March 21

Rosena Du ( East China Normal University )

Counting Labelled Trees with Given Indegree Sequence

Abstract: For a labelled tree on the vertex set $[n]:=\{1,2,\ldots, n\}$, define the direction of each edge $ij$ to be $i\rightarrow j$ if $i < j$. The indegree sequence of $T$ can be considered as a partition $\lambda \vdash n-1$. 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 $[n]$ with indegree sequence corresponding to a partition $\lambda$. In this talk I will give two proofs of Cotterill's conjecture: one is "semi-combinatorial" based on induction, the other is a bijective proof. (This is joint work with Jingbin Yin.)