  We extend the notion of consecutive pattern avoidance to considering sums
over all permutations where each term is a product of weights depending on each
consecutive pattern of a fixed length. We study the problem of finding the
asymptotics of these sums. Our technique is to extend the spectral method of
Ehrenborg, Kitaev and Perry. When the weight depends on the descent pattern we
show how to find the equation determining the spectrum. We give two length $4$
applications. First, we find the asymptotics of the number of permutations with
no triple ascents and no triple descents. Second, we give the asymptotics of
the number of permutations with no isolated ascents or descents. Our next
result is a weighted pattern of length $3$ where the associated operator only
has one non-zero eigenvalue. Using generating functions we show that the error
term in the asymptotic expression is the smallest possible.
