St. Petersburg Mathematical Olympiad Problem 2

Ten points, labelled \(1\) to \(10\) are chosen in the plane. Permutations of \(\{1,\dots ,10\}\) are obtained as follows: for each rectangular coordinate system in which the ten points have distinct first coordinates, the labels of the points are listed in increasing order of the first coordinates of the points. Over all sequences of \(10\) labelled points, what is the maximum number of permutations of \(\{1, \dots ,10\}\)obtained in this fashion?

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