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17
Two Nonholonomic Lattice walks in the Quarter Plane
, 2007
"... We present two classes of random walks restricted to the quarter plane whose generating function is not holonomic. The nonholonomy is established using the iterated kernel method, a recent variant of the kernel method. This adds evidence to a recent conjecture on combinatorial properties of walks w ..."
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We present two classes of random walks restricted to the quarter plane whose generating function is not holonomic. The nonholonomy is established using the iterated kernel method, a recent variant of the kernel method. This adds evidence to a recent conjecture on combinatorial properties of walks with holonomic generating functions. The method also yields an asymptotic expression for the number of walks of length n.
Overview of some general results in combinatorial enumeration
, 2008
"... This survey article is devoted to general results in combinatorial enumeration. The first part surveys results on growth of hereditary properties of combinatorial structures. These include permutations, ordered and unordered graphs and hypergraphs, relational structures, and others. The second part ..."
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This survey article is devoted to general results in combinatorial enumeration. The first part surveys results on growth of hereditary properties of combinatorial structures. These include permutations, ordered and unordered graphs and hypergraphs, relational structures, and others. The second part advertises five topics in general enumeration: 1. counting lattice points in lattice polytopes, 2. growth of contextfree languages, 3. holonomicity (i.e., Precursiveness) of numbers of labeled regular graphs, 4. frequent occurrence of the asymptotics cn −3/2 r n and 5. ultimate modular periodicity of numbers of MSOLdefinable structures. 1
Counting planar graphs and related families of graphs
 In Surveys in combinatorics 2009, 169–210
, 2009
"... In this article we survey recent results on the asymptotic enumeration of planar graphs and, more generally, graphs embeddable in a fixed surface and graphs defined in terms of excluded minors. We also discuss in detail properties of random planar graphs, such as the number of edges, the degree dist ..."
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In this article we survey recent results on the asymptotic enumeration of planar graphs and, more generally, graphs embeddable in a fixed surface and graphs defined in terms of excluded minors. We also discuss in detail properties of random planar graphs, such as the number of edges, the degree distribution or the size of the largest kconnected component. Most of the results we present use generating functions and analytic tools.
Culminating paths
, 2008
"... Let a and b be two positive integers. A culminating path is a path of Z 2 that starts from (0, 0), consists of steps (1, a) and (1, −b), stays above the xaxis and ends at the highest ordinate it ever reaches. These paths were first encountered in bioinformatics, in the analysis of similarity search ..."
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Cited by 5 (2 self)
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Let a and b be two positive integers. A culminating path is a path of Z 2 that starts from (0, 0), consists of steps (1, a) and (1, −b), stays above the xaxis and ends at the highest ordinate it ever reaches. These paths were first encountered in bioinformatics, in the analysis of similarity search algorithms. They are also related to certain models of Lorentzian gravity in theoretical physics. We first show that the language on a two letter alphabet that naturally encodes culminating paths is not contextfree. Then, we focus on the enumeration of culminating paths. A step by step approach, combined with the kernel method, provides a closed form expression for the generating function of culminating paths ending at a (generic) height k. In the case a = b, we derive from this expression the asymptotic behaviour of the number of culminating paths of length n. When a> b, we obtain the asymptotic behaviour by a simpler argument. When a < b, we only determine the exponential growth of the number of culminating paths. Finally, we study the uniform random generation of culminating paths via various methods. The rejection approach, coupled with a symmetry argument, gives an algorithm that is linear when a ≥ b, with no precomputation stage nor nonlinear storage required. The choice of the best algorithm is not as clear when a < b. An elementary recursive approach yields a linear algorithm after a precomputation stage involving O(n 3) arithmetic operations, but we also present some alternatives that may be more efficient in practice.
ANALYTIC ASPECTS OF THE SHUFFLE PRODUCT
, 2008
"... There exist very lucid explanations of the combinatorial origins of rational and algebraic functions, in particular with respect to regular and context free languages. In the search to understand how to extend these natural correspondences, we find that the shuffle product models many key aspects o ..."
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Cited by 4 (0 self)
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There exist very lucid explanations of the combinatorial origins of rational and algebraic functions, in particular with respect to regular and context free languages. In the search to understand how to extend these natural correspondences, we find that the shuffle product models many key aspects of Dfinite generating functions, a class which contains algebraic. We consider several different takes on the shuffle product, shuffle closure, and shuffle grammars, and give explicit generating function consequences. In the process, we define a grammar class that models Dfinite generating functions.
Automatic enumeration of regular objects
 J. Integer Sequences
"... Abstract. We describe a framework for systematic enumeration of families combinatorial structures which possess a certain regularity. More precisely, we describe how to obtain the differential equations satisfied by their generating series. These differential equations are then used to determine the ..."
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Abstract. We describe a framework for systematic enumeration of families combinatorial structures which possess a certain regularity. More precisely, we describe how to obtain the differential equations satisfied by their generating series. These differential equations are then used to determine the initial counting sequence and for asymptotic analysis. The key tool is the scalar product for symmetric functions and that this operation preserves Dfiniteness.
A NOTE ON NATURALLY EMBEDDED TERNARY TREES
"... Abstract. In this note we consider ternary trees naturally embedded in the plane in a deterministic way. The root has position zero, or in other words label zero, and the three children of a node with position j ∈ Z have positions j − 1, j, and j + 1. We derive the generating function of embedded te ..."
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Abstract. In this note we consider ternary trees naturally embedded in the plane in a deterministic way. The root has position zero, or in other words label zero, and the three children of a node with position j ∈ Z have positions j − 1, j, and j + 1. We derive the generating function of embedded ternary trees where all internal nodes have labels less than or equal to j, with j ∈ N. Furthermore, we study the generating function of the number of ternary trees of size n with a given number of internal nodes with label j. Moreover, we discuss generalizations of this counting problem to several labels at the same time. We also study a refinement of the depth of the external node of rank s, with 0 ≤ s ≤ 2n, by keeping track of the left, center, and right steps on the unique path from the root to the external node. The 2n + 1 external nodes of a ternary tree are ranked from the left to the right according to an inorder traversal of the tree. Finally, we discuss generalizations of the considered enumeration problems to embedded dary trees. 1.
COEFFICIENTS OF ALGEBRAIC FUNCTIONS: FORMULAE AND ASYMPTOTICS
, 2012
"... This paper studies the coefficients of algebraic functions. First, we recall the toolessknown fact that these coefficients fn always a closed form. Then, we study their asymptotics, known to be of the type fn â¼ CA n n Î±. When the function is a power series associated to a contextfree grammar ..."
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This paper studies the coefficients of algebraic functions. First, we recall the toolessknown fact that these coefficients fn always a closed form. Then, we study their asymptotics, known to be of the type fn â¼ CA n n Î±. When the function is a power series associated to a contextfree grammar, we solve a folklore conjecture: the appearing critical exponents Î± belong to a subset of dyadic numbers, and we initiate the study the set of possible values for A. We extend what Philippe Flajolet called the DrmotaâLalleyâWoods theorem (which is assuring Î± = â3/2 as soon as a âdependency graph â associated to the algebraic system defining the function is strongly connected): We fully characterize the possible singular behaviors in the nonstrongly connected case. As a corollary, it shows that certain lattice paths and planar maps can not be generated by a contextfree grammar (i.e., their generating function is not Nalgebraic). We give examples of Gaussian limit laws (beyond the case of the DrmotaâLalleyâWoods theorem), and examples of non Gaussian limit laws. We then extend our work to systems involving nonpolynomial entire functions (nonstrongly connected systems, fixed points of entire function with positive coefficients). We end by discussing few algorithmic aspects.
Exactly solved models of polyominoes and polygons
, 2008
"... This chapter deals with the exact enumeration of certain classes of selfavoiding polygons and polyominoes on the square lattice. We present three general approaches that apply to many classes of polyominoes. The common principle to all of them is a recursive description of the polyominoes which the ..."
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This chapter deals with the exact enumeration of certain classes of selfavoiding polygons and polyominoes on the square lattice. We present three general approaches that apply to many classes of polyominoes. The common principle to all of them is a recursive description of the polyominoes which then translates into a functional equation satisfied by the generating function. The first approach applies to classes of polyominoes having a linear recursive structure and results in a rational generating function. The second approach applies to classes of polyominoes having an algebraic recursive structure and results in an algebraic generating function. The third approach, commonly called the Temperley method, is based on the action of adding a new column to the polyominoes. We conclude by discussing some open questions.