tailieunhanh - Báo cáo toán học: "Hook Length Formulas for Trees by Han’s Expansion"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Hook Length Formulas for Trees by Han’s Expansion. | Hook Length Formulas for Trees by Han s Expansion William . Chen1 Oliver . Gao2 and Peter L. Guo3 Center for Combinatorics LPMC-TJKLC Nankai University Tianjin 300071 . China 1 chen@ 2oliver@ 3lguo@ Submitted Mar 19 2009 Accepted May 9 2009 Published May 15 2009 Mathematics Subject Classifications 05A15 05A19 Abstract Recently Han obtained a general formula for the weight function corresponding to the expansion of a series in terms of hook lengths of binary trees. In this paper we present weight function formulas for k-ary trees plane trees plane forests labeled trees and forests. We also find appropriate generating functions which lead to unifications of the hook length formulas due to Du and Liu Han Gessel and Seo and Postnikov. Keywords hook length formulas for trees k-ary trees plane trees labeled trees. 1 Introduction Recently Han developed an expansion technique for deriving hook length formulas for binary trees. He has shown that given any formal power series f x with f 0 1 one can determine the weight function p n that leads to a hook length formula for binary trees. In this paper we extend Han s technique and obtain the expansion formulas for k-ary trees plane trees plane forests labeled trees and forests. We find appropriate generating functions that can be used to derive new hook length formulas some of which can be viewed as unifications of the formulas due to Du and Liu 3 Han 6 7 8 Gessel and Seo 5 . Let us give a quick review of the background and terminology. For a tree or a forest T the hook length of a vertex u of T denoted by hu is the number of descendants of u in T under the assumption that u is counted as a descendant of itself. The hook length multi-set H T of T is defined to be the multi-set of hook lengths of the vertices u of T. Clearly the above definition of hook length applies to all kinds of trees and forests such as binary trees plane trees labeled trees plane forests and forests.