$p$-th order generalized Fibonacci cubes and maximal cubes in Fibonacci $p$-cubes
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| Format: | Preprint |
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2025
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| _version_ | 1866909698877292544 |
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| author | Mollard, Michel |
| author_facet | Mollard, Michel |
| contents | The Fibonacci cube $Γ_n$ is the subgraph of the hypercube $Q_n$ induced by vertices with no consecutive 1s. We study a one parameter generalization, p-th order Fibonacci cubes $Γ^{(p)}_n$, which are subgraphs of $Q_n$ induced by strings without p consecutive 1s. We show the link between vertices of $Γ^{(p)}_n$ and compositions of integers with parts in $\{1, 2, \ldots , p\}$. Among other eumerative properties, we study the order, size and cube polynomial of $Γ^{(p)}_n$ as well as their generating functions. Many of the given expressions are similar to those for Fibonacci cubes, where the $p$-nomial coefficients play the role of binomial coefficients. We also show that maximal induced hypercubes in Fibonacci $p$-cubes $Γ^p_n$ , another generalization of Fibonacci cubes, are connected to vertices of $(p + 1)$-th order Fibonacci cubes. We use this link to determine the maximal cube polynomial of Fibonacci $p$-cubes. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_16387 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $p$-th order generalized Fibonacci cubes and maximal cubes in Fibonacci $p$-cubes Mollard, Michel Combinatorics The Fibonacci cube $Γ_n$ is the subgraph of the hypercube $Q_n$ induced by vertices with no consecutive 1s. We study a one parameter generalization, p-th order Fibonacci cubes $Γ^{(p)}_n$, which are subgraphs of $Q_n$ induced by strings without p consecutive 1s. We show the link between vertices of $Γ^{(p)}_n$ and compositions of integers with parts in $\{1, 2, \ldots , p\}$. Among other eumerative properties, we study the order, size and cube polynomial of $Γ^{(p)}_n$ as well as their generating functions. Many of the given expressions are similar to those for Fibonacci cubes, where the $p$-nomial coefficients play the role of binomial coefficients. We also show that maximal induced hypercubes in Fibonacci $p$-cubes $Γ^p_n$ , another generalization of Fibonacci cubes, are connected to vertices of $(p + 1)$-th order Fibonacci cubes. We use this link to determine the maximal cube polynomial of Fibonacci $p$-cubes. |
| title | $p$-th order generalized Fibonacci cubes and maximal cubes in Fibonacci $p$-cubes |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2507.16387 |