Parametric decomposition of powers of ideals versus regularity of sequences

Let Q = ( a 1 , a 2 , ⋯ , a s )   ( ⊊ A ) Q = (a_{1}, a_{2}, \cdots , a_{s}) \ (\subsetneq A) be an ideal in a Noetherian local ring A A . Then the sequence a 1 , a 2 , ⋯ , a s a_{1}, a_{2}, \cdots , a_{s} is A A -regular if every a i a_{i} is a non-zerodivisor in A A and if Q n = ⋂ α ( a 1 α 1 , a 2 α 2 , ⋯ , a s α s ) Q^{n} = \bigcap _{\alpha } (a_{1}^{\alpha _{1}}, a_{2}^{\alpha _{2}}, \cdots , a_{s}^{\alpha _{s}}) for all integers n ≥ 1 n \geq 1 , where α = ( α 1 , α 2 , ⋯ , α s ) \alpha = (\alpha _{1}, \alpha _{2}, \cdots , \alpha _{s}) runs over the elements of the set Λ s , n = { ( α 1 , α 2 , ⋯ , α s ) ∈ Z s ∣ α i ≥ 1   for all   1 ≤ i ≤ s   and   ∑ i = 1 s α i = s + n − 1 } \Lambda _{s,n} = \{(\alpha _{1}, \alpha _{2}, \cdots , \alpha _{s}) \in {\mathbb {Z}}^{s} \mid \alpha _{i} \geq 1 \ \text {for all} \ 1 \leq i \leq s \ \text {and} \ \sum _{i=1}^{s}\alpha _{i} = s + n - 1\} .