2012/06/04 by Olivier Bernardi, Alejandro H. Morales, Bernardi, Olivier +1
Computer Science · Mathematics · Physics and Astronomy · #Combinatorics (math.CO) #FOS: Mathematics #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #Topological and Geometric Data Analysis #math.CO
paper · pdf · doi:10.48550/arxiv.1206.0598
17 pages, 7 figures
openalex publication_date 2012/06/04 · arxiv created 2013/04/05 · arxiv updated 2013/04/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove a new formula for the generating function of multitype Cayley trees counted according to their degree distribution. Using this formula we recover and extend several enumerative results about trees. In particular, we extend some results by Knuth and by Bousquet-Mélou and Chapuy about embedded trees. We also give a new proof of the multivariate Lagrange inversion formula. Our strategy for counting trees is to exploit symmetries of refined enumeration formulas: proving these symmetries is easy, and once the symmetries are proved the formulas follow effortlessly. We also adapt this strategy to recover an enumeration formula of Goulden and Jackson for cacti counted according to their degree distribution.