Local height in weighted Dyck models of random walks and the variability of the number of coalescent histories for caterpillar-shaped gene trees and species trees

Abstract

We examine combinatorial parameters of three models of random lattice walks with up and down steps. In particular, we study the height $$y_i$$ measured after i up-steps in a random weighted Dyck path of size (semilength) n. For a fixed integer $$w \in \{0,1,2\}$$ , the considered weighting scheme assigns to each Dyck path of size n a weight $$\prod _{i=1}^n y_i^w$$ that depends on the height of the up-steps of the path. We investigate the expected value $${\text{E}}_n(y_i)$$ of the height $$y_i$$ in a random weighted Dyck path of size n, providing exact formulas for $${\text{E}}_n(y_i)$$ and $${\text{E}}_n(y_i^2)$$ when $$w=0,1$$ , and estimates of the mean of $$y_i$$ for $$w=2$$ . Denoting by $$i^*(n)$$ the position i where $${\text{E}}_n(y_i)$$ reaches its maximum $${\text{m}}(n)$$ , our calculations indicate that, when n becomes large, the pair $$\big (i^*(n), {\text{m}}(n)\big )$$ grows like $$\big ( n/2 , 2\sqrt{n/\pi } \big )$$ if $$w=0$$ , $$\big ( 3n/4 , n/2 \big )$$ if $$w=1$$ , and $$\big ( (9+\sqrt{17})n/16 , (1+\sqrt{17})n/8 \big )$$ if $$w=2$$ . These results also contribute to the study of the variability of the number of “coalescent histories”: structures used in models of gene tree evolution to encode the combinatorially different configurations of a gene tree topology along the branches of a species tree. Relationships with other combinatorial and algebraic structures, such as alternating permutations and Meixner polynomials, are also discussed.