Anti-palindromic compositions

TL;DR

This paper introduces anti-palindromic compositions, explores their enumeration via a connection to the tribonacci sequence, and defines a new q-analogue related to their equivalence classes.

Contribution

It establishes a novel link between anti-palindromic compositions and the tribonacci sequence, and introduces a new q-analogue for these compositions.

Findings

Number of anti-palindromic compositions relates to tribonacci sequence

A new q-analogue of Fibonacci sequence is defined

Connections to equivalence classes of compositions are established

Abstract

A palindromic composition of $n$ is a composition of $n$ which can be read the same way forwards and backwards. In this paper we define an anti-palindromic composition of $n$ to be a composition of $n$ which has no mirror symmetry amongst its parts. We then give a surprising connection between the number of anti-palindromic compositions of $n$ and the so-called tribonacci sequence, a generalization of the Fibonacci sequence. We conclude by defining a new q-analogue of the Fibonacci sequence, which is related to certain equivalence classes of anti-palindromic compositions