Let $\alpha = (\alpha_{1}, \ldots, \alpha_{d}) \in \mathbb{Z}_{\geq 0}^{d}$ and let $|\alpha| = \alpha_{1} + \cdots + \alpha_{d}$. I have the following question about interchanging summations:
Is $$\sum_{j = 0}^{k}\sum_{\alpha: |\alpha| = j}\sum_{\substack{0 \leq \gamma_{i} \leq \alpha_{i}\\i = 1, 2, \ldots, d}} = \sum_{j = 0}^{k}\sum_{\gamma: |\gamma| = j}\sum_{\substack{\gamma_{i} \leq \alpha_{i} \leq k\\i = 1, 2, \ldots, d}}?$$ If not, how would I interchange the $\sum_{j = 0}^{k}\sum_{\alpha: |\alpha| = j}$ and $\sum_{\substack{0 \leq \gamma_{i} \leq \alpha_{i}\\i = 1, 2, \ldots, d}}$?
In the case of one dimension, above centered expression boils down to $$\sum_{\alpha = 0}^{k}\sum_{\gamma = 0}^{\alpha} = \sum_{\gamma= 0}^{k}\sum_{\alpha = \gamma}^{k}$$ which can easily be seen by looking at $\sum_{\alpha = 0}^{k}\sum_{\gamma = 0}^{k}1_{\gamma \leq \alpha}.$