A Hybrid of Darboux's Method and Singularity Analysis in Combinatorial Asymptotics

Flajolet, Philippe; Fusy, Éric; Gourdon, Xavier; Panario, Daniel; Pouyanne, Nicolas

doi:10.37236/1129

Cited by 24 publications

(38 citation statements)

References 29 publications

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“…The first term of the right-hand side of (7) represents the tree on one node, the second term represents all other trees as explained in the preceding paragraph, and the third term is the correction term for trees in which the two subtrees of the root are identical.…”

Section: 2mentioning

confidence: 99%

Isomorphism and Symmetries in Random Phylogenetic Trees

Bóna

Flajolet

2009

Journal of Applied Probability

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Abstract. The probability that two randomly selected phylogenetic trees of the same size are isomorphic is found to be asymptotic to a decreasing exponential modulated by a polynomial factor. The number of symmetrical nodes in a random phylogenetic tree of large size obeys a limiting Gaussian distribution, in the sense of both central and local limits. The probability that two random phylogenetic trees have the same number of symmetries asymptotically obeys an inverse square-root law. Precise estimates for these problems are obtained by methods of analytic combinatorics, involving bivariate generating functions, singularity analysis, and quasi-powers approximations.

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Section: 2mentioning

confidence: 99%

Isomorphism and Symmetries in Random Phylogenetic Trees

Bóna

Flajolet

2009

Journal of Applied Probability

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“…But k≥1 σ k x k /k is the weighted generating function of cycles, and we can apply the exponential formula. Thus P σ,x is a straightforward weighted generalization of the Boltzmann measure on labelled objects studied in [6,10]. We also retain the formulas from [6, Thm 2.1] for the expected size and the variance of the size of the objects chosen according to this measure.…”

Section: Boltzmann Samplingmentioning

confidence: 99%

Profiles of Permutations

Lugo

2009

Electron. J. Combin.

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This paper develops an analogy between the cycle structure of, on the one hand, random permutations with cycle lengths restricted to lie in an infinite set S with asymptotic density σ and, on the other hand, permutations selected according to the Ewens distribution with parameter σ. In particular we show that the asymptotic expected number of cycles of random permutations of [n] with all cycles even, with all cycles odd, and chosen from the Ewens distribution with parameter 1/2 are all 1 2 log n + O(1), and the variance is of the same order. Furthermore, we show that in permutations of [n] chosen from the Ewens distribution with parameter σ, the probability of a random element being in a cycle longer than γn approaches (1 − γ) σ for large n. The same limit law holds for permutations with cycles carrying multiplicative weights with average σ. We draw parallels between the Ewens distribution and the asymptotic-density case and explain why these parallels should exist using permutations drawn from weighted Boltzmann distributions.where c i (π) is the number of cycles of length i in π.2000 Mathematics Subject Classification. 05A15, 05A16, 60C05.

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For G a finite group, κ(G) is the probability that that σ, τ ∈ G are conjugate, when σ and τ are chosen independently and uniformly at random. Recently, Blackburn et al (2012) gave an elementary proof that κ(Sn) ∼ A/n 2 as n → ∞ for some constant A -a result which was first proved by Flajolet et al (2006). In this paper, we extend the elementary methods of Blackburn et al to show that κ(An) ∼ B/n 2 as n → ∞ for some constant B, given explicitly in this paper.

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confidence: 71%

Asymptotic behaviour of the conjugacy probability of the alternating group

Steinmetz

Whybrow

2015

Beitr Algebra Geom

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For G a finite group, κ(G) is the probability that σ, τ ∈ G are conjugate, when σ and τ are chosen independently and uniformly at random. Recently, Blackburn et al. (J Lond Math Soc 86(2):755-778, 2012) gave an elementary proof that κ(S n ) ∼ A/n 2 as n → ∞ for some constant A-a result which was first proved by Flajolet et al. (Electron J Comb 13(1):35, 2006). In this paper, we extend the elementary methods of Blackburn et al. to show that κ(A n ) ∼ B/n 2 as n → ∞ for some constant B, given explicitly in this paper.

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A Hybrid of Darboux's Method and Singularity Analysis in Combinatorial Asymptotics

Cited by 24 publications

References 29 publications

Isomorphism and Symmetries in Random Phylogenetic Trees

Isomorphism and Symmetries in Random Phylogenetic Trees

Profiles of Permutations

Asymptotic behaviour of the conjugacy probability of the alternating group

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