Disposition Polynomials and Plane Trees

Abstract

We define the disposition polynomial $R_{m}(x_1, x_2, ..., x_n)$ as $\prod_{k=0}^{m-1}(x_1+x_2+...+x_n+k)$. When $m=n-1$, this polynomial becomes the generating function of plane trees with respect to certain statistics as given by Guo and Zeng. When $x_i=1$ for $1\leq i\leq n$, $R_{m}(x_1, x_2, ..., x_n)$ reduces to the rising factorial $n(n+1)... (n+m-1)$. Guo and Zeng asked the question of finding a combinatorial proof of the formula for the generating function of plane trees with respect to the number of younger children and the number of elder children. We find a combinatorial interpretation of the disposition polynomials in terms of the number of right-to-left minima of each linear order in a disposition. Then we establish a bijection between plane trees on $n$ vertices and dispositions from ${1, 2,..., n-1}$ to ${1, 2,..., n}$ in the spirit of the Pr\"ufer correspondence. It gives an answer to the question of Guo and Zeng, and it also provides an answer to another question of Guo and Zeng concerning an identity on the plane tree expansion of a polynomial introduced by Gessel and Seo.