!#" $&%')(+*,,*.-0/2143568798:&;=<>? @ -Species and the A -Mehler Formula Hung Quang Ngo B Abstract In this paper, we present a bijective proof of the C -Mehler formula. The proof is in the same style as Foata’s proof of the Mehler formula. Since Foata’s proof was extended to show the Kibble-Slepian formula, a very general multilinear extension of the Mehler formula, we hope that the proof provided in this paper helps find some multilinear extension of the C -Mehler formula. The basic idea to obtain this proof comes from generalizing a result by Gessel. The generalization leads to the notion of species on permutations and the C -generating series for these species. The bijective proof is then obtained by applying this new exponential formula to a certain type of species on permutations and a weight preserving bijection relating this species to the C -Mehler formula. Some by-products of the C -exponential formula shall also be derived. 1 Introduction The Hermite polynomials are well-studied and have applications in diverse areas of Mathematics [2, 7]. They can be defined in terms of their generating function, whose bilinear extension is the well-known Mehler formula. The most general multilinear extension is known as the Kibble-Slepian formula [17,20]. Foata [10] discovered a combinatorial proof of the Mehler formula, which was later extended to show the Kibble-Slepian formula combinatorially by Foata and Garcia [12]. A D -analogue of the Hermite polynomials, called the D -Hermite polynomials, was introduced by Rogers [19], who used them to prove Rogers-Ramanujan identities. Up to rescaling, there are other variants of the D -Hermite polynomials [1, 8, 9, 15]. The Mehler formula for the D -Hermite is known, but no D -analogue of the Kibble-Slepian formula has been discovered yet. Similar to the normal Hermite polynomials, one hopes that a Foata-style combinatorial approach to the D -Mehler formula helps find a D -Kibble-Slepian. A few proofs of the D -Mehler formula is known (see, e.g., [4, 8, 15]), of which the one in [15] is combinatorial. However, the combinatorial objects were vector spaces over finite fields, which are difficult to be dealt with in bijective arguments. This paper presents a Foata-style proof of the D -Mehler formula. Along the way, we also introduce a new kind of species and a few consequences. Throughout this paper, we use E$FHGIKJ (avoiding confusion with ELIMJ0N ) to denote the set of integers from F to I . The following standard O Computer Science and Engineering Department, 201 Bell Hall, State University of New York at Buffalo, Amherst, NY 14260, U.S.A. hungngo@cse.buffalo.edu notations shall also be used: E !J N E8IKJ N D F F D I!#" F D F D D G I F &%+*, $ D F) D % F) D &%' (% - G 0/ + D D F D F. D N ! / I Most often, the subscript D is dropped when there is no potential confusion. Lastly, we use to denote the set of all matchings on I points E F=GIMJ . 1 2 Preliminaries 2.1 The Mehler formula and its extensions There are several variations of the Hermite polynomials, which are all the same up to rescaling. A typical definition of the Hermite polynomials and their generating function, respectively, are 435678 2 A F :9-;=<> 2 B C35D 9 IFE , > 3 @ 9 ?; < * ;+GH?GI< G (1) (2) To interpret the Hermite polynomials combinatorially, another variant43of the Hermite poly5 2 C35 2 nomials, denoted by J , was introduced. The definition of the J , their generating function and its bilinear extension are as follows, respectively: 2 A @ 2 B A @ 2 B , J , #35 2 J #35K ISE J WV Q IFE NO G * 9-;+TU?T < P C35RQ J C3,L?M P N 2 M * (3) G (4) N F F Q *XY[Z]\ Q 3^V * Q N C ) F * 43 Q * V * _ (5) Identity (5) is the celebrated Mehler’s formula, which was shown combinatorially by Foata [10]. For a discussion of this proof and its relation to other combinatorial results on orthogonal polynomials, the reader is referred to Stanton [21]. Carlitz [5, 6] found several multilinear extensions. Kibble [17], and later independently Slepian [20] found an extension, known as the Kibble-Slepian formula, whose specializations include all other extensions. Louck [18] proposed another extension which was proved combinatorially to be equivalent to the Kibble-Slepian formula by Foata [11]. To describe the Kibble-Slepian formula, let us first introduce some notation. For each integer N I , define a symmetric I` I matrix a by a bc)ed bfc D ji if g.h , otherwise, F 2 bcb c ? where D is an infinite sequence of indeterminates. Let 8 bc-G G i be a vector of I indeterminates. matrices ( F/ gG i/ I ) of order Rbb Let be the set of all symmetric bc I such that for all g/ I , and that is a non-negative integer for all g h . Also, for a fixed , let the g th row sum of be b b b * b[ R The Kibble-Slepian formula reads 2 J A + 2 J [ bfc D"!# $ b c Rbc bc F % M E N)( \ X=Y[Z X'& a F * ,+ a _ (6) Foata and Garsia [12] extended Foata’s proof [10] of the Mehler formula to give a combinatorial proof of the Kibble-Slepian formula. The left hand side of (6) was interpreted as the exponential generating function of the so-called I -involutionary graphs, while the right hand side could be written as the exponential of the series F F / % N.- N X'& a F A (0 b c bc bc a b c + (7) They showed that expression (7) is the generating function for the “connected components” of the I -involutionary graphs. Consequently, the exponential formula applies, proving (6). 2.2 The 1 -Mehler formula and its extensions A D -analogue of the Hermite polynomials called the D -Hermite polynomials, obtained from the so-called Rogers-Szegő polynomials [4, 9, 15, 19] can be defined by their generating function 3 2 G Q32 D as follows: 2 3 Q 2 D G4 A @ #3 2 B 2 To get the corresponding D -version 2 Q 2 D C35 J ; 2 2 * N3 , F % B D of 2 @ $ D C3 J #3 J D 2 , Q D J *,=3 2 D F3 2 J C3 D 2 % G 2 Q265 F 2 C3 2 (8) D . Define M F F PD 2 D G * D (9) F6 D D Q * * D , we also normalize the with the new three term recurrence: 2 F% 2 =43 J #3 2 2 D + (10) 2 D [15]. The interpreThis recurrence43relation yields a combinatorial interpretation for J 2 2 435 2 D as a D -analogue of the matching polynomials J tation gives J . Notice that each matching 78 1 can be viewed as an involution on E F=GIKJ . Define a new statistic on 7 as 9 follows: A 7 9 ;9 : 9 where the sum goes over all edges of 7 , and if 9 2 ! 2 ! g 5 i 5 3 G G g i# G and 7 i ,g 5 ! 5 , then i< 2 - ! 9 9 Pictorially, imagine putting I points F=G G I in this order on a horizontal line, then drawing all ! 9 edges of 7 on the upper half plane. The statistic for an edge is the number of points i lying between g and such that is eitheri a fixed point or an end-point of some edge 7 , both of whose end-points are onFthe left N of (see Figure 1). Let 2 7 2 denote the number of edges I 2 7 2 be the number of fixed points of 7 . It follows that in the matching 7 , and let 7 9 b % b % % b ( : 9 2 where C3 J J 7 : F 7 J 3 ( : - : c 8 , where A F D 2 9 9 % Figure 1: Illustration of c % ( : 9 9 9 c i ,g 5 . G (11) : ( D i# G g (12) On the same line of reasoning as in the previous section, one would hope that (11) helps combinatorially discover the D -analogues of the Mehler formula and its extensions. This turned out to be not easy. There are several known equivalent forms of the D -Mehler formula [4, 15]. 2 43 2 D , it reads In terms of A @ 2 B , C3 2 D 2 #V Q D 2 D $ @ %+B % , ( F Q D C3 2 D On the other hand, let % C3 F % B % 3 2 D , , then A 3^V N * * D Q % @ N3 F * , C3 2 D * NV % Q D 3^V % Q < < D + (13) be the generating function for the number of subspaces of N : @ B * Q CWV 2 D Q D D 3^V Q @ 3 Q @ Q WV * Q @ @ 43^V Q @ (14) It is this form of the D -Mehler formula which has the only known combinatorial proof [15] using the vector space interpretation. However, it does not seem to be possible to extend this proof along Foata and Garsia’s [12] line to find a multilinear extension of the D -Mehler formula. Firstly, we need a D -analogue of the exponential formula, which is not known in general. (A somewhat specialized D -analogue of the exponential formula was devised by Gessel [13], but I do not know how to use his method on linear spaces over finite fields.) Secondly, linear 4 subspaces, although very useful in enumeration arguments, are difficult to be dealt with in bijective arguments. Hence, beside needing a D -analogue of the exponential formula, we also need a different combinatorial proof of D -Mehler formula which uses some easier-to-describe combinatorial objects, which is precisely the main result presented in this paper. Several multilinear extensions have been found by Karande and Thakare [16], and Ismail and Stanton [14]. However, their formulas all involve an infinite sum, and not as general as the Kibble-Slepian formula. It would be interesting to have combinatorial proofs of their findings. 2.3 A 1 -exponential formula Gessel [13] gave a partial answer to the question raised near the end of the previous section. Let us briefly summarize here his main result which we shall make use of later. Define a D -analogue of the derivative: Suppose Let 9 F Q is a function with F Q D F) D Q (15) Q and D -exponential form Q @ A F B Q (16) IFEN , 9 E J be the D -analogue of the function where the coefficients Q Q , namely F @ A B , Q G IFEN (17) are defined recursively by Then, we can write *, d F % B , % &% if I if I % *, G F (18) as an infinite product: Q @ , F) B F D D F Q D Q H (19) 3 Main Results 3.1 Main theorem and approach The main result of this paper is to provide a D -analogue of Foata’s proof of the Mehler’s formula. This is done by first introducing another variant of43the D -Hermite polynomials denoted 2 C3 2 2 D . The new variant is just a rescaling of the old J 2 D , and can also be defined com 2 #3 2 D is then obtained by generalizing Gessel’s binatorially. The D -Mehler formula for the result summarized in Section 2.3. Specifically, define 3 2 43 2 D g D < 2 J \ .g M D 2 D _ Then, these new D -Hermite polynomials can be combinatorially defined as follows. 5 (20) Proposition 3.1. Let be a weight function on matchings defined by * 7 3 ( : - : (: - Then, 2 D 7 C3 A F D 2 : 7 2 Proof. Recall the combinatorial interpretation (11) of the 2 C3 2 D P D g D g * 2 J g : F : D C3 J D , we get 2 _ : ( - \ M (: 3 g _ D (: - 3 (: g * ( : - : ( : D 3 A : F D 2 D (22) 3 g M \ : ( : A < A (21) + D < < * ( : The following is the main theorem of the paper. Theorem 3.2. The polynomials A @ 2 B C3 , 2 43 D satisfy the following Mehler-type identity: 2 2 D 2 #V @ B %+ 2 D FQ IFEN , F Q * * D %+* ** Q D% * * Q @ F. D D ( F Q * * D %+* * 3^V Q D %+*, 43 * V * + (23) Note that (23) is the same as (13) and (14), up to a change of variables. Theorem 3.2 shall be shown in several steps. Notice that the right hand side of (23) looks similar to the right hand side of (19). We shall find a function so that the two are identical. The coefficients in the D -exponential expansion of shall be interpreted as enumerating a certain kind of new species (Section 3.3) called -species (Section 3.2). Another -species defined from is enumerated by a sequence which satisfes relation (18). The -species basically gives a combinatorial interpretation of relation (18). The last step is to bijectively 2 CWV 2 C3 2 D 2 D (Section 3.4). The species is completely analogous to the show that bicolored involutionary graphs introduced by Foata in his proof of the Mehler’s formula. Thus, our proof of Theorem 3.2 can be thought of as a D -analogue of Foata’s proof. A few simple by-products of the new kind of species shall also be derived. 3.2 Weighted -species In this section, we generalize Theorem 5.2 in [13] by introducing a new kind of species [3], which then gives a combinatorial interpretation of identity (18). V Let denote the symmetric group on E$FHGIKJ as usual. More generally, we use to denote the set of all permutations on a totally ordered set of size I . Each word g g 6 where Rg G i.e. V G 6 could be thought of as a permutation on written in one line notation, . The set is called the content of , and is denoted by I Q . For any , we use to denote the number of inversions in . For any two subsets and of , let G denote the number of inversions created by pairs of elements in and , namely i# i i g V G G 2 Gg G - Let be an integral domain and E LD G Q G Q * G J be a ring of formalpower series Q or of polynomials over on the variables D4G G . An -weighted set is a pair G where WC 2 2 g g < is a function associating a weight is a set and to each element . D Q Q* An -weighted set G is said to be summable if for each monomial , the WC number of elements whose weight contributes a non-zero coefficient to is finite. We are now ready to describe the new species. Definition 3.3. An -weighted -species is a rule which (i) to each totally ordered set , and each permutation weighted set E G J G , V , associates an - (ii) to each increasing bijection * , and each permutation associates a weight-preserving bijection where V E G J and * ( < ), E G J G E * G * J G G V * are derived from in the natural way. Moreover, these functions E G J must also satisfy the functorial properties: > ! E G J E# "$ G J > G E G J%" E& G J (24) (25) Basically, the functorial properties say that the weighted sets E G J G depend only on the fact that is totally ordered and on ’s cardinality. When 2 2 I we shall use ELI G J to denote E G J , and ELIMJ to denote ' ( ELI G J . Definition 3.4. Let be an -weighted -species with weight function . The -generating ) * Q32 D series of is the D -exponential formal power series with coefficients in defined by +) D Q32 A , 2 ELIMJ 2 ) Q ISEN G (26) ) where the D -inventory 2 E8IKJ 2 is defined by 2 E8IKJ 2 ) A A ( , ! WC D.- ( (27) Theorem 5.2 of [13] was about partitioning permutations into basic blocks with a multiplicative weight function on the blocks. We generalize this notion by defining the so-called permutation partition. Definition 3.5. %Given / G G such that , a permutation partition / of is a sequence of non empty words * % in one line notation, and that the largest elements of G G We shall write / 0 for “/ is a permutation partition of .” 7 % form an increasing sequence. We are now ready to define a -species whose “connected components” are structures of another -species. Definition Let be a weighted -species with weight function . Define the -species ) ) ) 3.6. with weight function as follows. For each totally ordered set and V , define 7 % % E G J J ` #` E G J G (28) E G where / G - G %R b , and 8 - G we associate I Q G % , for all g S E G b F=G J ` G . Moreover, for each % % E G #` - ! - J %R (29) This is the analogue of the multiplicative property in Theorem 5.2 of [13]. The fact that ) is a -species is easy to verify. At last, the promised generalization of Theorem 5.2 in [13] can now be stated: . Let Theorem weight function 3.7. ) Let be a -species of structures with 6 @ B , by , species defined as above. Define a sequence F and Let @ B 2 Then, F6 A @ B, Q IFEN %+*, A @ FJ Q 2 Q D 2 !G for 9 E , F B @ IFEN ) ! E be the - , and B, namely ) ! 2 9 E8IKJ 2 G I] F , be the sequence defined by , ) Q 2 F D D Q D J D Q H (30) (31) F relation (18). Recall that Proof. We only need to verify that the sequences E8I F J is a sequence of structures of :and satisfy each . Let be G G G G b b the corresponding permutations (or words) underlying G G ! . Let! , for each I Q , 2 2 g that I] F* . Suppose F , . Let F= G G I . Notice F , and E F=GI F J . Note that G G since I F . Furthermore, let E J be the structure of species obtained from by removing . For each structure of a -species, we use to denote the underlying permutation of . It is clear that H H and that G F 8 + G In order to form a then form I Consequently, A *, A D A A %+B A %+B I! A %+B , " BC% &% ` - D ( ( -- A - ` A A BC% ( - A - A BC% of E F=GIKJ ( / E J and ! I ), E J. / ( ( -- A %+B F - D , ( - D , ( - D , A *, ELI F J , we can first pick a -subset , and finally concatenate any pair of ! ( ( -- ` D ` D ( ( $-- D - ( ( -- D - A ( ( -- ( ( -- D.- % *, Example 3.8. Theorem 3.7 implies Theorem 5.2 in [13] and thus all its consequences as derived by Gessel. Example 3.9. Take F so that F in Theorem 3.7, we obtain A F @ B, where, IFEN Q @ B , ( F D @ Q N D Q 2 / ( D D @ Q ( - 2 / 0 2 D - A Q *, * * D + G (32) In fact, when D F , counts the number of sets of words on E F=GIKJ whose contents are disjoint and whose union of contents is exactly E$F=G IKJ . While, when D F the right hand side of (32) LC goes to XY[Z ( Q F. Q + . Thus, we could have proven easily identity (32) combinatorially when D F. Following Gessel’s line of derivation we can generalize the previous example as follows. 8 % Corollary 3.10. Let / G G be any permutation partition of number of words of size g of / . Define a weight function for / by / Then, ) D Q32 A F Q F ( / @ B where A @ B D - F , ( F A F) D D 3*, , 9 ( D ELI F JN Q . Let b 3 b # b( ( - / be the , and let (33) Q Q + G (34) Example 3.11. Write / 3 Q F D , so that % 0 if / 0 and hence ! and all words of / are of size at most . Set d F / ( 3 A F D 2 A ( if / 0 * otherwise, 43 Q < 3 ( - D - ( Corollary 3.10 gives ) 3 GQ 2D @ A B , #3 D 2 @ Q IFEN @ where ?b7T * ( * N N and Thus, A 2 A ( D F 3 < N F% !D Q D % * * G D Q (35) * F D . We now get g Q C3 F D D , F. %+B F , F. B F6 D g ( - D - ( - P * 2 J A : \ M g 3 F D D _ (36) F D : ( : - 3 ( : D G (37) an interesting combinatorial identity. 3.3 A 1 -analogue of the bicolored -involutionary graphs The previous section gives a combinatorial view of identity (18). As outlined at the end of section 3.1, the next step in the proof of Theorem 3.2 is to find a -species whose -generating ) Q.2 D series Q is such that the right hand side of (19) is the same as that of (23). The ) ) ) -species is a D -analogue of the bicolored I -involutionary graphs. We actually will start defining first. Definition 3.12. A graph G satisfies the following conditions: 1. 2. is called an ordered bicolored I -involutionary graph if has I vertices labeled by I distinct positive integers in . has no multiple edges, but can have loops. 3. The I vertices of line up on a horizontal line, so that we can speak of a vertex being on the-left or right of another, and so that the vertices of forms a permutation / V / / * / . 4. Each edge of 5. Each vertex of is colored either red or blue. N is incident to exactly edges of different colors. 6. A non-loop edge of can only connect some / 10 b to / b *, unless it completes a cycle of . 7. Let G G be the connected components of from left to right. Let denote must satisfy the the largest vertex number in a connected component of , then / 5 5 condition that . - 8. For each connected component , the vertex numbered blue edge incident to it. has to be on the left of the 9. If a connected component is a cycle, then the vertex numbered has to be the left most vertex among all vertices of . It is not difficult to see that the connected components of can only be in one of forms as shown in Figure 2. In the figure, the bold lines represent blue edges and the thin lines represent red edges. 9 9 ( ; ; (- (- (- ( ; ; Figure 2: Possible connected component types of an ordered bicolored I -involutionary graph. Let denote the set of all ordered bicolored I -involutionary graphs on , where is be the set of all graphs in which have exactly one an I -set of positive integers. Let connected component. When E F=GIMJ , and shall be used for convenience. 9 Let E F=GIMJ be the trivial one-to-one correspondence between and E$FHGIKJ which > preserves order. For each , let D denote the graph obtained from by renumber to denote the set of vertices of ing each vertex of by . Conversely, we also use . Definition 3.13. A weight function defined on with values over some commutative algebra over the rationals is said to be multiplicative if it satisfies the following conditions: (i) F - 9 D > H . % (ii) If G are ordered bicolored involutionG are the connected components % F of (which ary graphs themselves), then . The following theorem is obviously a very special case of Theorem 3.7 applied to the ordered bicolored involutionary graphs. Theorem 3.14. Supposed is a multiplicative function on @ B Let , @ B , be the sequence defined by , %+*, A A D ( ( -- - , and 11 D - ( (-- . For I , define a sequence ! for . Then, @ A B , Q A@ 9 ISEN Q ISEN B, 9 9 9 Definition 3.15. Let be a graph in . For each edge (respectively vertex g ) of , let 9 connected component containing (respectively g ). Define (respectively g ) denote the a weight function on each edge of D 9 D as follows: 9 V is a non-loop red edge G is non-loop, blue and to the left of is non-loop, blue and to the right of is a red loop G is a blue loop if 9 if 9 if 9 if 9 if F 3 9 H 9 G H G Let be a weight function defined on by: 9 G ( - 9 then obviously is multiplicative. We call an ordered bicolored I -involutionary graphs with the weight associated a bicolored D4G I -involutionary graph. Figure 3 shows an example of such a graph. In the figure, the largest ; N N B ( ; N > N < N W W (W W , > < * < N * ; ( % N *.- - "!$#%'&)(+*,-*/.0'*1* ( - B * Figure 3: An example of a bicolored D GI -involutionary graph. vertex number in each component has been put in bold face. @ B , be a sequence defined by , Lemma 3.16. Let be the function defined above, and and A D - ( (- - G when I] F Moreover, let D Q32 Then, Q)2 D F F. Q * * D B Q D * F) @ A Q IFEN , F * * Q D 12 Q * * $3^V D *Q F D Q D 43 * V * (38) Proof. Firstly, we claim that * %+*, ! 8 ! E J& E FJ D % * ! N 3^V N E %+*, ! the paths which To see this, let us consider Figure 2. The components in * can only be N F . Summing start and end with different colored loops, and have largest vertex number ( (- Dover all components which start with a blue loop and end with a red loop we get ! ! % the term * N 3^V E JD N G E while the components which start red and end blue introduce the term ! E F JD % * ! N N E 3^V The details are easy to be verified and hence omitted here. Secondly, we claim that %+* * % * * % * * D ! N * ! N ! F E N E * F JD % *, ! N ! ! % *, E Here, the term D F EN is from the cycle components, N * paths which start and end with a blue loop, and E F JD start and end with a red loop. By definition, D Q32 @ A B A ISEN @ A ! E J[ E , %+B ! N ! %+B + 3 * F J D V * F EN from the F E N from the paths which ! * * V %+*, Q , @ * 43 F E N , %+* * (D * FJ D % N ! N ! 3^V N E !Q N F E N E F JD F E N ! %+*, * * %+*, ! N F E N * 3 * V + N % * * !Q * N? N E Hence, Q42 D A F @ ! %+B ! E J& E , * FJ D % 3^V Q * % @ A %+* %+B * * D , %+*, Q @ A ! %+B E , F JD %+*, * * 3 * V Q * %+*, (39) Now, we calculate each term of (39) separately as follows: 3^V A @ %+B , ! E J& E ! FJ D * % Q * % 3^V Q * * D A Q * * D A Q * * D A F 3^V F 3^V F F 13 F ! E %+B , A b B cHB , @ Q Q Q b * * D *( D , b , % * % @ %+B % * FJ D * * $3^V D * * * D F Q D Q @ A @ c B % c , Q * - c * c c Q D * % (40) Similarly, @ A %+* %+B D , %+*, * * Q * D Q F Q * G D (41) and * 3 * V @ A ! % B E , F JD * %+*, %+*, Q * * 83 * 83 * V F A %+B * = Q * , V * * D b A F * Q , * Q * - c c %=c *( Q * - * c (42) c D D @ c B % c % * Q *( D , b b B cHB , @ A D Q * * D Q D , % @ A D cHB , % *Oc * A %+B Q 43 D Q D Q * V * 83 * V % @ A Combining (40), (41) and (42) yields (38). Corollary 3.17. Let be the function defined above, and A ( ( -- D.- @ B , be a sequence defined by (43) Then, Q)2 D F @ B %+ , F Q * * D % * ** where D% * * Q @ F D D ( F Q Q32 D A * * D %+* * $3^V Q D %+*, 3 * V * + G @ B Q , Q ISEN Proof. This is straightforward from Theorem 3.14, Lemma 3.16 and equation (19). 3.4 A bijection This section completes the last step of the proof of Theorem 3.2. We are left to demonstrate 2 CV 2 C3 2 2 . We shall show this relation combinatorially as formally put in that D D the following theorem. Theorem 3.18. Let Then, 2 be defined combinatorially by equation (22), and 2 43 2 D 2 CWV 2 D F by equation (43). [ *, 1 ` Proof. We want to finda*, weight-preserving bijection which maps a pair*,7 G7 *, *, . Let 7 G7 be a pair 1 1 `01 to a graph of matchings in , where the fixed 3 V GI0 F of 7 points of 7 are weighted by and of 7 by . As before, we view the vertices F=G and 7 as lying on a horizontal line from left to right in that order, with the edges drawn on the 14 upper half plane. Let G G , ( / I F ) be the sequence of vertices of 7 starting from the - 9 9 9 right which are not left end-points of 7 ’s edges. Similarly, let G G , be9 the corresponding - : sequence for 7 . Notice that I F . Let G G (respectively G G : ) be the set of edges of 7 (respectively 7 ) ordered by their right end-points starting from the right. Our idea is to start from the right, look simultaneously at and , * and * , ... determine the “right place” to stop and build up the right most connected component of based on the relative distribution of edges and points of 7 and 7 seen so far. Then, remove certain points and edges from 7 and 7 to get and respectively, and re-apply the method to get the next (from the right) connected component of , and so on. Looking at and , * and * , ... there will roughly be situations as follows. ! b 1. At some F , all of ! i0 respectively, and ! b 9 and , F / g / F , are right end-points of edges in 7 and 7 cF . F is the least integer such that *, ! %+*, 9 of 7 of 7 and then a fixed point where / . For this 2. We meet a fixed point 9 case to be disjoint from 1, it is necessary that all edges of 7 whose right end-points *, case . are on the right of have 3. We meet a fixed point *, of 7 % *, strictly before a fixed point of 7 . 4. Two fixed points of 7 are met before any fixed points of 7 . 5. Two fixed points of 7 are met before any fixed points of 7 . Note that similar to case 2, the cases 3, 4, and 5 need to be defined so that they are disjoint from case 1. These cases determine our “right place” to stop as mentioned above. Formally, we consider cases as follows. ! ! Case 1. There exists a , 0/ (i) i ! F (ii) For all i ( / * - , such that is the smallest ! 9 integer where FHG G F, c 9 c has right end-point The situation is depicted in Figure 4. Let c 9 . (i.e.9 and c (respectively c- for all ! i .) / has right end-point c . ) be the matching obtained 9 !#" Figure 4: Illustration of case 1. 9 %+*, 9 9 %+*, G by removing G and their end-points (respectively end G 7 G7 G suchandthattheir points) from 7 (respectively 7 ). We shall construct the last %+*, * I %+FH*,G g - G GHg , and that G forms connected component of is a cycle Rg * G G g be the set of the%+rest the rest of the components of . Let a of the *,, points on the cycle as shown. To do this, we need to pick a permutation g * g V %+*, * , where a a Rg G G g is a set of distinct integers in E F=GIKJ , such that the 9 of this cycle to the weight of is exactly equal 9to the contribution 9 9 contribution of G - G %+*, to the weight of 7 plus the contribution of G G %+*, to the 15 9 9 c c weight of 7 . Notice that removing the edges and total weights of the rest of edges of 7 and 7 . Let number of inversions created by pairs of numbers in GD `]a such that D . does not have any effect on the E F=GIMJ5 a , and G+a be the `a , namely the number of pairs As each red edge on is weighted D and each blue edge weighted F , it is easy to see that D - * * ( - ( - * %+*, D %+*, D - ( 9 - G D $ ( 9 $ $ $ %+*, D %+*, D G (44) (45) (46) Hence, we need to pick such that %+*, G+a A F %+*, 9 c B, A c 9 c B, ! c (47) Observe that 9 F/ %+*, 9 c 9 c / Q d I I g N G a the Q if Q if Q c % *, Gg* Q A I / i 2 T B, A 2 I ( T B, Q A i N-i N-i i0 G FHG T Q F N-i %+*, Q cHB, / F T jg , G! G - G F ! F=G G G HF G G i0 ! F Rg F=G G - N G g , working TU? GHg (51) ! T F so that g is well 9 c A cHB, * , and an , 9 / 16 I h N-i# + A cHB, ( I ! c G F 2 %+*, i + %+*, A ! i preceeds g , I ( cHB, Case 2. There exists a , 0/ F=G =F G F for all Q % ! , element by element starting from %+*, * which is exactly (47). i (48) (49) (50) ! - %+*, by c- 9 G 9 F i th smallest number in E$F=G IKJ / * g ! N G F Q It is easy to check that F defined. Moreover, - G N-i F G N-i F I / Then, recursively determine % *, toward g * as follows: T I / Q G on F=G Now, define a function / ! such that N-i NO N-i F + i 9 c c (i) For all F=G G , is the right end-point of 3 9 is a fixed point, which is weighted by . And, for all c+ right end-point of ! . 9 i F=G (ii) For all V weighted by . c G , 9 , and i0 c is the right end-point of c N . And, G %+*, The situation is depicted in Figure 5. This time, the last component *, ! . Moreover, c G F , is the is a fixed point of starts with Figure 5: Illustration of case 2. N F st point from the a red loop and ends with a blue loop. The point I F is the right. Let , a , be defined as in the previous case, then the corresponding , and are as follows: * * ( - ( - * - D -% (9 D % D $ $ (9 D D $ $ 3 D V %= * 3^V G (52) G (53) (54) Hence, we need to pick so that % For Q F=G G+a A F , the corresponding Q F I I I N Q Q Q Q 9 c B, ! N G % F. F. 9 9 9 c B, c ] (55) is: c if Q if Q if Q c 9 A c c N i N i N i i0 , i0 , F, i T FHG G G ! =F G G! G F=G G (56) T As in the previous case, g is defined by (51). To show that g is well defined and that they satisfy (55), we only need to observe that F / / / (ii) For all i 9 c FHG / ! G , G F N-i N-i / I F / I F / F=G I / c ! Case 3. There exists a , F i0 c 9 ! (i) For all 3 . 9 * , and an , c , i N-i / G G F G i F=G c . 17 G i0 - F=G G ! G F=G ! G G (57) (58) (59) ! / is the right end-point of 9 - 9 c F , such that , and %+*, is a fixed point weighted Figure 6: Illustration of case 3. i (iii) For all F=Gi V , and for all - G 9 c , *, 9 is-the right end-point of , is a fixed point weighted c c G G F is the right end-point of . N c ! * * ( - ( - * D -% - (9 D % D $ $ (9 D D $ $ ( The situation is depicted in Figure 6. In this case, we have Hence, we need to pick so that For Q F=G G+a I Q F I I I N Q Q V, F. 9 F. if Q if Q if Q if Q c 9 - 3^V G 9 c c c (60) G (61) (62) A 9 c B, c ] is: c F. Q 9 9 Q c B, Q 3 *, % A F , the corresponding D % * % ! N G *, N i N i N i N i i0 , i0 , F, F, F=G i T G G ! - =F G G G - F=G G G ! - F=G G . i (63) (64) T As in the previous case, g is defined by (51). To show that g is well defined and that they satisfy (63), we only need to observe that F / / / / 9 I F / I F c - * N-i F , and an G i0 , 0/ F=G (ii) For all FHG / c G F, c 9 c * * % ( - ( - * * - D (9 - 3 * D %+*, D $ G $ ( 9 - D D $ $ -% 18 9 , and N 9 c *,( c - G is the right end-point of D , such that G ! *, . Moreover, c F , is the right . The situation is depicted in Figure 7. In this case, we have ! (65) (66) (67) (68) ! (i) For all%+* F=G G , is the right 3 end-point i of * are 9 fixed points weighted . For all and c end-point of .! i G G ! N-i i0 - G =F G G G N-i i0 G F=G G G N-i i0 - F G F=G G F I / ( / I / c ! / c 9 Case 4. There exists a , 0/ i c 9 ! 9 3 * G (69) (70) (71) ! " Figure 7: Illustration of case 4. Hence, we need to pick so that For Q F=G G+a Q F I I I N Q Q Q 9 F F c A c 9 c B, c ] is: if Q if Q if Q c 9 Q c 9 9 c B, F , the corresponding %+*, A F ! N G % N-i N-i N-i i0 , i0 , i F, G G ! =F G G! G F=G G F T - F=G (72) (73) T As in the previous case, g is defined by (51). To show that g is well defined and that they satisfy (72), we only need to observe that 9 F c / 9 / / ! c 9 ! / - I / c Case 5. There exists a , 0/ i I / ( - * N-i F N-i , and an i G N-i F I / F F=G F G i G F=G - i (ii) For all (iii) For all FHG HF G G G , F 9, G G F 9 c ! (74) (75) (76) ! G G *, %+* * are . Moreover, and c F , is the right end-point of 9 is the right end-point of c F=G ! , such that / c c i0 (i) For all F=G G , V is the right end-point of i N 9 G G fixed points weighted . For all c . ! i G ! , 0/ - c . . The situation is depicted in Figure 8. In this case, we have " Figure 8: Illustration of case 5. D *, D % D * * ( - ( - * D.-% D - $ ( 9 $ $ 19 *, (9 $ D V G * * % *,( ( *, V * G (77) (78) (79) Hence, we need to pick so that %+*, For Q F=G G N F I F I I Q Q Q Q I N 9 9 9 F. F. 9 Q if Q if Q if Q if Q c F. c c c 9 c B, F , the corresponding Q ! G+a % A c A 9 c B, c ] (80) is: N i N i N i i0 , i0 , F, N i F, F=G G G ! =F G G G F=G G G - F=G G i0 i0 T (81) ! F T As in the previous case, g is defined by (51). To show that g is well defined and that they satisfy (80), we only need to observe that 9 F c / 9 / / / 9 I F / I F c / 9 c c I / / F I N-i i0 N-i G N-i N-i F F=G i0 G G i0 F G G G - =F G G F=G G G i0 ! F=G F=G - G ! (82) (83) (84) (85) Acknowledgements I would like to thank Prof. Dennis Stanton at the School of Mathematics, University of Minnesota, U.S.A., for many helpful discussions and insightful suggestions, which made this work possible. 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