A combinatorial proof of the multivariable Lagrange inversion formula
Journal of Combinatorial Theory Series A
Some combinatorial problems associated with products of conjugacy classes of the symmetric group
Journal of Combinatorial Theory Series A
European Journal of Combinatorics
Multivariable Lagrange inversion, Gessel-Viennot cancellation, and the Matrix tree theorem
Journal of Combinatorial Theory Series A
Journal of Combinatorial Theory Series A
Regular Article: Enumeration of m-Ary Cacti
Advances in Applied Mathematics
Theoretical Computer Science - Random generation of combinatorial objects and bijective combinatorics
Analytic Combinatorics
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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-Melou 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.