THE SUPER CATALAN NUMBERS S(m, m + s) FOR s 3 AND SOME INTEGER FACTORIAL RATIOS. 1. Introduction. = (2n)!

Transcription

1 THE SUPER CATALAN NUMBERS S(, + s FOR s 3 AND SOME INTEGER FACTORIAL RATIOS XIN CHEN AND JANE WANG Abstract. We give a cobinatorial interpretation for the super Catalan nuber S(, + s for s 3 using lattice paths and ae an attept at a cobinatorial interpretation for s = 4. We also exaine the integrality of soe factorial ratios. The Catalan nubers 1. Introduction C n = 1 ( 2n = (2n! n + 1 n n!(n + 1!, (1 are nown to be integers and have any cobinatorial interpretations. In 1874, E. Catalan [1] observed that the nubers S(, n = (2!(2n!!n!( + n!, (2 are also integers. Gessel [2] later referred to these nubers as the super Catalan nubers since S(1, n/2 gives the Catalan nuber C n. Gessel and Xin [3] presented cobinatorial interpretations for S(n, 2 and S(n, 3, but in general it reains an intriguing open proble to find a cobinatorial interpretation of the super Catalan nubers. In this paper, we give a cobinatorial interpretation for S(, + s for s A New Super Catalan Identity We first derive another identity for the super Catalan nubers fro the wellnown Von Szily identity ([2, Section 6] S(, n = ( ( 2n 2 ( 1. (3 n + We provide both a cobinatorial and an algebraic proof for this new identity. Proposition 2.1. For, s 0, the following identity for the super Catalan nubers holds: S(, + s = ( ( 2 2s ( 1. (4 s + 2 Cobinatorial Proof. We first interpret (3 in ters of lattice paths. We assue without loss of generality that n and note that for any, ( ( 2n 2 n + counts the nuber of lattice paths fro (0, 0 to ( + n, + n going though the point ( +, with unit right and up steps. 1

2 2 XIN CHEN AND JANE WANG We now define a sign-reversing involution φ on the lattice paths ((0, 0 ( +, ( + n, + n with the sign deterined by the parity of such that the nuber of fixed paths under this involution is the super Catalan nuber. We let P = (s 1, s 2,..., s 2+2n, an ordered set, be a path fro the point (0, 0 to ( + n, + n where each s i = R or U depending on whether it is a right step or an up step. We then find (if it exists the least i, 1 i 2, such that s i s 2+i and switch these two steps. In other words, we have the ap φ(p = P = (s 1, s 2,..., s 2+2n such that s 2+i, j = i s j = s i, j = 2 + i s j, otherwise. For such a path, since is the nuber of right steps in the first 2 steps inus, P and P will have s of opposite parity so φ is a sign-reversing involution. We notice that all the lattice paths with s s 2+ for soe 1 2 are cancelled under the involution φ. Therefore, the paths fixed under the involution are those fro the point (0, 0 to (, + to ( + n, + n such that s i = s 2+i for all 1 i 2. For any, we have ( ( 2 2n 2 n +2 such paths since by syetry there are ( ( 2 ways to choose the first 4 steps and 2n 2 n +2 ways to choose the last 2n 2 steps. Accounting for the parity of the s, we then have that S(, n = ( ( 2 2n 2 ( 1. n + 2 Letting s = n then gives us (4 as desired. Algebraic Proof. Equating the coefficients of x +n in we have that j (1 + x 2n (1 x 2 = (1 x 2 2 (1 + x 2n 2, ( 1 +j ( 2n n j ( 2 = + j ( 1 ( 2 ( 2n 2. + n 2 Applying Von Szily s identity (3 on the left hand side and replacing by on the right hand side, we get that ( 1 S(n, = ( ( 2 2n 2 ( 1. n + 2 The result follows by letting s = n. 3. A Cobinatorial Interpretation of S(, + s for s The Main Theore. In this subsection, we use the identity in Proposition 2.1 to provide a cobinatorial interpretation for S(, + s for s 3. To state our result, we first need to define four line segents, see Figure 1: l 1 connects (0, 1 and ( 1/2, + 1/2, l 2 connects (1, 0 and ( + 1/2, 1/2, l 3 connects ( 1, + 1 and ( + s 1, + s + 1, and l 4 connects ( + 1, 1 and ( + s + 1, + s 1.

3 S(, + s FOR s 3 AND SOME INTEGER FACTORIAL RATIOS 3 l 3 ( + s, + s (, l 4 l 1 l 2 (0, 0 Figure 1. l 1 through l 4 for the = 4, s = 3 case. Theore 3.1. For s 3, S(, + s counts the nuber of paths fro (0, 0 to ( + s, + s passing through (, that do not intersect both lines l 1 and l 4 or both l 2 and l 3. Proof. Fro (4, we have that for s 3, ( ( 2 2s S(, + s = s We notice that ( ( 2 2s ( 2 1 ( 2s s + 2 ( ( 2 2s. + 1 s 2 s counts the nuber of paths fro the point (0, 0 to (, to ( + s, + s and denote this set of paths by Path 0. Siilarly, ( ( 2 2s 1 s+2 counts the nuber of paths fro (0, 0 to ( 1, + 1 to ( + s + 1, + s 1 and ( 2 +1 ( 2s s 2 counts the nuber of paths fro (0, 0 to ( + 1, 1 to the point ( + s 1, + s + 1. We denote these sets of paths by Path 1 and Path 1, respectively. We define an injection fro Path 1 Path 1 to Path 0. To do so, first notice that any path P Path 1 ust intersect l 1. We can find the last such intersection and reflect the tail of P s (0, 0 to ( 1, + 1 segent over l 1. This will give us a segent fro (0, 0 to (,. We then translate the last 2s steps of P so that they start at (, and end at ( + s + 2, + s 2. This segent ust then intersect l 4. We can find the last such intersection and reflect the tail of this segent over l 4. Cobining the two new segents, we now have a path in Path 0, see Figure 2 for an exaple. Notice that this ap has a well-defined inverse. We define a siilar ap for paths in Path 1, but reflect over lines l 2 and l 3 instead of l 1 and l 4. We notice that paths in Path 1 cancel exactly with the paths in Path 0 that intersect both lines l 1 and l 4. Siilarly, we find that paths in Path 1 cancel out

4 4 XIN CHEN AND JANE WANG Figure 2. A apping fro Path 1 to Path 0 for = 4, s = 3. exactly with the paths in Path 0 that intersect both lines l 2 and l 3. Moreover, any path in Path 0 does not intersect both l 3 and l 4 since s 3. This guarantees that the paths in Path 0 that Path 1 and Path 1 cancel do not overlap. Therefore, S(, + s counts the nuber of paths in Path 0 that do not intersect both lines l 1 and l 4 or both l 2 and l Enueration of the Paths in Theore 3.1. Fro (2, we can find explicit expressions for the super Catalan nubers when s is sall. In this subsection, we count the paths reaining under the injection defined in Theore 3.1 and show how they atch with these explicit expressions. For s = 0, we have that S(, = (2!(2!!!( +! = ( 2, where the central binoial coefficient counts the paths fro (0, 0 to (,. These are exactly the paths specified in Theore 3.1 for s = 0. For s = 1, we have that S(, + 1 = (2!(2 + 2!!( + 1!(2 + 1! = 2 ( 2 which counts the nuber of paths fro (0, 0 to ( + 1, + 1 going through (,. Since no paths fro (, to ( + 1, + 1 intersect l 3 or l 4, these are also exactly the paths specified in Theore 3.1. For s = 2, S(, + 2 =, (2!(2 + 4!!( + 2!(2 + 2! = 2(2 + 3C. By Theore 3.1, the paths reaining in Path 0 under the injection either intersect only one of l 1 and l 2 or intersect both l 1 and l 2. We first count those that only intersect l 1. There are C ways to choose the first 2 steps since this segent of the path ust stay below the line y = x. Then, there are 5 possible paths fro (, to (+2, +2 that do not intersect l 4. Hence, we have totally 5C paths fro (0, 0 to (, to ( + 2, + 2 that intersect only l 1. By syetry, we have 5C paths that only intersect l 2.

5 S(, + s FOR s 3 AND SOME INTEGER FACTORIAL RATIOS 5 We now count the paths that intersect both l 1 and l 2. Consider the segent fro (0, 0 to (,. We notice that to get the nuber of paths that intersect both l 1 and l 2, we subtract the nuber of paths that intersect only l 1 or l 2 fro the total nuber of paths fro (0, 0 to (,. We have ( 2 2C = ( ( 2 = ( 1C such paths. Since there are 4 paths fro (, to ( + 2, + 2 that do not intersect l 3 or l 4, we have 4( 1C paths that intersect both l 1 and l 2. Therefore, the total nuber of paths is as desired. Finally, for s = 3, we have that 2 5C + 4( 1C = 2(2 + 3C, S(, + 3 = (2!(2 + 6!!( + 3!(2 + 3! = 4(2 + 5C. Applying a siilar ethod as in the s = 2 case, we can count 2 14C paths that only cross l 1 or l 2 and 8( 1C paths that cross both l 1 and l 2. It is clear that 2 14C + 8( 1C = 4(2 + 5C A Note on the q-analog. Warnaar and Zudilin [6] proved that the q-super Catalan nubers [2n]![2]! [S(n, ] = [n]![]![n + ]! are polynoials with positive coefficients, where the q-integer [n] = 1 + q + + q n 1 = 1 qn 1 q. Using our lattice paths interpretation, it is not hard to see why this is the case for s = n 3. For s = 0, S(, = ( 2 has the q-analog [S(, ] = [ 2 It is well-nown that [ ] 2 counts the nuber of unit squares in an box above the lattice paths that reain under our injection in Theore 3.1. For s = 1, S(, + 1 = 2 ( 2 has the q-analog [ ] 2 [S(, + 1] = (1 + q +1. For s = 2, ]. S(, + 2 = 2(5C + 2( 1C = 2(2 + 3C

6 6 XIN CHEN AND JANE WANG has the q-analog [2]![2 + 4]! [S(, + 2] = []![ + 2]![2 + 2]! [2 + 4] = [ + 2] [2 + 3][C ] where the q-analog of C, = (1 + q +2 ([2 2] + q 2 2 [5][C ] = (1 + q +2 ((1 + q 1 [ 1] + q 2 2 [5][C ], [C ] = [ ] 1 2, [ + 1] counts the ajor index of the length 2 words in the alphabet 1, 1, and for any given Catalan path, up and right steps correspond to 1 and 1, respectively (see exercise 6.34 in [5]. Siilarly, for s = 3, has the q-analog S(, + 3 = 2 2(2 + 5C [2]![2 + 6]! [S(, + 3] = []![ + 3]![2 + 3]! [2 + 6] [2 + 4] = [ + 3] [ + 2] [2 + 5][C ] = (1 + q +3 (1 + q +2 [2 + 5][C ] = (1 + q +3 (1 + q +2 ([2 2] + q 2 2 [7][C ] = (1 + q +3 (1 + q +2 ((1 + q 1 [ 1] + q 2 2 [7][C ]. 4. An Attept at s = 4 In general, the ethods used for finding a cobinatorial interpretation for the super Catalan nubers S(, + s for s 3 in Section 3 do not generalize nicely to higher s. In this section, we exaine the case of s = 4. We first define the following lines in addition to l 1, l 2, l 3, l 4 as defined in Section 3. These lines are pictured in Figure 3. l 5 connects ( 1/2, + 1/2 and ( + s 1/2, + s + 1/2, l 6 connects (, and ( + s, + s, and l 7 connects ( + 1/2, 1/2 and ( + s + 1/2, + s 1/2. Theore 4.1. For s = 4, S(, + s counts the nuber of paths fro (0, 0 to (, to ( + s, + s that do not touch both lines l 1 and l 4 or l 2 and l 3 such that if they have an intersection of l 1 before an intersection of l 2, they do not also reain between lines l 5 and l 6 or between l 6 and l 7. Proof. For s = 4, we notice that (4 gives us five ters corresponding to, 2 2. Interpreting these ters with lattice paths, we again have that for any i, 2 i 2, the = i ter counts the nuber of paths in Path i where Path i is defined as the set of lattice paths fro (0, 0 to (+i, i to (+s i, +s+i.

7 S(, + s FOR s 3 AND SOME INTEGER FACTORIAL RATIOS 7 l 3 l 5 l6 l 7 l 4 l 1 l 2 Figure 3. l 1 through l 7 for = 4, s = 4. We again ap paths in Path 1 and Path 1 to Path 0 using the apping defined in the proof of Theore 3.1. Under this apping however, we have double cancellation of paths that intersect all four lines l 1, l 2, l 3, and l 4. Therefore, Path 0 Path 1 Path 1 = {Paths that do not intersect both l 1, l 4 or l 2, l 3 } {Paths that intersect l 1, l 2, l 3, and l 4 }. We ay also ap the ters in Path 2 and Path 2 into the set Path 0. We first define l 8 to be the line segent fro (0, 3 to ( 3/2, + 3/2 and l 9 to be the line segent fro (3, 0 to ( + 3/2, 3/2. Then all paths in Path 2 ust intersect l 8. We find the last such intersection and reflect the tail end of these first 2 steps over line l 8. Now we have a segent that intersects l 8 and ends at ( 1, + 1. This segent ust also intersect l 1 before its last intersection of l 8. We find the last such intersection of l 1 and reflect the tail end of the segent over l 1. This then gives us a segent fro (0, 0 to (, that has an intersection of l 2 soewhere before an intersection of l 1. We note that this is also an invertible operation. Also, we transfor the last 2s steps of the path to the segent fro (, to ( + s, + s that intersects first l 3 and then l 4, see Figure 4 for an exaple. By syetry, we can ap the paths in Path 2 to the paths in Path 0 that have an intersection of l 1 before an intersection of l 2 and also hit lines l 4 and l 3 in that order. Hence, the paths in Path 2 and Path 2 are apped to exactly the paths in Path 0 that hit all four lines l 1, l 2, l 3, l 4, but at soe point hit l 1 before hitting l 2.

8 8 XIN CHEN AND JANE WANG l 3 l 8 l 1 l 4 l 2 l 9 Figure 4. A transforation fro Path 2 to Path 0 for = 4, s = 4. Adding this to our count for Path 0 Path 1 Path 1, we have S(, + 4 = {Paths that do not intersect both l 1, l 4 or l 2, l 3 } and {Paths that intersect l 1, l 2, l 3, and l 4, but never l 1 before l 2 }. But if we define the sets S 1 and S 2 where S 1 = {Paths that intersect l 1, l 2, l 3, and l 4, but not l 1 before l 2 } S 2 = {Paths that intersect l 1 and l 2, not l 1 before l 2 that stay between l 5 and l 6 or l 6 and l 7 }, we can find a bijection between the two sets as follows. For paths in S 1, we ap the last 2s steps of paths that intersect first l 4 and then l 3 to those that reain between l 5 and l 6, and the last 2s steps of paths that intersect first l 3 and then l 4 to those that reain between l 6 and l 7. Hence, we can find that S(, + 4 counts the nuber of paths fro (0, 0 to (, to ( + s, + s that do not touch both lines l 1 and l 4 or l 2 and l 3 such that if they have an intersection of l 1 before an intersection of l 2, they do not also reain between lines l 5 and l 6 or between l 6 and l 7. Rear 4.2. This is not a very satisfying interpretation of S(, + 4 because of the any conditions that it iposes on paths that are counted. The exaination of the case of s = 4, however, reveals soe of the difficulties of using this ethod for s 4, naely that we have to deal with double cancellation as well as the addition of ore paths. For these reasons, it is unliely that the ethods used in the proof of Theore 3.1 will generalize to higher s. 5. Connection to Annular Non-crossing Partitions In this section, we point out a connection between the nubers ( ( q 2p 2q P (p, q = 2(p + q p q (see Section 7 of [2] and annular non-crossing partitions.

9 S(, + s FOR s 3 AND SOME INTEGER FACTORIAL RATIOS 9 Definition 5.1. The set of annular non-crossing partitions of type B, denoted by NC (B (p, q, consist of set partitions of B(p, q = {1, 2,, p + q} { 1, 2,, (p + q} such that (1 If A is a bloc of the partition, then A is also a bloc of the partition. (2 We draw two circles: one with 2p points labeled 1, 2,, p, 1, 2,, p oriented clocwise on the outside, and the other with 2q points labeled p + 1, p+2,, p+q, (p+1, (p+2,, (p+q oriented counterclocwise on the inside. (3 We draw a non-self-intersecting clocwise closed contour (following the order of B(p, q for each bloc of the partition such that the region enclosed stays in the annulus. (4 Regions enclosed by different contours are utually disjoint. In other words, different blocs of the partition do not cross each other. (5 If a bloc A satisfies A = A, then A ust include points on both the inside and outside circles. We call such a bloc a zerobloc and notice that an annular non-crossing partition has at ost one zerobloc. Exaple 5.2. We refer the readers to [4, Fig. 1] for an exaple. We also need the following definition in order to state the connection between annular non-crossing partitions and the super Catalan nubers. Definition 5.3. An annular non-crossing partition is said to have connectivity c if it has c pairs of non-zero blocs with points on both the inside and outside circles. Such blocs are called connected blocs. Theore 5.4. The nuber of annular non-crossing partitions with connectivity greater than or equal to 1 is NC (B (p, q; c 1 = pq ( ( 2p 2q. p + q p q Rear 5.5. For the proof of the Theore, we refer the reader to [4, Theore 4.5]. With this result, we notice then that ( ( q 2p 2q P (p, q = = NC(B (p, q; c 1. (5 2(p + q p q 2p Gessel [2] gave a cobinatorial interpretation for P (p, q as the nuber of lattice paths fro (0, 0 to (p + q, p + q 1 with unit up and right steps such that the path never touches the points (q, q, (q + 1, q + 1, on the diagonal. It would also be interesting to find a cobinatorial interpretation of the connection between the nuber P (p, q and the nuber of annual non-crossing partitions with connectivity greater than or equal to 1 as stated in (5. Moreover, the super Catalan nubers can be obtained by choosing a subclass of the annular non-crossing partitions and put a signed weight using connectivity c, although it is not clear how we pic the subclass. 6. Other Integer Factorial Ratios The super Catalan nubers (2 are a natural extension of the Catalan nubers (1, which are in one variable, to an integer factorial ratio in two variables. In this

10 10 XIN CHEN AND JANE WANG section, we extend the idea of the super Catalan nubers to other integer factorial ratios in three variables. We then find a general faily of factorial ratios that are integral. Two natural extensions of the super Catalan nubers to three variables are defined in (6 and (7. We first give algebraic proofs of the integrality of these factorial ratios. Proposition 6.1. For a, b, c 0, is an integer. (3a!(3b!(3c! a!b!c!(a + b!(a + c!(c + b! Proof. To show that (6 is an integer, it suffices to show that the order (i.e. ultiplicity of any prie p in the expression is non-negative. To do this, we first recall a well-nown fact that the order (i.e. ultiplicity of a prie p in a! is a a a ord p a! = + p p 2 + p 3 +. Letting l = a p, = b p, n = c p, we see that to show is an integer, it suffices to prove that (3a!(3b!(3c! a!b!c!(a + b!(a + c!(c + b! 3l n l n l + l + n n + 0. Further letting l 0 = l l, 0 =, n 0 = n n, this inequality reduces to 3l n 0 l l 0 + n 0 n , which is not hard to verify by dividing into 4 cases. Proposition 6.2. For a, b, c 0, is an integer. 1 (4a!(4b!(4c! a!b!c!(a + b!(a + c!(c + b!(a + b + c! Proof. The proof follows using a siilar ethod as in the proof of Proposition 6.1. Moreover, Warnaar and Zudilin [6] observed that the above integrality of factorial ratios has interesting q-counterpart when we let [n] = 1 + q + + q n 1 = 1 qn 1 q. The integrality iplies that the q-analogs of the factorial ratios are polynoials with integer coefficients. We can also exaine the positivity of the coefficients of these q-analogs. They are positive for sall a, b, and c, which leads us to the following conjectures: (6 (7 1 The authors would lie to than Rohit Agrawal for suggesting this factorial ratio.

11 S(, + s FOR s 3 AND SOME INTEGER FACTORIAL RATIOS 11 Conjecture 6.3. For a, b, c Z 0, Conjecture 6.4. For a, b, c Z 0, [3a]![3b]![3c]! [a]![b]![c]![a + b]![a + c]![c + b]! N[q]. [4a]![4b]![4c]! [a]![b]![c]![a + b]![a + c]![c + b]![a + b + c]! N[q] Generalized Integer Factorial Ratios. We ay also generalize (7 to n variables x 1,..., x n as follows. We let X = {1,..., n} and P = {S X : S }. Then, we can consider whether the factorial ratio n (2 n 1 x i! i=1 ( x i! S P i S is integral. We can also interpret (8 as the factorial ratio on a Boolean algebra with the denoinator the product of factorial of each ato in the Boolean algebra poset and the nuerator the product of the factorial for the total nuber of each ato in the poset. We can confir the integrality of such factorial ratios for n 6 by coputer experientation. We can also consider the integrality of such ratios for generalized posets. To do so, we let a i, Z 0, 1 i l and r A i (x 1, x 2,..., x r = a i, x, where for each i, at least one a i, 0. Then, we can loo at factorial ratios of the for (( r l j=1 i=1 a i, x! l i=1 A i(x 1, x 2,..., x r!. (9 We note that not all such ratios are integral. For exaple, the factorial ratio 2 (8a!(6b!(6c! a!b!c!(a + b! 3 (a + c! 3 (b + c!(a + b + c! is not an integer for a = 6, b = 1, c = 1. Thus, an interesting question to consider is if there are sufficient conditions on the a i, for the integrality of ( An Attept a Generalization. In this section, we loo at the faily of integer factorial ratios where l i=1 a i, = for all, soe fixed. We attept to find a sufficient condition for the integrality of such ratios. We let a i, Z 0, 1 i l and r A i (x 1, x 2,..., x r = a i, x, (8 2 We would lie to than Prof. Dennis Stanton for showing us this exaple.

12 12 XIN CHEN AND JANE WANG where for each i, at least one a i, 0 and l i=1 a i, = n for all. Then, we loo at the factorial ratio (nx 1!(nx 2! (nx r! l i=1 A (10 i(x 1, x 2,..., x r! We first define y = x x and notice that by siilar reasoning as in the proof of Proposition 6.1, to show the integrality of (10, it suffices to show that r ny l r a i, y 0 for all {y 1,..., y r } where 0 y < 1 for all 1 r. But we have that r r ( (ny ny + ny a i, y a i, n i=1 b i + r a i, ny, n if b i = r a i, 1 where the last step follows since (ny ny < 1 for all y and r a i,y ust be an integer. Thus, we have that l r l ( bi + r a i, y a i, ny n i=1 i=1 l i=1 b i + ( r l i=1 a i, ny =. n Therefore, if l i=1 b i < n, we would have that r a i,y < 1 + r ny fro which it follows that since the su of floor functions is integral, we have that r l r ny a i, y 0. i=1 Hence, this ethod gives us that a sufficient condition for the integrality of (10 is that l i=1 b i < n. Rear 6.5. We notice that this says that l i=1 ( r a i, 1 = nr l < n, which is not strong enough of a condition to find a non-trivial faily of integer factorial ratios. This suggests that only looing at the su of a i, ters is not enough to find sufficient conditions for integrality and that we would need to loo at the interactions between a i, ters to find a non-trivial condition. 7. Acnowledgents This research was conducted at the 2012 suer REU (Research Experience for Undergraduates progra at the University of Minnesota, Twin Cities, funded by NSF grants DMS and DMS The first author was also partially supported by the Carleton Kolenow Reitz Fund. The authors would lie to than Profs. Dennis Stanton, Vic Reiner, Gregg Musier, and Pavlo Pylyavsyy for entoring the progra. We would also lie to express our particular gratitude to Prof.

13 S(, + s FOR s 3 AND SOME INTEGER FACTORIAL RATIOS 13 Stanton and Alex Miller for their guidance throughout the project as well as Jang Soo Ki for pointing out the connection to annular non-crossing partitions. References [1] E. Catalan, Question 1135, Nouvelles Annales de Mathéatiques (2 13 (1874, 207. [2] I. M. Gessel, Super ballot nubers, J. Sybolic Coput. 14 ( [3] I. M. Gessel; G. Xin, A cobinatorial interpretation of the nubers 6(2n!/n!(n+2!, J. Integer Seq. 8 (2005 Article ; arxiv:ath/ [4] I. P. Goulden; A. Nica; I. Oancea, Enuerative properties of NC(B(p,q, Ann. Cob. 15 (2011, no. 2, [5] R.P. Stanley, Enuerative Cobinatorics, Vol. 2, Cabridge University Press, Cabridge, (1999. [6] S. O. Warnaar; W. Zudilin, A q-rious positivity. Aequationes Math. 81 (2011, no. 1-2, Carleton College, Northfield, MN USA E-ail address: chenx@carleton.edu Princeton University, Princeton, NJ USA E-ail address: jywang@princeton.edu