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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 92 (21 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 15 (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.
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 ..."
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Cited by 13 (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(n&sup2;) operations (rather than O(n^4) operations for the naive method). This new algorithm uses techniques from algorithmic, discrete geometry and combinatorics on words.
Coinductive Counting With Weighted Automata
, 2002
"... A general methodology is developed to compute the solution of a wide variety of basic counting problems in a uniform way: (1) the objects to be counted are enumerated by means of an infinite weighted automaton; (2) the automaton is reduced by means of the quantitative notion of stream bisimulation; ..."
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Cited by 13 (2 self)
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A general methodology is developed to compute the solution of a wide variety of basic counting problems in a uniform way: (1) the objects to be counted are enumerated by means of an infinite weighted automaton; (2) the automaton is reduced by means of the quantitative notion of stream bisimulation; (3) the reduced automaton is used to compute an expression (in terms of stream constants and operators) that represents the stream of all counts.
On the Stanley–Wilf limit of 4231avoiding permutations and a conjecture of Arratia
, 2006
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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 (2 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.
Enumeration of convex polyominoes using the ECO method
 in Discrete Models for Complex Systems, DMCS’03 , Michel Morvan an Éric Rémila (eds.), Discrete Mathematics and Theoretical Computer Science Proceedings AB, 103
, 2003
"... ECO is a method for the enumeration of classes of combinatorial objects based on recursive constructions of such classes. In the first part of this paper we present a construction for the class of convex polyominoes based on the ECO method. Then we translate this construction into a succession rule. ..."
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Cited by 7 (3 self)
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ECO is a method for the enumeration of classes of combinatorial objects based on recursive constructions of such classes. In the first part of this paper we present a construction for the class of convex polyominoes based on the ECO method. Then we translate this construction into a succession rule. The final goal of the paper is to determine the generating function of convex polyominoes according to the semiperimeter, and it is achieved by applying an idea introduced in [11].
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 7 (4 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