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A " hybrid method " , dedicated to asymptotic coefficient extraction in combinatorial generating functions, is presented, which combines Darboux's method and singularity analysis theory. This hybrid method applies to functions that remain of moderate growth near the unit circle and satisfy suitable smoothness assumptions—this, even in the case when the unit… (More)

Let m be a positive integer, and p n (m) the proportion of permutations of the symmetric group S n that admit an m-th root. Calculating the exponential generating function of these permutations, we show the following asymptotic formula p n (m) ∼ n→+∞ π m n 1−ϕ(m)/m , where ϕ is the Euler function and π m an explicit constant.

- Nicolas Pouyanne
- 2006

Pólya processes are natural generalization of Pólya-Eggenberger urn models. This article presents a new approach of their asymptotic behaviour via moments, based on the spectral decomposition of a suitable finite difference transition operator on polynomial functions. Especially, it provides new results for large processes (a Pólya process is called small… (More)

Common assumptions on the source producing the words inserted in a suffix trie with n leaves lead to a log n height and saturation level. We provide an example of a suffix trie whose height increases faster than a power of n and another one whose saturation level is negligible with respect to log n. Both are built from VLMC (Variable Length Markov Chain)… (More)

Pólya processes are natural generalization of Pólya-Eggenberger urn models. This article presents a new approach of their asymptotic behaviour via moments, based on the spectral decomposition of a suitable finite difference transition operator on polynomial functions. Especially, it provides new results for large processes (a Pólya process is called small… (More)

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