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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.
Partially directed paths in a wedge
 Journal of Combinatorial Theory, Series A
"... The enumeration of lattice paths in wedges poses unique mathematical challenges. These models are not translationally invariant, and the absence of this symmetry complicates both the derivation of a functional recurrence for the generating function, and solving for it. In this paper we consider a mo ..."
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Cited by 12 (1 self)
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The enumeration of lattice paths in wedges poses unique mathematical challenges. These models are not translationally invariant, and the absence of this symmetry complicates both the derivation of a functional recurrence for the generating function, and solving for it. In this paper we consider a model of partially directed walks from the origin in the square lattice confined to both a symmetric wedge defined by Y = ±pX, and an asymmetric wedge defined by the lines Y = pX and Y = 0, where p> 0 is an integer. We prove that the growth constant for all these models is equal to 1+ √ 2, independent of the angle of the wedge. We derive functional recursions for both models, and obtain explicit expressions for the generating functions when p = 1. From these we find asymptotic formulas for the number of partially directed paths of length n in a wedge when p = 1. The functional recurrences are solved by a variation of the kernel method, which we call the “iterated kernel method”. This method appears to be similar to the obstinate kernel method used by BousquetMélou (see, for example, references [5, 6]). This method requires us to consider iterated compositions of the roots of the kernel. These compositions turn out to be surprisingly tractable, and we are able to find simple explicit expressions for them. However, in spite of this, the generating functions turn out to be similar in form to Jacobi θfunctions, and have natural boundaries on the unit circle.
Walks on the slit plane
, 2002
"... In the first part of this paper, we enumerate exactly walks on the square lattice that start from the origin, but otherwise avoid the halfline H = f(k; 0); k ^ 0g. We call them walks on the slit plane. We count them by their length, and by the coordinates of their endpoint. The corresponding three ..."
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Cited by 11 (2 self)
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In the first part of this paper, we enumerate exactly walks on the square lattice that start from the origin, but otherwise avoid the halfline H = f(k; 0); k ^ 0g. We call them walks on the slit plane. We count them by their length, and by the coordinates of their endpoint. The corresponding three variable generating function is algebraic of degree 8. Moreover, for any point (i; j), the length generating function for walks of this type ending at (i; j) is also algebraic, of degree 2 or 4, and involves the famous Catalan numbers. Our method is based on the solution of a functional equation, established via a simple combinatorial argument. It actually works for more general models, in which walks take their steps in a finite subset of Z 2 satisfying two simple conditions. The corresponding generating functions are always algebraic.
The siteperimeter of bargraphs
 Adv. in Appl. Math
"... The siteperimeter enumeration of polyominoes that are both column and rowconvex is a well understood problem that always yields algebraic generating functions. Counting more general families of polyominoes is a far more difficult problem. Here we enumerate (by their siteperimeter) the simplest fa ..."
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Cited by 9 (3 self)
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The siteperimeter enumeration of polyominoes that are both column and rowconvex is a well understood problem that always yields algebraic generating functions. Counting more general families of polyominoes is a far more difficult problem. Here we enumerate (by their siteperimeter) the simplest family of polyominoes that are not fully convex — bargraphs. The generating function we obtain is of a type that, to our knowledge, has never been encountered so far in the combinatorics literature: a qseries into which an algebraic series has been substituted. 1
Counting permutations with no long monotone subsequence via generating trees and the kernel method
 J. Algebraic Combin
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Enumeration of (k, 2)noncrossing partitions
 Discrete Math
"... A set partition is said to be (k, d)noncrossing if it avoids the pattern 12 · · ·k12 · · ·d. We find an explicit formula for the ordinary generating function of the number of (k, d)noncrossing partitions of {1, 2,..., n} when d = 1, 2. ..."
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Cited by 8 (2 self)
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A set partition is said to be (k, d)noncrossing if it avoids the pattern 12 · · ·k12 · · ·d. We find an explicit formula for the ordinary generating function of the number of (k, d)noncrossing partitions of {1, 2,..., n} when d = 1, 2.