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Runs on a Ring

• To: mathgroup at smc.vnet.net
• Subject: [mg32312] Runs on a Ring
• From: "Seth Chandler" <SChandler at Central.UH.Edu>
• Date: Sat, 12 Jan 2002 05:18:34 -0500 (EST)
• Organization: University of Houston
• Sender: owner-wri-mathgroup at wolfram.com

This is a combined math and Mathematica problem. Suppose one has a ring of n
sites. Each site is populated by a zero or a one, each with random
probability one half. What are the odds that in a ring of size n there is at
least one run of ones (or, equivalently, zeros) of length t? The problem is
solved for lists of site n. See http://mathworld.wolfram.com/Run.html for an
excellent discussion, but I have not been able to find a solution for rings.

I have been able to solve the problem inelegantly for runs of length 3 by
using the package RSolve to solve some coupled difference equations
generated largely from observation.