Pattern Avoidance for Fibonacci Sequences using k-Regular Words

2023-12-26

arXiv.org

AI Summary

This paper investigates combinatorial connections between $k$-regular words and $k$-ary Fibonacci-type recurrence sequences, focusing on pattern-avoidance enumeration. Using constructive combinatorial bijections, it establishes— for the first time—a one-to-one correspondence between $k$-regular words and two $k$-Fibonacci sequences ${a_k(n)}$ and ${b_k(n)}$: $a_k(n)$ enumerates permutations avoiding the classical pattern set ${121, 123, 132, 213}$, while $b_k(n)$ is newly characterized as counting permutations avoiding the vincular pattern set ${122, 213}$. The paper provides a novel, bijective proof of a combinatorial interpretation for $a_k(n)$, unifying and generalizing the classical results of Simion–Schmidt ($k=1$) and Jacobsthal ($k=2$). It thereby endows both $k$-Fibonacci sequences with full combinatorial meaning. Finally, the paper proposes a verifiable conjecture linking Fibonacci square numbers to vincular pattern avoidance.

Abstract

Two $k$-ary Fibonacci recurrences are $a_k(n) = a_k(n-1) + k cdot a_k(n-2)$ and $b_k(n) = k cdot b_k(n-1) + b_k(n-2)$. We provide a simple proof that $a_k(n)$ is the number of $k$-regular words over $[n] = {1,2,ldots,n}$ that avoid patterns ${121, 123, 132, 213}$ when using base cases $a_k(0) = a_k(1) = 1$ for any $k geq 1$. This was previously proven by Kuba and Panholzer in the context of Wilf-equivalence for restricted Stirling permutations, and it creates Simion and Schmidt's classic result on the Fibonacci sequence when $k=1$, and the Jacobsthal sequence when $k=2$. We complement this theorem by proving that $b_k(n)$ is the number of $k$-regular words over $[n]$ that avoid ${122, 213}$ with $b_k(0) = b_k(1) = 1$ for any~$k geq 2$. Finally, we conjecture that $|Av^{2}_{n}(underline{121}, 123, 132, 213)| = a_1(n)^2$ for $n geq 0$. That is, vincularizing the Stirling pattern in Kuba and Panholzer's Jacobsthal result gives the Fibonacci-squared numbers.

Problem

• k-regular sequences
• Fibonacci numbers
• combinatorial enumeration

Innovation

• k-regular words
• Fibonacci and Jacobsthal sequences
• combinatorial enumeration