Colored prufer codes and t-cores

Abstract

In section 1, a combinatorial bijection between k-edge colored trees and colored Prufer codes for labelled trees is established. This bijection gives a simple combinatorial proof for the number $k(n-2)!{nk-n \choose n-2}$ of k-edge colored trees with n vertices. We consider slightly different edge-coloring of labelled trees using colored Prufer codes. We also consider slightly different edge-coloring and slightly different vertex-coloring simultaneously of labelled trees using colored Prufer codes. In section 2, combinatorial proofs using Ferrers diagram are established in some cases of monotonicity conjecture of Stanton.