A direct encoding of Stoimenow's matchings as ascent sequences
AA DIRECT ENCODING OF STOIMENOWS MATCHINGS ASASCENT SEQUENCES
ANDERS CLAESSON, MARK DUKES, AND SERGEY KITAEV
Abstract.
In connection with Vassiliev’s knot invariants, Stoimenow (1998)introduced certain matchings, also called regular linearized chord diagrams.Bousquet-M´elou et al. (2008) gave a bijection from those matchings to un-labeled ( + )-free posets; they also showed how to encode the posets asso called ascent sequences. In this paper we present a direct encoding ofStoimenow’s matchings as ascent sequences. In doing so we give the rules forrecursively constructing and deconstructing such matchings. Introduction
To give upper bounds on the dimension of the space of Vassiliev’s knot invariantsof a given degree, Stoimenow [2] introduced what he calls regular linearized chorddiagrams. We call them
Stoimenow matchings . As an example, these are the 5Stoimenow matchings on the set { , , , , , } : In general, a matching of the integers { , , . . . , n } is a partition of that set intoblocks of size 2, often called arcs . We say that a matching is Stoimenow if thereare no occurrences of Type 1 or Type 2 arcs: π i Type 1: i +1 i +1 i +1 i +1 i π π i i Type 2: π In this paper we present a bijection between Stoimenow matchings on { , , . . . , n } and a collection of sequences of non-negative integers that we call ascent sequences .Given a sequence of integers x = ( x , . . . , x n ), we say that the sequence x has an ascent at position i if x i < x i +1 . The number of ascents of x is denoted by asc( x ).Let A n be the collection of ascent sequences of length n : A n = (cid:8) ( x , . . . , x n ) : x = 0 and 0 ≤ x i ≤ x , . . . , x i − ) for 1 < i ≤ n (cid:9) . These sequences were introduced in a recent paper by Bousquet-M´elou et al. [1].For example, A = { (0 , , , (0 , , , (0 , , , (0 , , , (0 , , } . Key words and phrases.
Combinatorial problem, encoding, matching, ascent sequences.The authors were supported by grant no. 090038011 from the Icelandic Research Fund. a r X i v : . [ m a t h . C O ] O c t A. CLAESSON, M. DUKES, AND S. KITAEV
Bousquet-M´elou et al. gave bijections between four classes of combinatorial objects,thus proving that they are equinumerous: Stoimenow matchings; unlabeled ( + )-free posets; permutations avoiding a specific pattern; and ascent sequences. Thefollowing diagram, in which solid arrows represents bijections given by Bousquet-M´elou et al., sums up the situation.Stoimenow matchingsunlabeled ( + )-free posets ascent sequences-avoiding permutationsΩ Ψ ΛΨ (cid:48) In particular, Ψ ◦ Ω is a bijection between Stoimenow matchings and ascent se-quences. The dashed arrow is the contribution of this paper. That is, we give adirect description of Ψ (cid:48) = Ψ ◦ Ω. Ascent sequences have an obvious recursive struc-ture. We unearth the corresponding recursive structure of Stoimenow matchings.It amounts to two functions, ϕ (cid:48) and ψ (cid:48) , which act on matchings in an identicalmanner to the functions ϕ and ψ of [1, §
3] acting on posets.2.
Stoimenow matchings and edge removal
Let I n be the collection of Stoimenow matchings with n arcs. Let S m be thecollection of all permutations of the set { , . . . , m } . Any Stoimenow matchings maybe written uniquely as a fixed point free involution π ∈ S n so that the numberpaired with i is π i . We shall abuse notation ever so slightly by considering π to bedually a matching in I n and an involution in S n .Given π ∈ I n let arcs ( π ) be the collection of all n arcs [ i, π i ] of π . Let us introducethe following labelling scheme label : arcs ( π ) → N of the arcs; for every arc in π ,call the left endpoint the opener and the right endpoint the closer .Label an arc with the number of runs of closersthat precede it. For example, consider the match-ing { [1 , , [2 , , [5 , } , or equivalently the involution π = 341265. The labels of the arcs are shown in thediagram to the right. To every Stoimenow matching we shall single out two (very important) arcs. Given π ∈ I n call maxarc ( π ) = [ π n , n ] the maximal arc of π and call the arc redarc ( π ) =[ π π n , π n ] the reduction arc of π .To every Stoimenow matching we shall associate two statistics: M ( π ) = label ( maxarc ( π )) and m ( π ) = label ( redarc ( π )) . For the matching { [1 , , [2 , , [5 , } we have redarc ( π ) = [5 ,
6] = maxarc ( π ) so that M (341265) = 1 and m (341265) = 1. In the diagrams that follow, vertices that areopeners are marked with a • and closers are marked with a (cid:3) . Example 1. (i) Consider π = 3 4 1 2 7 9 5 10 6 8 ∈ I .We have redarc ( π ) = [6 , maxarc ( π ) = [8 , m ( π ) = 1 and M ( π ) = 2. DIRECT ENCODING OF STOIMENOWS MATCHINGS AS ASCENT SEQUENCES 3 {{ (a) M z u vx * yi i M* x y z u vA A { {{{ {{ Y ZX j j j Y j Z j (b) X j Figure 1.
The removal rule
Rem3 .(ii) Consider π = 4 5 7 1 2 8 3 6 10 9 ∈ I .The labels of the arcs are shown in thediagram. We have redarc ( π ) = [9 ,
10] = maxarc ( π ) and so M ( π ) = 2 = m ( π ) =2. We begin with the removal operations for Stoimenow matchings. Let π ∈ I n where n ≥ i = m ( π ) be the label of the reduction arc redarc ( π ). In what followswe will remove the reduction arc in a very careful way so that we obtain σ ∈ I n − .Let L i ( π ) = { x ∈ arcs ( π ) : label ( x ) = i } be the set of arcs that have label i .( Rem1 ) If | L i ( π ) | > redarc ( π ).( Rem2 ) If | L i ( π ) | = 1 and i = M ( π ), then maxarc ( π ) = redarc ( π ) = [2 n − , n ] andwe remove this arc from π .( Rem3 ) If | L i ( π ) | = 1 and i < M ( π ) then do as follows (these steps are illustratedin Figure 1;(a) Let A be the collection of all closers between x and the next openerto its right. Move all points in A to between z and u while respectingtheir order relative to one-another.(b) For all j with 0 ≤ j < i , partition the collection of openers withlabel j into three segments X j , Y j and Z j where Y j is the collection ofopeners that have closers in A . Swap each of the sets Y j and Z j whilepreserving their respective internal order.(c) Remove the reduction arc. Example 2.
Three examples corresponding to the above removal operations.(i) In Example 1(i) we had i = m ( π ) = 1, M ( π ) = 2 and | L ( π ) | = 2 > Rem1 ) applies and we have σ : A. CLAESSON, M. DUKES, AND S. KITAEV
AAX Y Z
X Y Z
X Y Z * A u vu ** X X XY Y YZ Z Z u vu v
Figure 2.
Illustration of the 3 steps for
Rem3 .(ii) In Example 1(ii) we had i = m ( π ) = 2 = M ( π ) and | L ( π ) | = 1. Thus rule( Rem2 ) applies and we have σ :(iii) See Figure 2. Example 3.
See Figure 3 for an example of transforming a Stoimenow matchinginto an ascent sequence.We will now prove that the three types of removal operation give some σ ∈ I n − .If m ( π ) = i and the removal operation, when applied to π gives σ , then define ψ (cid:48) ( π ) = ( σ, i ). DIRECT ENCODING OF STOIMENOWS MATCHINGS AS ASCENT SEQUENCES 5 * AX Y Z ↓ ( Rem3 ) x = 1 * ↓ ( Rem1 ) x = 0 * ↓ ( Rem1 ) x = 0 * X Y Z A ↓ ( Rem3 ) x = 1 * ↓ ( Rem1 ) x = 0 ↓ ( Rem2 ) x = 1 Figure 3.
Using the removal operations to go from the Stoimenowmatching π = (5 , , , , , , , , , , , , , ∈ I to theascent sequence x = (0 , , , , , , A. CLAESSON, M. DUKES, AND S. KITAEV
Lemma 4. If n ≥ , π ∈ I n and ψ (cid:48) ( π ) = ( σ, i ) then σ ∈ I n − and ≤ i ≤ M ( π ) .Also, M ( σ ) = (cid:26) M ( π ) if i ≤ m ( σ ) , M ( π ) − if i > m ( σ ) . Proof.
In this proof we show that each of the three removal operations, when appliedto a Stoimenow matching, produce another Stoimenow matching. The removal ofan arc from a Stoimenow matching produces a matching, but it is necessary to showthe matching is Stoimenow, i.e. does not contain type 1 or type 2 arcs.In both
Rem1 and
Rem2 we are simply deleting the reduction arc. Thus the onlyneighbouring points to check the Stoimenow property (no type 1 or type 2 nestings)are the pairs of points adjacent to the left and right endpoints of redarc ( π ). Howeverfor Rem3 the situation is slightly more complicated.In the diagrams, lozenge vertices ♦ correspond to points which could be openers orclosers and the reduction arc is indicated by (cid:63) .( Rem1 ) In this case | L i ( π ) | >
1. We mustcheck that the removal of the reduction arc redarc ( π ) does not introduce a type 1 or 2 arcin σ . If m ( π ) = M ( π ) then we have the situa-tion as indicated to the right. MM {{ A Bx w y z * If the set of points A is empty then the set B must be empty, for otherwise the arcswith endpoints y and z are type 1. By the same argument, if B is empty then sois A . This gives the following situation if A = B = ∅ : x M x M M*
The point x must be a closer sincethere are no available points toits right. Removing the reduc-tion arc preserves the Stoimenowproperty.Alternatively, A is not empty iff B is not empty. In fact all arcs with opener in A have a closer in B . Similarly, all closers in B have openers in A (for otherwisea type 1 arc would appear). Also, x must be a closer, for otherwise a type 2 arcarises with x and w as endpoints. We have the following situation: M x zy MM x * y zw It is straightforward to see that the removal of the opener of the reduction arcpreserves the Stoimenow property.If m ( π ) < M ( π ) then there are at least 2 arcswith label m ( π ). Consequently, at least oneof x and y in the following diagram must bean opener. Also, note that there must be acloser between the openers of redarc ( π ) and maxarc ( π ). M x m y u z v * First note that the Stoimenow property is preserved at the newly adjacent points u and v once z is removed. Next, if x is a closer then y must be an opener. Thusremoving the opener of redarc ( π ) preserves the Stoimenow property at (the newlyadjacent points) x and y . DIRECT ENCODING OF STOIMENOWS MATCHINGS AS ASCENT SEQUENCES 7 If x is an opener, then y can be an opener or a closer. In the case that y is acloser, then the Stoimenow property is preserved. If x and y are both openersthen the endpoint of y must be to the left of z since it is a Stoimenow matching.Similarly, the endpoint of x must be to the left of u . Thus the Stoimenow propertyis preserved at the newly adjacent points x and y . In the arguments above, thelabel of the maximal arc remains unchanged, hence M ( π ) = M ( σ ).( Rem2 ) In this case | L i ( π ) | = 1 and i = M ( π ). There is a unique arc [2 n − , n ]in π that has maximal label M ( π ). Removing this arc of course yields σ ∈ I n − .Since this arc does not cross any other arcs in the diagram, its removal cannotinduce a type 1 or type 2 arc. It was the only arc with label M ( π ) so we have M ( σ ) = M ( π ) − Rem3 ) In this case | L i ( π ) | = 1 and i < M ( π ). We must check that the matchingobtained after performing operations (a), (b) and (c) is Stoimenow. It is not neces-sarily true that the matching is Stoimenow after performing (a). The combinationof (a), (b) and (c) is needed to ensure the Stoimenow property. See Figure 4 for anillustration of Rem3 .Note that 0 ≤ j < i . Let A be the run of closers between the opener of the reductionarc and the next opener to its right. Let B be the segment whose leftmost point isthe opener to the right of A and whose rightmost point is the opener of the maximalarc. Let C be the run of closers that is to the right of the closer of the reductionarc, and to the left of the rightmost closer.There are only certain places in σ where the Stoimenow property may have beenbroken. The segments of openers W j = ( X j , Y j , Z j ) in π are separated by closers.Thus no two arcs from two different W k ’s (where k < i ) can form a type 2 pairin σ . Hence we may restrict our attention to one segment of openers W j and theaction of steps (a), (b) and (c) on this segment and its interaction with A , B and C .After Rem3 has been applied, the internal order of each of the X j , Y j and Z j segments is unaltered. Thus the Stoimenow property cannot be broken within eachof these segments. However, the order in which the segments Y j and Z j appear in σ has been transposed. Similarly, the segments A , B and C retain their internalorder so that the Stoimenow property is not violated within each.By this reasoning there are only six cases to consider where the Stoimenow propertymay be broken. These are indicated by roman numerals in Figure 4.The adjacent points in cases III, IV and V are such that one point is an opener andthe other is a closer, thereby preserving the Stoimenow property at these positions.(I) If X j is empty then there is no opener immediately to the left of Z j in σ with which to form a type 2 arc. Otherwise X j is not empty and in σ the closers corresponding to X j are located to the left of a , whereasclosers corresponding to Z j are in B which is to the right of a . Hence theStoimenow property is preserved.(II) If both Z j and X j are empty then there is no opener immediately to theleft of Y j . If Z j is empty and X j is not empty then the closers of X j are tothe left of a and the closers of Y j are to the right of a . If Z j is not emptythen the closers corresponding to Z j are in B . The closers correspondingto Y j are in A . Since A is to the right of B in σ , the new neighbors do notform a prohibited type 2 arc.(VI) If C is empty then we have two adjacent closers at the end of σ . Theopener corresponding to the rightmost opener of A is to the left of b so A. CLAESSON, M. DUKES, AND S. KITAEV {{ {{ { {{{ {{ {{ Y j Z j AB CCX j M a b I II VIIII IV V πσ Y ZX j j j * M
A B ba
Figure 4. the Stoimenow property is preserved. Otherwise C is non-empty and theopeners corresponding to A are Y j , whereas the openers corresponding to C are in B . Since Y j is to the left of B , the new neighbors do not form theprohibited type I arc.The arc that was removed was the only arc in π with label i , so M ( σ ) = M ( π ) − (cid:3) Adding an edge to a Stoimenow matching
We now define the addition operation for Stoimenow matchings. Given σ ∈ I n − and 0 ≤ i ≤ M ( σ ), let ϕ (cid:48) ( σ, i ) be the Stoimenow matching π obtained from σ according to the following addition rules( Add1 ) If i ≤ m ( σ ) then partition the segment of openers with label i into two(possibly empty) segments: let A be the contiguous segment of openerswhich do not intersect the maximal arc and let B be the contiguous segmentof openers that do intersect the maximal arc. Note the A is always to theleft of B . Insert an arc by introducing a new point between A and B , andanother new point immediately to the right of π n − . (See Figure 5.)( Add2 ) If i = 1 + M ( σ ) then introduce the arc [2 n − , n ] to σ .( Add3 ) If m ( σ ) < i ≤ M ( σ ) then do as follows (each of these steps is illustrated inFigure 6)(a) Locate the first opener of σ with label i and call it d . Insert an imagi-nary vertical line L in the diagram just before d . Let A be the contigu-ous segment of closers immediately right of the opener of the maximalarc, c , whose openers lie to the right of L . Let C be the segment ofpoints after A and before the rightmost point of σ . Insert two newpoints: one where the line L crosses the diagram, a , and another in-between A and C , b . Join these points by an arc.(b) For all 0 ≤ j < i , partition the segments of openers with label j intothree segments X j , Y j and Z j . The arcs from X j have closers that lieto the left of L . The arcs from Y j have closers that are in A . Z j is whatremains. Swap the segments Z j and Y j for each j while preserving theinternal order of the openers. DIRECT ENCODING OF STOIMENOWS MATCHINGS AS ASCENT SEQUENCES 9 { B L { B { A { A ii i i iii i
Mm * Mm ba * c c Figure 5.
The addition rule
Add1 .(c) Finally, move the segment of closers A in-between the points a and d . Lemma 5. If n ≥ , σ ∈ I n − , ≤ i ≤ M ( σ ) and π = ϕ (cid:48) ( σ, i ) then π ∈ I n .Also, M ( π ) = (cid:40) M ( σ ) if i ≤ m ( σ ) , M ( σ ) − if i > m ( σ ) . Proof.
The proof of this requires examining the three addition operations sepa-rately.For the first case i ≤ m ( σ ) and Add1 is used. This is illustrated in Figure 5. Itintroduces a new arc (the end points of this arc are a and b in the figure) whichhas label i and serves as the new reduction arc. Since arcs with openers in B haveclosers to the right of c , and arcs with openers in A have closers to the left of c ,the Stoimenow property is preserved. It is easy to see from the diagram that theStoimenow property is preserved. Furthermore, since this new arc is essentially acopy of arcs with label i , M ( π ) = M ( σ ).If i = 1 + M ( σ ) then Add2 is used. A new arc is added to the matching on theright hand side. This arc does not meet any other arcs so it retains the propertyof being Stoimenow. Also, the label of this arc will be one more than M ( σ ) so that M ( π ) = M ( σ ) + 1.If m ( σ ) < i ≤ M ( σ ) then Add3 is used. The details of this part of the proof are thesame as the final part of [1, Lemma 4], re-written in the language of matchings asin Lemma 4. (cid:3)
The machinery has now been set up so that we can see that the recursive structureof Stoimenow matchings is isomorphic to that of ascent sequences. We omit theformal proof by induction of the following result which gives the compatibility ofthe removal and addition operations for Stoimenow matchings.
Lemma 6.
For any Stoimenow matching σ and integer i such that ≤ i ≤ M ( Q ) we have ψ (cid:48) ( ϕ (cid:48) ( σ, i )) = ( σ, i ) . If σ has more than one element then we also have ϕ (cid:48) ( ψ (cid:48) ( σ )) = σ . { {{ { {{{ { { {{ { * Y j Z j X j B A M i CL B AY j Z j X j M C (a) ia bd c { {{{ {{ Y ZX j j j Y j Z j X j (b) { { (c) i M* A M* i Aa d c b a d c b Figure 6.
The addition rule
Add3 .4.
Stoimenow matchings to ascent sequences
Define the map Ψ (cid:48) : I n → A n as follows. For n = 1 we associate the only Stoimenowmatching in I with the sequence (0). Let n ≥ π ∈ I n gives ψ (cid:48) ( π ) = ( σ, i ). Then the sequence associatedwith π is Ψ (cid:48) ( π ) := ( x , . . . , x n − , i ) where ( x , . . . , x n − ) = Ψ (cid:48) ( σ ). Combining theprevious lemmas, we have the following theorem that is easily proved by induction. Theorem 7.
The map Ψ (cid:48) is a one-to-one correspondence between Stoimenowmatchings with n arcs and ascent sequences of length n . References [1] M. Bousquet-M´elou, A. Claesson, M. Dukes and S. Kitaev, ( + )-free posets, ascent se-quences and pattern avoiding permutations, arXiv:0806.0666 . DIRECT ENCODING OF STOIMENOWS MATCHINGS AS ASCENT SEQUENCES 11 [2] A. Stoimenow, Enumeration of chord diagrams and an upper bound for Vassiliev invariants,
J. Knot Theory Ramifications no. 1 (1998) 93–114.no. 1 (1998) 93–114.