Three \'Etudes on a sequence transformation pipeline

TL;DR

This paper investigates a sequence transformation pipeline that connects rational generating functions to permutation-based combinatorial sequences, revealing links to geometric objects like associahedra and permutahedra via the $ ext{ extbackslash T}$ transform.

Contribution

It introduces a novel sequence transformation pipeline and demonstrates its connections to important combinatorial and geometric structures.

Findings

Sequences with rational generating functions can be transformed into permutation-based families.

The $ ext{ extbackslash T}$ transform links these sequences to simplicial objects.

Many number triangles relate to associahedra and permutahedra.

Abstract

We study a sequence transformation pipeline that maps certain sequences with rational generating functions to permutation-based sequence families of combinatorial significance. Many of the number triangles we encounter can be related to simplicial objects such as the associahedron and the permutahedron. The linkages between these objects is facilitated by the use of the previously introduced $\mathcal{T}$ transform.