# On Restricted Ternary Words and Insets

## TL;DR

This paper explores the combinatorial properties of insets related to restricted ternary words, providing new interpretations of known sequences and deriving generating functions with applications to various lattice and sequence configurations.

## Contribution

It introduces new combinatorial interpretations of integer sequences via insets and restricted ternary words, and derives generating functions for these structures.

## Key findings

- New combinatorial interpretations of Chebyshev, Fibonacci, and Delannoy numbers.
- Derived three generating functions with two variables constant.
- Enumerated 40 combinatorial configurations related to the words.

## Abstract

We investigate combinatorial properties of a kind of insets we defined in an earlier paper, interpreting them now in terms of restricted ternary words. This allows us to give new combinatorial interpretations of a number of known integer sequences, namely the coefficients of Chebyshev polynomials of both kinds, Fibonacci numbers, Delannoy numbers, asymmetric Delannoy numbers, Sulanke numbers, coordinating sequences for some cubic lattices, crystal ball sequences for some cubic lattices, and others. We also obtain several new properties of said insets. In particular, we derive three generating functions when two of three variables are constant. At the end, we state 40 combinatorial configurations counted by our words.