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Let $\alpha \models 2n$ be a composition of length $k$, i.e. it is an integer sequence $\alpha = (\alpha_1, \alpha_2, … , \alpha_k)$ such that $\alpha_1 + \alpha_2 + … + \alpha_k = 2n$.

I want to count the number of compositions $\beta \models n$ of length $k$ such that $0 \leq \beta_i \leq \alpha_i$. Is there a way to do that?

eti902
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    This is equivalent to counting the number of positive integer solutions to $\beta_1+\dots+\beta_k=n$, subject to $0\le \beta_i\le \alpha_i$, which in turn is answered by https://math.stackexchange.com/questions/553960/extended-stars-and-bars-problemwhere-the-upper-limit-of-the-variable-is-bounded – Mike Earnest Feb 20 '22 at 22:35

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