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I actually know that the answer is $N(k-1)$ (where $k$ is the minimum between number of rows and number of columns).

However, I can not seem to find a simple proof for why the statistic is bounded by this. Any suggestions (or references?)

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    $\begingroup$ Tal, can you state the problem in more details and add some motivation? $\endgroup$
    – Gil Kalai
    Commented Jul 31, 2011 at 18:35
  • $\begingroup$ Sure. The motivation is to understand Cramer's V statistic: en.wikipedia.org/wiki/Cram%C3%A9r's_V $\endgroup$
    – Tal Galili
    Commented Jul 31, 2011 at 19:39

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