Some properties of random staircase tableaux

Transcription

1 Some properties of ranom staircase tableaux Sanrine Dasse Hartaut Pawe l Hitczenko Downloae /4/7 to Reistribution subject to SIAM license or copyright; see Abstract We escribe a probabilistic approach to a relatively new combinatorial object calle staircase tableaux Our approach allows us to analyze several parameters of a ranomly chosen staircase tableau of a given size Introuction A new combinatorial structure, calle staircase tableaux, was introuce in recent work of Corteel an Williams [8, ] They are relate to the asymmetric exclusion process on an one-imensional lattice with open bounaries, the ASEP This is an important an heavily stuie particle moel in statistical mechanics we refer to [8] for some backgroun information on several versions of that moel an their applications an connections to other branches of science) The stuy of the generating function of the staircase tableau has given a formula for the steay state probability of the ASEP In the same work staircase tableaux were also use to give a combinatorial formula for the moments of the weight function of the) Askey-Wilson polynomials; for a follow up work see [7] The authors of [8] calle for further investigation of the staircase tableaux because of their combinatorial interest an their potential connection to geometry In this note, we take up that issue an stuy some basic properties of staircase tableaux More precisely, we analyze the istribution of appearances of Greek letters α, β,, an in a ranomly chosen staircase tableau of size n or on the iagonal of such tableau We refer to the next section or to [8, Section ] for the necessary efinitions an the precise meaning of these symbols The work of the first author was carrie out while she hel an ANR Gamma internship at LIPN, Université Paris Nor, uner the irection of Fréérique Bassino LIPN) an Sylvie Corteel LIAFA) She woul like to thank both of them for their guiance, the members of LIPN for their hospitality, an ANR Gamma for the support The secon author was partially supporte in part by NSA Grant #H Most of his work was one uring his stay at LIPN in July 00 He woul like to thank Fréérique Bassino LIPN) for the invitation an acknowlege the hospitality of LIPN LIAFA, Université Paris Dierot Paris 7, F-7505 Paris, France Department of Mathematics, Drexel University, Philaelphia, PA904, USA Here, we only mention that in the context of the ASEP these letters correspon to the probabilities of particles entering or leaving the lattice from either irection) Staircase tableaux are generalizations of permutation tableaux see eg [5, 9, 0, 7] an references therein for more information on these objects an their connection to a version of ASEP referre to as the partially asymmetric exclusion process; PASEP) For permutation tableaux, the authors of [5] evelope a probabilistic approach that later allowe the erivation of the limiting an even exact) istributions of various parameters of the permutation tableaux Our goal here is the same: we will evelop a probabilistic approach parallel to that of [5] that will allow us to systematically compute generating functions of various quantities associate with staircase tableaux an, as a consequence, obtain their exact or limiting istributions A few of those statements coul be also obtaine by combinatorial approach base on an involution on staircase tableaux an on transformation of the parameters uner that involution Nonetheless, asie of giving new insights, our probabilistic approach allows for a more systematic an universal analysis, an thus has a value in itself, we think As we will see below, one of the parameters we stuy, namely the number of letters α or or, equivalently, the number of letters β or ) on the iagonal of a tableau turns out to coincie with a generalization of Eulerian numbers see [0, sequence A06087]) relate to Whitney numbers of Dowling lattices It seems that the first trace of the sequence [0, sequence A06087] in the literature goes back to MacMahon s paper [8], an that fairly recently this sequence has been stuie in a wier context in [6] We refer to [0, sequences A06087, A4590, A09775] an [4,,, 4] for efinitions an further information on the numbers we mentione, an the relations between them This rather unexpecte an intriguing connection between the parameter we stuy here an the generalize Eulerian numbers has not been explaine an merits, perhaps, further stuies One consequence of our work is that the triangle of generalize Eulerian numbers [0, sequence A06087], when suitably normalize, satisfies the central limit theorem As far as we can tell this result is new although it is an easy consequence of a general Copyright 0 SIAM Unauthorize reprouction is prohibite 58

2 Downloae /4/7 to Reistribution subject to SIAM license or copyright; see theorem of Bener []) Limit theorems for a relate sequence [0, A4590] are establishe in [4] This link may have unravele connection to geometry allue to in [8] as the sequences A06087, A4590, an A09775 from [0] all have a very strong geometrical flavor Due to space limitation, we will not give the full etails here, referring instea to the full version of the paper, now in preparation We will confine ourselves to giving a etaile escription of our approach, state our results, an inclue a sample proof to illustrate how our approach works in practice Definitions an notation Staircase tableaux We recall the following concept first introuce in [8, ]: A staircase tableau of size n is a Young iagram of shape n, n,,, ) whose boxes are fille accoring to the following rules: each box is either empty or contains one of the letters α, β,, or; no box on the iagonal is empty; all boxes in the same row an to the left of a β or a are empty; all boxes in the same column an above an α or a are empty An example of a staircase tableau of size 7 is given in Figure below β β α Figure : A staircase tableau of size 7; its top row is inexe by β, the next one by α α We enote the set of all staircase tableaux of size n by S n, n It is known that the carinality of S n is 4 n n! There are several proofs of this statement cf [7] for one of them an for references to further proofs) All these proofs are base on combinatorial approaches an we wish to mention that a probabilistic technique that we evelop in this paper may be use to provie yet another proof of that fact We present our proof in Section 4 below α Connection to ASEP Staircase or earlier permutation) tableaux were introuce an stuie in the connection with ASEP Because of the importance of this connection we briefly recall its nature The ASEP is a Markov chain on wors of size n on an alphabet A {, } consisting of two letters Each such wor represents an one-imensional lattice of length n with some sites occupie by particles represente by ), an others not represente by ) A particle can only hop to the right or the left with the probabilities u an q, respectively), provie that the ajacent site is unoccupie, or enter or quit the lattice Entering from the left right) happens with the probability α resp, ) ifthe first last) site is unoccupie Exiting to the left right) happens with the probability resp β) if the first last) site is occupie At a given time one of the n + possible locations for a move is selecte uniformly at ranom) an, if possible, a transition escribe above is performe with the given probability We refer to [,, 5] or [8] for more etaile escription an further references To escribe the connection to staircase tableaux, efine the type of a staircase tableau S of size n to be a wor of the same size on the alphabet {, } obtaine by reaing the iagonal boxes from northeast NE) to southwest SW) an writing for each α or, an for each β or Thus a type of a tableau is a possible state for the ASEP) Figure shows a tableau with its type β α α α β Figure : A staircase tableau an its type ) We also nee a weight of a tableau S To compute it, we first label the empty boxes of S with u s an q s as follows: first, we fill all the boxes to the left of a β with u s, an all the boxes to the left of a with q s Then, we fill the boxes above an α or a with u s, an the boxes above a β or a with q s When the tableau is fille, its weight, wts), is a monomial of egree nn + )/ in α, β,,, u an q, which is the prouct of labels of the boxes of S Figure shows a tableau fille with u s an q s Its weight is α β u 8 q 9 Corteel an Williams [8, ] have shown that the Copyright 0 SIAM Unauthorize reprouction is prohibite 59

3 Downloae /4/7 to Reistribution subject to SIAM license or copyright; see u β u u α q q u α u u q q q q q u α q q u β Figure : A staircase tableau with u s an q s steay state probability that the ASEP is in state σ is Z σ α, β,,, q, u) Z n α, β,,, q, u), where Z n α, β,,, q, u) Z σ α, β,,, q, u) S of type σ S of size n wts) wts) an Further efinitions an notation We now efine some parameters that will be object of our stuy Let be a subset of the set of symbols {α, β,, } We say that a row of a staircase tableau is inexe by if the leftmost entry in that row is in For the sake of brevity we will refer to rows inexe by simply as rows Thus, for example, the number of α/ rows is the number of rows inexe by α or The tableau in Figure has two α/ rows, the secon from the top inexe by α) an the bottom inexe by ) For a given staircase tableau S S n we enote this quantity by r n S) an we occasionally will skip the subscript n if there is no risk of confusion As we will see below this parameter will play a funamental role in our approach Other parameters we will consier are: the total number of entries β or β/ for short), the total number of entries α or α/), the number of entries β/ on the iagonal of the tableau, an the number of entries α/ on the iagonal For a given tableau S S n these parameters will be enote by Δ n S), Γ n S), B n S), an A n S), respectively For a tableau in Figure they are: Δ 7 S) 5,Γ 7 S) 6,B 7 S), an A 7 S) 4 Our results may be summarize as follows: all these parameters, after suitable normalization, are asymptotically normal More precisely, if Y n is any of the statistics, r n,δ n S), Γ n S), B n S), or A n S)thenasn Y n EY n N0, ), varyn ) where N0, ) enotes the stanar normal variable an the convergence is in istribution for a precise statement see Theorem 5 below) We will also obtain the exact expressions for the expecte value an the variance of Y n in each of the five cases Furthermore, inthecaseofr n,δ n,anγ n we will obtain a simple escription of the exact istribution of Y n from which it reaily follows that they are asymptotically normal) Because of the symmetries in staircase tableaux, these five cases are not inepenent of one another In fact, it suffices to consier one of r n,δ n,γ n an either A n or B n see Remark in Section 5 for the explanation), but it is interesting to know that each of these cases can be treate by our approach inepenently of other cases Also, our results o not istinguish α from an β from an so we coul replace each pair by one symbol an use only two nontrivial entries in our tableaux However, each of them has its own interpretation in terms of ASEP an this symmetry in α/ an β/ breaks in that context For this reason we ecie to follow the same notation as in the literature on ASEP As we mentione earlier our viewpoint will be probabilistic Thus, we will equip the set S n with the uniform probability measure enote by P n That means that for each S S n we have P n S) 4 n n! As is customary we will refer to a tableau chosen accoring to that measure as a ranom tableau of size n We will enote the integration with respect to the measure P n by E n With this unerstaning, the quantities r n,δ n,γ n, B n,ana n being functions on a probability space, are ranom variables from now on referre to as statistics) an we will analyze their probabilistic properties like expecte values, variances, an exact or limiting istributions) In orer to o that we will evelop a technique that is analogous to what has been one in the case of permutation tableaux see [5] or [7]) Let us recall at this point that permutation tableaux have been use to give a combinatorial escription of a stationary istribution for the PASEP We refer to eg [9, 0, 5, 7] for the efinition, connections to PASEP, further properties an etails Just as PASEP is a particular case of ASEP, permutation tableaux of size n are in bijection with a subset of staircase tableaux of that size corresponing to the case 0 Let us recall that the approach use in [5, 7] for the permutation tableaux was to ientify a funamental parameter, trace its evolution as the size of a tableau is increase by, an then use successively conitioning to reuce the size of a tableau We refer the reaer to either [5] or [7] for more etails, here we only recall that this funamental parameter was the number of unrestricte rows in a permutation tableau, an that its conitional istribution was given by + BinU n, /) This is to Copyright 0 SIAM Unauthorize reprouction is prohibite 60

4 Downloae /4/7 to Reistribution subject to SIAM license or copyright; see mean that if a size of permutation tableau with U n unrestricte rows was increase from n ton, then the number of unrestricte rows in this extension ha the conitional) istribution + BinU n, /) As we will see in the forthcoming section, in the case of staircase tableaux the role of a funamental parameter will be playe by the number of α/ rows Set-up To escribe our approach we nee to briefly recall the evolution process of staircase tableaux escribe in [6] Let S S n be a tableau with r n r n S) α/ rows To exten its size by, we a a new column of length n at the left en an we nee to fill it accoring to the rules If r n 0thenalln rowsofs are β/ rows an hence the top n boxes of the new column have to be empty since no entries are allowe to the left of a β/ inthesamerow Thus,ifr n 0weobtain four ifferent tableaux of size n by putting one of the four symbols in the bottom box of the new column If r n then we can either put one of the symbols α/ in the bottom corner an then we are force to leave all other boxes in that column empty), or we can put a β/ in the bottom box of the new column In that case, we nee to fill all boxes in the new column corresponing to one of the r n α/ rows Accoring to the rules, if we put an α/ in any of them, then we nee to leave all boxes above it empty, otherwise we have a complete freeom It is thus seen that any tableau of size n with r n α/ rows gives 4 rn ifferent staircase tableaux of size n Inee, if r n 0 then there are 4 extensions an if r n then there are rn )+ rn ) extensions Here, the first is from putting an α/ in the bottom box of the new column, the next is from putting a β/ in that box, the term i, i r n is from putting the first counting from bottom up in the new column) α/ in the ith row as then there are i ways of filling the earlier i boxeswithsymbols β,, or leaving them empty, an finally, the rn term comes from not putting an α/ in any of the r n rows an thus filling these r n rows with β/ s or leaving them empty Summing the above gives ) + rn + rn + rn + rn )4 rn, as claime The above iscussion may be phrase in a more probabilistic language using what Shiryaev [9, Chapter I, 8 ] refers to as ecompositions which is just a special case of conitioning with respect to σ algebra) To escribe the ecomposition of S n that we will be using, note that every tableau from S n is an extension of a unique tableau S from S n Therefore, enoting by D S the set of all tableaux from S n which are obtaine from S by the process escribe above as we just iscusse, there are 4 rn S) such tableaux), we can write S n D S, S S n where D S ) S Sn are pairwise isjoint subsets of S n We enote this ecomposition of S n by D n We will nee to be able to compute the conitional probabilities P D n ) an the conitional expectations E D n ) with respect to this ecomposition To o that, let S S n be a particular tableau with r n r n S) α/ rows an let r be the number of such rows in any of its extensions to a tableau of size n We wish to know the conitional) istribution of r Clearly, the possible values for r are r n +,r n,r n,,, 0 an we nee to know the probabilities for each of these possibilities First, Pr r n + D S )PC α/ D S ) ) 4 rn, rn where the symbol C enotes the event that we put one of the symbols in the bottom box of the new column as we exten the tableau Next, for k 0,,,r n we compute Pr r n k D S ) Since k is the number of β/ s that we put in the r n allowable ie corresponing to α/ rows of S) boxes,r r n k means that we put a β/ in the bottom box an aitional kβ/ s in the r n allowable boxes above it If we o not put an α/ in any of those k boxes, we have k r n ) k possibilities is for putting a β/ at the bottom, an the rest accounts for putting β/ s in any k of the allowable r n boxes) If we o put an α/ in one of the boxes we nee to pick k + of the allowable r n boxes, put an α/ in the topmost of them an put β/ s in the remaining k of them an at the bottom) This gives k+ r n ) k+ possibilities Thus, Pr r n k D S ) k+ r n ) k+ + rn ) k ) ) 4 rn This together with ) completely escribes the conitional istribution of r n given D n an leas, in particular, to the following basic relation: For a complex number z an n with the unerstaning that Copyright 0 SIAM Unauthorize reprouction is prohibite 6

5 Downloae /4/7 to Reistribution subject to SIAM license or copyright; see r 0 0) ) Ez rn D n ) z + ) rn z + Since we will prove a more general statement later see a comment following 65)) we will not justify ) here We wish to comment, however, that calculations similar to those giving ) are typical for our approach In fact, the ability to compute the conitional expectations of various quantities will turn up to be one of the two main ingreients of our metho We now iscuss the secon ingreient Focus on S n, the set of all staircase tableaux of size n There are two natural probability measures on S n to consier One is the uniform measure enote by P n ) an the other is a measure obtaine from the uniform probability measure on S n by collapsing all the elements of S n that are extensions of the same element S S n We will enote the latter measure P n S) there is an apparent ambiguity of notation here, however, it isappears once we remember whether S is in S n or S n ) The relationship between these two measures on S n is straightforwar to fin: since a tableau S S n with r n α/ rows gives 4 rn tableaux in S n we have 4) P n S) 4 rn 4 rn S n 4 S n rn P n S) S n Consequently, for any ranom variable X n on S n we have E n X n E n 4 S n rn X n 4 S n 5) E n rn X n Here we have use the same convention as above; for a ranom variable X on S n, E n X enotes the expectation with respect to the uniform measure on S n while E n X enotes the expectation with respect to the measure that is inuce on S n by the uniform measure on S n The relations ) an 5) are key an will allow us to analyze the istributions of the various statistics on S n Note that 4) an 5) are true regarless of whether we know the carinalities of S n an S n or not As a matter of fact, one can use 5) to provie a yet another argument that 4 n n! We refer the reaer to the full version of the paper for the etails, here we just note that once this is known, 4) an 5) simplify to 6) an 7) respectively P n S) rn n P n S) E n X n n E n rn X n, 4 Illustration: the number of staircase tableaux To illustrate how ) an 5) work we re-erive the expression for the total number of staircase tableaux of size n Proposition 4 Let S n be the set of all staircase tableaux of size n Then 4 n n! Proof Since a tableau S of size n with r n S) α/ rows gives 4 rns) tableaux of size n +, grouping the elements of S n+ accoringly, we have: S n+ 4 rns) 4 rns) S S n S S n Now observe that S S n rns) is simply the expecte value of rn compute over the uniform probability measure on S n Therefore, S n+ 4E n rn By basic properties of the conitional expectation see eg [9, Formula 6), p 79]) the expectation on the right han sie is equal to E n E rn D n ) Using this an then ) with z we see that the right han sie above is ) rn + + 4E n E rn D n )4E n 4 S n E n ) rn 5 We now use 5) with X n 5/) rn to get S n+ 4 S n 4 S n 5 E n rn 4 S n E n 5 rn ) rn Copyright 0 SIAM Unauthorize reprouction is prohibite 6

6 Downloae /4/7 to Reistribution subject to SIAM license or copyright; see This can be iterate an gives that S n+ is equal to S n k 4k k ) ) k+)e n kk+)+) r n k With k n this becomes S n+ S 4n n n n!e n +) r Since S 4anE n +) r n +)+ we get S n+ 4 n n ++ n! 4 n+ n +)!, which proves the statement an inepenently confirms the count of staircase tableaux 5 Main results Our technique allows us to obtain the istribution sometimes exact, sometimes only asymptotic) of the statistics iscusse above Before we state our results let us recall that if X an Y are two ranom variables not necessarily efine on the same probability space) then X Y means equality in istribution Moreover if X n ) is a sequence of ranom variables which may be efine on ifferent probability spaces) then X n X enotes the convergence in istribution That means that as n then PX n x) PX x) for all x at which the function on the right is continuous We will be ealing exclusively with the case when X is the stanar normal ranom variable, N0, ) an we recall that its istribution function is given by Φx) π x We can now state our results e t / t Theorem 5 Consier the set S n with the uniform probability measure P n The following are true: i) For every n we have r n n J k, where J k s are inepenent an J k is a ranom variable which is with probability /k) an 0 with the remaining probability In particular, n E n r n k H n ln n, varr n ) n ) k k H n H) n 4 ln n, where H n n k an H) n n k are harmonic numbers of the first an secon orer, respectively Moreover, as n, r n ln n ln n ii) For every n we have 58) Δ n N0, ) n J k ), where J k ) are as in part i) In particular, we have 59) E n Δ n n H n, 50) varδ n ) H n H) n 4, an, as n, 5) iii) For every n Δ n n + ln n ln n Γ n Δn N0, ) In particular, 58) 5) hol with Δ n replace by Γ n iv) The expecte value an the variance of the number A n of α/ on the iagonal of a ranom staircase tableau of size n are, respectively, 5) E n A n n an vara n ) n + Furthermore, for n, 5) A n n/ n/ v) For every n we have N0, ) B n An In particular, 5) an 5) hol for B n in place of A n Copyright 0 SIAM Unauthorize reprouction is prohibite 6

7 Downloae /4/7 to Reistribution subject to SIAM license or copyright; see Remark Several statements of the above theorem may be euce from others Inee, there is an involution φ on the set of staircase tableaux of a given size This involution has the following properties: if S, T S n are such that T φs) then a) Δ n S) Γ n T )anγ n S) Δ n T ); b) B n S) A n T )ana n S) B n T ) Moreover, since any row is either inexe by α/ or contains a β/, one has c) r n S)+Δ n S) n for any S S n Thus a) an c) combine imply that any of the parts i), ii), iii) of Theorem 5 implies the other two Furthermore, b) implies that part v) of that theorem follows from iv) or vice versa) Nonetheless, as far as we know, neither any of i)-iii) nor iv) or v) were known before, an so the results presente in Theorem 5 are new Besies, we think it is it is worth emphasizing that our probabilistic approach oes provie a unifie an systematic approach that in particular allows one to prove all parts of that theorem irectly an without a nee to appeal to any further information 6 Sample proof Detaile proofs of the results given in Theorem 5 will be inclue in the full version of the paper Here, to illustrate how the technique works we will just prove one part of this theorem In orer to avoi overlap with what will be presente in the paper an also to show that each part of Theorem 5 may be obtaine irectly ie without appealing to the involution on staircase tableaux) we will prove part iii) More precisely, we will show irectly that Γ n satisfies 58) 5) of part ii) of Theorem 5 To this en, we compute the joint probability generating function E n t Γn z rn of Γ n an r n We write Γ n n g k where g k is if an α/ is inserte when the kth column to the tableau is ae an is 0 otherwise recall that there can be at most one α/ in any column) We then have n E n t Γn z rn E n t g k z rn E n t Γn t gn z rn By the funamental properties of the conitional expectation see eg [9, Formulas 6), p 79 an 7), p 80]) the quantity on the right han sie is equal to 64) E n Et Γ n t gn z rn D n ) ) E n t Γ n Et gn z rn D n ) ) To procee we nee to compute the conitional expectation Et gn z rn D n ) We let F k enote the event that as we a the nth column we put kβ/ s in the r n boxes corresponing to α/ rows of a tableau of size n that we are extening Then Et gn z rn D n ) r n k0 r n tz rn k Pg n,f k D n ) + z rn k Pg n 0,F k D n ) k0 Note that the first sum is only to r n sincethe event {g n,f rn } is impossible) Consier the secon sum The event {g n 0,F k } means that we put a β/ in the bottom box of the nth column an aitional kβ/ s in the r n rows inexe by α/ s Thus we see that the secon sum is r n k0 ) z rn k k+ rn k 4 rn z +)rn 4 rn ) rn z + As for the first sum, {g n,f 0 } means that we either put an α/ in the bottom box of the nth column or we put a β/ in that box an no β/ in the r n allowable boxes above it Splitting the probability of this event accoringly, we see that the first sum is tz rn + rn r n ) +t z rn k k+ rn k + 4 rn k0 tz z ) rn + tz r n ) z rn k+) k+ rn 4 rn k + k0 tz z ) rn tz + z rn +)rn z rn ) tz ) rn z + Combining, we obtain that 65) Et gn z rn D n ) tz + ) rn z + Note that putting t gives ) an that the above calculation oes not use anything but ) an ) so Copyright 0 SIAM Unauthorize reprouction is prohibite 64

8 Downloae /4/7 to Reistribution subject to SIAM license or copyright; see it is a self containe proof of )) Substituting 65) into equation 64) we obtain ) rn tz + z + E n t Γn z rn E n t Γn Now using 7) with we further get E n t Γn z rn X n t Γn z + ) rn tz + z + n E n rn t Γn tz + n E n t Γn z +) rn ) rn This can be iterate an upon further iteration yiels E n t Γn z rn Since n k0 tz +k)+) n E t Γ z +n )) r n! E t Γ z +n )t) r tz +n )) + ) tz +n )) +, we finally obtain E n t Γn z rn n Putting z wegetthat n tk ) + E n t Γn k n k0 tz +k)+) n n! tz +k )) + k n k )t + ) k The kth factor on the right han sie is the probability generating function of a ranom variable I k that is with probability /k) an is 0 with the remaining probability Since the prouct correspons to aing inepenent ranom variables, we see that Γ n n But this is just what 58) states The rest follows immeiately since n n E n Γ n EI k k )n H n, I k an varγ n ) n vari k ) n k k )H n H n 4 Finally, since I k EI k are uniformly boune an variances of the partial sums go to infinity, the Lineberg s conition for the central limit theorem see eg [9, Chapter III, 4]) hols trivially, thus giving 5) References [] E A Bener Central an local limit theorems applie to asymptotic enumeration J Combin Theory Ser A, 5:9, 97 [] M Benoumhani On Whitney numbers of Dowling lattices Discrete Math, 59-):, 996 [] M Benoumhani On some numbers relate to Whitney numbers of Dowling lattices Av in Appl Math, 9):06 6, 997 [4] L Clark Limit theorems for associate Whitney numbers of Dowling lattices J Combin Math Combin Comput, 50:05, 004 [5] S Corteel an P Hitczenko Expecte values of statistics on permutation tableaux In 007 Conference on Analysis of Algorithms, AofA 07, Discrete Math Theor Comput Sci Proc, AH, pages 5 9 Assoc Discrete Math Theor Comput Sci, Nancy, 007 [6] S Corteel, D Stanton, an L Williams Enumeration of staircase tableaux, 007 Preprint [7] S Corteel, D Stanton, an R Stanley L K Williams Formulae for Askey Wilson moments an enumeration of staircase tableaux arxiv: [8] S Corteel an L K Williams Tableaux combinatorics for the asymmetric exclusion process an Askey Wilson polynomials arxiv: [9] S Corteel an L K Williams A Markov chain on permutations which projects to the PASEP Int Math Res Notes, 7, 007 Art ID rnm055, 7pp [0] S Corteel an L K Williams Tableaux combinatorics for the asymmetric exclusion process Av Appl Math,, 9:9 0, 007 [] S Corteel an L K Williams Staircase tableaux, the asymmetric exclusion process, an askey- wilson polynomials Proc Natl Aca Sci, 075): , 00 [] B Derria, E Domany, an D Mukamel An exact solution of a one imensional asymmetric exclusion process with open bounaries J Statist Phys, 69-4): , 99 [] B Derria, M R Evans, V Hakim, an V Pasquier Exact solution of a D asymmetric exclusion moel using a matrix formulation J Phys A, 67):49 57, 99 [4] T A Dowling A class of geometric lattices base on finite groups J Combinatorial Theory Ser B, 4:6 86, 97 Erratum: J Combinatorial Theory Ser B, 5:, 97) Copyright 0 SIAM Unauthorize reprouction is prohibite 65

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