2022/08/09 by Gao, Zhicheng · 1 citation
#05C88 #05C89 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2208.04801
Let Cn,g be the number of rooted cubic maps with 2n vertices on the orientable surface of genus g. We show that the sequence (Cn,g:g≥ 0) is asymptotically normal with mean and variance asymptotic to (1/2)(n-ln n) and (1/4)ln n, respectively. We derive an asymptotic expression for Cn,g when (n-2g)/ln n lies in any closed subinterval of (0,2). Using rotation systems and Bender's theorem about generating functions with fast-growing coefficients, we derive simple asymptotic expressions for the numbers of rooted regular maps, disregarding the genus. In particular, we show that the number of rooted cubic maps with 2n vertices, disregarding the genus, is asymptotic to \frac3π n!6n.