DOI 10.1007/s11083-006-9034-0
The Phagocyte Lattice of Dyck Words
J. L. Baril·J. M. Pallo
Received: 26 September 2005 / Accepted: 25 May 2006
© Springer Science + Business Media B.V. 2006
Abstract We introduce a new lattice structure on Dyck words. We exhibit efficient algorithms to compute meets and joins of Dyck words.
Key words lattice·Dyck words·noncrossing partitions
1. Introduction
A large number of various classes of combinatorial objects are enumerated by the well-known Catalan numbers. It is the case, among others, of ballot sequences, planar trees, Young tableaux, nonassociative products, stack sortable permutations, and so on. A list of over 60 types of such combinatorial classes of independent interest has been compiled by Stanley [26]. A certain number of explicit bijections between these Catalan classes can also be found in the literature.
From an order theoretic point of view, there are some partial ordering relations on Catalan sets which endow them with a lattice structure. Of much interest are the following lattices. First, the so-called Tamari lattice can be obtained equivalently in three different ways. The coverings correspond to elementary transformations in three Catalan classes, namely to reparenthesizations of letters products [6], to rotations on binary trees [17,24], and to diagonal flips in triangulations [10,24]. See references in [18]. Second, the lattice of noncrossing partitions is equipped with the refinement order [11]. Under this partial ordering≤, two partitionsπandπ0satisfy π≤π0 if every block of π is a subset of some block of π0. See [3,4, 20,22] and numerous references in the exhaustive survey [23]. More recently, a link between
J. L. Baril (
B
)·J. M. PalloLE2I, UMR 5158, Université de Bourgogne, B.P. 47870, F21078 Dijon-Cedex, France e-mail: [email protected]
J. M. Pallo
e-mail: [email protected]
Dyck paths, noncrossing partitions and321-avoiding permutations allows to define a new distributive lattice structure on these Catalan sets [1,5].
In this paper, we introduce a new lattice structure on Dyck words (also called well- formed parentheses strings). We define a ‘phagocyte’ mutation on Dyck words in the following way. When in a Dyck word a Dyck subwordwoccurs immediately at the left of a word of the form((· · ·(())· · ·)),wis inserted at the center of this word. Thus we obtain ((· · ·((w))· · ·)). Thereby, whenw((· · ·(())· · ·))occurs in a Dyck word, thenwis ‘phagocyted’ by((· · ·(())· · ·)), and so is changed to((· · ·((w))· · ·)).whas been ‘gobbled’ by a series of nested parentheses. This elementary mutation endows the Catalan set of Dyck words with a lattice structure. We show how to compute meets and joins of Dyck words efficiently. After Grätzer, which cites Tamari lattices
“to dispel the impression that it is always easy to prove that a poset is a lattice”
[9, p. 18], we should like to say that the currently studied lattice falls into the same category.
2. Dyck Words and Noncrossing Partitions
The setDof Dyck words over{(, )}is the language generated by the grammarS→ λ|(S)|SS, i.e. the set of legal strings of matched parentheses. Let us denoteDnthe set of Dyck words withnopen andnclose parentheses. The cardinality ofDnis then-th Catalan number:car d(Dn)=(2n)!/(n!(n+1)!). The open (resp. close) parentheses ofw∈Dnare numbered from1tonby traversingwfrom left to right.
A partition b1/b2/· · ·/bk of [1,n] = {1,2,· · ·,n} into blocks bi is called non- crossing if there do not exist four numbers p<q<r<s such that p,r∈bi and q,s∈bj withi 6= j. Thus146/23/5is a noncrossing partition of[1,6](ncp in short) while135/2/46is crossing. Let us denote NCnthe set of all ncp of [1,n]. We have car d(NCn)=car d(Dn). A ncp will be written by listing the elements in each block in increasing order, and the blocks in increasing order of their minima. There are various ways to represent a ncp. For example, in the linear representation,[1,n]
appears as usual on the real line and successive elements in the same block are joined by an arc in the first quadrant. In this paper, we will use what we call the canonical representation. Given a ncpπ∈NCn, let us define the relationRπon[1,n] byi Rπjiffi is the smallest element of the block ofπwhich contains j (i ≤ j). The canonical representation ofπis obtained by drawing the directed graph ofRπon the first quadrant. In these two representations, the noncrossing property of the partition corresponds to the fact that arcs do not intersect.
Sincecar d(Dn)=car d(NCn), let us exhibit an explicit bijection between Dnand NCn[7,8,22,23]. Given a ncpπ∈NCnand its canonical representation, for each visited elementi ∈ [1,n](forigoing from1ton), we writeropen parentheses (r≥0) if there arerarcs starting iniand next we write one close parenthesis since there is always exactly one arc ending ini. Figure1illustrates this bijection.
w =((()(()))(()())()) πw =149/23/57/6/8
cw =(1,2,2,1,5,6,5,8,1)
1 2 3 4 5 6 97 8 Figure 1 The canonical representation of a Dyck wordwofD9.
We denote byπw the ncp corresponding to the Dyck wordw. Let us remark that the numbernbl(πw)of blocks of πw is equal to the number of occurrences of ‘()’
inw. Givenw∈Dn, we define the vectorcwwherecw[i]is the smallest element of the block ofπwwhich containsi(1≤i≤n).
In the sequel, the strict (respectively, large) inclusion relation will be denoted by the symbol⊂(respectively,⊆).
3. The Phagocyte Transformation
DEFINITION 1.The phagocyte transformation−→on Dnis defined byw−→w0if we havew=u0v(n)nu00andw0=u0(nv)nu00, wherevis a non-empty Dyck word, n≥1 and u0,u00∈ {(, )}∗. Let−→∗ denote the reflexive transitive closure of−→.
LEMMA 1.For allw, w0∈Dn,we havew−→w0iffπwandπw0hold:
– one among the blocks ofπw0 is the union of two blocks b1 andb2 ofπw where min(b1) <min(b2)andb2= [min(b2),max(b2)]is an interval,
– every block ofπwdifferent from b1and b2is also a block ofπw0.
Proof.This result is a simple application of Definiton 1. Remark that in Figure2, the covering1/24/3→124/3is inNC4but is not inD4since24is not an interval.
()()()()
14/23
1/2/3/4
()(())() (()())()
(()()()) ()(()())
()()(()) (())()()
()((())) (()(()))
134/2 ((()()))
12/34
(())(()) ((())()) ((()))()
1234 (((())))
12/3/4 13/2/4 14/2/3 1/23/4 1/24/3 1/2/34
123/4 124/3 1/234
Figure 2 The phagocyte latticeD4.
PROPOSITION 1. (Dn, −→∗ ) is a poset with zero 0=()() . . . () and unit 1= ((. . . () . . .)), which is graded by the rank function r(w)=n−nbl(πw).
Proof. The integer valued function r defined by r(w)=n−nbl(πw) verifies
obviouslyr(w0)=r(w)+1ifw−→w0.
See the diagram ofD4in Figure2. In order to prove that the poset(Dn,−→∗ )is a lattice, we first give a Lemma on the structure of noncrossing partitions, and then we characterize the reflexive transitive closure of the phagocyte transformation.
LEMMA 2.Letπ∈NCn. If b is a block ofπwhich is not an interval, then there exists a block b0ofπverifying the following two conditions:
– b0⊂ [min(b),max(b)]
– b0is an interval possibly reduced to a singleton.
Proof.Given a blockbofπ, let us consider in[min(b),max(b)]the leftmost largest intervalI= [x,y]such thatI∩b= ∅. IfIis a block ofπ, the result holds. IfIis not a block ofπ, then let us consider the blockb1ofπsuch thatx=min(b1). Remark that b1exists by the noncrossing property. We haveb1⊂Iand we repeat this process with the new blockb1ofπ. So we construct a finite sequence of non-empty blocksbi(1≤ i≤r) such thatbi+1⊂ [min(bi),max(bi)]for eachi. The last blockbrof the sequence is an interval possibly reduced to a singleton. Moreoverbrverifies the condition:br⊂
[min(b),max(b)].
THEOREM 1.For allw, w0∈Dn, we havew−→∗ w0iff the following two conditions (C1)and(C2)hold:
– (C1)every block ofπwis a subset of some block ofπw0,
– (C2) for all blocks b1,b2∈πw, b0∈πw0 such that b1∪b2⊂b0 and min(b1) <
min(b2)then[min(b2),max(b2)] ⊂b0.
Remark 1.Condition (C1) is the well-known refinement order on partitions [11,23].
Condition (C2) means that any blockb0∈πw0 is made up of one blockbi∈πw and possibly disjoint intervals[k,l]for which one blockbj ∈πw (j>i) exists such that k=min(bj)andl=max(bj).
Proof.First let us suppose thatw−→∗ w0and let us apply the phagocyte transfor- mation on the Dyck subwordv((. . . () . . .))ofw∈Dnin order to obtain the Dyck wordw1. Leti≥0be the number of close parentheses of w before the first open parenthesis ofvand let j>i be the number of close parentheses ofw before the first open parenthesis of the subword((. . . () . . .))defined above. So there are inπw
a blockbcontainingi+1and a block[j+1,j+k]which is an interval withk≥1.
The result is a Dyck wordw1which contains the Dyck subword((. . . (v) . . .)). Thus
1 2
1
3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
i=1 V j=5
W=
W =
Figure 3 A phagocyte mutationw−→w1;πw=19/23/45/678andπw1 =19/23678/45.
πw1has a block which containsband[j+1,j+k]. See Figure3. So conditions(C1) and(C2)are satisfied forwandw1such thatw−→w1. By applying the phagocyte mutation several times, conditions(C1)and(C2)are also satisfied forwandw0such
thatw−→∗ w0.
Conversely, suppose that w and w0 verify the two conditions (C1) and (C2), and denote it by w≺w0. Let us consider w, w0∈Dn such that w6=w0 and w≺ w0. Consider alsoπw=b1/b2/ . . .andπw0 =b01/b02/ . . .withmin(bi) <min(bi+1)and min(b0i) <min(b0i+1) for all i ≥1. Let us denote by m the smallest integer such thatbm6=b0m. Since b1=b01, . . . ,bm−1=b0m−1, we havemin(bm)=min(b0m)=i0. By condition (C1), we have bm⊂bm0 since i0∈bm∩b0m. Now let j0 be the smallest integer such that j0∈/bm and j0∈b0m. Letbp(p≥m+1) be the block ofπw which contains j0. Since j0∈bp∩b0m, we havebp⊆bm0 and by the condition(C2)we have [j0=min(bp),max(bp)] ⊆b0m. Then there are two cases to consider:
– (a):bpis an interval.
Let denote j0=min(bp). Letvbe the largest Dyck subword ofw, the last close parenthesis of which has the number j0−1 and the first open parenthesis of which has the numbery≥i0. So there exists inw a Dyck subwordv(. . . () . . .) such that the first open (resp. last close) parenthesis of(. . . () . . .)has the number j0=min(bp) (resp. max(bp)). If we apply on w the phagocyte transformation v(. . . () . . .)−→(. . . (v) . . .), we obtain a Dyck wordw1. We havew1≺w0. Indeed πw and πw1 have the same blocks except one block which is the union of bm
andbp.
– (b):bpis not an interval, i.e.bp⊂
min(bp),max(bp)
. Let denote j0=min(bp).
According to Lemma 2, there exists a block bl (l≥p) of πw which is an intervalbl= [r,s]included in
min(bp),max(bp)
. Observe thatr=scan occur.
We choose the rightmost block bl. Let b be the block of πw which contains r−1 and x=mi n(b). Let v be the largest Dyck subword of w the last close parenthesis of which has the number r−1 and the first open parenthesis of which has the number y≥x. Then, we apply on w the phagocyte mutation v(. . . () . . .)−→(. . . (v) . . .)where the first open (resp. last close) parentheses of the nested parenthesis((. . . () . . .))has numberr (resp.s). We obtain the Dyck wordw1which verifiesw−→w1andw1≺w0as above.
By repeating this process, we find a finite sequence of Dyck wordswksuch that wk≺w0 for all k and w−→w1−→w2. . .−→wl=w0. By transitivity w−→∗ w0 follows.
Remark 2.This proof is constructive since we exhibit a minimal length path between wandw0ifw−→∗ w0.
THEOREM 2.For all n, the poset(Dn,−→∗ )is a lattice.
Proof. It suffices to show that any two elements ofDnhave a greatest lower bound.
The existence of least upper bounds then follows automatically since Dnis finite.
Givenw, w0∈Dnwithw6=w0, let us considerb∈Dnsuch that b−→∗ w,b−→∗ w0 andbmaximal, i.e. there does not existb0∈Dn(b06=b) such thatb−→∗ b0,b0−→∗ w, b0−→∗ w0. Assume that there existsa∈Dnmaximal such thata−→∗ w,a−→∗ w0and a6=b. We denote:
πa=A1/A2/ . . . , πb=B1/B2/ . . . , πw =W1/W2/ . . . , πw0=W01/W02/ . . . .
Letmbe the smallest integer such thatAm6= Bm. Obviously there existsxsuch that x∈Amandx∈/ Bmorx∈/Amandx∈Bm. Let us defineWk(respectively,W0l) as the block ofπw(respectively,πw0) which containsx. We have Am⊆Wkand Am⊆W0l. Moreover,Bm⊂WkandBm⊂W0lsinceAm∩Bm6= ∅andmin(Am)=min(Bm). By Remark 1 we have:
Wk= Ai0∪[
i∈I
[ri,~si]and Wk= Bi0∪[
j∈J
[uj,~vj]
wherei0≤m, and for each i∈I (resp. j∈ J)ri andsi (resp. uj and vj) are the minimum and the maximum of the same block ofπa (resp. πb). We have as well:
W0l=Ai00∪S
i0∈I0[r0i0,~s0i0]andW0l=Bi00∪S
j0∈J0[u0j0,~v0j0].
Sincex∈/ BmandAi0=Bi0for alli0<m,xbelongs to an interval[uj0, vj0],j0∈J, whereuj0 andvj0 are in a same block Bx ofπb(x∈Bx). Similarly xbelongs to an interval[u0j00, v0j00]with j00∈J0.
Sincebverifies the noncrossing property, we have necessarily
uj0, vj0
⊆
u0j00, v0j00
or u0j00, v0j00
⊆ uj0, vj0
Assume that the first case holds.
If[u0j00, v0j00] 6= [uj0, vj0]then[u0j00, v0j00] \ [uj0, vj0]has some elements below and above the interval[uj0, vj0]. Let us remark that the block containingu0j00 surrounds the blockBx.
In order to transformbintow, we must merge the blockBxwith the blockBi0. So the only possibility is either that the block which containsu0j00is inBi0or that we also must mergeBxwithBi0. This proves that the interval[u0j00, v0j00]is included intoWk.
Hence, each block included in[u0j00, v0j00]is also included intoWkandW0l, which is trivially verified if[u0j00, v0j00] = [uj0, vj0]. We can builtb0∈Dnsuch thatπb0andπb
have the same blocks except the block containingu0j00which is replaced by its union with each block ofπbincluded in[uj0, vj0]. Sob0∈Dnverifiesb6=b0andb−→∗ b0. By constructionb0−→∗ wandb0−→∗ w0, which contradicts the maximality ofb.
In the following propositions, we enumerate the join- and meet-irreducible ele- ments ofDn. Recall thatx∈Dnis a join (resp. meet)-irreducible element ifx=a∨b (resp.x=a∧b) impliesx=aorx=b. In other words, join (resp. meet)-irreducible elements are the elements that have an unique lower (resp. upper) cover. The proofs are easily obtained.
PROPOSITION 2.The number of join-irreducible elements in the phagocyte lattice Dnis n(n−1)/2.
Proof.It is enough to count all ncps consisting of one block with two elements and
all other blocks having one element.
PROPOSITION 3.The number of meet-irreducible elements in the phagocyte lattice Dnis2n−1−1.
Proof.Let denote X= {(p)p,p≥1}and Y= {(q,q≥1}. Then elements we are interested in are exactly alternative sequences of element inXandYfollowed by two elements inXand some), i.e. the sequences of the formx1y2x3. . .yk−2xk−1xk)rwhere xi∈X,yi∈Yandr≥0. Thus the number of these sequences equals to the number
of non-trivial compositions ofni.e.2n−1−1.
4. Computing Meets and Joins
Using Theorem 1, we exhibit algorithms for computing the meetw∧w0and the join w∨w0of two Dyck wordsw, w0∈Dn.
Meet algorithm:
fori :=1tondocw∧w0[i] :=0enddo fori :=1tondo
ifcw∧w0[i] =0then
ifcw[i]<iorcw0[i]<ithen j0:=i;
for j :=itondo
ifcw∧w0[j] =0andcw[j] =cw[i]andcw0[j] =cw0[i]then cw∧w0[j] := j0;
else j0:= j;endif endo
else
for j :=itondo
ifcw[j] =iandcw0[j] =ithen cw∧w0[j] :=i;endif enddo
endif endif enddo
Proof and analysis of the meet algorithm.Suppose the current elementcw∧w0[i]has been computed for alli <i0. For computingcw∧w0[i0]there are two cases to consider:
– (1):cw[i0] =i0andcw0[i0] =i0.
So there exist two blocks b∈πw andb0∈πw0 such thati0=min(b)=min(b0). We putcw∧w0[i0] =i0. Now forj ≥i0, letbj ∈πw(respectively,b0j∈πw0) be the block ofπw (respectively,πw0) which contains j. If we havei0∈bj andi0∈b0j then we put alsocw∧w0[j] =i0. We repeat this process for allj ≥i0.
– (2):cw[i0]<i0orcw0[i0]<i0.
Assume that cw[i0]<i0. Thus there exists a block b∈πw such that min(b)= cw[i0]<i0 andi0∈b. Sincecw∧w0[i0] =0is not determined by the algorithm,i0 does not belong to the blockb0ofπw∧w0which containscw[i0]<i0. To construct a path between w∧w0 and w, we must merge the block b00 of πw∧w0 which containsi0 with the block b0 sincei0 andcw[i0]are in the same blockb∈πw. Using Theorem 1, we obtain[min(b00),max(b00)] ⊆b. Sob00is the greatest interval b00=[i0,k]withk≥i0verifyingb00⊆bandb00 is also included in a block ofπw0. We repeat this process for all integers greater thani0.
Letσ be the noncrossing partition corresponding to the vectorcw∧w0obtained by the above algorithm. Thenσverifies the followings properties:
First by construction, each block ofσ is included into a block ofπw and a block ofπw0.
Second if there exist a blockbσ inσ and a blockbπinπw(respectively,πw0) such thatmin(bσ)=min(bπ), thenπw\bσis by construction an union of intervalsIverifying (i)I is also a block ofσ andmin(I)≥min(σ), (ii) Iis included into a block ofπw
(respectively,πw0). From Theorem 1, we obtainσ−→∗ πwandσ−→∗ πw0.
Moreover, we determine at each step of algorithm the greatest block verifying the two previous properties. This means thatσis the greatest noncrossing partition such thatσ −→∗ πw and σ−→∗ πw0. Indeed in the case (1), we consider the greatest set containingi0 which is included into a block ofπw andπw0 and in the case (2), we consider the greatest interval containingi0which is also included into a block ofπw
andπw0. So,σcorresponds to the noncrossing partition ofw∧w0.
The space complexity isO(n). The time complexity isO(n2)because computing nbl(πw)requiresO(n)time and the meet algorithm requires two nested loops.
We can easily define a recursive algorithm to compute the joinπtπ0of two ncpπ andπ0in the classical lattice of noncrossing partitions. So we can suppose thatcwtw0
has been computed for allw, w0∈Dn. Assume thatcwandcw0are determined. Letdw (respectively,dw0) be the vector defined bydw[i](respectively,dw0[i]) is the greatest element of the block ofπw(respectively,πw0) which containsi.
Join algorithm.
fori:=1tondo ifcw∨w0[i] =0then
for j :=itonif
ifcwtw0[j] =cwtw0[i]then cw∨w0[j] :=cwtw0[i];
ifcw[j] 6=cw[i]then
fork:= j+1todw[j]do cw∨w0[k] :=cwtw0[i];enddo endif
ifcw0[j] 6=cw0[i]then
fork:= j+1todw0[j]do cw∨w0[k] :=cwtw0[i];enddo endif
endif enddo endif enddo
Proof of the join algorithm.Suppose the current elementcw∨w0[i]has been computed for alli<i0. So we putcw∨w0[i0] =i0and for all j>i0such that j andi0are in the same block inπwtw0, we putcw∨w0[j] =cwtw0[i0]. We discuss the two following cases (1) and (2):
– (1): Ifcw[j] 6=cw[i0]:i0andjdo not belong to the same block inπw. To construct a path between w and w∨w0, we must include the interval
j+1,dw[j]
in the block ofπw∨w0 which containsi0. So ifk∈
j+1,dw[j]
, we putcw∨w0[k] = cwtw0[i0].
– (2): Ifcw0[j] 6=cw0[i0]. The proof is similar to the proof of (1).
Letσ(respectively,τ) be the noncrossing partition obtained bycw∨w0(respectively, cwtw0). By construction, every block ofσis included into a block ofτ, so every block ofπwandπw0is included into a block ofσ.
Moreover, if there exist a blockbτ inτ and a blockbπ inπw (respectively,πw0) such thatmin(bτ)=min(bπ)thenbτ\bπis by construction an union of blocksbinπw
(respectively,πw0) withmin(b)≥min(bτ). In order to obtainσfromτ, it is necessary to merge the blockbπ with the intervals[min(b),max(b)](by Theorem 1). So at the end of the algorithm,σverifies the conditions (C1) and (C2) of Theorem 1. Thus we obtainπw ∗
−→σandπw0 ∗
−→σ.
The minimality ofσ holds since at each step of this algorithm we run only the necessary operations. Thusσ =πw∨w0.
For example, ifπw=149/23/57/6/8andπw0=147/23/56/89, the algorithms give πw∧w0 =14/23/5/6/7/8/9andπw∨w0 =1456789/23.
5. Conclusion
In this paper, a new lattice structure has been defined on the Catalan sets of Dyck words via a natural transformation. The simple and natural definition of the phagocyte transformation is unfortunately at odds with the rather complex theorem which characterizes this transformation.
The greedy transformation v()→(v) defined in [7] is a particular case of the phagocyte transformation. However the poset obtained by the greedy transformation is not an effective lattice. Nevertheless, this poset is a graded lower semi-modular meet-semilattice. This property allows to compute the corresponding shortest path metric [12].
Some problems remain to be solved.
Is there an algorithm to compute the Möbius function of the phagocyte lattice of Dyck words as in [16]?
Computer experiments show that the number of coveringscov(n)ofDnis equal to 3(2n+2)!/((n+1)!(n+4)!)for small values ofn(n≤6). We obtain the first terms of the sequenceA003517=[1,6,27,110,· · · ]included in the Sloane Encyclopedia [25].
This sequence enumerates two combinatorial classes of objects: (a) The set P I Sn
of permutations on[n]with exactly one increasing subsequence of length three [13]
and (b) the set NI Pn of pairs of non-intersecting paths of lengthn and distance three [21]. In order to prove the equalitycov(n)=car d(P I Sn+1)=car d(NI Pn+1), can we exhibit two explicit bijections between the coverings of Dnand two above combinatorial classes? Is there a polynomial time algorithm to compute the minimal path length distance between Dyck words in the phagocyte lattice [12]? If so, a new shortest-path-type metric could be obtained, and could be added to the existing metrics on Catalan sets [2,7,14,15,17,19,24]. Let us recall that we still do not know if the rotation distance on binary trees can be computed in polynomial time.
Acknowledgments We would like to thank the anonymous referees and Jean-Paul Auffrand for their constructive remarks which have greatly improved this paper.
References
1. Barcucci, E., Bernini, A., Ferrari, L., Poneti, M.: A distributive lattice structure connecting Dyck paths, noncrossing partitions and312-avoiding permutations. Order (2005), available online
2. Bonnin, A., Pallo, J.M.: A shortest path metric on unlabeled binary trees. Pattern Recogn. Lett.
13, 411–415 (1992)
3. Edelman, P.H.: Chain enumeration and noncrossing partitions. Discrete Math. 31, 171–180 (1980)
4. Edelman, P.H., Simion, R.: Chains on the lattice of noncrossing partitions. Discrete Math.126, 107–119 (1994)
5. Ferrari, L., Pinzani, R.: Lattices of lattice paths. J. Stat. Plan. Inference135, 77–92 (2005) 6. Friedman, H., Tamari, D.: Problèmes d’associativité: une structure de treillis fini induite par une
loi demi-associative. J. Comb. Theory2, 215–242 (1967)
7. Germain, C., Pallo, J.M.: Two shortest path metrics on well-formed parentheses strings. Inf.
Process. Lett.60, 283–287 (1996)
8. Germain, C., Pallo, J.M.: The number of coverings in four Catalan lattices. Int. J. Comput. Math.
61, 19–28 (1996)
9. Grätzer, G.: General Lattice Theory. Birkäuser Verlag, Basel (1998)
10. Hanke, S., Ottmann, T., Schuierer, S.J.: The edge-flipping distance of triangulations. J. Univers.
Comput. Sci.2, 570–579 (1996)
11. Kreweras, G.J.: Sur les partitions noncroisées d’un cycle. Discrete Math.1, 333–350 (1972) 12. Monjardet, B.: Metrics on partially ordered sets – A survey. Discrete Math.35, 173–184 (1981) 13. Noonan, J.: The number of permutations containing exactly one increasing subsequence of length
three. Discrete Math.152, 307–313 (1996)
14. Pallo, J.M.: On the rotation distance in the lattice of binary trees. Inf. Process. Lett.25, 369–373 (1987)
15. Pallo, J.M.: A distance metric on binary trees using lattice-theoretic measures. Inf. Process. Lett.
34, 113–116 (1990)
16. Pallo, J.M.: An algorithm to compute the Möbius function of the rotation lattice of binary trees.
RAIRO Informatique Théorique et Applications27, 341–348 (1993)
17. Pallo, J.M.: An efficient upper bound of the rotation distance of binary trees. Inf. Process. Lett.
73, 87–92 (2000)
18. Pallo, J.M.: Generating binary trees by Glivenko classes on Tamari lattices. Inf. Process. Lett.85, 235–238 (2003)
19. Pallo, J.M.: Right-arm rotation distance between binary trees. Inf. Process. Lett.87, 173–177 (2003)
20. Reiner, V.: Noncrossing partitions for classical reflection groups. Discrete Math.177, 195–222 (1997)
21. Shapiro, L.W.: A Catalan triangle. Discrete Math.14, 83–90 (1976)
22. Simion, R., Ullman, D.: On the structure of lattice of noncrossing partitions. Discrete Math.98, 193–206 (1991)
23. Simion, R.: Noncrossing partitions. Discrete Math.217, 367–409 (2000)
24. Sleator, D.D., Tarjan, R.E., Thurston, W.P.: Rotation distance, triangulations and hyperbolic geometry. J. Am. Math. Soc.1, 647–681 (1988)
25. Sloane, N.J.A.: The On-Line Encyclopedia of Integer Sequences, published electronically at http://www.research.att.com/~njas/sequences/
26. Stanley, R.P.: Enumerative Combinatorics, vol. 2. Cambridge University Press, Cambridge (1999)
