On Stahl's conjectures about the region distributions of bouquets

Abstract

AbstractLet denote the graph with one vertex and loops, and be the number of embeddings of with regions in an orientable surface. Stahl conjectured that both the mode and the median of the sequence are within one unit of the harmonic number . In this paper we will confirm the conjecture about the median and disprove the conjecture about the mode. We will show that the mode of the sequence belongs to the set , and each of these three values can be attained by infinitely many .