Pattern Avoidance for Fibonacci Sequences using $k$-Regular Words
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914381545078784 |
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| author | Downing, Emily Hartung, Elizabeth Lucido, Cody Williams, Aaron |
| author_facet | Downing, Emily Hartung, Elizabeth Lucido, Cody Williams, Aaron |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_16052 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Pattern Avoidance for Fibonacci Sequences using $k$-Regular Words Downing, Emily Hartung, Elizabeth Lucido, Cody Williams, Aaron Combinatorics Discrete Mathematics 05 (Primary) 68 (Secondary) G.2.1; G.4 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. |
| title | Pattern Avoidance for Fibonacci Sequences using $k$-Regular Words |
| topic | Combinatorics Discrete Mathematics 05 (Primary) 68 (Secondary) G.2.1; G.4 |
| url | https://arxiv.org/abs/2312.16052 |