Central and local limit theorems applied to asymptotic enumeration IV: multivariate generating functions

AbstractFlajolet and Soria (1989, 1990) discussed some general combinatorial structures in which central limit theorem and exponential tail results hold. In this paper, we shall use Flajolet and Odlyzko's “transfer theorems” (1990) to extend Bender and Richmond's (1983) central and local limit theorems to a wider class of generating functions which will cover the above-mentioned combinatorial structures. The local limit theorem provides more accurate asymptotic information and implies the superexponential tail results

Multi-dimensional Boltzmann Sampling of Languages

This paper addresses the uniform random generation of words from a context-free language (over an alphabet of size $k$), while constraining every letter to a targeted frequency of occurrence. Our approach consists in a multidimensional extension of Boltzmann samplers \cite{Duchon2004}. We show that, under mostly \emph{strong-connectivity} hypotheses, our samplers return a word of size in $[(1-\varepsilon)n, (1+\varepsilon)n]$ and exact frequency in $\mathcal{O}(n^{1+k/2})$ expected time. Moreover, if we accept tolerance intervals of width in $\Omega(\sqrt{n})$ for the number of occurrences of each letters, our samplers perform an approximate-size generation of words in expected $\mathcal{O}(n)$ time. We illustrate these techniques on the generation of Tetris tessellations with uniform statistics in the different types of tetraminoes.Comment: 12p