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

Let $\mathbf{a}:=(a_1,\dots ,a_r)$ be a sequence of positive integers, $d\ge 2$ and $j\ge 1$, some integers. We study the functions $p_{\mathbf{a},d}(n):=$ the number of integer solutions $(x_1,\dots ,x_r)$ of ${\sum }_{i=1}^r{a}_i{x}_i=n$, with $x_i\ge 0$ and $x_i\equiv 0,1(d)$, for all $1\le i\le r$, and $p_{\mathbf{a},d}(n;j):=$ the number of $(x_1,\dots ,x_r)$ as above which satisfy also the condition ${\sum }_{i=1}^r(x_i-(d-2)\lfloor \frac{x_i}{d}\rfloor )=j$. 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.