Pattern statistics in faro words and permutations

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Discrete Mathematics

www.elsevier.com/locate/disc

Pattern statistics in faro words and permutations

Jean-Luc Baril

^a

^,∗ , Alexander Burstein

^b

, Sergey Kirgizov

^a

aLIB,UniversitédeBourgogneFranche-Comté,B.P.47870,21078DijonCedex,France bHowardUniversity,DepartmentofMathematics,Washington,DC20059,USA

a rt i c l e i nf o a b s t ra c t

Articlehistory:

Received13October2020

Receivedinrevisedform22April2021 Accepted3May2021

Availableonlinexxxx

Keywords:

Word Permutation Faroshuffle Patternstatistics Popularity Equidistribution

Westudythedistributionandthepopularityofsomepatternsink-aryfarowords,i.e.words over the alphabet {¹,2,. . . ,k} ^{obtained by} interlacing the lettersoftwo nondecreasing wordsoflengthsdifferingby atmostone.We presentabijection betweenthesewords and dispersed Dyck paths(i.e. Motzkinpaths with alllevel stepsonthe x-axis) witha givennumberofpeaks.Weshowhowthebijectionmapsstatisticsofconsecutivepatterns offarowordsintolinearcombinationsofotherpatternstatisticsonpaths.Then,wededuce enumerativeresultsbyprovidingmultivariategeneratingfunctionsforthedistributionand the popularityofpatterns oflengthatmostthree.Finally,weconsider someinteresting subclassesoffarowordsthatarepermutations,involutions,derangements,orsubexcedent words.

©^{2021ElsevierB.V.Allrightsreserved.}

1. Introductionandnotations

The faroshuffleis awell-known technique toshuffle a deckofcards.The deckissplitin twoatthe middle,andthe cardsfromthetwohalvesarecombinedbackbytakingalternativelythebottomsofstacks.Certainmathematicalquestions aboutthefaroshuffleare consideredforexampleintheworksofMorris [23],Diaconis,GrahamandKantor [15].Inspired bythesestudiesandasolidbodyofmoderncombinatorialliterature(seeforinstanceLothaire [21],Stanley [27],Bóna [13]

andKitaev [19] books)thatexploresenumerativeandbijectiveaspectsofpatternsinvariousdiscretestructures,thepresent paperconsidersanunexpectedlyoverlookedcombinatorialobjects,whichwecallfarowords.Theyarespecialkindofword shuffles,whichareimportantinseveralalgorithmicandcombinatorialsettings(seeforexampleBarneswork [7] andrefer- encestherein).Inthispaper,wepresentenumerativeresultsandshowhowfarowordsandpatternsthereinarerelatedto otherstructuressuchasDyckpaths,MotzkinpathsandDumontpermutations.

1.1. Farowordsandpermutations

Wedealwithk-arywordsu1u2

. . .

un overtheintegeralphabet

[

¹

,

k

] = {

¹

,

2

, . . . ,

k

}

^{endowedwiththeusualtotalorder.}

Ak-arywordiscallednondecreasingifui^ui+¹foralli

∈ [

¹

,

n

−

¹

]

^.

Definition1.1.Fortwo k-arywords u and v such that 0

|

^u

| − |

^v

|

^{1,the faroshuffle ofu and v isthe k-ary wordof} length

|

^u

| + |

^v

|

^obtainedbyinterlacingthelettersofuandv asfollows:u1v1u2v2u3v3

. . .

Ak-aryfarowordisafaroshuffle oftwonondecreasingk-arywords.

*

Correspondingauthor.

E-mailaddresses:[email protected](J.-L. Baril),[email protected](A. Burstein),[email protected](S. Kirgizov).

https://doi.org/10.1016/j.disc.2021.112464 0012-365X/©^{2021ElsevierB.V.Allrightsreserved.}

Let Sn,k be the set of k-ary faro words of length n. Its cardinality equals the product of two binomial coefficients _n/2+k−1

k−¹

_n/2+k−1

k−¹

,eachofthembeing,respectively,thenumberofm-multisetsof

[

¹

,

k

]

^form

=

ⁿ₂

^andm

=

ⁿ₂

^.For example,wehaveS4,2

= {

¹¹¹¹

,

1112

,

1121

,

1122

,

1212

,

1222

,

2121

,

2122

,

2222

}

^and

|S

4,2

| =

^9.

Definition1.2.Afaropermutationoflengthnisan n-aryfarowordoflengthn thatcontains everyletterin

[

¹

,

n

]

^exactly once.

LetPn bethesetoflengthnfaropermutations.Forinstance,wehaveP3

= {

¹²³

,

132

,

213

}

^.Sinceafaropermutationis entirelydeterminedbythechoiceofitsvaluesontheoddindices,thecardinalityofPn is _n

n/2

.

A k-ary word w

=

^w1w₂

. . .

w_n avoids a classical pattern (resp. consecutive pattern) p

=

^p1-p₂-

· · ·

^-pk (resp. p

=

p1p2

. . .

p_k)iftheredoesnotexistastrictlyincreasingsequenceofindicesi1i2

. . .

i_k(resp.withi_j+1

=

ⁱj

+

^{1 for1 jk}

−

¹⁾ suchthat wi1wi₂

. . .

wi_kisorder-isomorphictop(see[19] forinstance).Obviously,anyfarowordavoidstheclassicalpattern 3-2-1.LetA vn

( σ )

denotethesetofpermutationsavoidingaclassicalpattern

σ

,thenwehavePn

⊆

^{A v}n

(

3-2-1

)

forn^0, andPn

=

^{A v}n

(

3-2-1

)

forn^{3 since}

(

n

−

¹

)

n12

. . . (

n

−

²

) ∈

^{A v}n

(

3-2-1

)

isnotafaroword.Notethatafaropermutationcan containallclassicalpatternsoflength3 except3-2-1 (e.g.,31425).

Remark1.3.Ak-arywordw

=

^w1w2

. . .

wnisafarowordifandonlyifwi^wi+² foranyi

∈ [

¹

,

n

−

²

]

^{,whichmeansthat} faropermutationsarepreciselythoseavoidingthethreeconsecutivepatterns231,321 and312.

1.2. DyckanddispersedDyckpaths

Inordertostudythedistributionofpatternsinfarowords,wewillexhibitone-to-one correspondencesbetweenthese objectsandsome specificlatticepathsinthefirstquadrantoftheplane.Hence,weprovidebasicnecessarydefinitionson latticepaths.

Definition1.4.DispersedDyckpaths(see[17])arelattice paths starting at

(

0

,

0

)

,endingat

(

n

,

0

)

, consistingoflevelsteps F

= (

1

,

0

)

,upstepU

= (

1

,

1

)

anddownsteps D

= (

1

, −

¹

)

,andnevergoingbelowthex-axisandwherealllevelstepsare onthex-axis.

LetBn bethesetofdispersedDyckpathsoflengthn(or,equivalently,consistingofnsteps)andsetB

= ∪

n⁰Bn,where the empty path isdenoted by

. A Dyckpath of semilengthn^{0 is a dispersed Dyck path of length 2n with no level} steps. Let Dn be the setof Dyck paths of semilengthn and let D

=

n0Dn.Dispersed Dyck paths oflength n are in straightforwardbijectionwithprefixesofDyckpaths oflengthn,alsoknownasballotpaths [8,28].Indeed,wecanobtain a ballot path froma dispersed Dyck path by replacingall levelsteps with up steps.Dyck anddispersed Dyck paths are countedbytheCatalanandballot numbers,respectively(seeA000108andA001405intheOnlineEncyclopediaofInteger SequencesofN.J.A.Sloane [26],wherethegeneraltermsarec_n

=

_n+¹_12n

n

andb_n

=

_n

ⁿ/2

,respectively).

Apath P avoids apattern X ifandonlyif P doesnotcontain X asasequenceofconsecutivesteps (see forinstance [14,22]).Notethatother patterndefinitionsexistintheliteraturewherestepsarenotnecessarilyconsecutive [3].Wealso need some notationssimilar to Kleene starand plus symbolsof formal language theory. Fora nonempty pattern X, an occurrenceofthepattern X⁺ ina path P isamaximalsequence ofconsecutiverepetitionsof X,i.e.amaximal subword oftheform X^kfork^{1.ThepatternX∗ willbeeitheranemptypatternorapatternX+.Moregenerally,fortwopossibly} empty patternsY and Z suchthat Y doesnotendwithX and Z doesnot startwith X,thepatternY X⁺Z (resp.Y X^∗Z) corresponds to an occurrence obtained by concatenation of Y, X⁺ and Z (resp. Y, X^∗ and Z). For instance, the path F U DU D F F U D F containstwooccurrencesofthepatternF

(

U D

)

⁺F andthreeoccurrencesofF

(

U D

)

^∗F.

1.3. Statisticsonwordsandlatticepaths

Definition1.5.Astatisticsisaninteger-valuedfunctionfromasetA^{ofwordsorpaths.}

Toagivenpattern p,we associatethepatternstatisticp

:

A

→

N such thatp

(

a

)

isthe numberofoccurrences ofthe pattern p in theobject a

∈

A ^{(we usethe boldface todenote} statistics). Forexample,the statisticgiving thenumberof occurrencesoftheconsecutivepattern123 (resp. U DU D)inaword(resp.alatticepath)isdenotedby123(resp.UDUD).

Wedenoteby1ˆ (resp.2ˆ

,

n)ˆ theconstantstatisticreturningthevalue1 (resp.2,n).

Definition1.6.The popularity of apattern p inA ^{is the totalnumber of}occurrences of p overall objects of A^{, that is} p

(A) =

a∈Ap

(

a

)

(see[5,10,18,19]).

For instance, for a dispersed Dyck path P

=

F F U D F U U DU U U D D D D we have FF

(

P

) =

^{1, DDD}

(

P

) =

^{2, UD}

(

P

) =

^3, UUUU

(

P

) =

^{0 and}1ˆ

(

P

) =

^{1.Moreover,if}A

= {

^{U U D D}

,

U DU D

}

^{thenthepopularityofthepatternU Din}A^isUD

(

A

) =

^3.

LetT_AbethesetofallstatisticsdefinedonasetA.Foranypairofstatisticss

,

t

∈

^T_A^{,wedefine thestatistics}

+

^tby

(

s

+

^t

)(

a

) =

^s

(

a

) +

^t

(

a

)

foranya

∈

A^{,whichendowsT}_A^withaZ-modulestructure.LetB^beasetofcombinatorialobjects, andlet T_B bethecorresponding setofstatistics.We saythat twostatistics s

∈

^T_A ^andt

∈

^T_B ^havethesamedistribution, orareequidistributed,ifthereexistsabijection f

:

A

→

B^suchthats

(

a

) =

^t

(

f

(

a

))

foranya

∈

A^{.Inthiscase,withaslight} abuseofthenotationalreadyusedin[4],wewriteshortly f

(

s

) =

^tors

=

^{twhenever f istheidentity.Asabyproduct,for} anyconstantstatisticn,ˆ wehave f

(ˆ

n

) =

n.ˆ

1.4. Outlineofthepaper

Thepaperisorganizedasfollows.InSection2,wepresentaconstructivebijection f betweenthesetSn,kofk-aryfaro words of length n andthe setof dispersed Dyck paths oflength n

+

^2k

−

^{2 with k}

−

^{1 peaks. We show where pattern} statisticsaretransportedby f,whichprovidesamoresuitablegroundforstudyingthedistributionofconsecutivepatterns.

Thus,wederiveenumeratingresultsonthedistributionandpopularityofpatternsinSn,k bygivingmultivariategenerating functionswhere the coefficient ofxⁿy^kz^t is thenumber ofk-ary farowords oflength n having exactlyt occurrences of a givenpattern. In Section 3,we presenta similarstudy for faropermutations. Moreprecisely, we provide abijection g betweenPn andthesetofdispersedDyckpaths oflengthnandshowhow g actsonpatternstatisticsoflengthatmost three.Consequently,wededuceenumerativeresultsforthedistributionandthepopularityofthesepatternsinPn.Wealso presentabijectionbetweenPn andinvolutionsavoidingtheclassicalpattern3-2-1.Finally,inSection4,weprovethatthe set ofsubexcedent words inSn,n is relatedto ternarytrees andDumont permutationsofthe second kind [12] avoiding theclassicalpattern2-1-4-3,andweshowwhyfaroinvolutionsandfaroderangementsarerespectivelyenumeratedbythe FibonacciandCatalannumbers.

2. Patternsinfarowords

Inthissectionweconstructabijection f betweenthesetSn,kofk-aryfarowordsoflengthnandasubsetofdispersed Dyck paths,andshow how f transportspatternstatistics.Then, we deducegeneratingfunctionsforthedistribution and popularityofsomepatterns.

Apair ina faroword w isan occurrencew_iw_i+1 withw_i

>

w_i+1.Remark 1.3impliesthat alettercannot be partof two pairssincea farowordavoidsthe consecutivepattern321.Asingletonin w isaletter w_i notinanypairof w.Any farowordcan be uniquely decomposedasasequence ofpairs andsingletons,whichare calledblocksof farowords.For instance,theblockdecompositionof111212131333 is1³

(

21

)

²

(

31

)

3³.

LetLkbethesetofallpossibleblocksofadecompositionofak-aryfaroword,thatis Lk

= {

¹

,

2

, . . . ,

k

} ∪ {

^ji

:

¹

ⁱ

<

j

^k

} .

Definition2.1.Wedefineanorderrelation

^onLk asfollows:forg

,

h

,

i

,

j

∈ {

¹

,

2

, . . . ,

k

}

^,

⎧ ⎪

⎪ ⎨

⎪ ⎪

⎩

i

^j

,

ifi

^j

,

i

^jh

,

ifi

^h

<

j

,

ig

j

,

ifg

<

i

j

,

ig

^jh

,

ifg

<

i

^jandg

^h

<

j

.

Remark2.2.Theorderrelation

canbedefinedlesstechnicallyasfollows:forp

,

q

∈

Lk, p

^q

⇐⇒

^pqis a faro word different from a pair.

ThisorderrelationendowsthesetLkwithaposetstructure,whichwecallfaroposet.SeeFig.2.1foranillustrationof theHassediagramof

(L

k

, )

^.

Amultichaininaposetisachain,i.e.atotallyorderedsubset,withrepetitionsallowed.Duetothesimplestructureofthe faroposet,weeasilydeducethefollowingremarks.

Remark2.3.Thereisaone-to-onecorrespondencebetweenk-aryfarowordsandthemultichainsofLk.Indeed,Remark2.2 implies that the block decomposition of a k-ary faro word w into pairs and singletons w

=

^b1b2

. . .

b unambiguously corresponds to the multichain b₁

^b2

· · ·

^b in Lk, and vice versa. For instance, the faro word 11313232343

=

11

(

31

)(

32

)(

32

)

3

(

43

)

correspondstothemultichain1

¹

³¹

³²

³²

³

^{43 (seeFig.2.1).}

Remark2.4.Ifa k-aryfaroword w containsa singletonxinits decompositionintoblocks,thenitsatisfiesthefollowing property:thesetofpairsoftheformab,b

<

a^{x,equalsthesetofpairsoftheformcd,dx}

−

^1.

1 21 31 41 . . . _k1

2 32 42 . . . _k2

3 43 . . . _k3

. .. . . . ...

k−1 k(k−1) k

Fig. 2.1.Thefaroposet(Lk,).Redblocksrepresentthemultichainassociatedtothek-aryfaroword11313232343=¹²(31)(32)²3(43).(Forinterpretation ofthecolorsinthefigure(s),thereaderisreferredtothewebversionofthisarticle.)

2.1. AbijectiontothesetofdispersedDyckpaths

AsmentionedbyE.Deutschin[26] (seesequenceA124428),thenumberofdispersedpathsoflengthnwithkpeaks(a peakisanoccurrenceofthepatternU D)isgivenby

|

Bn,k

| =

n

2

k

n 2

k

.

Thus,wepresentabijection f fromthesetSn,kofk-aryfarowordsoflengthntothesetB_{n+2(k−1),k−1} ofdispersedDyck pathsoflengthn

+

²

(

k

−

¹

)

withexactlyk

−

^{1 peaks.Foragivenw}

∈

Sn,k,weset

f

(

w

) =

F^T0U^T1D^T2F^T3

. . .

F^T3(k−2)U^T3(k−2)+1D^T3(k−2)+2F^T3(k−1)

,

whereTiisdefinedfor0ⁱ³

(

k

−

¹

)

asfollows:

– ifi

=

³

(

x

−

¹

)

then T_iisthenumberofoccurrencesofthesingletonxinw;

– ifi

=

³

(

x

−

¹

) −

^{1 thenT}i isoneplusthenumberofpairsxy, y

<

x,inw;

– ifi

=

³

(

x

−

¹

) +

^{1 thenT}i isoneplusthenumberofpairs yx,y

>

x,inw.

It is worth noting that the image ofa faro word w

∈

Sn,k dependson the arityk that we consider. Indeed,the im- age of the empty word

is U D when k

=

^{2, while f}

( ) =

^{U DU D for k}

=

^{3. We refer to Fig. 2.2 for one detailed} example of this bijection, while Fig. 2.3 provides more additional examples. For instance, the images by f of the 5- arywords

,

12345

,

3141

,

111111212222 are,respectively,U DU DU DU D, F U D F U D F U D F U D F,U U U DU D DU D DU D and F F F F F F U U D D F F F F U DU DU D.

Remark2.5.Clearly, thevalues Ti,0ⁱ³

(

k

−

¹

)

,can be obtainedfrom w by reading itfromleft to rightandby de- terminingifthecurrententryxbelongstoeitherapairxy or yx,orasingletonx.Moreover,valuesofT atindicesi

=

⁰ mod 3 correspondtothelengthsofmaximalrunsofconsecutivelevelsteps,andvaluesatindicesi

=

1 mod 3 (resp.i

=

² mod 3) correspondtothe lengthsofmaximal runsofconsecutiveup(resp. down)steps, whichmeansthat thesequence T

=

^T0T₁

. . .

T_3(k−1)isarun-length-like encodingofthepath f

(

w

)

.Thus, f

(

w

)

canbe constructedfromw usingalinear timealgorithm.

Lemma2.6.Thepathf

(

w

)

isnecessarilyadispersedDyckpathoflengthn

+

²

(

k

−

¹

)

withexactlyk

−

^1peaks.

Proof. Since for any i

=

^{0 mod 3, 1i3}

(

k

−

¹

) −

^{1 wehave T}i^{1, the path w contains exactly k}

−

^{1 peaks U D.}

Interpreting Remark2.4onthe path f

(

w

)

,thenumberofup steps beforeagivenlevelstep equalsthenumberofdown steps before the samelevel step, whichimpliesthat any levelstep belongs tothe x-axis. Letdx

=

x+²

i=²T3(i−¹)−¹ (resp.

u_x

=

_x+1

i=¹T_3(x−1)+1) be thetotal numberofdown steps (resp.up steps) inthe first x

+

^{1 maximal runsofdown steps} (resp.up steps).Due tothe definitionof f,dx equals thenumberofpairs i j, 1^j

<

i^x

+

^{2,in w,and u}x equals the

Fig. 2.2.The image byf of the 5-ary faro wordw=11313232343 isf(w)=F F U U DU U U D D D D F U U D DU D.

numberof pairsi j, 1 ^jx

+

^{1, ij}

+

^{1,which impliesthatd}x^ux. Alsobydefinition,the totalnumberofup steps (resp.downsteps)in f

(

w

)

equalsthetotalnumberofpairsinw,whichcompletestheproof.

Theorem2.7.Themap f isabijectionfromSn,ktothesetBn+2(k−1),k−1ofdispersedDyckpathsoflengthn

+

²

(

k

−

¹

)

withexactly k

−

^1peaks.

Proof. Letus prove that if w and w are two distinct k-ary faro words then we have f

(

w

) =

^f

(

w

)

. Let i^{1 be the} smallestpositiveintegersuch that wi

=

^w_i^{.Without lossof}generality, weassume wi

<

w_i.Letusconsiderthepositions ofwiandw_i intheblockdecompositionof w.

Ifw_iandw_iarebothinthepairs w_iw_i+1andw_iw_i+1,thenRemark2.3impliesthatapairw_ix,w_i

>

x,cannotappear totherightof w_i inw,whichimpliesthatT3(wi−¹)−¹

=

^T_{3(wi−1)−1}^{,andthus f}

(

w

) =

^f

(

w

)

.

Thereremainthefollowingcases:

(i) w_iorw_iisasingletoninw,

(ii) wiandw_iarebothinthepairs wi−¹wi andw_i−1w_i

=

^wi−¹w_i, (iii) w_ibelongstothepair w_i−1w_iandw_ibelongstothepair w_iw_i+1, (iv) wiandw_iarebothinthepairs wiwi+¹ andw_i−1w_i.

Thefactthatafarowordavoids231 incase(i)andRemark2.3forcases(ii),(iii)(iv),implythat w_i cannotappeartothe rightof w_i in w.Then the numberof wi in w, i.e.T3(w_i−¹)

+

^T3(w_i−¹)+¹

+

^T3(w_i−¹)−¹,is differentfromthe numberof wi inw,whichis T_3(w

i−¹)

+

T_3(w

i−¹)+¹

+

T_3(w

i−¹)−¹.Therefore,thereis

δ ∈ {−

1

,

0

,

1

}

suchthat T3(wi−¹)+δ

=

T_3(w

i−¹)+δ, whichimpliesthat f

(

w

) =

^f

(

w

)

.

Thus, f isan injectivemap,andusingacardinalityargument(seeA124428 in[26]),we concludethat f isabijection fromSn,ktoBn+²(k−¹),k−¹.

Althoughitisnotusedinthepaper,we couldprovethat fromagivendispersedDyckpath P

∈

Bn+²(k−¹),k−¹, f⁻¹

(

P

)

canbeobtainedafterapplyingthefollowingprocedure.WerefertoFig.2.3forseveralexamples.

Wesets

=

^{1 astheinitialvalue.Wemarkall D-stepsprecededbyanU-stepandalltheotherD-stepsareleftunmarked.}

ReadingthestepsofP fromlefttoright:

– Ifa D-stepisencountered,thenskipit.

– Ifan F-stepisencountered,thenwritethesingletons.Ifthenextstepisnotan F-step,thenupdates

=

^s

+

^1.

– Ifan U-stepisencounteredintheithrunofU-steps,thenwedistinguishtwocases:

(i) thenextstepis D;thenweskipthisU D-patternbycontinuingfromthestepafter D,ifitexists.

(ii) thenextstepisU;thenwewritethepair ji,where j istheleastintegersuchthatthe

(

j

−

¹

)

-thrun ofD-steps hasatleastoneunmarked D-step.Markthefirstunmarked D-stepfromthe

(

j

−

¹

)

-thrunof D-steps.

2.2. Distributionandpopularityofpatterns

Inthispart,wefirst showhowthe bijection f transportspatternstatistics onSn,k intothecontextofdispersed Dyck paths.After,wededucemultivariategeneratingfunctionsforthedistributionandthepopularityofpatternsoflengthtwo byexploitingtheclassicrecursivedecompositionofdispersedDyckpaths.

Fig. 2.3.Images of several 5-ary words under bijection f.

Theorem2.8.Forn^{0,thebijection f from}Sn,ktoBn+2(k−1),k−1mapsstatisticsassociatedtopatternsoflength2asfollows:

f

(

11

) =

^FF

,

f

(

21

) =

^UU

=

^DD

,

f

(

12

) =

^DD

(

UD

)

^∗UU

+

^DD

(

UD

)

^∗D

+

^DD

(

UD

)

^∗F

+

^F

(

UD

)

⁺F

+

^F

(

UD

)

^∗UU

=

n

ˆ −

1

ˆ −

^UU

−

^FF

.

Proof. ByRemark2.3,anyoccurrenceofthepattern11 inafarowordwisformedbytwoconsecutivesingletonsxx.From thedefinitionofthebijection f,itfollowsthatthenumberofoccurrencesof11 inw equalsthenumberofoccurrencesof F F in f

(

w

)

,thatis f

(

11

) =

^FF.

Anoccurrenceofthepattern21 inwisnecessarilyapairinthedecompositionofw.Sincethelengthofamaximalrun ofconsecutiveupstepsisequaltooneplusthenumberofpairsyxinwforagivenx

∈ [

¹

,

n

]

^,thenumberofoccurrencesof 21 inwequalsthenumberofoccurrencesofU U in f

(

w

)

.Ontheotherhand,anynonemptydispersedDyckpathP isofthe formeitherP

=

^{F RorP}

=

^{U Q D RwhereQ isaDyckpathandRadispersedDyckpath.Reasoningbyinduction,weobtain} thatthenumberofoccurrencesof D D equalsthoseofU U inanydispersedDyckpath,whichimplies f

(

21

) =

^UU

=

^DD.

Now,letusprovetheequation f

(

12

) =

^DD(UD)^∗UU

+

^DD(UD)^∗D

+

^DD(UD)^∗F

+

^F(UD)⁺F

+

^F(UD)^∗UU.Anoccurrencexy ofthepattern12 occursinwasasubblockofoneofthefollowing:

(i) twodistinctconsecutivepairs

(

ax

)(

yb

)

, (ii) twoequalconsecutivepairs

(

yx

)(

yx

)

, (iii) apairfollowedbyasingleton

(

ax

)(

y

)

, (iv) asingletonfollowedbyapair

(

x

)(

ya

)

,

(v) twodistinctsingletons

(

x

)(

y

)

. Forthecase(i),wedistinguishthreesubcases.

Subcase1. Theoccurrencexy appearsina factoroftheform

(

ax

)(

yb

)

withb^{a.Thisimpliesthatneithera singleton} s

∈ [

^a

,

b

]

^{nora pairpq with p}

∈ (

a

,

b

]

^orq

∈ [

^a

,

b

)

canappearin w.Therefore,T_3(s−1)

=

^{0 fors}

∈ [

^a

,

b

]

^,T3(p−1)−1

=

^{1 for} p

∈ (

a

,

b

]

^andT3(q−¹)+¹

=

^{1 foranyq}

∈ [

^a

,

b

)

.Thus,betweentherunofD-stepsassociatedtoT3(a−¹)−^{12 andtherunof} U-stepsassociatedtoT3(b−¹)+^{12,therearenolevelsteps,andtherunsofD-stepsandU-stepsareoflengthone,which} createsm

=

^b

−

^{a0 peaksU D.Hence,theoccurrencexyisassociatedtoanoccurrenceofthepatternD D}

(

U D

)

^∗U U.

Subcase2.The occurrencexy appears inafactorofthe form

(

ax

)(

yb

)

withb

<

a anda

<

y.Thisimpliesthatneither a singleton x

∈ [

^a

,

y

)

nor a pair pq with p

∈ (

a

,

y

)

or q

∈ [

^a

,

y

)

can appear in the word w. Therefore, T3(x−¹)

=

^{0 for}

x

∈ [

^a

,

y

[

^{, T}3(p−¹)−¹

=

^{1 for p}

∈ (

a

,

y

)

andT3(q−¹)+¹

=

^{1 foranyq}

∈ [

^a

,

y

)

.Thus,betweentherunof D-steps associatedto T_3(a−1)−1^{2 andthe run of D-stepsassociated to T}3(y−1)+1^{2,there are nolevel steps, andtheruns of D-steps and} U-stepsareoflengthone,whichcreatesm

=

^y

−

^a

>

0 peaksU D.Hence,theoccurrencexyisassociatedtoanoccurrence ofthepatternD D

(

U D

)

⁺D.

Subcase3.Theoccurrencexyappearsinafactoroftheform

(

ax

)(

yb

)

withb

<

aanda^{y.Bydefinitionofafaroword,} we necessarilyhavea^{y.Thus, wededucea}

=

^{y.So,we haveT}3(a−1)−1^{3,whichcountsall} consecutivepairsaz

,

a

>

z in w.DuetoRemark2.3,allthesepairsappearconsecutivelyin w.Thus,thenumberofoccurrencesoftheform

(

ax

)(

ab

)

, forx

,

b suchthatx^b

<

a isequaltothenumberofD D D

=

^{D D}

(

U D

)

⁰D patternsinthe

(

a

−

¹

)

-thrunofD-stepsinthe corresponding dispersed Dyckpath.Combining tothe subcase2,the occurrencexy is associatedtoan occurrenceof the patternD D

(

U D

)

^∗D.

Inthecase(ii),wehaveafactoroftheform

(

ax

)(

yb

)

witha

=

^{y andx}

=

^{b andtheargumentfromSubcase3ofcase(i)} applies.Fortheremainingcases,(iii)through(v), theoccurrencexy ofthepattern12 iseithercreatedby apairfollowed byasingleton

(

ax

)(

y

)

,orbyasingletonfollowedbyapair

(

x

)(

ya

)

,orbytwodifferentsingletons

(

x

)(

y

)

.Argumentssimilar totheonesgivenabove,allowustoprovethatanoccurrencexyinw correspondstoanoccurrenceof:

– D D

(

U D

)

^y−aF forthecase

(

ax

)(

y

)

, – F

(

U D

)

^a−xU U forthecase

(

x

)(

ya

)

,and – F

(

U D

)

^y−xF forthecase

(

x

)(

y

)

.

Finally,inanyn-length wordwe haven

−

1 occurrencesof2-lengthpatterns,thusnˆ

−

1ˆ

=

¹¹

+

²¹

+

^{12.Applying the} bijection f tobothpartsoftheequation,weobtain f

(

12

) =

ⁿˆ

−

1ˆ

−

^f

(

11

) −

^f

(

21

) =

ⁿˆ

−

1ˆ

−

^UU

−

^FF.

Theorem2.9.Forp

∈ {

¹¹

,

12

,

21

}

^{,thetrivariategeneratingfunctionsF}p

(

x

,

y

,

z

)

wherethecoefficientatxⁿy^kz^tisthenumberof k-aryfarowordsoflengthn containingexactlyt occurrencesofthepatternp are:

F₁₁

(

x

,

y

,

z

) =

^2y

(

xz

−

^x

−

¹

)

−

^xyz

+

^xy

+

^x3z

−

^x3

+

^y

−

^x2

+

^xz

+

^x

−

¹

+ (

xz

−

^x

−

¹

)

A₁

,

F21

(

x

,

y

,

z

) =

^2y

−

^y

+

^x2z

−

^2x

+

¹

+

^A2

,

F₁₂

(

x

,

y

,

z

) =

^y

x³z²

−

^x3z

+

^x2z

+

^xyz

−

^xy

−

^3xz

+

^x

+

^y

−

¹

+ (

xz

−

^x

+

¹

)

A2

x³z²

−

^x3z

+

^x2z

−

^xyz

+

^xy

−

^xz

−

^x

−

^y

+

¹

+ (

xz

−

^x

+

¹

)

A₂

(−

¹

+

^y

)

z

+

^y 1

−

^y

,

whereA₁

=

x⁴

−

^2x2y

−

^2x2

+

^y2

−

^2y

+

^1andA2

=

x⁴z²

−

^2x2yz

−

^2x2z

+

^y2

−

^2y

+

^1.

Proof. Wehave f

(S

n,k

) =

Bn+²(k−¹),k−¹.Thus,foranypattern p, thetrivariate generatingfunction F_p

(

x

,

y

,

z

)

isgivenby y

·

^Bp

(

x

,

_x^y₂

,

z

)

where Bp

(

x

,

y

,

z

)

isthetrivariate generatingfunctionwhosecoefficientatxⁿy^kz^t isequaltothenumberof dispersedDyckpaths P

∈

Bn,ksuchthatq

(

P

) =

^t,whereq

=

^f

(

p

)

.

Forp

=

^{21,Theorem2.8has f}

(

21

) =

^{UU.Therefore,wedecomposetheset}D^{ofDyckpathsasfollows:}

D

=

^{U D}D

^U

(

D

\ )

DD

.

WealsodecomposethesetB^{ofdispersedpathsasfollows:}

B

=

FB

^{U D}B

^U

(

D

\ )

DB

.

IfD

(

x

,

y

,

z

)

isthegeneratingfunctionwherexⁿy^kz^t isthenumberofDyckpathsoflengthnwithkpeaksandt occurrences ofU U,thentheabove algebraicequationyields D

(

x

,

y

,

z

) =

¹

+

^{x2y D}

(

x

,

y

,

z

) +

^x2z

(

D

(

x

,

y

,

z

) −

¹

)

D

(

x

,

y

,

z

)

.If B21

(

x

,

y

,

z

)

isthegeneratingfunctionwhosecoefficientatxⁿy^kz^t isthenumberofdispersedDyckpathsoflengthnwithk peaksand t occurrencesofU U,thentheabovedecompositionofB ^{yieldsthefunctionalequation}

B21

(

x

,

y

,

z

) =

1

+

xB21

(

x

,

y

,

z

) +

x²y B21

(

x

,

y

,

z

) +

x²z

(

D

(

x

,

y

,

z

) −

1

)

B21

(

x

,

y

,

z

),

which,inturn,yieldsthedesiredresult.

Forp

=

^{11,Theorem2.8has f}

(

11

) =

^{FF.Therefore,wedecomposetheset}DofDyckpathsasfollows:

D

=

U DD

^U

(

D

\ )

DD

.

WealsodecomposethesetB^{ofdispersedDyckpathsasfollows:}

B

=

F

F^{U D}B

F^U

(

D

\ )

DB

^{U D}B

^U

(

D

\ )

DB

,