Duplicial algebras and Lagrange inversion
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by
Jean-Christophe Novelli, Jean-Yves Thibon
2012
Abstract
We provide operadic interpretations for two Hopf subalgebras of the algebra
of parking functions. The Catalan subalgebra is identified with the free
duplicial algebra on one generator, and the Schr\"oder subalgebra is
interpreted by means of a new operad, which we call triduplicial.
The noncommutative Lagrange inversion formula is then interpreted in terms of
duplicial structures. The generic solution of the noncommutative inversion
problem appears as the formal sum of all parking functions. This suggests that
combinatorial generating functions derived by functional inversion should be
obtainable by evaluating a suitable character on this generic solution. This
idea is illustrated by means of the Narayana polynomials, of which we obtain
bivariate "super-analogues" by lifting to parking functions a classical
character of the algebra of symmetric functions. Other characters, such as
evaluation of symmetric functions on a binomial element, are also discussed.
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