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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)
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)
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)
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)
Endometriosis, defined as the presence of endometrial cells outside of the uterine cavity, is a major cause of infertility and pelvic pain, afflicting more than 10% of reproductive age women. Endometriosis is a chronic inflammatory disease and lipopolysaccharide promotes the proliferation and invasion of endometriotic stromal cells. Cysteine-rich secretory(More)