| | |
| | |
Stat |
Members: 3645 Articles: 2'506'133 Articles rated: 2609
26 April 2024 |
|
| | | |
|
Article overview
| |
|
A refinement of Cayley's formula for trees | Ira M. Gessel
; Seunghyun Seo
; | Date: |
24 Jul 2005 | Subject: | Combinatorics MSC-class: 05A15 | math.CO | Abstract: | A proper vertex of a rooted tree with totally ordered vertices is a vertex that is less than all its proper descendants. We count several kinds of labeled rooted trees and forests by the number of proper vertices. Our results are all expressed in terms of the polynomials P_n(a,b,c)= c(a+(n-1)b+c)(2a+(n-2)b+c)...((n-1)a+b+c) which reduce to (n+1)^{n-1} for a=b=c=1. Our study of proper vertices was motivated by A. Postnikov’s hook length formula for binary trees (arXiv:math.CO/0507163), which was also proved by W. Y. C. Chen and L. L. M. Yang (arXiv:math.CO/0507163) and generalized by R. R. X. Du and F. Liu (arXiv:math.CO/0501147). Our approach gives a new proof of Du and Liu’s results and gives new hook length formulas. We also find an interpretation of the polynomials P_n(a,b,c) in terms of parking functions: we count parking functions according to the number of cars that park in their preferred parking spaces. | Source: | arXiv, math.CO/0507497 | Services: | Forum | Review | PDF | Favorites |
|
|
No review found.
Did you like this article?
Note: answers to reviews or questions about the article must be posted in the forum section.
Authors are not allowed to review their own article. They can use the forum section.
browser Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)
|
| |
|
|
|
| News, job offers and information for researchers and scientists:
| |