Waiting Time Distribution for the Emergence of Superpatterns

Consider a sequence X_1, X_2,... of i.i.d. uniform random variables taking values in the alphabet set {1,2,...,d}. A k-superpattern is a realization of X_1,...,X_t that contains, as an embedded subsequence, each of the non-order-isomorphic subpatterns of length k. We focus on the non-trivial case of d=k=3 and study the waiting time distribution of tau=inf{t>=7: X_1,...,X_t is a superpattern}