New optima for the deletion shadow
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866909834547298304 |
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| author | Shaw, Benedict Randall |
| author_facet | Shaw, Benedict Randall |
| contents | For a family $\mathcal{F}$ of words of length $n$ drawn from an alphabet $A=[r]=\{1,\dots,r\}$, Danh and Daykin defined the deletion shadow $Δ\mathcal{F}$ as the family containing all words that can be made by deleting one letter of a word of $\mathcal{F}$. They asked, given the size of such a family, how small its deletion shadow can be, and answered this with a Kruskal-Katona type result when the alphabet has size $2$. However, Leck showed that no ordering can give such a result for larger alphabets. The minimal shadow has been known for families of size $s^n$, where the optimal family has form $[s]^n$. We give the minimal shadow for many intermediate sizes between these levels, showing that families of the form 'all words in $[s]^n$ in which the symbol $s$ appears at most $k$ times' are optimal. Our proof uses some fractional techniques that may be of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_01131 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | New optima for the deletion shadow Shaw, Benedict Randall Combinatorics 05D05 For a family $\mathcal{F}$ of words of length $n$ drawn from an alphabet $A=[r]=\{1,\dots,r\}$, Danh and Daykin defined the deletion shadow $Δ\mathcal{F}$ as the family containing all words that can be made by deleting one letter of a word of $\mathcal{F}$. They asked, given the size of such a family, how small its deletion shadow can be, and answered this with a Kruskal-Katona type result when the alphabet has size $2$. However, Leck showed that no ordering can give such a result for larger alphabets. The minimal shadow has been known for families of size $s^n$, where the optimal family has form $[s]^n$. We give the minimal shadow for many intermediate sizes between these levels, showing that families of the form 'all words in $[s]^n$ in which the symbol $s$ appears at most $k$ times' are optimal. Our proof uses some fractional techniques that may be of independent interest. |
| title | New optima for the deletion shadow |
| topic | Combinatorics 05D05 |
| url | https://arxiv.org/abs/2505.01131 |