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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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Tal, can you state the problem in more details and add some motivation? –  Gil Kalai Jul 31 '11 at 18:35
Sure. The motivation is to understand Cramer's V statistic: en.wikipedia.org/wiki/Cram%C3%A9r's_V –  Tal Galili Jul 31 '11 at 19:39

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