RESEARCH STATEMENT. Sergi Elizalde

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1 RESEARCH STATEMENT Sergi Elizalde My research interests are in the field of enumerative and algebraic combinatorics. Much of my work during the past years has been on enumeration problems, bijective combinatorics and generating functions. On the other hand, recently I am becoming more and more interested in the connections of combinatorics with other branches of mathematics, particularly with commutative algebra, computational biology, discrete geometry, algebraic topology, and probability and algorithms. I enjoy learning new problems in which I can apply my knowledge in combinatorics. I finished my Ph.D. thesis at MIT in June of 2004, under the supervision of Professor Richard Stanley. The topic of my thesis is pattern-avoiding permutations (also called restricted permutations), and more specifically, the study of the distribution of statistics on them. For example, one of my results is that the pair of statistics number of fixed points and number of excedances has the same joint distribution in both 321-avoiding and 132-avoiding permutations. In order to prove this and other results combinatorially, we introduce new statistics on the sets of Dyck and Motzkin paths, and construct bijections between them and restricted permutations. This transforms problems about permutations to the enumeration of lattice paths with respect to certain statistics that are easier to deal with. The subject of pattern avoidance has blossomed in the past decade, and it has proved to be useful in a variety of seemingly unrelated problems. From the computational point of view, pattern avoidance appears naturally when sorting through a stack [21] or several stacks [34]. In a more algebraic context, the vexillary permutations, which index smooth Schubert cells in the study of flag manifolds in algebraic geometry, are those which avoid On the other hand, the study of statistics on restricted permutations started developing very recently, and the interest in this topic is currently growing. Here are the basic definitions. Let n, m be two positive integers with m n, and let π = π 1 π 2 π n S n and σ = σ 1 σ 2 σ m S m be two permutations. We say that π contains σ if there exist indices i 1 < i 2 <... < i m such that π i1 π i2 π im is in the same relative order as σ 1 σ 2 σ m. If π does not contain σ, we say that π is σ-avoiding. Denote by S n (σ) the set of σ-avoiding permutations in S n. It is a long-studied and very hard problem to determine S n (σ) for given σ. The first nontrivial case is when σ has length 3. It is known [21] that regardless of the pattern σ S 3, S n (σ) = C n = 1 2n ) n+1( n, the n-th Catalan number. For patterns of length 4 one already finds an unsolved case: no formula for S n (1324) is known. Regarding asymptotic enumeration, a recent breakthrough [23] (see also [20, 1]) has been the proof of the so-called Stanley-Wilf conjecture, which gives an exponential upper bound on the number of permutations avoiding any given pattern. Another important topic in combinatorics is that of permutation statistics. Recall that i is a fixed point of a permutation π if π i = i, and that i is an excedance of π if π i > i. Denote by fp(π) and exc(π) the number of fixed points and the number of excedances of π respectively. Let lis(π) be the length of the longest increasing subsequence of π. Define the rank of π, denoted rank(π), to be the largest k such that π i > k for all i k. An unexpected connection between pattern avoidance and permutation statistics was recently found by Robertson, Saracino and Zeilberger [28]. They prove that the number of 321-avoiding permutations π S n with fp(π) = i equals the number of 132-avoiding permutations π S n with fp(π) = i, for any 0 i n. This refinement of the classical result that S n (321) = S n (132) inspired a large part of my work in this area. First, because at that time there was no combinatorial proof of that result, which would give a good understanding of why the 1

2 2 distribution of fixed points in S n (321) and S n (132) is the same. None of the several known bijections between these two sets preserved the number of fixed points. Second, because it was natural to ask whether similar results for other permutations statistics would hold as well. In [11] I give a generalization of this result, proving that it still holds when we fix not only the number of fixed points but also the number of excedances. Theorem 1 ([11]). The number of 321-avoiding permutations π S n with fp(π) = i and exc(π) = j equals the number of 132-avoiding permutations π S n with fp(π) = i and exc(π) = j, for any 0 i, j n. The original proof given in [11] is analytical, and it uses a new technique involving diagonals of non-rational generating functions. If A(x, y) = a i,j x i y j, the diagonal of A is defined by (diag A)(z) := a n,n z n. A general method for obtaining diagonals of rational functions is described in [30, chapter 6], but it does not apply to the case of non-rational functions. I believe my approach gives the first instance of an application of the computation of diagonals of non-rational generating functions to solve a combinatorial problem. The problem of finding a combinatorial proof of Theorem 1 is solved in [16]. The bijection that we give preserves additional statistics. Theorem 2 ([16]). There is a bijection Θ : S n (321) S n (132) satisfying fp(θ(π)) = fp(π), exc(θ(π)) = exc(π), and rank(θ(π)) = n lis(π). The bijection Θ is a composition of two slightly modified known bijections into Dyck paths, and the result follows from a new analysis of these bijections. The Robinson-Schensted-Knuth correspondence is a part of one of them, and from it stems the difficulty of the analysis. From the fact that Θ maps involutions into involutions we obtain the following corollary. Corollary 3 ([16]). The number of 321-avoiding involutions π S n with fp(π) = i, exc(π) = j and lis(π) = k equals the number of 132-avoiding involutions π S n with fp(π) = i, exc(π) = j and rank(π) = n k, for any 0 i, j, k n. While the case of permutations avoiding a single pattern has attracted much attention, the case of multiple pattern avoidance remains less investigated. A systematic enumeration of permutations avoiding simultaneously any subset of patterns of length 3 was given in [29]. For any set Σ k 1 S k of patterns, the distribution of the statistics fp and exc in permutations avoiding simultaneously all the patterns in Σ is described by the generating function F Σ (x, q, z) := x fp(π) q exc(π) z n. n 0 π S n(σ) In [12] I complete the enumeration of fixed points and excedances in permutations avoiding any subset Σ S 3 with Σ 2, by giving explicit expressions for F Σ (x, q, z). In some instances the results are generalized to the case in which one of the patterns has arbitrary length. In particular, I find the expressions for the generating functions F Σ (x, q, z) for Σ = {312, k(k 1) 1}, Σ = {132, k(k 1) 1} and Σ = {132, 12 k}, for all k 1. In the same paper I study fixed points and excedances in permutations avoiding a single pattern of length 3, solving all the cases except for the pattern 123. I also prove a result that was conjectured in [28], namely that for all n 4, s 0 n(132) < s 0 n(123), where s 0 n(σ) is the number of permutations in S n (σ) with no fixed points. Dyck paths are closely related to the study of patterns of length 3. It has been essential for my developments to introduce a new kind of statistics on Dyck paths. Regarding a path as a sequence D of up-steps u and down-steps d (i.e., a Dyck word), we define a tunnel to be

3 a decomposition D = AuBdC, where B is a Dyck word. A tunnel is centered if A = C, and it is a right tunnel if A > C. Under certain bijections, our statistics in permutations are mapped to centered and right tunnels in Dyck paths. The statistics number of centered tunnels and number of right tunnels are not easy to deal with because they are not defined locally. However, their enumeration becomes simpler using the statistic-preserving bijections introduced in [13]. These bijections have consequences in enumeration of pattern-avoiding permutations with respect to several statistics: Theorem 4 ([13]). Let L(x, q, p, z) := 1 + n 1 π S n(132) where des(π) is the number of descents of π. Then (1) L(x, q, p, z) = x fp(π) q exc(π) p des(π)+1 z n, 2(1 + xz(p 1)) 1 + (1 + q 2x)z qz 2 (p 1) 2 + f 1 (q, z), where f 1 (q, z) = 1 2(1 + q)z + [(1 q) 2 2q(p 1)(p + 3)]z 2 2q(1 + q)(p 1) 2 z 3 + q 2 (p 1) 4 z 4. These bijections also provide combinatorial proofs of some new results like the following. Theorem 5 ([13]). Fix r, n 0. For any π S n, define α r (π) = #{i : π i = i + r}, β r (π) = #{i : i > r, π i = i}. Then, the number of 321-avoiding permutations π S n with β r (π) = k equals the number of 132-avoiding permutations π S n with α r (π) = k, for any 0 k n. 3 Generalized patterns. An extension of the concept of pattern avoidance was introduced in [2]. A generalized pattern allows the requirement that two adjacent letters in a pattern must be adjacent in the permutation. In particular, if we require that all the elements occur in consecutive positions, we obtain the so-called consecutive patterns. These patterns were considered for the first time in [15]. There we solve the problem of enumerating occurrences of patterns of several shapes by obtaining the corresponding bivariate exponential generating functions as solutions of certain linear differential equations with polynomial coefficients. We give the generating function for the increasing consecutive pattern σ = 123 m, and also for any pattern of the form σ = 12 (a 1)a τ (a + 1), where 1 a m 2 and τ is any permutation of the numbers {a + 2, a + 3,..., m}. We have also studied [14] another class of restricted permutations involving generalized patterns. We define a Motzkin permutation π to be a 132-avoiding permutation in which there do not exist indices a < b such that π a < π b < π b+1. We give a bijection from this class of permutations to the set of Motzkin paths, which allows us to enumerate Motzkin permutations with respect to several statistics, such as the length of the longest increasing and decreasing subsequences and the number of rises. Plane trees. Another problem I have done research on is related to plane trees. Many enumerative results about plane (or ordered) trees appear throughout the literature. For example, a well-known interpretation of the Narayana numbers is that they count the number of plane trees with a fixed number of leaves. We can distinguish between two kinds of leaves. A leaf of a plane tree is called an old leaf if it is the leftmost child of its parent, and it is called a young leaf otherwise. In [6] we enumerate plane trees with a given number of old leaves and young leaves. The formula is obtained combinatorially by presenting two bijections between plane trees and 2-Motzkin paths with very convenient properties. They map young leaves to red horizontal steps, and old leaves to up steps plus one. We derive some implications to the enumeration of

4 4 restricted permutations with respect to double descents, pairs of consecutive deficiencies, and ascending runs. Another application of our bijection appears in [7], where Chen, Yan and Yang use it to give combinatorial interpretations of two identities involving the Narayana numbers and Catalan numbers, due to Coker [9], solving the two open problems left in [9]. In [6] we show that a more detailed analysis of the bijection and its properties gives refinements of the two identities of Coker, as well as bijective proofs of these refinements. Representation theory of the symmetric group. During my years as a graduate student at MIT I have also learned about problems in other areas of combinatorics. One such area is representation theory of the symmetric group. Working on a conjecture of Stanley [31], I was able to show that the leading coefficients of the polynomial functions expressing the normalized characters of S n, χ σ (k, 1 n k ), in terms of the lengths of the sides of the rectangles that form the shape σ are nonnegative (see [31] for details). Current directions of research Pattern avoidance. Here are some research projects I have recently started or plan to pursue in the future. The proof of the Stanley-Wilf conjecture settles one of the big open questions in pattern avoidance, but it also gives rise to new ones. For example, it follows now that lim n n S n (σ) is a constant that only depends on the pattern σ. It is a challenging problem to find the value of this limit for a given arbitrary pattern. There is a conjectured [1] upper bound of (k 1) 2, where k is the size of σ. The first instance of a pattern for which this limit is not an integer was found very recently by Bóna [5]. Most questions about classical pattern avoidance have a natural generalization to generalized patterns. Some work has been done regarding the exact enumeration of permutations avoiding generalized patterns [8, 15]. However, an interesting problem that has received almost no attention is the study of the asymptotic behavior of α n (σ) := S n (σ) as n goes to infinity, when σ is a generalized pattern. The case of classical patterns (generalized patterns with dashes between any two elements) is solved because it reduces to the Stanley-Wilf conjecture, which states that there is a constant λ such that α n (σ) < λ n. At the other end of the spectrum we have consecutive patterns (generalized patterns with no dashes). In this case I have been able to show that, given σ, there exist constants c and d such that c n n! α n (σ) d n n!. This immediately raises the question of what the asymptotic behavior is for all the remaining generalized patterns, that is, those that have dashes only in some positions. Experimental evidence gives a range of possible behaviors. I have obtained results for some patterns, and one of my goals is to give a complete classification of all generalized patterns according to the asymptotic behavior of α n (σ) as n goes to infinity. The developing topic of statistics on restricted permutations has opened up a good number of enumerative questions. One can wonder whether there are results similar to Theorem 1 involving other statistics and perhaps longer patterns. Note that before [28], concepts concerning permutations regarded as words (such as pattern avoidance) and concepts that regard the permutation as a bijection from [n] to itself (such as fixed points) had never been studied together. From this point of view theorems like this one seem to open a new line of results. One possible next step would be to enumerate permutations avoiding a pattern of length 4 with respect to the number of fixed points. Let us mention also another question. In the bijection of Theorem 2 and in [10], the RSK correspondence appears quite naturally. It would be interesting to investigate if

5 RSK can be used to obtain a more general result on pattern avoidance, or if this theorem has some connection with representation theory of S n. A new approach to the study of permutation statistics and consecutive patterns has recently been developed by Mendes and Remmel [25, 24]. It uses relationships between symmetric functions as a tool to gather facts about the symmetric group. The idea is to define a homomorphism mapping the elementary symmetric functions to a polynomial ring (this method was introduced by Brenti), and then combinatorially construct a involution to cancel terms with opposite sign. In collaboration with Remmel I will try to use this approach to enumerate occurrences of consecutive patterns in permutations, for patterns for which the previously used methods fail. We have already obtained results for some new patterns. Commutative Algebra. Another problem that I am currently interested in, and in which I am planning to do further research, is the following conjecture of Neil White [35] from It is closely related to work in commutative algebra by Herzog and Hibi [19]. Suppose that M is a matroid on {1, 2,..., n} of rank d. Then the basis monomial ring of M is the subalgebra of the polynomial ring C[x 1, x 2,..., x n ] which is generated by the squarefree monomials i B x i, where B runs over all bases of M. We introduce a new unknown y B for each basis B, and we consider the polynomial ring S M = C[y B : B basis of M] in these unknowns. There is a canonical C-algebra homomorphism from S M onto the basis monomial ring of M. Let I M be the kernel of that homomorphism. White s conjecture states that for any matroid M, the toric ideal I M is generated by quadrics. The problem has a combinatorial formulation as well. We say that two multisets of bases of a matroid are equivalent if each element of the ground set appears the same number of times in both multisets. Two multisets of bases A 1 and A 2 are connected by a local move if A 2 is obtained from A 1 by taking two bases of A 1 and exchanging an element from one with an element from the other. Clearly, multisets connected by local moves are equivalent. In this setting, White s conjecture is equivalent to the fact that any two equivalent multisets of bases are connected by a sequence of local moves. In [19] the conjecture is generalized to discrete polymatroids, and shown to be equivalent to White s conjecture. In [32], Sturmfels shows that for varieties of Veronese type the toric ideal has a quadratic Gröbner basis. It follows that the conjecture holds for uniform matroids and uniform polymatroids. Herzog and Hibi [19] use the same idea to prove that it also holds for any discrete polymatroid whose set of bases satisfies the strong exchange property. I have recently proved that the conjecture is also true for a more general class of discrete polymatroids, which in general do not satisfy the strong exchange property. These polymatroids are those obtained as transversals of sets of the form {a i, a i + 1, a i + 2,..., b i }, with 1 = a 1 a 2 a k and b 1 b 2 b k = n, where each element j {1, 2,..., n} can appear at most s j times. Setting all s j = 1 we obtain a subclass of transversal matroids, and setting a i = 1, b i = n for all i we obtain the uniform ones. The next step will be to try to extend this result to all transversal matroids, and after that I will try to solve the question for graphic matroids. Computational Biology. During the last few months I have become particularly interested in the applications of combinatorics to computational biology. I have recently made some progress on a problem of Bernd Sturmfels and Lior Pachter, regarding the number of inference functions of a graphical model. A graphical model (or Bayesian network) is a statistical model used for inferring information about the genetic code, for comparing the genomes of different species, and for many other applications in computational biology. An inference function takes each observation to its corresponding explanation, which determines what are the most likely values 5

6 6 of the hidden random variables for a particular choice of the parameters. Examples of such functions are gene finding functions, which, given a DNA sequence, output where the exons and introns are located. In [26, Corollary 10] they prove that the number of inference functions scales at most exponentially in the complexity of the graphical model. I have recently been able to show the following. Theorem 6. The number of inference functions grows polynomially in the complexity of the graphical model. This important result opens up new possibilities, because it is now feasible to compute all the inference functions of a given model. Here is another problem in the field, in which I am currently working as well. Given two strings S and T of length n, an alignment is a pair of equal length strings A = (S, T ) where S (resp. T ) is obtained by inserting special space characters into S (resp. T ), in such a way that there is no position in which both S and T have spaces. A match is a position where S and T have the same character, a mismatch is a position where S and T have different characters, anda gap is a position in which one of S and T has a space. Let α and β be arbitrary positive parameters denoting mismatch and gap penalties respectively. Define the score of an alignment with w matches, x mismatches, and y gaps to be w xα yβ. Different choices of the parameters α and β yield different optimum alignments. For any pair of strings, the (α, β) plane is decomposed into convex polyhedral optimality regions such that for all the points in the interior of a region, the set of optimal alignments is the same. It was shown in [18] that the number of optimality regions is O(n 2/3 ). Giving bounds on the maximum number of optimality regions is important because they determine the running time of the algorithms used to build the decomposition of the parameter space induced by two sequences. All known such algorithms run in time proportional to the number of regions (see for example [26]). In [17] they prove that when the alphabet is unbounded, the exact upper bound is 3(n/2π) 2/3 +O(n 1/2 log n). However, the biologically meaningful case is when the alphabet has four letters A, C, G and T, because in this case the alignment is a comparison of two actual DNA sequences. All that is known for finite alphabets is a lower bound of Ω(n 1/2 ) on the number of regions for the binary alphabet. It is an open problem to determine whether there exist two binary sequences for which the plane is divided in up to Θ(n 2/3 ) regions. We conjecture that no such sequences exist. More precisely, our project is to prove that for the binary alphabet, O(n 1/2 ) is also an upper bound on the number of regions. Once we show this, the next step will be to find the exact bound for the four-letter alphabet. This is a summary of my current and future projects. I enjoy working in many areas of combinatorics, and to find applications of combinatorics to other fields. I am currently a member of the program on Hyperplane Arrangements and Applications at MSRI. This has given me the opportunity to interact with researchers from other areas such as algebra and topology, and to broaden my perspective by studying hyperplane arrangements from points of view other that the purely combinatorial one. I enjoy learning new problems where I can apply my skills. This spring I will be a member of the program on Probability, Algorithms and Statistical Physics at MSRI. And after that I will do a postdoctoral stay at the Institut Mittag-Leffler in Sweden during the months of May and June, as part of the special program in Algebraic Combinatorics. I am positive that these experiences will help me broaden my view of combinatorics even more, and give me new ideas for my research. I believe that a mathematician should keep an open mind and be able to work on different topics. I am excited to use an academic position to collaborate with faculty on their combinatorial and related problems.

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