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37
Generating Functions for Generating Trees
 PROCEEDINGS OF 11TH FORMAL POWER SERIES AND ALGEBRAIC COMBINATORICS
, 1999
"... Certain families of combinatorial objects admit recursive descriptions in terms of generating trees: each node of the tree corresponds to an object, and the branch leading to the node encodes the choices made in the construction of the object. Generating trees lead to a fast computation of enumerati ..."
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Cited by 68 (18 self)
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Certain families of combinatorial objects admit recursive descriptions in terms of generating trees: each node of the tree corresponds to an object, and the branch leading to the node encodes the choices made in the construction of the object. Generating trees lead to a fast computation of enumeration sequences (sometimes, to explicit formulae as well) and provide efficient random generation algorithms. We investigate the links between the structural properties of the rewriting rules defining such trees and the rationality, algebraicity, or transcendence of the corresponding generating function.
Finitely labeled generating trees and restricted permutations
 Journal of Symbolic Computation
, 2006
"... Abstract. Generating trees are a useful technique in the enumeration of various combinatorial objects, particularly restricted permutations. Quite often the generating tree for the set of permutations avoiding a set of patterns requires infinitely many labels. Sometimes, however, this generating tre ..."
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Cited by 12 (5 self)
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Abstract. Generating trees are a useful technique in the enumeration of various combinatorial objects, particularly restricted permutations. Quite often the generating tree for the set of permutations avoiding a set of patterns requires infinitely many labels. Sometimes, however, this generating tree needs only finitely many labels. We characterize the finite sets of patterns for which this phenomenon occurs. We also present an algorithm — in fact, a special case of an algorithm of Zeilberger — that is guaranteed to find such a generating tree if it exists. 1.
Motzkin Paths and Reduced Decompositions for Permutations with Forbidden Patterns
, 2003
"... We obtain a characterization of (321; 3 142)avoiding permutations in terms of their canonical reduced decompositions. This characterization is used to construct a bijection for a recent result that the number of (321; 3 142)avoiding permutations of length n equals the nth Motzkin number, d ..."
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Cited by 8 (3 self)
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We obtain a characterization of (321; 3 142)avoiding permutations in terms of their canonical reduced decompositions. This characterization is used to construct a bijection for a recent result that the number of (321; 3 142)avoiding permutations of length n equals the nth Motzkin number, due to Gire, and further studied by Barcucci, Del Lungo, Pergola, Pinzani and Guibert. Similarly, we obtain a characterization of (231; 4 132)avoiding permutations. For these two classes, we show that the number of descents of a permutation equals the number of up steps on the corresponding Motzkin path. Moreover, we nd a relationship between the inversion number of a permutation and the area of the corresponding Motzkin path.
An Algorithm for Deciding If a Polyomino Tiles the Plane by Translations
"... We explain a fast algorithm to decide if a polyomino tiles the plane by translations. More precisely, if the polyomino has a boundary word of length n then the algorithm decides if the polyomino tiles the plane by translations in O(n²) operations (rather than O(n^4) operations for the naive met ..."
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Cited by 7 (2 self)
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We explain a fast algorithm to decide if a polyomino tiles the plane by translations. More precisely, if the polyomino has a boundary word of length n then the algorithm decides if the polyomino tiles the plane by translations in O(n²) operations (rather than O(n^4) operations for the naive method). This new algorithm uses techniques from algorithmic, discrete geometry and combinatorics on words.
Generating functions for the area below some lattice paths
 In Discrete random walks (Paris, 2003), Discrete Math. Theor. Comput. Sci. Proc., AC
, 2003
"... We study some lattice paths related to the concept of generating trees. When the matrix associated to this kind of trees is a Riordan array D ¡ d ¡ t¢¤ £ h ¡ t¢¥ ¢ , we are able to find the generating function for the total area below these paths expressed in terms of the functions d ¡ t ¢ and h ¡ ..."
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Cited by 5 (1 self)
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We study some lattice paths related to the concept of generating trees. When the matrix associated to this kind of trees is a Riordan array D ¡ d ¡ t¢¤ £ h ¡ t¢¥ ¢ , we are able to find the generating function for the total area below these paths expressed in terms of the functions d ¡ t ¢ and h ¡ t¢¤¦
Binary words excluding a pattern and proper Riordan arrays
 Discrete Mathematics
"... We study the relation between binary words excluding a pattern and proper Riordan arrays. In particular, we prove necessary and sufficient conditions under which the number of words counted with respect to the number of zeroes and ones bits are related to proper Riordan arrays. We also give formulas ..."
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Cited by 5 (3 self)
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We study the relation between binary words excluding a pattern and proper Riordan arrays. In particular, we prove necessary and sufficient conditions under which the number of words counted with respect to the number of zeroes and ones bits are related to proper Riordan arrays. We also give formulas for computing the generating functions (d(x), h(x)) defining the Riordan array. 1
A survey on certain pattern problems
"... Abstract. The paper contains all the definitions and notations needed to understand the results concerning the field dealing with occurrences of patterns in permutations and words. Also, this paper includes a historical overview on the results obtained in this subject. The authors tried to collect a ..."
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Cited by 4 (3 self)
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Abstract. The paper contains all the definitions and notations needed to understand the results concerning the field dealing with occurrences of patterns in permutations and words. Also, this paper includes a historical overview on the results obtained in this subject. The authors tried to collect all the currently existing references to the papers directly related to the subject. Moreover, a number of basic approaches to study the pattern problems are discussed.
ECO method and hillfree generalized Motzkin paths
 SÉMINAIRE LOTHARINGIEN DE COMBINATOIRE 46 (2001), ARTICLE B46B
, 2001
"... In this paper we study the class of generalized Motzkin paths with no hills and prove some of their combinatorial properties in a bijective way; as a particular case we have the Fine numbers, enumerating Dyck paths with no hills. Using the ECO method, we define a recursive construction for Dyck path ..."
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Cited by 4 (0 self)
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In this paper we study the class of generalized Motzkin paths with no hills and prove some of their combinatorial properties in a bijective way; as a particular case we have the Fine numbers, enumerating Dyck paths with no hills. Using the ECO method, we define a recursive construction for Dyck paths such that the number of local expansions performed on each path depends on the number of its hills. We then extend this construction to the set of generalized Motzkin paths.