Pattern Avoiding Partitions, Sequence A and the Kernel Method

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1 Avalable at Appl Appl Math ISSN: Vol 6 Issue (Decembe ) pp Applcatos ad Appled Mathematcs: A Iteatoal Joual (AAM) Patte Avodg Pattos Sequece A5439 ad the Keel Method Touf Masou ad Ma Shattuc Depatmet of Mathematcs Uvest of Hafa 395 Hafa Isael touf@mathhafaacl; maaos@exctecom Receved:Mach 6 ; Accepted: August 8 Abstact Sequece A5439 OEIS whch we wll deote b a couts a ceta two-patte avodace class of the pemutatos of sze I ths pape we povde addtoal combatoal tepetatos fo these umbes tems of fte set pattos I patcula we detf sx classes of the pattos of sze all of whch have cadalt a ad each avodg two classcal pattes We use both algebac ad combatoal methods to establsh ou esults I oe appaetl moe dffcult case to show the esult we mae use of the eel method solvg a sstem of thee fuctoal equatos whch ases afte a ceta paamete s toduced We also defe a algothmc becto betwee the avodace class ths case ad aothe whch sstematcall eplaces the occueces of a gve patte wth those of aothe havg the same legth Kewods: Patte avodace set patto eel method MSC No: 5A5 5A8 Itoducto If the let s deote the sequece (see Bacucc et al ()) whch couts the pemutatos of sze avodg the pattes 3ad ( )( 3) ( ) Lettg va poduces dffeet sequeces the ad cases fo example coespodg to the Motz umbes ad to eumeatos of Motz left factos (whch was show Bacucc et al ()) Lettg go to ft poduces the Catala sequece ad so thee s a ``dscete 397

2 398 Touf Masou & Ma Shattuc cotut'' betwee the Motz ad Catala sequeces as oted Bacucc et al () I the peset coespodece we ae coceed wth the case 3 the tems of whch we wll deote b a I patcula we wll show that a also couts ceta avodace classes of set pattos The sequece a occus as A5439 Sloae ad has geeatg fucto gve b x a x x 3x x 3x () The fst few a values statg wth ae gve b If the a patto of [ ] { } s a collecto of o-empt pawse dsot subsets called blocs whose uo s [] (If the thee s a sgle empt patto whch has o blocs) Let P deote the set of all pattos of [] A patto s sad to be stadad fom f t s wtte as B / B / whee the blocs ae aaged ascedg ode accodg to the sze of the smallest elemets Oe ma also epeset B / B/ P expessed the stadad fom equvaletl as whee B called the caocal sequetal fom; ad such case we wll wte Fo example the patto 5/37/4/68 has the caocal sequetal fom 344 Note that possesses the estcted gowth popet (see eg Stato ad Whte (986) o Wage (996) fo detals) meag that t satsfes the followg thee codtos: () () s oto [] fo some ad () max{ } fo all I what follows we wll epeset set pattos as wods usg the caocal sequetal foms ad cosde some patcula cases of the geeal poblem of coutg the membes of a patto class havg vaous estctos mposed o the ode of the lettes m A classcal patte s a membe of [] whch cotas all of the lettes [] We sa that a wod [] cotas the classcal patte f cotas a subsequece somophc to Othewse we sa that avods Fo example a wod avods the patte 3 f t has o subsequece wth < < ad < < ad avods the patte f t has o subsequece wth < The patte avodace questo has bee the topc of much eseach eumeatve combatocs statg wth Kuth (974) ad Smo ad Schmdt (985) o pemutatos ad cosdeed moe ecetl o futhe stuctues such as wods ad compostos The avodace poblem ca be exteded to set pattos upo cosdeg the questo o the assocated caocal sequetal foms We efe the eade to the papes b Klaza (996) Saga () ad Jelíe ad Masou (8) ad to the efeeces thee We wll use the followg otato If { w w } s a set of classcal pattes the let P ( w w ) be the subset of P whch avods all of the pattes whose cadalt we wll deote b p w w ) (

3 AAM: Ite J Vol 6 Issue (Decembe ) 399 I ths pape we detf sx classes of the pattos of [] each avodg a classcal patte of legth fou ad aothe of legth fve ad each eumeated b the umbe a I addto to povdg ew combatoal tepetatos fo a sequece ths addesses specfc cases of a geeal questo ased b Got (8) fo example egadg the eumeato of avodace classes of pattos coespodg to two o moe pattes Aalogous esults coceg the avodace of two pattes b a pemutato have bee gve fo example b West (995) Ou ma esult s the followg theoem whch we pove the ext secto as a sees of popostos Theoem If the p ( u v) a fo the followg sets ( u v) : () () () () (3) () (4) (3) (5) () (6) () To show ths we gve algebac poofs fo cases () (3) (4) ad (6) ad fd oe-to-oe coespodeces betwee cases () ad () ad (5) ad (6) To establsh (4) ad (6) we mae use of the eel method (see Badee et al () ad Hou ad Masou ()) the latte case to solve a sstem of fuctoal equatos whch ases oce a ceta paamete has bee toduced Ou becto betwee cases (5) ad (6) s of a algothmc atue ad sstematcall eplaces occueces of wth oes of wthout toducg Poof of the Ma Result The cases {} {} ad {} I ths secto we cosde the cases of avodg {} {} ad {} Poposto The geeatg fucto fo p ( ) ad p ( ) whee s gve b x x 3x x 3x Poof: Note fst that each o-empt patto P () ma be decomposed as ethe whee cotas o 's o as fo some whee the cota o 's The must avod {} f wth avodg {}

4 4 Touf Masou & Ma Shattuc Futhemoe ote that each lette a occuece of Let f ( p () x Fom the foegog obsevatos we obta the elato o x f ( g( f ( xf ( xg( s geate tha each lette ad f > ode to avod p x g ( () xg( f ( x xg( () To compute g ( obseve that P () must be of the fom o whee ad cota o 's ad avod {} Futhemoe all of the lettes of ae geate tha all of the lettes of the secod case Ths mples o g( xg( x g ( x g ( x 3x x (3) Substtutg (3) to () ad smplfg elds the fst case above To compute the geeatg fucto h ( fo p ( ) we use the same cases as we dd fo fdg p () I the secod case howeve cosde futhe whethe o ot cotas a epeated lette Note that f t does the must avod {} Futhemoe o lettes geate tha oe occug afte the thd ca be epeated Ths elds the elato o x x h ( xh( h( x x x x x g( h( 3x x x ( g( fom whch the desed esult follows fom (3) x g( h( x x

5 AAM: Ite J Vol 6 Issue (Decembe ) 4 Thee s a dect becto showg the equvalece of the pas {} ad {} Poposto If the p () p() Poof: Let A P () ad B P () We wll defe a becto f betwee the sets A ad B a ductve mae the cases 3 clea Suppose 4 ad A If the B ad let f ( ) If the let m be the smallest lette of whch occus at least twce Thus we ma wte ( m ) mm m whee ad the cota ol lettes { m m } but ae othewse ust as the poof of Poposto above Gve a fte wod w o the alphabet of postve teges havg m dstct lettes let sta( w ) deote the equvalet wod o [m] havg all of the same elatve compasos wth egad to ts o postos (ofte called the stadadzato of w ) Let f ( sta ( )) ad be obtaed fom b addg m to each lette If the let ' be obtaed fom b addg to each lette whee s the umbe of dstct lettes of Let f ( ) be gve b o f ( ) ( m ) m m ' m ' m' o The f ( ) belogs to B sce avods {} ad each ' avods {} Oe ma vef that the mappg f s a becto whch completes the poof Rema: The mappg f s see to peseve the umbe of blocs; thus the membes of P () ad P ( ) havg the same pescbed umbe of blocs ae equumeous The Case {3} Let deote a patto of [] epeseted caocall Recall that empt sums tae the value zeo b coveto To establsh ths case we dvde up the set of pattos questo accodg to a ceta statstc amel the oe whch ecods the legth of the maxmal ceasg tal u To do so gve let f ( deote the geeatg fucto fo the umbe of pattos of [] havg at least lettes ad avodg the pattes ad 3 such that wth (f thee s a ( ) -st lette) We have the followg elato volvg the geeatg fuctos f (

6 4 Touf Masou & Ma Shattuc Lemma 3 If the f ( x x x f ( x f ( x f ( ) (4) wth tal codtos f ( x ) ad f ( ) ( ) x x x f x whee f ( f ( Poof: Note that f x x f ( ) sce a patto ths case ma ust have oe lette o stat If ( x the a patto eumeated b f ( x ) must ethe be o stat wth o whch mples f x x f ( x f ( ( Note that the secod case the secod lette s extaeous (coceg possble occueces of 3 o ) ad theefoe ca be emoved wthout affectg the eumeato whle the thd case the lettes ad at the begg ae extaeous If 3 the we cosde the followg cases coceg the pattos eumeated b f ( : ( ) ( ) ( ) ( v) whee ( ) The fst case cotbutes x Note that the secod case the wod cotas o lettes [ ] fo othewse thee would be a occuece of f t cotaed a lette [ ] o a occuece of 3 f t cotaed the lette Thus the lettes ( )( ) tae togethe compse a patto of the fom eumeated b f ( whch mples the cotbuto ths case s x f ( Smla easog the thd case elds a cotbuto of x f ( sce ca cota o lettes [ ] ad thus ( ) s a patto of the fom eumeated b f ( x ) (the facto of x accouts fo the lettes ( ) as well as the secod occuece of ) Fall the fouth case o membe of [ ] ca occu wth the secod extaeous whch mples a cotbuto of x f ( x ) Combg all of the cases elds (4) whch s also see to hold the case as well Poposto 4 The geeatg fucto fo p (3) s gve b

7 AAM: Ite J Vol 6 Issue (Decembe ) 43 x x 3x x 3x Poof: Defe the geeatg fucto f ( x F ( x ) Fst ote that f f ( f ( F( x) f x x f ( xf( ) ad ( f F( x) f ( ( F( x) ( ( x Multplg (4) b ad summg ove elds F( x f ( x x f ( x f ( x f ( F( x) [ F( x) ] [( F( x) ] x f ( x F( x) [ F( x) ] f ( x F( x) [ F( x) ] f ( F( x) [ F( x) ] ( F( x) f( ) ( F( x f( whch mples x x ( ) F( x F( x) ( x( ( x( (5)

8 44 Touf Masou & Ma Shattuc Ths tpe of fuctoal equato ca be solved sstematcall usg the eel method (see Badee et al ()) I ths case f we assume that (5) whee satsfes x ( )( e ) x x 3x x( the ( )( x ) p(3) x F( x) ( x x x 3x x3x as equed (Note that F () dctates ou choce of oot fo ) Rema: Substtutg the expesso above fo F (x) to (5) ecoves the expesso fo F ( x fom whch oe ca compute a explct fomula fo the coeffcet of x 3 The Cases {} ad {} Fo these cases we show fst that p () p() though a dect becto ad the show that the geeatg fucto fo p () s gve b () above Befoe defg the becto we wll eed the followg two lemmas Lemma 5 Suppose P () has at least oe occuece of the patte Let b be the smallest lette such that thee exsts a [ b ] fo whch thee s a subsequece gve b abaab The () the elemet b occus exactl twce ad () the elemet a s uquel detemed Poof: Fo () suppose to the cota that b occus at least thee tmes Let deote a occuece of the subsequece abaab ; we ma assume that a ad b coespod to tal occueces of lettes of the d If a addtoal b occus to the ght of the thd a the thee would be a occuece of whch s ot allowed If a addtoal b occus to the left of the thd a the thee would be a occuece of whch s also ot allowed Thus thee ae exactl two b 's ad the coespod to a occuece of Note that the mmalt of b s ot eeded fo ths pat

9 AAM: Ite J Vol 6 Issue (Decembe ) 45 To show () suppose to the cota that thee exsts a subsequece volvg the two b 's gve b cbccb fo some c [ b ] c a Suppose fst c < a If the thd c of occus to the ght of the secod a above the thee would be a occuece of ad f t occus to the left of the secod a the thee would be a occuece of Thus thee ae o occueces of cbccb wth c < b ( fact ths shows that a lette c < a ca ol occu po to the left-most b ) Now suppose a < c < b ad let aga deote a possble occuece of cbccb If the thd c of occus afte the thd a the thee would be a occuece of of the fom acaac wth c < b cotadctg the assumed mmalt of b O the othe had f the thd c of occus befoe the thd a of the thee would be a occuece of of the fom acca Thus o elemet of [ b ] ca fom a occuece of wth b whch completes the poof of () Lemma 6 Suppose P () cotas at least oe occuece of the patte ad let a ad b be as defed Lemma 5 above Wte WbWbW3 whee W s possbl empt The we have the followg: () ol lettes geate tha b ca occu 3 W wth the excepto of a ad o lette othe tha a ca occu moe tha oce W ; ad () a lette occug W ca occu at most oce W 3 all of whose lettes ae geate tha b Poof: To pove () wte W a a a fo some 3 whee the ae possbl empt ad cota o a 's Suppose to the cota that c < b occus W whee c a If c < a s fo some the thee would be a occuece of wheeas f t s the thee would be a occuece of If a < c < b ad c belogs to fo some < the thee would be a occuece of wth acca wheeas f c belogs to the thee would be a occuece of of the fom acaac wth c < b cotadctg the mmalt of b Thus ol lettes geate tha b ca occu W wth the excepto of a ad each lette geate tha b ca occu W at most oce so as to avod Statemet () s also a cosequece of belogg to P () We ow establsh the equvalece of avodg {} ad {} Poposto 7 If the p () p()

10 46 Touf Masou & Ma Shattuc Poof: Let A P () ad B P () We wll descbe a becto f betwee the sets A ad B algothmcall as follows Suppose A If B the let f ( ) Othewse cotas at least oe occuece of the patte Let b deote the smallest lette b fo whch thee exsts a subsequece abaab fo some a < b ad let a deote the coespodg lette a whch s uquel detemed b Lemma 5 Suppose we wte as Wb WbW 3 as Lemma 6 above whee W a a a Let Wb W ' bw 3 whee W ' a b b whch we have chaged all but the fst a occug W to b Note that ths eplaces all of the occueces of volvg a ad b wth oes of Usg Lemma 6 oe ca vef that o occueces of ae toduced If has o occueces of the let f ( ) Othewse let b deote the smallest lette b fo whch thee exsts a subsequece abaab fo some a < b Oe ca vef b > b B easog smla to that used the poof of Lemma 5 above oe ca also show that a lette a < b fo whch the subsequece ab aab occus s uquel detemed whch we'll deote b a Let deote the patto obtaed b chagg all of the lettes a except the fst comg afte the leftmost b to b No subsequeces ae toduced whch follows fom the mmalt of b Now epeat the above pocess cosdeg Sce b < b < b < the pocedue descbed must ed a fte umbe sa t of steps wth the esultg patto t belogg to P () Let f ( ) t Note that the lagest b fo whch thee exsts a < b such that ababb occus t s b b t wheeve t Ths follows fom the fact oe ca vef that o occueces of whch the coespods to a lette geate tha b ae toduced the tasto fom to fo all If t the oe ca also vef that the lagest a < b t fo whch thee s a subsequece of the fom ab tabt bt t s a a t So to evese the algothm we fst cosde the lagest lette b (f t exsts) fo whch ababb occus t fo some a < b ad the cosde the lagest such a coespodg to ths b Oe ca the chage the lettes accodgl to evese the fal step of the algothm descbg f ad the othe steps ca be smlal evesed gog fom last to fst Note that the above becto peseves the umbe of blocs Below we povde a example whe 5 : f ( ) 3 4 5

11 AAM: Ite J Vol 6 Issue (Decembe ) 47 We ow fd a explct fomula fo the geeatg fucto fo the umbe of pattos P () I ode to acheve ths we wll cosde the followg thee tpes of geeatg fuctos: Fo all let F ( be the geeatg fucto fo the umbe of pattos P () such that ad We defe ( F Fo all let H ( be the geeatg fucto fo the umbe of pattos P () such that ad Fo all let G ( be the geeatg fucto fo the umbe of pattos P () such that ad fo all We defe the futhe geeatg fuctos F ( x F ( H ( x H ( ad G ( x G ( Ou goal wll be to fd F (x) whch s the geeatg fucto fo the sequece p () The ext thee lemmas povde elatos whch we wll eed betwee these geeatg fuctos Lemma 8 We have Poof: F( x F( x) H ( x Let a ( be the geeatg fucto fo the umbe of pattos P () such that whee s some wod B the deftos we have Hece F ( F( xa ( xx F ( xx( F( x) ) ( ) x x H ( x ( F( x) ) F ( x a ( x x a ( x a (

12 48 Touf Masou & Ma Shattuc F( x whch s equvalet to H ( F( x F( x) H ( x as equed ( F( x) ) Lemma 9 We have Poof: H ( x G( x ( H ( x) H ( x ) Let us wte a equato fo the geeatg fucto H ( Suppose s a membe of P () such that Cosde the followg two cases: () fo all 3 o () thee exsts at least oe dex > such that Cleal the fst case cotbutes G ( Fo the secod case we wte obsevg that sce avods thee exsts some such that ( ) Sce the membes of P () of the fom ( ) ae oe-to-oe coespodece wth the membes of P () of the fom we see that the secod case cotbutes x H ( Thus H ( G ( xh ( Multplg ths elato b ad summg ove all elds H ( x G( x x whch s equvalet to H ( H ( x G( x ( H ( x) H ( x )

13 AAM: Ite J Vol 6 Issue (Decembe ) 49 as equed Lemma We have Poof: x G( x ( ( x x x x ( x G( x) F( x) H ( x) ( ( x ( ( x ( ( x Let a ( (espectvel ( b ) be the geeatg fucto fo the umbe of pattos P () such that (espectvel ad fo all ) Fom the deftos we have ( ) ( ) ( ) ( ) ( ) G x x b x b x b x ( ) x xg ( x F( x) b ( ad fo all b ( ) ( x b ( ) ( b( ) ( b ( ) ( ) ( ( ) ( ) x xg ( xh ( x ( F( x) ) b ( Hece b summg ove all we obta x x x G ( x F( x) ( F( x) ) H( x) x x x x x x x x G ( G ( G ( 3 x x Multplg the above ecuece b ad summg ove all elds

14 4 Touf Masou & Ma Shattuc G ( ) Fx ( ) ( Fx ( ) ) ( ( x ( ( x 3 3 H( x) G( x G( x ( ( x 3 ( G( x) G ( )) Gx ( ) ( ( ( ( x whch leads to the equed esult Now we ae ead to fd a explct fomula fo the geeatg fucto F (x) fo the umbe of pattos P () Lemmas 8 9 ad gve se to a sstem of fuctoal equatos whch we wll solve usg the eel method (see Hou ad Masou ()) Lemma 8 mples x F( x) H ( x) x x (6) ad eplacg b x Lemma 9 gves H ( x) ( G x (7) x Replacg fst b x x 3x x( ad the b Lemma gves x ( Gx ( ) Fx ( ) Hx ( ) ( ( ) ( ( ) (8) ( x Gx F( x) H( x) G( x) x 3x 3x 3x 3x (9) Solvg the sstem of equatos (6)-(9) elds the followg esult Poposto The geeatg fucto fo p () s gve b F( x) x x 3x x 3x

15 AAM: Ite J Vol 6 Issue (Decembe ) 4 3 Cocluso I ths pape we have detfed sx subsets of the pattos of sze each avodg a classcal patte of legth fou ad aothe of legth fve ad each eumeated b the sequece a We have theeb obtaed ew classes of combatoal stuctues eumeated b a appaetl the fst such examples elated to set pattos Futhemoe umecal evdece shows that thee ae o othe membes of the (4 5) Wlf-equvalece class fo set pattos coespodg to the sequece a We have used both algebac ad combatoal methods to establsh ou esults I a couple of the seemgl moe dffcult cases we mae use of the eel method to solve the fuctoal equatos that ae satsfed b the elated geeatg fuctos It would be teestg to see f a of the othe sequeces the ''dscete cotut'' metoed the toducto eumeate estcted subsets of pattos whe > 3 (such as those avodg thee o moe pattes) Fall t seems that the techque of toducg a auxla paamete ad solvg the fuctoal equatos whch ase as a esult usg the eel method would have wde applcablt to othe questos of avodace ot ol fo set pattos but also fo othe fte dscete stuctues such as -aa wods o pemutatos REFERENCES Badee C Bousquet-Mélou M Dese A Flaolet P Gad D ad Gouou- Beauchamps D () Geeatg fuctos fo geeatg tees (Fomal Powe Sees ad Algebac Combatocs Baceloa 999) Dscete Math 46: Bacucc E Lugo AD Pegola E ad Pza R () Fom Motz to Catala pemutatos Dscete Math Got A (8) Avodace of pattos of a thee-elemet set Adv Appl Math Hou QH ad Masou T () Keel method ad sstems of fuctoal equatos wth seveal codtos J Comput Appl Math 35:5 5- Jelíe V ad Masou T (8) O patte-avodg pattos Electo J Comb 5 # R39 Klaza M (996) O abab -fee ad abba -fee set pattos Euopea J Comb Kuth DE (974) The At of Compute Pogammg Vol's ad 3 Addso-Wesle Readg Massachusetts Saga BE () Patte avodace set pattos As Comb Smo R ad Schmdt FW (985) Restcted pemutatos Euopea J Comb Sloae NJ The O-Le Ecclopeda of Itege Sequeces Stato D ad Whte D (986) Costuctve Combatocs Spge New Yo Wage C (996) Geealzed Stlg ad Lah umbes Dscete Math West J (995) Geeatg tees ad the Catala ad Schöde umbes Dscete Math