2021/12/11 by Xin, Guoce, Zhang, Zihao
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2112.05936
For any integer m≥ 2 and r ∈ \1,…, m\, let fnm,r denote the number of n-Dyck paths whose peak's heights are im+r for some integer i. We find the generating function of fnm,r satisfies a simple algebraic functional equation of degree 2. The r=m case is particularly nice and we give a combinatorial proof. By using the Sulanke and Xin's continued fraction method, we calculate the Hankel determinants for fnm,r. The special case r=m of our result solves a conjecture proposed by Chien, Eu and Fu. We also enriched the class of eventually periodic Hankel determinant sequences.