A family of Bell transformations

TL;DR

This paper introduces a family of Bell polynomial-based sequence transformations, providing algebraic and combinatorial insights, inverse relations, and applications to enumerating combinatorial structures like Dyck paths and planar maps.

Contribution

It systematically studies Bell polynomial-based transformations, deriving functional equations, inverse relations, and demonstrating their use in enumerating various combinatorial objects.

Findings

Derived functional equations for generating functions.

Established inverse relations for the transformations.

Applied transformations to enumerate combinatorial structures.

Abstract

We introduce a family of sequence transformations, defined via partial Bell polynomials, that may be used for a systematic study of a wide variety of problems in enumerative combinatorics. This family includes some of the transformations listed in the paper by Bernstein & Sloane, now seen as transformations under the umbrella of partial Bell polynomials. Our goal is to describe these transformations from the algebraic and combinatorial points of view. We provide functional equations satisfied by the generating functions, derive inverse relations, and give a convolution formula. While the full range of applications remains unexplored, in this paper we show a glimpse of the versatility of Bell transformations by discussing the enumeration of several combinatorial configurations, including rational Dyck paths, rooted planar maps, and certain classes of permutations.