23 research outputs found

    Counting Humps in Motzkin paths

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    In this paper we study the number of humps (peaks) in Dyck, Motzkin and Schr\"{o}der paths. Recently A. Regev noticed that the number of peaks in all Dyck paths of order nn is one half of the number of super Dyck paths of order nn. He also computed the number of humps in Motzkin paths and found a similar relation, and asked for bijective proofs. We give a bijection and prove these results. Using this bijection we also give a new proof that the number of Dyck paths of order nn with kk peaks is the Narayana number. By double counting super Schr\"{o}der paths, we also get an identity involving products of binomial coefficients.Comment: 8 pages, 2 Figure

    Counting Labelled Trees with Given Indegree Sequence

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    For a labelled tree on the vertex set [n]:={1,2,...,n}[n]:=\{1,2,..., n\}, define the direction of each edge ijij to be iβ†’ji\to j if i<ji<j. The indegree sequence of TT can be considered as a partition λ⊒nβˆ’1\lambda \vdash n-1. The enumeration of trees with a given indegree sequence arises in counting secant planes of curves in projective spaces. Recently Ethan Cotterill conjectured a formula for the number of trees on [n][n] with indegree sequence corresponding to a partition Ξ»\lambda. In this paper we give two proofs of Cotterill's conjecture: one is `semi-combinatorial" based on induction, the other is a bijective proof.Comment: 10 page

    Reduction of mm-Regular Noncrossing Partitions

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    In this paper, we present a reduction algorithm which transforms mm-regular partitions of [n]={1,2,...,n}[n]=\{1, 2, ..., n\} to (mβˆ’1)(m-1)-regular partitions of [nβˆ’1][n-1]. We show that this algorithm preserves the noncrossing property. This yields a simple explanation of an identity due to Simion-Ullman and Klazar in connection with enumeration problems on noncrossing partitions and RNA secondary structures. For ordinary noncrossing partitions, the reduction algorithm leads to a representation of noncrossing partitions in terms of independent arcs and loops, as well as an identity of Simion and Ullman which expresses the Narayana numbers in terms of the Catalan numbers