Mathematics – Combinatorics
Scientific paper
2011-04-04
Mathematics
Combinatorics
26 pages, 12 figures
Scientific paper
By rewriting the famous hook-content formula it easily follows that there are $\prod\limits_{1 \le i < j \le n} \frac{k_j - k_i + j -i}{j-i}$ semistandard tableaux of shape $(k_n,k_{n-1},...,k_1)$ with entries in $\{1,2,...,n\}$ or, equivalently, Gelfand-Tsetlin patterns with bottom row $(k_1,...,k_n)$. In this article we introduce certain sequences of labeled trees, the signed enumeration of which is also given by this formula. In these trees, vertices as well as edges are labeled, the crucial condition being that each edge label lies between the vertex labels of the two endpoints of the edge. This notion enables us to give combinatorial explanations of the shifted antisymmetry of the formula and its polynomiality. Furthermore, we propose to develop an analog approach of combinatorial reasoning for monotone triangles and explain how this may lead to a combinatorial understanding of the alternating sign matrix theorem.
No associations
LandOfFree
Sequences of labeled trees related to Gelfand-Tsetlin patterns does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Sequences of labeled trees related to Gelfand-Tsetlin patterns, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Sequences of labeled trees related to Gelfand-Tsetlin patterns will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-318041