Special restricted partition functions for the stable sheaf cohomology on flag varieties
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908532932083712 |
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| author | Cimpoeas, Mircea |
| author_facet | Cimpoeas, Mircea |
| contents | Let $\mathbf a:=(a_1,\ldots,a_r)$ be a sequence of positive integers, $d\geq 2$ and $j\geq 1$, some integers. We study the functions $p_{\mathbf a,d}(n):=$ the number of integer solutions $(x_1,\dots,x_r)$ of $\sum_{i=1}^r a_ix_i=n$, with $x_i\geq 0$ and $x_i \equiv 0,1(\bmod\;d)$, for all $1\leq i\leq r$, and $p_{\mathbf a,d}(n;j):=$ the number of $(x_1,\ldots,x_r)$ as above which satisfy also the condition $\sum_{i=1}^r \left(x_i-(d-2)\left\lfloor \frac{x_i}{d} \right\rfloor\right) =j$.
We give formulas for $p_{\mathbf a,d}(n)$ and its polynomial part $P_{\mathbf a,d}(n)$, and also for $p_{\mathbf a,d}(n;j)$. As an application, we compute the dimensions of the stable cohomology groups for certain line bundles associated to flag varieties, defined over an algebraically closed field of positive characteristic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_09418 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Special restricted partition functions for the stable sheaf cohomology on flag varieties Cimpoeas, Mircea Combinatorics 11P81, 11P83, 14M15 Let $\mathbf a:=(a_1,\ldots,a_r)$ be a sequence of positive integers, $d\geq 2$ and $j\geq 1$, some integers. We study the functions $p_{\mathbf a,d}(n):=$ the number of integer solutions $(x_1,\dots,x_r)$ of $\sum_{i=1}^r a_ix_i=n$, with $x_i\geq 0$ and $x_i \equiv 0,1(\bmod\;d)$, for all $1\leq i\leq r$, and $p_{\mathbf a,d}(n;j):=$ the number of $(x_1,\ldots,x_r)$ as above which satisfy also the condition $\sum_{i=1}^r \left(x_i-(d-2)\left\lfloor \frac{x_i}{d} \right\rfloor\right) =j$. We give formulas for $p_{\mathbf a,d}(n)$ and its polynomial part $P_{\mathbf a,d}(n)$, and also for $p_{\mathbf a,d}(n;j)$. As an application, we compute the dimensions of the stable cohomology groups for certain line bundles associated to flag varieties, defined over an algebraically closed field of positive characteristic. |
| title | Special restricted partition functions for the stable sheaf cohomology on flag varieties |
| topic | Combinatorics 11P81, 11P83, 14M15 |
| url | https://arxiv.org/abs/2509.09418 |