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Counting Labelled Trees with Given Indegree Sequence

Abstract

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 iji\to j if i<ji<j. The indegree sequence of TT can be considered as a partition λn1\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 λ\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

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