A $\emph{leader}$ of a tree $T$ is a vertex which has no smaller descendants in $T$. Gessel and Seo showed that
$\displaystyle\sum_{T \in T_n}u^{(\# of leaders in T)}c^{(degree of 1 in T)} = u P_{n-1}(1,u,cu),$
which is a generalization of Cayley``s formula, where $T_n$ is the set of trees on $[n]$ and
$P_n(a,b,c) = c \displaystyle\prod_{i=1}^{n-1}(ia+(n-i)b+c).$
Using a variation of the Pr$\ddot{u}$fer code which is called an $\em{RP-code}$, we give a simple bijective proof of Gessel and Seo``s formula.
A car in a parking function is called $\emph{lucky}$ if it succeeds to park in its preferred space. Gessel and Seo also showed that
$\displaystyle\sum_{P \in PF_n} u^{(\# of lucky cars in P)} = P_n(1,u,u),$
but this proof was not combinatorial. We construct the bijection $\varphi$ from forests to parking functions and give a bijective proof of it. We generalize it further using the bijection $\varphi$.

- Advisors
- Kim, Dong-Su
*researcher*; 김동수*researcher*

- Description
- 한국과학기술원 : 수리과학과,

- Publisher
- 한국과학기술원

- Issue Date
- 2007

- Identifier
- 268707/325007 / 020015159

- Language
- eng

- Description
학위논문(박사) - 한국과학기술원 : 수리과학과, 2007. 8, [ v, 42 p. ]

- Keywords
refinement; numbers; trees; parking functions; 세분화; 개수; 수형도; 주차 함수; refinement; numbers; trees; parking functions; 세분화; 개수; 수형도; 주차 함수

- Appears in Collection
- MA-Theses_Ph.D.(박사논문)

- Files in This Item
- There are no files associated with this item.

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