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A combinatorial bijection between k-edge colored trees and colored Prüfer codes for labelled trees is established. This bijection gives a simple combinatorial proof for the number k(n − 2)! nk−n n−2 of k-edge colored trees with n vertices.

For a labeled tree on the vertex set {1, 2,. .. , 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 λ = 1 e1 2 e2. .. of a tree on the vertex… (More)

BACKGROUND
Cells coordinate their metabolism, proliferation, and cellular communication according to environmental cues through signal transduction. Because signal transduction has a primary role in cellular processes, many experimental techniques and approaches have emerged to discover the molecular components and dynamics that are dependent on cellular… (More)

A leader of a tree T on [n] is a vertex which has no smaller descendants in T. Gessel and Seo showed that T ∈T n u (# of leaders in T) c (degree of 1 in T) = uP 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 n−1 i=1 ia + (n − i)b + c. Using a variation of the Prüfer code which is… (More)

It is well known that the (−1)-evaluation of the enumerator polynomials of permutations (resp. derangements) by the number of excedances gives rise to tangent numbers (resp. secant numbers). Recently, two distinct q-analogues of the latter result have been discovered by Foata and Han, and Josuat-Vergès, respectively. In this paper, we will prove some… (More)

- Heesung Shin, Jiang Zeng
- 2009

A further correspondence between (bc, ¯ b)-parking functions and (bc, ¯ b)-forests Abstract. For a fixed sequence of n positive integers (a, ¯ b) := (a, b, b,. .. , b), an (a, ¯ b)-parking function of length n is a sequence (p1, p2,. .. , pn) of positive integers whose nondecreasing rearrangement q1 ≤ q2 ≤ · · · ≤ qn satisfies qi ≤ a + (i − 1)b for any i =… (More)

- HEESUNG SHIN
- 2008

In 1980, G. Kreweras [Kre80] gave a recursive bijection between forests and parking functions. In this paper we construct a nonrecursive bijection from forests onto parking functions, which answers a question raised by R. Stanley [Sta07, Exercise 6.4]. As a by-product, we obtain a bijective proof of Gessel and Seo's formula for lucky statistic on parking… (More)