Descents of Permutations in a Ferrers Board

Transcription

1 Desces of Permuaios i a Ferrers Board Chuwei Sog School of Mahemaical Scieces, LMAM, Pekig Uiversiy, Beijig 0087, P. R. Chia Caherie Ya Deparme of Mahemaics, Texas A&M Uiversiy, College Saio, TX Mahemaics Subjec Classificaio: 05A05, 05A5, 05A30 Absrac The classical Euleria polyomials are defied by seig A ) σ S +desσ) k A,k k where A,k is he umber of permuaios of legh wih k desces. Le A, q) π S +desπ) q ivπ) be he iv q-aalogue of he classical Euleria polyomials whose geeraig fucio is well kow: 0 u A, q) [] q! ) k u k. 0.) [k] q! k I his paper we cosider permuaios resriced i a Ferrers board ad sudy heir desce polyomials. For a geeral Ferrers board F, we derive a formula i he form of permae for he resriced q-euleria polyomial A F, q) : σ F +desσ) q ivσ). If he Ferrers board has he special shape of a square wih a riagular board of size s removed, we prove ha A F, q) is a sum of s + erms, each saisfyig a equaio ha is similar o 0.). The we apply our resuls o permuaios wih bouded drop or excedace) size, for which he desce polyomial was firs sudied by Chug e al. Europea J. Combi., 37) 200): ). Our mehod preses a aleraive approach. Iroducio Le S deoe he symmeric group of order. Give a permuaio σ S, le Desσ) be he desce se of σ, i.e., Desσ) {i σ i > σ i+, i }, ad le desσ) Desσ) deoe he umber of desces of σ. For D {, 2,..., }, we deoe by α D) he umber Suppored i par by NSF Chia gra #07260 ad by he Scieific Research Foudaio for ROCS, Chiese Miisry of Educaio. Suppored i par by NSF gra #DMS ad NSA gra #H

2 of permuaios π S whose desce se is coaied i D, ad by β D) he umber of permuaios π S whose desce se is equal o D. I symbols, α D) : {σ S : Desσ) D}, β D) : {σ S : Desσ) D}. Le D {d, d 2,..., d k } where d < < d k. For coveiece, also le d 0 0 ad d k+. The he followig formulas for α D) ad β D) are well-kow see, for example, [4, p.69]): ) α D).) d, d 2 d,..., d k [ ] [ )] di β D)! de de,.2) d j+ d i )! d j+ d i where i, j) [0, k] [0, k] i he marix of equaio.2). A q-aalogue of he above formulas is give by cosiderig he permuaio saisic ivσ), where ivσ) i<j χσ i > σ j ). By coveio, he symbol χp ) is equivale o if he saeme P is rue ad 0 if o. See [4, Example 2.2.5]. Explicily, le α D, q) q ivπ), β D, q) q ivπ). π S :Desπ) D π S :Desπ)D The [ ] []! α D, q).3) d, d 2 d,..., d k [d ]![d 2 d ]! [ d k ]! [ ] [[ ]] di β D, q) []! de de,.4) [d j+ d i ]! d j+ d i where i, j) [0, k] [0, k] as before. Here we use he sadard oaio [ ] [] : q [] q! )/ q), []! : [][2] [], : k [k]![ k]! for he q-aalogue of he ieger, he q-facorial, ad he q-biomial coefficie, respecively. Someimes i is ecessary o wrie he base q explicily as i [] q, [] q!, ad [ ] k, ec., bu we q omi q i his paper as we do o use he aalogues of ay oher variables. The classical Euleria polyomials are defied by seig A ) σ S +desσ) A,k k, where A,k is called he Euleria umber ha deoes he umber of permuaios of legh wih k desces. Le A 0 ). The polyomials A ) have he geeraig fucio see e.g. Riorda [2]) k A ) u! 0 ) k u k k! k..5) eu ) 2

3 Le A, q) π S +desπ) q ivπ) be he iv q-aalogue of he Euleria polyomials. Saley [3] showed ha 0 u A, q) []! Eu ); q),.6) where Ez; q) 0 z []!. By simple maipulaios we ca see ha a equivale form of.6) is 0 u A, q) []! ) k u k..7) [k]! k Aleraive proofs of.7) have bee give by Gessel [9] ad Garsia [8]. I his paper we cosider permuaios wih resriced posiios, ad exed he above resuls o desce polyomials of permuaios i a Ferrers board. Tradiioally a permuaios σ S is also represeed as a 0-fillig of a by square board: Readig from lef o righ ad boom o op, we simply pu a i he ih row ad he jh colum wheever σ i j for i,...,. Give iegers 0 < r r 2 r, he Ferrers board of shape r,..., r ) is defied by F {i, j) : i, j r i }. I he followig we ideify a permuaio σ wih is 0-fillig represeaio, ad say ha σ is i a Ferrers board F if all he cells i, σ i ) are i F. I Secio 2 we exed he formulas.3) ad.4) o he se of permuaios o a fixed Ferrers shape wih rows ad colums, ad derive a permae formula for he resriced q-euleria polyomial A F, q) : +desσ) q ivσ). σ F I Secio 3 we focus o he Ferrers board ha is obaied from he square by removig a riagular board of size s, ad prove ha he resriced q-eularia polyomial is a sum of s + erms, each deermied by a equaio ha geeralizes.7). Fially i Secio 4, we apply our resuls o permuaios wih bouded drop or excedace) size, for which he desce polyomial was firs sudied by Chug, Claesso, Dukes ad Graham [4]. Our mehod preses a aleraive approach o he resuls i [4]. Noaio o laice pah Here we recall some oaio ad resuls abou he couig of laice pahs wih a geeral righ boudary. These resuls offer he mai ool o describe permuaios resriced i a Ferrers board. For a referece o laice pah couig, see Mohay []. A laice pah P is a pah i he plae wih wo kids of seps: a ui orh sep N or a ui eas sep E. If x is a posiive ieger, a laice pah from he origi 0, 0) o he poi x, ) ca be coded by a legh o-decreasig sequece x, x 2,..., x ), where 0 x i x ad x i is he x-coordiae of he ih orh sep. For example, le x 5 ad 3. The he pah EENENNEE is coded by 2, 3, 3). 3

4 I geeral, le s be a o-decreasig sequece wih posiive ieger erms s, s 2,..., s. A laice pah from 0, 0) o x, ) is oe wih he righ boudary s if x i < s i for i. If x s, he he umber of laice pahs from 0, 0) o x, ) wih he righ boudary s does o deped o x. Le P ah s) be he se of laice pahs from 0, 0) o s, ) wih he righ boudary s, ad LP s) be he cardialiy of P ah s). For a give sequece s s, s 2,..., s ), le LP s; q) q areap ), P P ah s) where areap ) i x i is he area eclosed by he pah P, he y-axis, ad he lie y. Hece LP s) LP s; ). I his paper we will also allow he eries s i o saisfy s s 2 s, i which case [ ] s + LP s; q) LP s, s,..., s ); q). I paricular LP +, +,, + ); q) [ 2 ]. I is also easy o see ha LP, 2,, ); q) C q), where C q) is Carliz-Riorda s q-caala umber [2]. 2 Desces of permuaios i Ferrers boards Le F be a Ferrers board wih rows ad colums, which is aliged o he op ad lef. Idex he rows from boom o op, ad colums from lef o righ. Le r i be he size of row i. Hece r r 2 r. For a se D {d, d 2,..., d k } wih d < < d k, le β F D) be he umber of permuaios i F wih he desce se D, ad α F D) be he umber of permuaios i F whose desce se is coaied i D. The iv q-aalogues of α F D) ad β F D) are defied by α F D, q) σ F :Desσ) D q ivσ), β F D, q) σ F :Desσ)D q ivσ). Clearly α F D, ) α F D) ad β F D, ) β F D). The Iclusio-Exclusio Priciple implies ha α F D, q) β F T, q), β F D, q) T D T D ) D T α F T, q). We shall show ha α F D, q) ad β F D, q) ca be expressed i erms of LP s, q), he area eumeraor of laice pahs wih proper righ boudaries ad leghs. Le s firs compue α F D, q). To ge a permuaio σ i F saisfyig Desσ) D, we firs choose x < x 2 < < x d such ha x i r i, ad pu a i he cell x i, i) for i d. The choose x d + < x d +2 < < x d2 such ha x i r i, ad pu a i he cell x i, i) for d < i d 2, ad so o. We say ha he cell i, j) is a -cell if i is filled wih a. I is clear ha a iversio of σ correspods o a souheas chai of size 2 i he fillig, i.e. a pair of -cells { x i, i ), x i2, i 2 ) } such ha i < i 2 while x i > x i2. 4

5 For i d, he -cell i he ih row i.e. y i) has exacly x i i may oher -cells lyig above i ad o is lef. Hece he -cell i he ih row coribues x i i o he saisic ivσ), ad all he -cells i he firs d rows coribued x ) + x 2 2) + + x d d ) o he saisic ivσ). Noe ha 0 x x 2 2 x d d, ad x i i < r i i +. Hece he umber of choices for he sequece x,..., x d ) is exacly he umber of laice pahs from 0, 0) o r d d +, d ) wih he righ boudary r, r 2,..., r d d + ), ad d i x i i) is he area of he correspodig laice pah. Therefore he firs d rows of F coribue a facor of LP d h,..., h d ); q) o α F D, q). Le h h, h 2,..., h ) where h i r i i +. Le he i-h block of F cosis of rows d i + o d i. Applyig he above aalysis o he i-h block of he Ferrers board F for i 2,..., k +, we ge ha Theorem 2. α F D, q) σ F :Desσ) D q ivσ) k LP di+ d i h di +,..., h di+ ); q) 2.) i0 where we use he coveio ha d 0 0 ad d k+. Accordigly, β F D, q) T D ) D T α F T, q) i <i 2 < <i j k ) k j f0, i )fi, i 2 )... fi j, k + ) where LP dj d i h di +,..., h dj ); q) if i < j fi, j) if i j 0 if i > j. 2.2) Followig he discussio of Saley [4, p.69], we obai ha Theorem 2.2 β F D, q) is he deermia of a k + ) k + ) marix wih is i, j) ery fi, j + ), 0 i, j k. Tha is, where fi, j) is give by 2.2). β F D, q) de[f i,j+ ] k 0 2.3) Whe he Ferrers board F is a square, h i i +, ad [ ] i LP j i h i+,..., h j ); q). j i 5

6 Theorems 2. ad 2.2 reduce o he classical resuls ha [ ] q ivπ) d, d 2 d,..., d k ad π S Desπ) D π S Desπ)D [[ ]] k q ivπ) di de. d j+ d i 0 For a geeral Ferrers board wih rows ad colums, le s check wo exreme cases: D {, 2,..., } ad D. Case. D {, 2,..., }: Theorem 2. yields he ideiy q ivσ) α F {, 2,..., }, q) σ F LP h i ); q) i [h i ]. 2.4) Noe ha permuaio filligs of a Ferrers board wih rows ad colums correspod o complee machigs of {,, 2} wih fixed ses of lef edpois ad righ edpois, ad a iversio of he permuaio is exacly a esig of he machig; See de Mier [6] ad Kasraoui [0]. To see his, for a give Ferrers board F wih rows ad colums, oe raverses he pah from he lower-lef corer o he op-righ corer, ad records he pah by is seps a, a 2,..., a 2 where a i E if he ih sep is Eas, ad a i N if he ih sep is Norh. Le L {i : a i E} ad R {i : a i N}. The 0-filligs of F cosidered here are i oe-o-oe correspodece wih he machigs of {,, 2} for which he se of lef edpois is L ad he se of righ edpois is R. For example, i he followig Ferrers board F, raversig from he lower-lef corer o he op-righ corer, we ge he sequece EENENENN. Thus L, 2, 4, 6 ad R 3, 5, 7, 8. The fillig give i he figure correspods o he machig {, 7), 2, 3), 4, 5), 6, 8)}. i I follows ha equaio 2.4) is exacly he geeraig fucio of he saisic e 2 M), which is he umber of esigs i a machig M, coued over all he machigs wih give ses of lef ad righ edpois. Tha is, q e2m) which maches he kow resuls i [7, 0]. M [h i ], i 6

7 Case 2, D : We have α F, q) β F, q) LP h,..., h ), q) Noe ha h, hece LP h,..., h ), q) iff r i i for all i, where he oly permuaio i he Ferrers board F wih o desces is he ideiy permuaio; oherwise P ah h,..., h ) ad LP h,..., h ), q) 0. Theorems 2. ad 2.2 ca be used o ge a formula for he joi disribuio of desσ) ad ivσ) over permuaios i F. Le A F, q) σ F +desσ) q ivσ). 2.5) Theorem 2.3 A F, q) ) perm), 2.6) where M is a marix whose i, j)-ery is give by LP j i+h i h j ); q) if i j M ij if i j + 0 if i > j +, ad perm) is he permae of he marix M. Proof. We have +desσ) q ivσ) σ F D {,2,..., } D {,2,..., } T {,2,..., } ) + D β F D, q) + D α F T, q) T {,..., k } < ) perm), T :T D D:T D ) D T α F T, q) ) D T + T + D T )+k T LP h)) where M is a marix as described i Theorem, ad D LP h)) deoes he righ-had side of 2.). Remark 2. We remark ha desces of permuaios i a Ferrers board provide a example of oe-depede deermiaal poi processes, as sudied by Borodi, Diacois ad Fulma []. Le χ be a fiie se. A poi process o χ is a probabiliy measure P o he 2 χ subses of χ. Oe simple way o specify P is via is correlaio fucios ρa), where for A χ, ρa) P {S : S A}. 7

8 A poi process is deermiaal wih kerel Kx, y) if ρa) de[kx, y)] x,y A. I is oe-depede if ρx Y ) ρx)ρy ) wheever disx, Y ) 2. Borodi e al. showed ha may examples from combiaorics, algebra ad group heory are deermiaal oe-depede poi processes, for example, he carries process, he desce se of uiformly radom permuaios, ad he desce se i Mallows model []. For hese hree cases, he poi processes are saioary, while he desce se of permuaios i a Ferrers board correspods o a deermiaal oe-depede poi process ha is o saioary. Explicily, for ay se D {d,..., d k } wih d < < d k, le P F D) β F D)/ i h i). Usig [, Theorem 7.5] we obai ha P F is a deermiaal, oe-depede process wih correlaio fucios ρd) α F D) de[kd i, d j )] k i,j ad wih correlaio kerel Kx, y) δ x,y + E ) x,y+, where E is he upper riagular marix E [ei, j)] i,j whose eries are give by LP j i h i+,..., h j ) if i < j ei, j) if i j 0 if i > j. 3 Permuaios i he rucaed board s For a geeral o-decreasig sequece of posiive iegers s, LP s, q) ca be compued by a deermia formula see, for example, []). Bu here is o simple closed formula. I he special cases ha he Ferrers board F is obaied from rucaig he square board by a riagular board i he corer, we ca describe he joi disribuio of he saisics desσ) ad ivσ) by ideiies of heir bi-variae geeraig fucios. Le s be he riagular board wih row size s, s,..., ). For s, le Λ,s be he rucaed board s cosisig of cells ha are lyig i 0 x, y ad above he lie y x s). I oher words, Λ,s is he Ferrers board whose row leghs are s, s +,...,,..., ). See he followig figure for Λ,s wih wih 7 ad s 4. Now le D {d,, d k } wih d < < d k. We shall compue he joi disribuio of des ad iv over all permuaios i Λ,s usig he formulas obaied i Secio 8

9 2. Agai le d 0 0 ad d k+. Le d i d i for i,..., k +, ad assume ha j is he paricular idex o make d j s < d j+ occur. Firs we compue α Λ,s D, q). Le r i be he size of row i i Λ,s. The Le h i r i i +. The r i { s + i if i s if s < i.. For 0 i < j, [ ] s + δi+ LP di+ d i h di +,..., h di+ ), q) LP δi+ s,..., s), q). 2. For i j, LP dj+ d j h dj +,..., h di+ ), q) LP δj+ s) s d j, s,..., d j+ + ), q) [ ] dj+ + δ j+ δ j+ [ ] dj. 3. For i > j, δ j+ LP di+ d i h di +,..., h di+ ), q) LP δi+ d i, d i,..., d i+ + ), q) [ ] di [ ] di. Summig over all permuaios σ wih Desσ) D i he Ferrers board Λ,s, we obai α Λ,s D, q) σ Λ,s Desσ) D q ivσ) + j [ ] s + δi i j [ s + δi i where D j ) represes he sequece δ j+,..., δ k+. Hece he Priciple of Iclusio-Exclusio leads o β Λ,s I, q) σ Λ,s Desσ)I q ivσ) D I [ ] j ) I D dj D j ) i ] k [ ] di ij + [ ] dj, D j ) [ s + δi + ]. 3.) 9

10 Le F,s q, ) be he bi-variae geeraig fucio of he saisics iv ad des over all permuaios i he board Λ,s. Tha is, F,s q, ) +desσ) q ivσ) σ Λ s + I I {,2,..., } σ Λ,s Desσ)I q ivσ). Le a; q) a) aq) aq ) ad a, q) i0 aqi ). Our mai resul here is a aalog of he formula.7). Explicily, we show ha F,s q, ) ca be expressed as a liear combiaio of s + erms, each of which saisfies a q-ideiy similar o.7). Theorem 3. For s, F,s q, ) 0. For > s, we have F,s q, ) θ 0 F 0),s q, ) + θ F ),s q, ) + + θ s F s),sq, ), 3.2) where θ k s are defied by he formal power series θ k z k k0 ) ), 3.3) z; q) s ad for each i 0,,..., s, he erm F i),sq, ) is give by he ideiy s+ z [ i]! F i),sq, ) ) k s+ i z k [k]! z k. 3.4) [k]! Proof. As before assume D {d, d 2,..., d k } wih d 0 0 ad d k+. Le j be he idex uiquely decided by d j s < d j+. For s +, by he equaio 3.) we have F,s q, ) [ ] j [ ] + I ) I D dj s + δi D j ) δ I {,2,..., } D I i i [ ] j [ ] dj s + δi ) I D + I D j ) D {,2,..., } D {,2,..., } i [ ] j dj D j ) +k ) k k0 i [ s + δi δ +δ δ k+ δ +...+δ j s<δ +...+δ j+ k I:D I ] ) D + D [ d j ]! j i [ s + ]! [δ ]! [δ k+ ]![ s ]!) j. 0

11 Le d i l ad γ F,s q, ) ) s l0 k 0 s l0 γ +k j δ +...+δ j l. The, δ +δ δ k+ lδ +...+δ j s<δ +...+δ j+ γ j j [ ] s + δi i Le θ 0 ad for l,..., s, θ l : j δ +...+δ j l [ l]! j i [ s + ]! [δ ]! [δ k+ ]![ s ]!) j. γ j k;δ j+ s+ l δ j+ +δ j δ k+ l j [ ] s + δi, i γ k+ j [ l]! [δ j+ ]! [δ k+ ]! 3.5) ad F l),s : ) k;τ 0 s+ l τ 0 +τ +...+τ k l γ k+ [ l]! [τ 0 ]! [τ k ]!. 3.6) The F,s q, ) θ 0 F 0),s q, ) + θ F ),s q, ) + + θ s F s),sq, ). We show ha θ l ad F,sq, l) ) saisfy 3.3) ad 3.4). Firs, observe ha for l > 0, θ l is he coefficie of z l i he formal power series [ ] ) j s + k γ z k 3.7) k j0 k Usig he q-aalog of he biomial heorem k0 a; q) k q; q) k z k az; q) z; q), where a; q) a) aq) aq ) ad a, q) i0 aqi ), we have ) j θ l z l [ s + k ][ s + k 2] [ s] γ z k [k]! l0 j0 k ) j γ q s z; q) ) z; q) j0 ) γ ). z; q) s

12 This proves he formula 3.3). To ge he formula 3.4), observe ha 3.6) ca be wrie as F,s l) [ l]! ) z [z l ] γ τ 0 [τ 0 ]! γ k z τ [τ]! )k. This leads o he ideiy s+ z l [ l]! τ 0 s+ l F l),sq, ) ) k0 k s+ l τ z k [k]! z k. [k]! I he case ha s 0, F,s q, ) F,s 0) q, ) A, q), ad equaio 3.4) reduces o he well-kow ideiy.7) by leig u z. k 4 Permuaios wih bouded drop or excedace size Permuaios wih bouded drop size is relaed o he bubble sor ad sequeces ha ca be raslaed io jugglig paers [5], whose eumeraio was firs sudied by Chug, Claesso, Dukes, ad Graham [4]. For a permuaio σ, we say ha i is a drop of σ if σ i < i ad he drop size is i σ i. Similarly, we say ha i is a excedace of σ if σ i > i, ad he excedace size is σ i i. I is well-kow ha he umber of excedaces is a Euleria saisic, i.e., has he same disribuio as des over he se of permuaios. Followig [4], we use maxdropσ) o deoe he maximum drop of σ, maxdropσ) : max{i σ i i }, ad similarly, maxexcσ) o deoe he maximum excedace size of σ, maxexcσ) : max{σ i i i }. Le B,k {σ S maxdropσ) k}. I is easy o see ha B,k k!k + ) k : Jus oe ha here are k + ) k ways o deermie σ,, σ k+ i he correc order, oe afer aoher, ad he remaiig is clear e.g., see [5, Thm.]). I [4], Chug e al. defied he k-maxdrop desce polyomials B,k ) : desσ). σ B,k ad obaied recurreces ad a formula for he geeraig fucio B k, z) : 0 B,k)z. I his secio, we will use he aalysis i he previous secio o derive a varia geeraig fucio for B k, z). Explicily, we ge a exac formula for E k, z) : B,k )z desσ) z. 4.) k k σ B,k 2

13 Firs, le B,k {σ S maxexcσ) k}. I is clear ha he map a a 2... a + a ) + a )... + a ) is a bijecio from B,k o B,k ha preserves he saisic desσ) ad ivσ). I follows ha B,k ) desσ) ad hece E k, z) k desσ) z. σ B,s σ B,k Noe ha B,k is he se of permuaios σ S saisfyig σ i i + k. I is easy o check ha i is exacly he se of permuaios o he rucaed board Λ, k. Hece for k +, we have σ B,k +desσ) q ivσ) F, k q, ) ad Theorem 3. wih s k gives a descripio of F, k. To obai he ordiary geeraig fucio for B,k ), se q. As before, le γ / ). The Formula 3.5) becomes he followig equaio for k + : k B,k ) ) l0 j δ +...+δ j l γ j j ) k + δi i p;δ j+ > k l δ j+ +δ j δ p+ l Noe ha from he aalysis, he upper limi of l ca iclude k. ) Le θ k) 0 ad for l, θ k) l : j;δ +...+δ j l >0 γ j j ) k + δi, i γ p+ j l)! δ j+! δ k+!. ad c k) l : τ 0 > k l ) l γ τ 0 τ + +τp l τ 0 τ i ) l γ p τ0, for k > 0. τ,..., τ p For k 0, le c 0) δ,0. The for ay fixed k 0 ad k +, we have k B,k ) ) l0 θ k) l c k) l, 4.2) Leig q ad k s i equaio 3.3), we obai Θ k z) θ k) z ) ). 4.3) z) k+ 0 For he coefficie c k) i, usig Formula 3.5) wih s 0, we ge ha ) γ p A ) τ,..., τ p ), τ + +τp τ i 3

14 where A ) is he classical Euleria polyomial defied by A ) σ S +desσ), > 0. By coveio, we se A 0 ). I follows ha for k > 0, c k) γ p> k ) k A p ) p ) p γ ) Ap ) p ) p. 4.4) p0 Wriig as a geeraig fucio, we obai ha for k > 0, C k z) k c k) k z γ p0 A p ) ) p k ) z p which leads o C k z) k p0 A p ) ) p z p k z) p+ p ) )z. 4.5) p For k 0, C 0 z). Observe ha equaio 4.2) is rue for k as well. I fac i is equivale o he ideiy k ) k A k ) ) k p A p ), p p0 which ca be readily checked by usig he followig expressio of he Euleria polyomial, see, for example [3, Lemma 4., p.57], Therefore for all k, A ) ) + j j, 0. B,k ) j0 ) k l0 θ k) l c k) l, Muliplyig boh sides by z ad summig over k, we have obaied he geeraig fucio E k, z). Theorem 4. Le E k, z) B,k )z k k σ B,k desσ) z. The E 0, z) / z) ad for k, E k, z) Θ k )z)c k )z), 4.6) 4

15 where Θ k z) ad C k z) are give i formulas 4.3) ad 4.5). Explicily, E k, z) k A p ) p0 z p k )z) p+ Example 4. For he case k, formula 4.7) gives E, z) p ) ) ) p z p )z) k+. 4.7) )z )z) 2 z )z) z2 )z). Comparig wih equaio 5) i [4], ad oig ha he summaio of B k z, y) i [4] sars from 0, oe checks easily ha he wo formulas agree wih each oher. Refereces [] Alexei Borodi, Persi Diacois, ad Jaso Fulma, O addig a lis of umbers ad oher oe-depede deermiaal processes, Bull. Amer. Mah. Soc. N.S.). America Mahemaical Sociey. Bullei. New Series, 474): , 200. [2] Leoard Carliz ad Joh Riorda. Two eleme laice permuaio umbers ad heir q-geeralizaio. Duke Mah. J., 3:37 388, 964. [3] Charalambos A. Charalambides. Eumeraive combiaorics. CRC Press Series o Discree Mahemaics ad is Applicaios. Chapma & Hall/CRC, Boca Rao, FL, [4] Fa Chug, Aders Claesso, Mark Dukes, ad Roald Graham. Desce polyomials for permuaios wih bouded drop size. Europea J. Combi., 37): , 200. [5] Fa Chug ad Ro Graham. Primiive jugglig sequeces. Amer. Mah. Mohly, 53):85 94, [6] Aa de Mier. O he symmery of he disribuio of k-crossigs ad k-esigs i graphs. Elecro. J. Combi., 3):Noe 2, 6 pp. elecroic), [7] M. de Saie-Caherie. Couplages e pfaffies e combiaoire, physique e iformaique. Maser s hesis, Uiversiy of Bordeaux I, 983. [8] Adriao M. Garsia. O he maj ad iv q-aalogues of Euleria polyomials. Liear ad Muliliear Algebra, 8):2 34, 979/80. [9] Ira M. Gessel. Geeraig fucios ad eumeraio of sequeces. PhD hesis, M.I.T., 977. [0] Aisse Kasraoui. Asces ad desces i 0-filligs of moo polyomioes. Europea J. Combi., 3):87 05, 200. [] Sri Gopal Mohay. Laice pah couig ad applicaios. Academic Press [Harcour Brace Jovaovich Publishers], New York, 979. Probabiliy ad Mahemaical Saisics. 5

16 [2] Joh Riorda. A iroducio o combiaorial aalysis. Wiley Publicaios i Mahemaical Saisics. Joh Wiley & Sos Ic., New York, 958. [3] Richard P. Saley. Biomial poses, Möbius iversio, ad permuaio eumeraio. J. Combiaorial Theory Ser. A, 203): , 976. [4] Richard P. Saley. Eumeraive combiaorics. Vol., volume 49 of Cambridge Sudies i Advaced Mahemaics. Cambridge Uiversiy Press, Cambridge, 997. Wih a foreword by Gia-Carlo Roa, Correced repri of he 986 origial. 6