Descents of Permutations in a Ferrers Board

Transcription

1 Desces of Permuaos a Ferrers Board Chuwe Sog School of Mahemacal Sceces, LMAM, Pekg Uversy, Bejg 0087, P. R. Cha Cahere Ya y Deparme of Mahemacs, Texas A&M Uversy, College Sao, T Submed: Sep 6, 20; Acceped: Dec 20, 20; Publshed: Ja 6, 202 Mahemacs Subjec Classcao: 05A05, 05A5, 05A30 Absrac The classcal Eulera polyomals are deed by seg A () +des() A ;k k 2S k where A ;k s he umber of permuaos of legh wh k desces. Le A (; q) P 2S +des() q v() be he v q-aalogue of he classcal Eulera polyomals whose geerag fuco s well kow: u A (; q) [] q! 0 k ( ) k u k : (0.) I hs paper we cosder permuaos resrced a Ferrers board ad sudy her desce polyomals. For a geeral Ferrers board F, we derve a formula he form of permae for he resrced q-eulera polyomal A F (; q) : +des() q v() : 2F If he Ferrers board has he specal shape of a square wh a ragular board of sze s removed, we prove ha A F (; q) s a sum of s + erms, each sasfyg a equao ha s smlar o (0.). The we apply our resuls o permuaos wh bouded drop (or excedace) sze, for whch he desce polyomal was rs suded by Chug e al. (Europea J. Comb., 3(7) (200): ). Our mehod preses a alerave approach. Suppored par by NSF Cha gra #07260 ad by he Scec Research Foudao for ROCS, Chese Msry of Educao. [k] q! y Suppored par by NSF gra #DMS ad NSA gra #H he elecroc joural of combaorcs 9 (202), #P7

2 Iroduco Le S deoe he symmerc group of order. Gve a permuao 2 S, le Des() be he desce se of,.e., Des() fj > + ; g, ad le des() jdes()j deoe he umber of desces of. For D f; 2; : : : ; g, we deoe by (D) he umber of permuaos 2 S whose desce se s coaed D, ad by (D) he umber of permuaos 2 S whose desce se s equal o D. I symbols, (D) : jf 2 S : Des() Dgj; (D) : jf 2 S : Des() Dgj: Le D fd ; d 2 ; : : : ; d k g where d < < d k. For coveece, also le d 0 0 ad d k+. The he followg formulas for (D) ad (D) are well-kow (see, for example, [4, p.69]): (D) (.) d ; d 2 d ; : : : ; d k (D)! de (d j+ d )! de d d j+ d ; (.2) where (; j) 2 [0; k] [0; k] he marx of equao (.2). A q-aalogue of he above P formulas s gve by cosderg he permuao sasc v(), where v() ( <j > j ). By coveo, he symbol (P ) s f he saeme P s rue ad 0 f o. See [4, Example 2.2.5]. Explcly, le (D; q) q v() ; (D; q) q v() : 2S :Des()D 2S :Des()D The (D; q) (D; q) []! de d ; d 2 d ; : : : ; d k [d j+ d ]! []! [d ]![d 2 d ]! [ d k ]! (.3) de d ; (.4) d j+ d where (; j) 2 [0; k] [0; k] as before. Here we use he sadard oao [] [] : ( q q! )( q); []! : [][2] []; : k [k]![ k]! for he q-aalogue of he eger, he q-facoral, ad he q-bomal coece, respecvely. Somemes s ecessary o wre he base q explcly as [] q ; [] q!, ad, k q ec., bu we om q hs paper as we do o use he aalogues of ay oher varables. The classcal Eulera polyomals are deed by seg A () +des() A ;k k ; 2S k he elecroc joural of combaorcs 9 (202), #P7 2

3 where A ;k s called he Eulera umber ha deoes he umber of permuaos of legh wh k desces. Le A 0 (). The polyomals A () have he geerag fuco (see e.g. Rorda [2]) 0 A () u! ( ) k u k k! k : (.5) u( ) e Le A (; q) P 2S +des() q v() be he v q-aalogue of he Eulera polyomals. Saley [3] showed ha 0 u A (; q) []! E(u( ); q) ; (.6) where E(z; q) 0 z []! : By smple mapulaos we ca see ha a equvale form of (.6) s 0 u A (; q) []! : (.7) ( ) k u k [k]! k Alerave proofs of (.7) have bee gve by Gessel [9] ad Garsa [8]. I hs paper we cosder permuaos wh resrced posos, ad exed he above resuls o desce polyomals of permuaos a Ferrers board. Tradoally a permuao 2 S s also represeed as a 0-llg of a by square board: Readg from lef o rgh ad boom o op, we smply pu a he h row ad he jh colum wheever j for ; : : : ;. Gve egers 0 < r r 2 r, he Ferrers board of shape (r ; : : : ; r ) s deed by F f(; j) : ; j r g: I he followg we defy a permuao wh s 0-llg represeao, ad say ha s a Ferrers board F f all he cells (; ) are F. I Seco 2 we exed he formulas (.3) ad (.4) o he se of permuaos o a xed Ferrers shape wh rows ad colums, ad derve a permae formula for he resrced q-eulera polyomal A F (; q) : 2F +des() q v() : I Seco 3 we focus o he Ferrers board ha s obaed from he square by removg a ragular board of sze s, ad prove ha he resrced q-eulara polyomal s a sum of s + erms, each deermed by a equao ha geeralzes (.7). he elecroc joural of combaorcs 9 (202), #P7 3

4 Fally Seco 4, we apply our resuls o permuaos wh bouded drop (or excedace) sze, for whch he desce polyomal was rs suded by Chug, Claesso, Dukes ad Graham [4]. Our mehod preses a alerave approach o he resuls [4]. Noao o lace pah Here we recall some oao ad resuls abou he coug of lace pahs wh a geeral rgh boudary. These resuls oer he ma ool o descrbe permuaos resrced a Ferrers board. For a referece o lace pah coug, see Mohay []. A lace pah P s a pah he plae wh wo kds of seps: a u orh sep N or a u eas sep E. If x s a posve eger, a lace pah from he org (0; 0) o he po (x; ) ca be coded by a legh o-decreasg sequece (x ; x 2 ; : : : ; x ), where 0 x x ad x s he x-coordae of he h orh sep. For example, le x 5 ad 3. The he pah EENENNEE s coded by (2; 3; 3). I geeral, le s be a o-decreasg sequece wh posve eger erms s ; s 2 ; : : : ; s. A lace pah from (0; 0) o (x; ) s oe wh he rgh boudary s f x < s for. If x s, he he umber of lace pahs from (0; 0) o (x; ) wh he rgh boudary s does o deped o x. Le P ah (s) be he se of lace pahs from (0; 0) o (s ; ) wh he rgh boudary s, ad LP (s) be he cardaly of P ah (s). For a gve sequece s (s ; s 2 ; : : : ; s ), le LP (s; q) q area(p ) ; P 2P ah (s) P where area(p ) x s he area eclosed by he pah P, he y-axs, ad he le y. Hece LP (s) LP (s; ). I hs paper we wll also allow he eres s o sasfy s s 2 s, whch case s + LP (s; q) LP ((s ; s ; : : : ; s ); q) : 2 I parcular LP (( + ; + ; : : : ; + ); q). I s also easy o see ha LP ((; 2; : : : ; ); q) C (q); where C (q) s Carlz-Rorda's q-caala umber [2]. 2 Desces of permuaos Ferrers boards Le F be a Ferrers board wh rows ad colums, whch s alged o he op ad lef. Idex he rows from boom o op, ad colums from lef o rgh. Le r be he sze of row. Hece r r 2 r. For a se D fd ; d 2 ; : : : ; d k g wh d < < d k, le F (D) be he umber of permuaos F wh he desce se D, ad F (D) be he umber of permuaos he elecroc joural of combaorcs 9 (202), #P7 4

5 F whose desce se s coaed D. The v q-aalogues of F (D) ad F (D) are deed by F (D; q) 2F :Des()D q v() ; F (D; q) 2F :Des()D q v() : Clearly F (D; ) F (D) ad F (D; ) F (D). The Icluso-Excluso Prcple mples ha F (D; q) T D F (T; q); F (D; q) T D( ) jd T j F (T; q): We shall show ha F (D; q) ad F (D; q) ca be expressed erms of LP (s; q), he area eumeraor of lace pahs wh proper rgh boudares ad leghs. Le's rs compue F (D; q). To ge a permuao F sasfyg Des() D, we rs choose x < x 2 < < x d such ha x r, ad pu a he cell (x ; ) for d. The choose x d + < x d +2 < < x d2 such ha x r, ad pu a he cell (x ; ) for d < d 2, ad so o. We say ha he cell (; j) s a -cell f s lled wh a. I s clear ha a verso of correspods o a souheas cha of sze 2 he llg,.e. a par of -cells f (x ; ); (x 2 ; 2 ) g such ha < 2 whle x > x 2. For d, he -cell he h row (.e. y ) has exacly x may oher -cells lyg above ad o s lef. Hece he -cell he h row corbues x o he sasc v(), ad all he -cells he rs d rows corbued (x ) + (x 2 2) + + (x d d ) o he sasc v(). Noe ha 0 x x 2 2 x d d, ad x < r +. Hece he umber of choces for he sequece (x ; : : : ; x d ) s exacly he umber of lace pahs from P (0; 0) o (r d d + ; d ) wh he rgh boudary (r ; r 2 ; : : : ; r d d + ), ad d ) s he area of he correspodg lace pah. Therefore he rs d rows of F corbue a facor of LP d ((h ; : : : ; h d ); q) o F (D; q). Le h (h ; h 2 ; : : : ; h ) where h r +. Le he -h block of F coss of rows d + o d. Applyg he above aalyss o he -h block of he Ferrers board F for 2; : : : ; k +, we ge ha Theorem 2. F (D; q) 2F :Des()D q v() ky 0 LP d+ d ((h d +; : : : ; h d+ ); q) (2.) where we use he coveo ha d 0 0 ad d k+. he elecroc joural of combaorcs 9 (202), #P7 5

6 Accordgly, F (D; q) ) T D( jd T j F (T; q) < 2 < < j k ( ) k j f(0; )f( ; 2 ) : : : f( j ; k + ) where f(; j) 8 < : LP dj d (h d +; : : : ; h dj ); q) f < j f j 0 f > j: (2.2) Followg he dscusso of Saley [4, p.69], we oba ha Theorem 2.2 F (D; q) s he deerma of a (k + ) (k + ) marx wh s (; j) ery f(; j + ), 0 ; j k. Tha s, where f(; j) s gve by (2.2). F (D; q) de[f ;j+ ] k 0 (2.3) Whe he Ferrers board F s a square, h +, ad LP j ((h + ; : : : ; h j ); q) : j Theorems 2. ad 2.2 reduce o he classcal resuls ad 2S Des()D 2S Des()D q v() d ; d 2 d ; : : : ; d k k q v() de d : d j+ d 0 For a geeral Ferrers board wh rows ad colums, le's check wo exreme cases: D f; 2; : : : ; g ad D ;. Case. D f; 2; : : : ; g: q v() F ( f; 2; : : : ; g; q) Theorem 2. yelds he dey 2F Y LP ((h ); q) Y [h ]: (2.4) Noe ha permuao llgs of a Ferrers board wh rows ad colums correspod o complee machgs of f; : : : ; 2g wh xed ses of lef edpos ad rgh edpos, ad a verso of he permuao s exacly a esg of he he elecroc joural of combaorcs 9 (202), #P7 6

7 machg; See de Mer [6] ad Kasraou [0]. To see hs, for a gve Ferrers board F wh rows ad colums, oe raverses he pah from he lower-lef corer o he op-rgh corer, ad records he pah by s seps a ; a 2 ; : : : ; a 2 where a E f he h sep s Eas, ad a N f he h sep s Norh. Le L f : a Eg ad R f : a Ng. The 0-llgs of F cosdered here are oe-o-oe correspodece wh he machgs of f; : : : ; 2g for whch he se of lef edpos s L ad he se of rgh edpos s R. For example, he followg Ferrers board F, raversg from he lower-lef corer o he op-rgh corer, we ge he sequece EENENENN. Thus L ; 2; 4; 6 ad R 3; 5; 7; 8. The llg gve he gure correspods o he machg f(; 7); (2; 3); (4; 5); (6; 8)g. I follows ha equao (2.4) s exacly he geerag fuco of he sasc e 2 (M), whch s he umber of esgs a machg M, coued over all he machgs wh gve ses of lef ad rgh edpos. Tha s, M q e 2(M ) whch maches he kow resuls [7, 0]. Y [h ]; Case 2, D ;: We have F (;; q) F (;; q) LP ((h ; : : : ; h ); q) Noe ha h, hece LP ((h ; : : : ; h ); q) r for all, where he oly permuao he Ferrers board F wh o desces s he dey permuao; oherwse P ah (h ; : : : ; h ) ; ad LP ((h ; : : : ; h ); q) 0. Theorems 2. ad 2.2 ca be used o ge a formula for he jo dsrbuo of des() ad v() over permuaos F. Le A F (; q) 2F +des() q v() : (2.5) Theorem 2.3 A F (; q) ( ) per(m); (2.6) he elecroc joural of combaorcs 9 (202), #P7 7

8 where M s a marx whose (; j)-ery s gve by M j 8 < : LP j +((h ; : : : ; h j ); q) f j f j + 0 f > j + ; ad per(m) s he permae of he marx M. Proof. We have 2F +des() q v() Df;2;:::; g +jdj F (D; q) +jdj Df;2;:::; g T :T D F (T; q) ( ) jd T j F (T; q) ( ) jd T j +jt j+jd T j T f;2;:::; g D:T D +k ( ) T (LP (h)) T f ;:::; k g < ( ) per(m); where M s a marx as descrbed Theorem, ad D (LP (h)) deoes he rgh-had sde of (2.). 2 Remark 2. We remark ha desces of permuaos a Ferrers board provde a example of oe-depede deermaal po processes, as suded by Borod, Dacos ad Fulma []. Le U be a e se. A po process o U s a probably measure P o he 2 juj subses of U. Oe smple way o specfy P s va s correlao fucos (A), where for A U, (A) P fs : S Ag: A po process s deermaal wh kerel K(x; y) f (A) de[k(x; y)] x;y2a : I s oe-depede f ( [ Y ) ()(Y ) wheever ds(; Y ) 2. Borod e al. showed ha may examples from combaorcs, algebra ad group heory are deermaal oe-depede po processes, for example, he carres process, he desce se of uformly radom permuaos, ad he desce se Mallows model []. For hese hree cases, he po processes are saoary, whle he desce se of permuaos a Ferrers board correspods o a deermaal oe-depede po process ha s o saoary. Explcly, Q for ay se D fd ; : : : ; d k g wh d < < d k, le P F (D) F (D)( h ). Usg [, Theorem 7.5] we oba ha s a deermaal, oe-depede process wh correlao fucos P F (D) F (D) de[k(d ; d j )] k ;j he elecroc joural of combaorcs 9 (202), #P7 8

9 ad wh correlao kerel K(x; y) x;y + (E ) x;y+ ; where E s he upper ragular marx E [e( ; j)] ;j whose eres are gve by e(; j) 8 < : LP j (h + ; : : : ; h j ) f < j f j 0 f > j: 3 Permuaos he rucaed board s For a geeral o-decreasg sequece of posve egers s, LP (s; q) ca be compued by a deerma formula (see, for example, []). Bu here s o smple closed formula. I he specal cases ha he Ferrers board F s obaed from rucag he square board by a ragular board he corer, we ca descrbe he jo dsrbuo of he sascs des() ad v() by dees of her b-varae geerag fucos. Le s be he ragular board wh row sze (s; s ; : : : ; ). For s, le ;s be he rucaed board s cossg of cells ha are lyg 0 x; y ad above he le y x ( s). I oher words, ;s s he Ferrers board whose row leghs are ( s; s+; : : : ; ; : : : ; ). See he followg gure for ;s wh wh 7 ad s 4. Now le D fd ; : : : ; d k g wh d < < d k. We shall compue he jo dsrbuo of + des ad v over all permuaos ;s usg he formulas obaed Seco 2. Aga le d 0 0 ad d k+. Le d d for ; : : : ; k +, ad assume ha j s he parcular dex o make d j s < d j+ occur. Frs we compue ;s (D; q). Le r be he sze of row ;s. The s + f s r f s < : Le h r +. The. For 0 < j, s + + LP d+ d ((h d +; : : : ; h d+ ); q) LP + (( s; : : : ; s); q) + : he elecroc joural of combaorcs 9 (202), #P7 9

10 2. For j, LP dj+ d j ((h dj +; : : : ; h d+ ); q) LP j+ ((( s) s d j ; s ; : : : ; d j+ + ); q) dj+ + j+ j+ dj : 3. For > j, j+ LP d+ d ((h d +; : : : ; h d+ ); q) LP + ((( d ; d ; : : : ; d + + ); q) d d : + Summg over all permuaos wh Des() D he Ferrers board ;s, we oba jy s + ky d ;s (D; q) q v() 2 ;s Des()D jy s + where (D j ) represes he sequece j+ ; : : : ; k+. Hece he Prcple of Icluso-Excluso leads o ;s (I; q) 2 ;s Des()I q v() DI ( ) jij jdj dj (D j ) j Y j + dj (D j ) s ; + : (3.) Le F ;s (q; ) be he b-varae geerag fuco of he sascs v ad des over all permuaos he board ;s. Tha s, F ;s (q; ) 2 s +des() q v() If;2;:::; g +jij 2 ;s Des()I q v() : Le (a; q) ( a)( aq) ( aq ) ad (a; q) Q 0 ( aq ). Our ma resul here s a aalog of he formula (.7). Explcly, we show ha F ;s (q; ) ca be expressed as a lear combao of s+ erms, each of whch sases a q-dey smlar o (.7). he elecroc joural of combaorcs 9 (202), #P7 0

11 Theorem 3. For s, F ;s (q; ) 0. For > s, we have F ;s (q; ) 0 F (0) ;s (q; ) + F () ;s (q; ) + + s F (s) ;s(q; ); (3.2) where he k 's are deed by he formal power seres k z k k0 (z; q) s ad for each 0; ; : : : ; s, he erm F () ;s(q; ) s gve by he dey s+ z [ ]! F () ;s(q; ) ( ) ks+ k z k [k]! z k [k]! ; (3.3) : (3.4) Proof. As before assume D fd ; d 2 ; : : : ; d k g wh d 0 0 ad d k+. Le j be he dex uquely decded by d j s < d j+. For s +, by he equao (3.) we have F ;s (q; ) +jij ( ) jij jdj dj Y j s + (D j ) If;2;:::; g DI Y j s + Df;2;:::; g Df;2;:::; g k0 Le d l ad F ;s (q; ) ( ) s l0 k0 0 s l0 +k j +:::+ j l dj (D j ) dj (D j ) +k ( ) k. The, + 2 +:::+ k+ l +:::+ j s< +:::+ j+ j jy j Y s + s :::+ k+ +:::+ j s< +:::+ j+ I:DI [ l]! Q j [ s + ]! [ ]! [ k+ ]!([ s ]!) j : 0 C ( ) jij jdj +jij ( ) jdj +jdj [ d j ]! Q j [ s + ]! [ ]! [ k+ ]!([ s ]!) j : k; j+ s+ l j+ + j+2 +:::+ k+ l C k+ j [ l]! [ j+ ]! [ k+ ]! A (3.5) he elecroc joural of combaorcs 9 (202), #P7

12 Le 0 ad for l ; : : : ; s, l : j +:::+ j l j jy s + ; ad F (l) ;s : ( ) k; 0 s+ l 0 + +:::+ k l k+ [ l]! [ 0 ]! [ k ]! : (3.6) The F ;s (q; ) 0 F (0) ;s (q; ) + F () ;s (q; ) + + s F (s) ;s(q; ): We show ha l ad F ;s(q; (l) ) sasfy (3.3) ad (3.4). Frs, observe ha for l > 0, l s he coece of z l he formal power seres! j s + k z k : (3.7) k j0 k Usg he q-aalog of he bomal heorem k0 (a; q) k (q; q) k z k (az; q) (z; q) ; we have l0 l z l j0 j0 k! j [ s + k ][ s + k 2] [ s] z k [k]! ( (q s z; q) ) (z; q) ( ) (z; q) s j : Ths proves he formula (3.3). To ge he formula (3.4), observe ha (3.6) ca be wre as F (l) ;s [ l]! ( ) [z l ] 0 s+ l z 0 [ 0 ]! k0 k (! z []! )k : he elecroc joural of combaorcs 9 (202), #P7 2

13 Ths leads o he dey s+ z l [ l]! F (l) ;s(q; ) ( ) ks+ l I he case ha s 0, F ;s (q; ) F (0) ;s (q; ) A (; q), ad equao (3.4) reduces o he well-kow dey (.7) by leg u z. 2 k z k [k]! z k [k]! : 4 Permuaos wh bouded drop or excedace sze Permuaos wh bouded drop sze s relaed o he bubble sor ad sequeces ha ca be raslaed o jugglg paers [5], whose eumerao was rs suded by Chug, Claesso, Dukes, ad Graham [4]. For a permuao, we say ha s a drop of f < ad he drop sze s. Smlarly, we say ha s a excedace of f >, ad he excedace sze s. I s well-kow ha he umber of excedaces s a Eulera sasc,.e., has he same dsrbuo as des over he se of permuaos. Followg [4], we use maxdrop() o deoe he maxmum drop of, maxdrop() : maxf j g; ad smlarly, maxexc() o deoe he maxmum excedace sze of, maxexc() : maxf j g: Le B ;k f 2 S jmaxdrop() kg. I s easy o see ha jb ;k j k!(k + ) k : Jus oe ha here are (k + ) k ways o deerme ; ; k+ he correc order, oe afer aoher, ad he remag s clear (e.g., see [5, Thm.]). I [4], Chug e al. deed he k-maxdrop desce polyomals B ;k () : 2B ;k des() ad P obaed recurreces as well as a formula for he geerag fuco B k (; z) : B 0 ;k()z. I hs seco, we wll use he aalyss he prevous seco o derve a vara geerag fuco for B k (; z). Explcly, we ge a exac formula for E k (; z) : k B ;k ()z k 2B ;k des() A z : (4.) he elecroc joural of combaorcs 9 (202), #P7 3

14 Frs, le B 0 f 2 S ;k jmaxexc() kg. I s clear ha he map a a 2 : : : a 7! ( + a )( + a ) : : : ( + a ) s a bjeco from B ;k o B 0 ;k ha preserves he sasc des() ad v(). I follows ha B ;k () des() ad hece E k (; z) k des() A z : 2B 0 ;s 2B 0 ;k Noe ha B 0 ;k s he se of permuaos 2 S sasfyg + k. I s easy o check ha s exacly he se of permuaos o he rucaed board ; k. Hece for k +, we have 2B ;k +des() q v() F ; k (q; ) ad Theorem 3. wh s k gves a descrpo of F ; k (q; ). To oba he ordary geerag fuco for B ;k (), se q. As before, le ( ). The formula (3.5) becomes he followg equao for k + : 0 k B ;k () ( ) l0 j +:::+ j l j jy k + 0 C p; j+ > k l j+ + j+2 +:::+ p+ l (Noe ha from he aalyss, he upper lm of l ca clude k. ) Le (k) 0 ad for l, jy k + (k) l : j ; ad c (k) l : For k 0, le c (0) 0 > k l Leg q ad k s l 0 j; +:::+ j l >0 + +p l 0 p+ j ( l)! j+! k+! l p 0 ; for k > 0: ; : : : ; p ;0. The for ay xed k 0 ad k +, we have k (z) 0 k B ;k () ( ) l0 equao (3.3), we oba (k) z C A : (k) l c (k) ; (4.2) l ( z) k+ : (4.3) he elecroc joural of combaorcs 9 (202), #P7 4

15 For he coece c (k), usg formula (3.5) wh s 0, we ge ha p A () ; : : : ; p ( ) ; + +p where A () s he classcal Eulera polyomal deed by A () 2S +des() ; > 0: By coveo, we se A 0 (). I follows ha for k > 0, k A c (k) p () p ( ) Ap () p p ( ) : (4.4) p p> k p0 Wrg as a geerag fuco, we oba ha for k > 0, whch leads o C k (z) C k (z) k k p0 k c (k) z p0 A p () ( ) p k p A p () ( ) p z p ( z) p+ k p z! z : (4.5) p For k 0, C 0 (z). Observe ha equao (4.2) s rue for k as well. I fac s equvale o he dey k k A k () ( ) k p A p (); p p0 whch ca be readly checked by usg he followg expresso of he Eulera polyomal, see, for example [3, Lemma 4., p.57], A () ( ) + j0 j j ; 0: Therefore for all k, B ;k () ( ) k l0 (k) l c (k) ; l Mulplyg boh sdes by z ad summg over k, we have obaed he geerag fuco E k (; z). he elecroc joural of combaorcs 9 (202), #P7 5

16 Theorem 4. Le E k (; z) k The E 0 (; z) ( z) ad for k, B ;k ()z k 2B ;k des() A z : E k (; z) k(( )z)c k (( )z); (4.6) where k (z) ad C k (z) are gve formulas (4.3) ad (4.5). Explcly, k z p k! A p () ( ) p z ( ( )z) p+ p p0 p E k (; z) : (4.7) ( ( )z) k+ Example 4. For he case k, formula (4.7) gves E (; z) ( )z ( ( )z) 2 z( ( )z) z(2 ( )z) : Comparg wh equao (5) [4], ad og ha he summao of B k (z; y) [4] sars from 0, oe checks easly ha he wo formulas agree wh each oher. Ackowledgeme The auhors wsh o hak a aoymous referee for he very careful readg ad may helpful suggesos. Refereces [] Alexe Borod, Pers Dacos, ad Jaso Fulma. O addg a ls of umbers (ad oher oe-depede deermaal processes). Bull. Amer. Mah. Soc. (N.S.), 47(4):639{670, 200. [2] Leoard Carlz ad Joh Rorda. Two eleme lace permuao umbers ad her q-geeralzao. Duke Mah. J., 3:37{388, 964. [3] Charalambos A. Charalambdes. Eumerave combaorcs. CRC Press Seres o Dscree Mahemacs ad s Applcaos. Chapma & Hall/CRC, Boca Rao, FL, [4] Fa Chug, Aders Claesso, Mark Dukes, ad Roald Graham. Desce polyomals for permuaos wh bouded drop sze. Europea J. Comb., 3(7):853{867, 200. [5] Fa Chug ad Ro Graham. Prmve jugglg sequeces. Amer. Mah. Mohly, 5(3):85{94, he elecroc joural of combaorcs 9 (202), #P7 6

17 [6] Aa de Mer. O he symmery of he dsrbuo of k-crossgs ad k-esgs graphs. Elecro. J. Comb., 3():Noe 2, 6 pp. (elecroc), [7] M. de Sae-Cahere. Couplages e pfaes e combaore, physque e formaque. Maser's hess, Uversy of Bordeaux I. [8] Adrao M. Garsa. O he \maj" ad \v" q-aalogues of Eulera polyomals. Lear ad Mullear Algebra, 8():2{34, 979/80. [9] Ira M. Gessel. Geerag fucos ad eumerao of sequeces. PhD hess, M.I.T., 977. [0] Asse Kasraou. Asces ad desces 0-llgs of moo polyomoes. Europea J. Comb., 3():87{05, 200. [] Sr Gopal Mohay. Lace pah coug ad applcaos. Academc Press [Harcour Brace Jovaovch Publshers], New York, 979. Probably ad Mahemacal Sascs. [2] Joh Rorda. A roduco o combaoral aalyss. Wley Publcaos Mahemacal Sascs. Joh Wley & Sos Ic., New York, 958. [3] Rchard P. Saley. Bomal poses, Mobus verso, ad permuao eumerao. J. Combaoral Theory Ser. A, 20(3):336{356, 976. [4] Rchard P. Saley. Eumerave combaorcs. Vol., volume 49 of Cambrdge Sudes Advaced Mahemacs. Cambrdge Uversy Press, Cambrdge, 997. Wh a foreword by Ga-Carlo Roa, Correced repr of he 986 orgal. he elecroc joural of combaorcs 9 (202), #P7 7