Special restricted partition functions for the stable sheaf cohomology on flag varieties

Abstract

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.