Enumerative combinatorics
A combinatorial proof of the multivariable Lagrange inversion formula
Journal of Combinatorial Theory Series A
Incidence algebra antipodes and Lagrange inversion in one and several variables
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
Regular Article: Enumeration of m-Ary Cacti
Advances in Applied Mathematics
Combinatorial Enumeration
Counting trees using symmetries
Journal of Combinatorial Theory Series A
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Goulden and Kulkarni (J. Combin. Theory Ser. A 80 (2) (1997) 295) give a bijective proof of an arborescent form of the Good-Lagrange multivariable inversion formula. This formula was first stated explicitly by Bender and Richmond (Electron. J. Combin. 5 (1) (1998) 4pp) but is implicit in Goulden and Kulkarni (1997). In this paper, we propose two new simple bijective proofs of this formula and we illustrate the interest of these proofs by applying them to the enumeration and random generation of colored rooted trees and rooted m-ary cacti.