On Stahl's conjectures about the region distributions of bouquets

Abstract LetB ndenote the graph with one vertex andnloops, andb n , kbe the number of embeddings ofB nwithkregions in an orientable surface. Stahl conjectured that both the mode and the median of the sequence( b n , k : k ≥ 1 , 2 ∤( n − k ) )are within one unit of the harmonic numberH 2 n ≔ ∑ j = 1 2 n1 j. 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( b n , k : k ≥ 1 , 2 ∤( n − k ) )belongs to the set{ ⌊ H 2 n ⌋ − 1 , ⌊ H 2 n ⌋ , ⌊ H 2 n ⌋ + 1 }, and each of these three values can be attained by infinitely manyn.