A Refinement of Cayley's Formula for Trees

@article{Gessel2006ARO,
  title={A Refinement of Cayley's Formula for Trees},
  author={Ira M. Gessel and Seunghyun Seo},
  journal={Electr. J. Comb.},
  year={2006},
  volume={11}
}
A proper vertex of a rooted tree with totally ordered vertices is a vertex that is less than all its proper descendants. We count several kinds of labeled rooted trees and forests by the number of proper vertices. Our results are all expressed in terms of the polynomials P n (a, b, c) = c n−1 i=1 (ia + (n − i)b + c), which reduce to to (n + 1) n−1 for a = b = c = 1. Our study of proper vertices was motivated by Postnikov's hook length formula (n + 1) n−1 = n! 2 n T v 1 + 1 h(v) , where the sum… CONTINUE READING

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