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15
Generating Trees and the Catalan and Schröder Numbers
 DEPARTMENT OF MATHEMATICS, STOCKHOLMS UNIVERSITET, S106 91
, 1995
"... A permutation 2 Sn avoids the subpattern iff has no subsequence having all the same pairwise comparisons as , and we write 2 Sn ( ). We present a new bijective proof of the wellknown result that jS n (123)j = jS n (132)j = c n , the nth Catalan number. A generalization to forbidden pattern ..."
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Cited by 103 (3 self)
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A permutation 2 Sn avoids the subpattern iff has no subsequence having all the same pairwise comparisons as , and we write 2 Sn ( ). We present a new bijective proof of the wellknown result that jS n (123)j = jS n (132)j = c n , the nth Catalan number. A generalization to forbidden patterns of length 4 gives an asymptotic formula for the vexillary permutations. We settle a conjecture of Shapiro and Getu that jS n (3142; 2413)j = s n\Gamma1 , the Schröder number, and characterize the dequesortable permutations of Knuth, also counted by s n\Gamma1 .
Synchronous binarization for machine translation
 In Proc. HLTNAACL
, 2006
"... Systems based on synchronous grammars and tree transducers promise to improve the quality of statistical machine translation output, but are often very computationally intensive. The complexity is exponential in the size of individual grammar rules due to arbitrary reorderings between the two langu ..."
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Cited by 35 (10 self)
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Systems based on synchronous grammars and tree transducers promise to improve the quality of statistical machine translation output, but are often very computationally intensive. The complexity is exponential in the size of individual grammar rules due to arbitrary reorderings between the two languages, and rules extracted from parallel corpora can be quite large. We devise a lineartime algorithm for factoring syntactic reorderings by binarizing synchronous rules when possible and show that the resulting rule set significantly improves the speed and accuracy of a stateoftheart syntaxbased machine translation system. 1
Binarization of Synchronous ContextFree Grammars
"... Systems based on synchronous grammars and tree transducers promise to improve the quality of statistical machine translation output, but are often very computationally intensive. The complexity is exponential in the size of individual grammar rules due to arbitrary reorderings between the two langu ..."
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Cited by 24 (5 self)
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Systems based on synchronous grammars and tree transducers promise to improve the quality of statistical machine translation output, but are often very computationally intensive. The complexity is exponential in the size of individual grammar rules due to arbitrary reorderings between the two languages. We develop a theory of binarization for synchronous contextfree grammars and present a lineartime algorithm for binarizing synchronous rules when possible. In our largescale experiments, we found that almost all rules are binarizable and the resulting binarized rule set significantly improves the speed and accuracy of a stateoftheart syntaxbased machine translation system. We also discuss the more general, and computationally more difficult, problem of finding good parsing strategies for nonbinarizable rules, and present an approximate polynomialtime algorithm for this problem. 1.
On the Complexity of Generating Optimal Plans with Cross Products (Extended Abstract)
 In PODS Conference
, 1997
"... In modern advanced database systems the optimizer is often faced with the problem of finding optimal evaluation strategies for queries involving a large number of joins. Examples are queries generated by deductive database systems and path expressions in objectoriented database systems. The be ..."
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Cited by 21 (1 self)
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In modern advanced database systems the optimizer is often faced with the problem of finding optimal evaluation strategies for queries involving a large number of joins. Examples are queries generated by deductive database systems and path expressions in objectoriented database systems. The best plan can be found in the very large search space of bushy trees where plans are allowed to contain cross products. A general question arises: For which (sub) problems can we expect to find polynomial algorithms generating the best plan? We attack this question from both ends of the spectrum. First, we show that we cannot expect to find any polynomial algorithm for any subproblem as long as optimal bushy trees are to be generated. More specifically, we show that the problem is NPhard independent of the query graph.
Simple permutations and algebraic generating functions
 In preparation
, 2006
"... A simple permutation is one that never maps a nontrivial contiguous set of indices contiguously. Given a set of permutations that is closed under taking subpermutations and contains only finitely many simple permutations, we provide a framework for enumerating ..."
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Cited by 20 (9 self)
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A simple permutation is one that never maps a nontrivial contiguous set of indices contiguously. Given a set of permutations that is closed under taking subpermutations and contains only finitely many simple permutations, we provide a framework for enumerating
A Combinatorial Interpretation of the Area of Schröder Paths
 Electronic J. Combinatorics
, 1999
"... An elevated Schroder path is a lattice path that uses the steps (1, 1), (1, 1), and (2, 0), that begins and ends on the xaxis, and that remains strictly above the xaxis otherwise. The total area of elevated Schroder paths of length 2n + 2 satisfies the recurrence f n+1 =6f nf n1 ,n#2, with ..."
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Cited by 6 (2 self)
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An elevated Schroder path is a lattice path that uses the steps (1, 1), (1, 1), and (2, 0), that begins and ends on the xaxis, and that remains strictly above the xaxis otherwise. The total area of elevated Schroder paths of length 2n + 2 satisfies the recurrence f n+1 =6f nf n1 ,n#2, with the initial conditions f 0 =1,f 1 =7. A combinatorial interpretation of this recurrence is given, by first introducing sets of unrestricted paths whose cardinality also satisfies the recurrence relation and then establishing a bijection between the set of these paths and the set of triangles constituting the total area of elevated Schroder paths. 1 Introduction In the plane ZZ ZZ, we will use lattice paths with three steps types: a rise step defined by (1, 1), a fall step defined by (1, 1), and a horizontal step defined by (2, 0). A Schroder path is a sequence of rise, fall and horizontal steps running from (0, 0) to (2n,0) and remaining weakly above the xaxis. These paths are coun...
A Bijection Between Permutations and Floorplans, and its Applications
"... A floorplan represents the relative relations between modules on an integrated circuit. Floorplans are commonly classified as slicing, mosaic, or general. Separable and Baxter permutations are classes of permutations that can be defined in terms of forbidden subsequences. It is known that the number ..."
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Cited by 4 (0 self)
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A floorplan represents the relative relations between modules on an integrated circuit. Floorplans are commonly classified as slicing, mosaic, or general. Separable and Baxter permutations are classes of permutations that can be defined in terms of forbidden subsequences. It is known that the number of slicing floorplans equals the number of separable permutations and that the number of mosaic floorplans equals the number of Baxter permutations [17]. We present a simple and efficient bijection between Baxter permutations and mosaic floorplans with applications to integrated circuits design. Moreover, this bijection has two additional merits: (1) It also maps between separable permutations and slicing floorplans; and (2) It suggests enumerations of mosaic floorplans according to various structural parameters.
Permutations, Parenthesis Words, and Schröder Numbers ∗
, 2005
"... A different proof for the following result due to J. West is given: the Schröder number sn−1 equals the number of permutations on {1, 2,..., n} that avoid the pattern (3, 1, 4, 2) and its dual (2, 4, 1, 3). ..."
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Cited by 3 (0 self)
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A different proof for the following result due to J. West is given: the Schröder number sn−1 equals the number of permutations on {1, 2,..., n} that avoid the pattern (3, 1, 4, 2) and its dual (2, 4, 1, 3).
The REVERE Project: Experiments with the application of probabilistic NLP to Systems Engineering
 in Proceedings of 5th International Conference on Applications of Natural Language to Information Systems (NLDB'2000
, 2000
"... We investigate the number of different ways in which a rectangle containing a set of n noncorectilinear points can be partitioned into smaller rectangles by n (nonintersecting) segments, such that every point lies on a segment. We show that when the relative order of the points forms a separable pe ..."
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Cited by 2 (0 self)
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We investigate the number of different ways in which a rectangle containing a set of n noncorectilinear points can be partitioned into smaller rectangles by n (nonintersecting) segments, such that every point lies on a segment. We show that when the relative order of the points forms a separable permutation, the number of rectangulations is exactly the (n + 1)st Baxter number. We also show that no matter what the order of the points is, the number of guillotine rectangulations is always the nth Schröder number, and the total number of rectangulations is O(20 n /n 4). 1
Enumeration of Factorizable MultiDimensional Permutations
"... A ddimensional permutation is a sequence of d + 1 permutations with the leading element being the identity permutation. It can be viewed as an alignment structure across d+1 sequences, or visualized as the result of permuting n hypercubes of (d+1) dimensions. We study the hierarchical decomposition ..."
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Cited by 1 (1 self)
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A ddimensional permutation is a sequence of d + 1 permutations with the leading element being the identity permutation. It can be viewed as an alignment structure across d+1 sequences, or visualized as the result of permuting n hypercubes of (d+1) dimensions. We study the hierarchical decomposition of ddimensional permutations. We show that when d ≥ 2, the ratio between nondecomposable or simple dpermutations and all dpermutations approaches 1. We also prove that the growth rate of the number of dpermutations that can be factorized into kary branching trees approaches � � k d e as k grows. 1