On Han's Hook Length Formulas for Trees
نویسندگان
چکیده
Recently, Han obtained two hook length formulas for binary trees and asked for combinatorial proofs. One of Han’s formulas has been generalized to k-ary trees by Yang. Sagan has found a probabilistic proof of Yang’s extension. We give combinatorial proofs of Yang’s formula for k-ary trees and the other formula of Han for binary trees. Our bijections are based on the structure of k-ary trees associated with staircase labelings.
منابع مشابه
Hook Length Formulas for Trees by Han's Expansion
Recently Han obtained a general formula for the weight function corresponding to the expansion of a generating function in terms of hook lengths of binary trees. In this paper, we present 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, Ges...
متن کاملDiscovering Hook Length Formulas by an Expansion Technique
— We introduce a hook length expansion technique and explain how to discover old and new hook length formulas for partitions and plane trees. The new hook length formulas for trees obtained by our method can be proved rather easily, whereas those for partitions are much more difficult and some of them still remain open conjectures. We also develop a Maple package HookExp for computing the hook ...
متن کاملA Unifying Approach for Proving Hook-Length Formulas for Weighted Tree Families
We propose an expansion technique for weighted tree families, which unifies and extends recent results on hook-length formulas of trees obtained by Han [10], Chen et al. [3], and Yang [19]. Moreover, the approach presented is used to derive new hook-length formulas for tree families, where several hook-functions in the corresponding expansion formulas occur in a natural way. Furthermore we cons...
متن کاملNew hook length formulas for binary trees
— We find two new hook length formulas for binary trees. The particularity of our formulas is that the hook length h v appears as an exponent. Consider the set B (n) of all binary trees with n vertices. It is well-known that the cardinality of B (n) is equal to the Catalan number (see, e.g., [9, p.220]): (1) T ∈B(n) 1 = 1 n + 1 2n n. For each vertex v of a binary tree T ∈ B (n) the hook length ...
متن کاملTwo kinds of hook length formulas for complete m-ary trees
Yidong Sun and Huajun Zhang Department of Mathematics, Dalian Maritime University, 116026 Dalian, P.R. China Department of Mathematics, Zhejiang Normal University, 321004 Jinhua, P.R. China [email protected], [email protected] Abstract. In this paper, we define two kinds of hook length for internal vertices of complete m-ary trees, and deduce their corresponding hook length formulas, which...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Electr. J. Comb.
دوره 18 شماره
صفحات -
تاریخ انتشار 2011