Abstract
Consider a sequence \(\{X_{n}\}_{n=1}^{\infty }\) of i.i.d. uniform random variables taking values in the alphabet set {1, 2,…, d}. A k-superpattern is a realization of \(\{X_{n}\}_{n=1}^{t}\) 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 \(\tau =\inf \{t\ge 1:\{X_{n}\}_{n=1}^{t}\ \text {is\ a\ superpattern}\}\). Our restricted set-up leads to proofs that are very combinatorial in nature, since we are essentially conducting a string analysis.
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Godbole, A.P., Liendo, M. Waiting Time Distribution for the Emergence of Superpatterns. Methodol Comput Appl Probab 18, 517–528 (2016). https://doi.org/10.1007/s11009-015-9439-6
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DOI: https://doi.org/10.1007/s11009-015-9439-6