Let a $n-$length pauli string represent any tensor product of finitely many pauli matrices, Ex: $X\otimes Z\otimes \mathbb{I}\otimes Y\otimes \mathbb{I}\otimes X\otimes\cdots\otimes Z$ where the number of matrices (involved in the tensor product) are $n$.
We know that all $n-$length pauli strings form a group, the Pauli Group of size $4^n$.
Let the weight of an $n-$length pauli string be equal to the number of non-identity pauli matrices in it. The number of $n-$length pauli strings with weight $w$ is just: $\binom{n}{w}3^{w}$.
Can we have a bound (different from the one above) for the number of commuting pauli strings of weight $w$ ?
If someone could give me some hints for estimating this, it'll be nice.