Derek's permutations

Let \(N\) be the number of permutations \(\tau\) of \( (1,4,9,16,25,36,49) \) such that

\[\tau(1) + \tau(9) + \tau(25) > \tau(4) + \tau(16) + \tau(36).\]

What are the last 3 digits of \(N\)?

This problem is posed by Derek K.

Details and assumptions

A permutation \( \tau\), on a set \(S\), is a bijection from \(S\) onto itself. \( \tau(s) \) denotes the image of the element \(s\) under this map.


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