Q-grammars and wall polyominoes

Duchon, Philippe

doi:10.1007/bf01608790

Cited by 32 publications

(66 citation statements)

References 10 publications

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“…Given the functional equation of the staircase polygon half-perimeter and area generating function, this result is a special case of Theorem 4 below, which is stated in [23].…”

Section: Remarks ([31]mentioning

confidence: 96%

“…iii) For models with counting parameters different from area, or for models in higher dimensions, a modified assumption holds, with the growth condition i) being replaced by m > cn k 0 and n > cm k 1 for appropriate values of k 0 and k 1 . The condition p m,n = 0 for n > cm k appears also as a so-called rank k condition [23].…”

Section: Polygon Modelsmentioning

confidence: 99%

“…Definition 3 (Algebraic q-difference equation [23,78]). An algebraic qdifference equation is an equation of the form…”

Section: Algebraic Q-difference Equationsmentioning

confidence: 99%

“…In fact, p m (q) is a polynomial in q. The growth of its degree in m is not larger than cm 2 for some positive constant c. Such counting parameters are also called rank 2 parameters [23]. For polygons models, such a bound is expected, since the area of a polygon grows at most quadratically with its perimeter.…”

Section: Algebraic Q-difference Equationsmentioning

confidence: 99%

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Limit Distributions and Scaling Functions

Richard

2009

Polygons, Polyominoes and Polycubes

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We discuss the asymptotic behaviour of models of lattice polygons, mainly on the square lattice. In particular, we focus on limiting area laws in the uniform perimeter ensemble where, for fixed perimeter, each polygon of a given area occurs with the same probability. We relate limit distributions to the scaling behaviour of the associated perimeter and area generating functions, thereby providing a geometric interpretation of scaling functions. To a major extent, this article is a pedagogic review of known results.

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“…Given the functional equation of the staircase polygon half-perimeter and area generating function, this result is a special case of Theorem 4 below, which is stated in [23].…”

Section: Remarks ([31]mentioning

confidence: 96%

Section: Polygon Modelsmentioning

confidence: 99%

“…Definition 3 (Algebraic q-difference equation [23,78]). An algebraic qdifference equation is an equation of the form…”

Section: Algebraic Q-difference Equationsmentioning

confidence: 99%

Section: Algebraic Q-difference Equationsmentioning

confidence: 99%

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Limit Distributions and Scaling Functions

Richard

2009

Polygons, Polyominoes and Polycubes

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“…See Louchard's works [16], [17]. The Airy distribution occurs in related problems relative to the analysis of dynamic data structures (like stacks under a Markovian model and priority queues under Knuth's model), to the busy period of an M/G/1 queue, and to the area of various classes of polyominoes; see [12] on the dynamic aspects and [8] for applications to polyominoes. 2.…”

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confidence: 99%

Analytic Variations on the Airy Distribution

2001

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Abstract. The Airy distribution (of the "area" type) occurs as a limit distribution of cumulative parameters in a number of combinatorial structures, like path length in trees, area below walks, displacement in parking sequences, and it is also related to basic graph and polyomino enumeration. We obtain curious explicit evaluations for certain moments of the Airy distribution, including moments of orders −1, −3, −5, etc., as well as + 1 3 , − 5 3 , − 11 3 , etc. and − 7 3 , − 13 3 , − 19 3 , etc. Our proofs are based on integral transforms of the Laplace and Mellin type and they rely essentially on "non-probabilistic" arguments like analytic continuation. A byproduct of this approach is the existence of relations between moments of the Airy distribution, the asymptotic expansion of the Airy function Ai(z) at +∞, and power symmetric functions of the zeros −α k of Ai(z).Key Words. Brownian excursion area, Airy function, Parking problem, Linear probing hashing.Introduction. For probabilists, the Airy distribution considered here is nothing but the distribution of the area under the Brownian excursion. The name is derived from the connection between Brownian motion and the Airy function, a fact discovered around 1980 by several authors; see [16] and [20]. For combinatorialists and theoretical computer scientists, this Airy distribution (of the "area type") arises in a surprising diversity of contexts like parking allocations, hashing tables, trees, discrete random walks, mergesorting, etc.The most straightforward description of the Airy distribution is by its moments themselves defined by a simple nonlinear recurrence. We follow here the notations and the normalization of [11].

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Random maps, coalescing saddles, singularity analysis, and Airy phenomena

Banderier

Flajolet

Schaeffer

et al. 2001

Random Struct Algorithms

128

224

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A considerable number of asymptotic distributions arising in random combinatorics and analysis of algorithms are of the exponential-quadratic type, that is, Gaussian. We exhibit a class of "universal" phenomena that are of the exponential-cubic type, corresponding to distributions that involve the Airy function. In this article, such Airy phenomena are related to the coalescence of saddle points and the confluence of singularities of generating functions. For about a dozen types of random planar maps, a common Airy distribution (equivalently, a stable law of exponent 3 2 ) describes the sizes of cores and of largest (multi)connected components. Consequences include the analysis and fine optimization of random generation algorithms for a multiple connected planar graphs. Based on an extension of the singularity analysis framework suggested by the Airy case, the article also presents a general classification of compositional schemas in analytic combinatorics.

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Q-grammars and wall polyominoes

Cited by 32 publications

References 10 publications

Limit Distributions and Scaling Functions

Limit Distributions and Scaling Functions

Analytic Variations on the Airy Distribution

Random maps, coalescing saddles, singularity analysis, and Airy phenomena

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