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I was reading a paper

On the page 38, under the topic The vertex hierarchy I have doubt. From where did the following term come?

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I am getting no idea how they did it? Any hint will be appreciated. Thanks a lot for help.

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You want to count vertices that has zeros at coordinates $< k$ and are non-zero at exactly $k$-th coordinate and anything at bigger coordinates. Hence, you have exactly $1$ possibility for $i < k$ (namely, zero), $n_i - 1$ possibilities for $i = k$ (anything but a zero) and $n_i$ possibilities for $i > k$ (no constraints). Multiplying it all will get you the aforementioned formula. – dtldarek Aug 22 '13 at 10:38
thanks a lot :) I got it. – monalisa Aug 22 '13 at 10:42

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