the EG-treeof w . Since P k − has one fewer NW elbows than P k , the bumpless pipedream P has noNW elbows and hence is the Rothe pipedream of w . Set u (0) = w . By the constructionof the sequence (5.2) together with the fact that the empty boxes of P m form a partitionat the northwest corner of the n × n grid, it is not hard to check that ( i , j ) is the largestbox in the set pivot( w ) under the total order defined in (4.8). In view of Proposition 4.7,the empty boxes of P are also the empty boxes of the Rothe pipedream of u (1) for some u (1) ∈ Φ( wt p,q , p ). Along the same line, we can deduce that for 2 ≤ k ≤ m , the emptyboxes of P k form the empty boxes of the Rothe pipedream of some u ( k ) . In particular,for 1 ≤ k ≤ m , u ( k ) is a child of u ( k − in the modified LS-tree of w . Hence, the sequence w = u (0) → u (1) → · · · → u ( m ) forms a path in the modified LS-tree of w from the root to a leaf u ( m ) . This shows that(5.2) is a path from a leaf to the root in the EG-tree of w . Now we see that P is thelabel of a leaf in the EG-tree of w . This completes the proof.It is easy to verify that the construction of (5.2) is the reverse process of the con-struction of (5.1). Hence we arrive at the following conclusion. Corollary 5.4. For a permutation w , the labels of leaves in the EG-tree of w are inbijection with the EG-pipedreams of w . By Theorem 4.5 and Corollary 5.4, we obtain an alternative proof of Theorem 3.2. In this section, we establish the promised bijection between reduced word tableaux andEG-pipedreams. For a permutation w , let RT( w ) denote the set of reduced word tableaux23or w , namely, the set of increasing tableaux whose column reading words are reducedwords of w . Let EG( w ) denote the set of EG-pipedreams of w . Theorem 6.1. There is a shape preserving bijection between RT( w ) and EG( w ) . By Theorem 5.2 and Corollary 5.4, we need only to establish a bijection betweenthe set RT( w ) and the set of leaves in the modified LS-tree of w . However, we shalldirectly construct a shape preserving bijection between RT( w ) and EG( w ). Of course,such a construction implies a bijection between the set RT( w ) and the set of leaves inthe modified LS-tree of w .We first define two maps Γ : RT( w ) −→ EG( w ) . and e Γ : EG( w ) −→ RT( w ) . Then we show that they are the inverses of each other.We need to employ the Little map for the transition in Theorem 4.3. In this case, theLittle map can be written as θ p,w q , since q is the unique element of S ( wt p,q , p ) defined in(2.7). Note that ( p, w q ) is the maximum empty box in the Rothe pipedream of w whichhas a pivot. Moreover, for a reduced word a of a permutation in Φ( wt p,q , p ), we have θ − p,w q ( a ) = ( θ n +1 − p,n +1 − w q ( a c )) c , which is a reduced word of w .Let T ∈ RT( w ) be a reduced word tableau for w with shape λ . We first construct apath of permutations w = w (0) → w (1) → · · · → w ( m ) (6.1)in the modified LS-tree of w from the root w (0) = w to a leaf w ( m ) . Then there is a pathin the EG-tree of w which corresponds to the path in (6.1). Let P be the EG-pipedreamcorresponding to w ( m ) in the EG-tree of w . Define Γ( T ) = P .To construct (6.1), we need a path of reduced words τ = τ (0) → τ (1) → · · · → τ ( m ) (6.2)such that τ ( i ) is a reduced word of w ( i ) for 0 ≤ i ≤ m . Let τ (0) = column( T ). Bydefinition, column( T ) is a reduced word of w . Note that T is the insertion tableau ofcolumn( T ) rev under the Edelman-Greene algorithm. For 0 ≤ i ≤ m − 1, let τ ( i +1) beobtained from τ ( i ) by applying the Little map θ p i ,w ( i ) qi , where ( p i , w ( i ) q i ) is the maximumempty box in the Rothe pipedream of w ( i ) which has a pivot.Conversely, let P be an EG-pipedream of w with m NW elbows, say( i m , j m ) < · · · < ( i , j )in the order (4.8). Assume that w ′ is the permutation in the modified LS-tree of w which corresponds to P . Then w ′ is a dominant permutation with a unique reduced24ord tableau, say T ′ , namely, the frozen tableau of w ′ as mentioned in Section 2.3. Takethe column reading word column( T ′ ) of w ′ . Let w ( P ) be the reduced word of w definedby w ( P ) = θ − i ,j ◦ · · · ◦ θ − i m ,j m (column( T ′ )) . (6.3)Let e T be the insertion tableau of w ( P ) rev by the Edelman-Greene algorithm. Define e Γ( P ) = e T . By Theorem 2.1, the row reading word of e T is a reduced word of w − . Bythe proof of Theorem 2.3, the column reading word of e T is a reduced word of w , andthus e T is a reduced word tableau for w . Proof of Theorem 6.1 . We show that e Γ is the inverse of Γ. Let P = Γ( T ) where T ∈ RT( w ). We need to show that e T = e Γ( P ) = T . For 0 ≤ i ≤ m , let T i be theinsertion tableau of ( τ ( i ) ) rev for the reduced words in (6.2). Note that T = T . ByTheorem 2.5, the tableaux T i have the same shape. Moreover, by Theorem 2.6, thecolumn reading word of T i is τ ( i ) . By the construction of the EG-tree of w and in viewof the construction of e T , it is easy to check that e T = T . In the same manner, it is alsoeasy to verify that Γ is the inverse of e Γ. So the proof is complete.For example, let w = 231654 and let T be the following reduced word tableau of w : T = . We see that column( T ) = (5 , , , , w ) is (5 , θ , to column( T ) =(5 , , , , θ , (5 , , , , 5) = (5 , , , , , which is a reduced word for w (1) = 241635. The maximum box in pivot( w (1) ) is (4 , θ , to 53124 to yield θ , (5 , , , , 4) = (4 , , , , , which is a reduced word of w (2) = 251436. The maximum box in pivot( w (2) ) is (4 , θ , to 43124 to obtain θ , (4 , , , , 4) = (4 , , , , , which is a reduced word of w (3) = 253146. The maximum box in pivot( w (3) ) is (2 , θ , to (4 , , , , 3) to yield θ , (4 , , , , 3) = (3 , , , , , which is a reduced word of w (4) = 423156. Since w (4) is a dominant permutation,the Rothe diagram Rothe( w (4) ) is a partition, and we stop. Therefore, the path ofpermutations in the modified LS-tree of w is231654 → → → → . T ) = .By the EG-tree of w displayed in Figure 5.16 , the EG-pipedream corresponding to theleaf 423156 isConversely, let P be the second EG-pipedream in the bottom row of Figure 5.16. Theset of NW elbows of P is { (2 , , (4 , , (4 , , (5 , } < . The leaf in the modified LS-treeof w corresponding to P is w ′ = 423156, which has a unique reduced word tableau T ′ = . Then column( T ′ ) = (3 , , , , 3) and w ( P ) = θ − , ◦ θ − , ◦ θ − , ◦ θ − , (3 , , , , . We have the following calculations: θ − , (3 , , , , 3) = ( θ − , (3 , , , , c = (2 , , , , c = (4 , , , , θ − , (4 , , , , 3) = ( θ − , (2 , , , , c = (2 , , , , c = (4 , , , , θ − , (4 , , , , 4) = ( θ − , (2 , , , , c = (1 , , , , c = (5 , , , , θ − , (5 , , , , 4) = ( θ − , (1 , , , , c = (1 , , , , c = (5 , , , , . Therefore, w ( P ) = (5 , , , , w ( P ) rev = (5 , , , , 5) by the Edelman-Greene algorithm to obtain T = , which is a reduced word tableau for w = 231654. Acknowledgments. Part of this work was completed during Neil Fan was visitingthe Department of Mathematics at the University of Illinois at Urbana-Champaign, hewishes to thank the department for its hospitality and thank Alexander Yong for helpfulconversations. This work was supported by the 973 Project, the PCSIRT Project of theMinistry of Education, the National Science Foundation of China.