Multiplicity of powers of squarefree monomial ideals

Let I be an arbitrary nonzero squarefree monomial ideal of dimension d in a polynomial ring \(S = \textrm{k}[x_1,\ldots ,x_n]\). Let \(\mu \) be the number of associated primes of S/I of dimension d. We prove that the multiplicity of powers of I is given by

$$\begin{aligned} e_0(S/I^s) = \mu \left( {\begin{array}{c}n-d+s-1\\ s-1\end{array}}\right) \end{aligned}$$

for all \(s \ge 1\). Consequently, we compute the multiplicity of all powers of path ideals of cycles.