2 research outputs found
Evaluating the generalized Buchshtab function and revisiting the variance of the distribution of the smallest components of combinatorial objects
Let and be the random variable representing the size of the
smallest component of a combinatorial object generated uniformly and randomly
over elements. A combinatorial object could be a permutation, a monic
polynomial over a finite field, a surjective map, a graph, and so on. It is
understood that a component of a permutation is a cycle, an irreducible factor
for a monic polynomial, a connected component for a graph, etc. Combinatorial
objects are categorized into parametric classes. In this article, we focus on
the exp-log class with parameter (permutations, derangements, polynomials
over finite field, etc.) and (surjective maps, -regular graphs,
etc.) The generalized Buchshtab function plays an important role
in evaluating probabilistic and statistical quantities. For , Theorem
from \cite{PanRic_2001_small_explog} stipulates that
for some and
sufficiently large . We revisit the evaluation of using
different methods: analytic estimation using tools from complex analysis,
numerical integration using Taylor expansions, and computation of the exact
distributions for using the recursive nature of the counting
problem. In general for any , Theorem from \cite{BenMasPanRic_2003}
connects the quantity for with the asymptotic
proportion of -objects with large smallest components. We show how the
coefficients of the Taylor expansion of for depends on those for . We use this family of coefficients to evaluate
.Comment: 16 pages, 2 tables, 15 reference