Charles Kenney, Rutgers University

Abstract

Let b=(b_1, b_2, b_3, ...) be a sequence of 0s and 1s. We say b is stopped at time T if, for every index t in (T/2, T], b_t = 0. The stopping time of b is the smallest T such that b is stopped at time T, if it exists; otherwise b has infinite stopping time. For instance, the sequence

(1,0,1,1,0,0,0,0,1)

is stopped at times 2 and 8, and has stopping time 2. The Narayana-Zidek-Capell (NZC) numbers, A002083 in the OEIS, were introduced by Capell and Narayana (1970) to enumerate what they called ``random knock-out tournaments." We give two proofs that the set of sequences with stopping time 2n is enumerated by the NZC sequence, and then discuss related problems and generalizations (some new, some well-known), such as ``how likely is a random binary sequence to have finite stopping time?" and ``what can we say about natural numbers whose binary expressions have certain stopping times?"

Presented via Zoom - Meeting ID: 984 4140 9199

https://rutgers.zoom.us/j/98441409199

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