For a labeled tree on the vertex set $set{1,2,ldots,n}$, the local direction of each edge $(i,j)$ is from $i$ to $j$ if $i<j$. For a rooted tree, there is also a natural global direction of edges towards the root. The number of edges pointing to a vertex is called its indegree. Thus the local (resp. global) indegree sequence $lambda = 1^{e_1}2^{e_2} ldots$ of a tree on the vertex set $set{1,2,ldots,n}$ is a partition of $n-1$. We construct a bijection from (unrooted) trees to rooted trees such that the local indegree sequence of a (unrooted) tree equals the global indegree sequence of the corresponding rooted tree. Combining with a Prufer-like code for rooted labeled trees, we obtain a bijective proof of a recent conjecture by Cotterill and also solve two open problems proposed by Du and Yin. We also prove a $q$-multisum binomial coefficient identity which confirms another conjecture of Cotterill in a very special case.
تحميل البحث