q-Series Arising From The Study of Random Graphs

Abstract

This paper deals with q{series arising from the study of the transitiveclosure problem in random acyclic digraphs. In particularit presents an identity involvingdivisor generatingfunctions which allows to determine the asymptotic behavior of polynomials de ned by a general class of recursive equations, including the polynomials for the mean and the variance of the size of the transitive closure in random acyclic digraphs.

D⋅E⋅K=(1000)8

“…Prodinger's crisp note shows that Rice's integral representations are susceptible to the q-analogue process. Some of the sums studied by Prodinger, in fact, originate in the analysis of transitive closure algorithms in a natural randomness model for acyclic digraphs developed by Andrews, Crippa, and Simon [2]; others are evocative of the Prodingerian q-analogues that smoothly connect sequence statistics to permutation statistics [69].…”

“…It consists in seeking not only a characterization of average-case behavior (this was Knuth's original approach), but also the asymptotic forms of the probability distributions that arise in algorithms and random discrete structures. 2 The analytic school has developed its own tools for this, basing itself largely on the pioneering work of Ed Bender and his coauthors (see [3,4]). A nodal point is constituted by Hwang's thesis [31] defended in 1994 and many subsequent articles by Hwang [32][33][34].…”

The Asymptotic Behavior of Certain Birth Processes

“…In this case, the birth process has combinatorial interpretations in terms of heaps [12] and random graphs [1]. We recall the usual q-series notation…”

“…It was independently rediscovered by Andrews-Crippa-Simon [1] along with (1.16) in a less explicit form. In fact, as pointed out in [1] and Dilcher [3], (1.16) can be easily derived from (1.15).…”

“…It was independently rediscovered by Andrews-Crippa-Simon [1] along with (1.16) in a less explicit form. In fact, as pointed out in [1] and Dilcher [3], (1.16) can be easily derived from (1.15). Dilcher also discovered finite analogs of (1.16) which were subsequently generalized by Prodinger [8] and Fu-Lascoux [5].…”