Non-P-recursiveness of numbers of matchings or linear chord diagrams with many crossings

Klazar, Martin

doi:10.1016/s0196-8858(02)00528-6

Cited by 23 publications

(23 citation statements)

References 40 publications

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“…If we temporarily forget the parameter z codifying crossings, we know that [Kla03] and proved that it is not D-finite (i.e. the solution of a differential equation with polynomial coefficients).…”

Section: Connected Matchingsmentioning

confidence: 99%

Analytic combinatorics of chord and hyperchord diagrams with k crossings

Pilaud

Rué

2014

Advances in Applied Mathematics

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Using methods from Analytic Combinatorics, we study the families of perfect matchings, partitions, chord diagrams, and hyperchord diagrams on a disk with a prescribed number of crossings. For each family, we express the generating function of the configurations with exactly k crossings as a rational function of the generating function of crossing-free configurations. Using these expressions, we study the singular behavior of these generating functions and derive asymptotic results on the counting sequences of the configurations with precisely k crossings. Limiting distributions and random generators are also studied. Contents 42 4.2. Configurations on surfaces of higher genus 46Acknowlegdments 47

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Section: Connected Matchingsmentioning

confidence: 99%

Analytic combinatorics of chord and hyperchord diagrams with k crossings

Pilaud

Rué

2014

Advances in Applied Mathematics

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“…The numbers of strongly-irreducible words and strongly-irreducible palindromes have been previously studied within different contexts (see, for example, [17]). In [10] strongly-irreducible palindromes are called ''connected linked diagrams'' and the following formula is obtained. …”

Section: Rigid Vertices All Classes G Nmentioning

confidence: 99%

Four-regular graphs with rigid vertices associated to DNA recombination

Burns¹,

Dolzhenko²,

Jonoska³

et al. 2013

Discrete Applied Mathematics

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a b s t r a c tGenome rearrangement and homological recombination processes have been modeled by Angeleska et al. [A. Angeleska, N. Jonoska, M. Saito, DNA recombinations through assembly graphs, Discrete Appl. Math. 157 (2009) 3020-3037] as 4-regular spacial graphs with rigid vertices, called assembly graphs. These graphs can also be represented by double occurrence words called assembly words. The rearranged DNA segments are modeled by certain types of paths in the assembly graphs called polygonal paths. The minimum number of polygonal paths visiting all vertices in a graph is called an assembly number for the graph.In this paper, we give formulas for counting certain types of assembly graphs and assembly words. Some of these formulas produce sequences not previously reported at the Online Encyclopedia of Integer Sequences (http://oeis.org). We provide a sharp upper bound for the number of polygonal paths in Hamiltonian sets of polygonal paths, and present a family of graphs that achieves this bound. We investigate changes in the assembly numbers as a result of graph compositions. Finally, we introduce a polynomial invariant for assembly graphs and show properties of this invariant.

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“…The first few values of the free cumulants of the Gaussian distribution are This sequence of the numbers of irreducible diagrams of 2n nodes has been well-studied from a combinatorial point of view, see, e.g., [41,40]; it appears for example as sequence A000699 in Sloane's Encyclopedia of Integer Sequences [37]. For a recent bibliography of this sequence see [28] where it is shown that the sequence is not holonomic, i.e., it does not satisfy a linear recurrence with polynomial coefficients. However, positivity questions for this sequence have never been considered.…”

Section: Combinatorial Considerationsmentioning

confidence: 99%

The normal distribution is ⊞-infinitely divisible

Belinschi

Bożejko

Lehner

et al. 2011

Advances in Mathematics

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We prove that the classical normal distribution is infinitely divisible with respect to the free additive convolution. We study the Voiculescu transform first by giving a survey of its combinatorial implications and then analytically, including a proof of free infinite divisibility. In fact we prove that a subfamily Askey-Wimp-Kerov distributions are freely infinitely divisible, of which the normal distribution is a special case.The Cauchy-Stieltjes transform of a measure µ on the real line is defined bywhen µ is a positive measure, this function maps C + into the lower half-plane. For details we refer to the excellent book of Akhieser [1]. For c = 0, we have µ 0 = γ, the normal distribution [27], and one can easily check that one can extend this family continuously to c = −1 by letting µ −1 = δ 0 , the probability giving mass one to {0}. This family, introduced in [4], plays an important role in [27]; we shall call its members the Askey-Wimp-Kerov distributions. It will turn out from our proof that {µ c : c ∈ [−1, 0]} are freely infinitely divisible. Numerical computations show that for several values of c > 0, µ c is not freely infinitely divisible. Numerical evidence seems also to indicate that µ c is classicaly infinitely divisible only when c = 0 or c = −1. An interesting interpolation between the normal and the semicircle law was constructed by Bryc, Dembo and Jiang [18] and further investigated by Buchholz [19]. This leads to a generalized Brownian motion, given by a weight function (0 < b < 1)

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Non-P-recursiveness of numbers of matchings or linear chord diagrams with many crossings

Abstract: In order to prove irrationality of √ 2 by using only decimal expansions (and not fractions), we develop in detail a model of real numbers based on infinite decimals and arithmetic operations with them.

Cited by 23 publications

References 40 publications

Analytic combinatorics of chord and hyperchord diagrams with k crossings

Analytic combinatorics of chord and hyperchord diagrams with k crossings

Four-regular graphs with rigid vertices associated to DNA recombination

The normal distribution is ⊞-infinitely divisible

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