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Points $a_1, a_2, \dots, a_n$ on a line form a set from $n(n-1)/2$ distances between them. Suppose all that distances are different, numerating them from the shortest to the longest one we obtain some permutation on $n(n-1)/2$ elements.

How many permutations can be obtained by this way?

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    $\begingroup$ In other words, you want to count the number of regions in the hyperplane arrangement $\{x_i-x_j,x_i+x_j-x_k-x_l\}$ (and divide it by $n!$). I doubt that a formula exists. $\endgroup$
    – Fedor Petrov
    Commented Feb 15, 2020 at 16:17
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    $\begingroup$ Can you do it for a few small values of $n$, Arseniy, and then consult the Online Encyclopedia of Integer Sequences? $\endgroup$
    – Gerry Myerson
    Commented Feb 15, 2020 at 22:04
  • $\begingroup$ @GerryMyerson, I will try if I found how to program it. $\endgroup$
    – Arseniy Akopyan
    Commented Feb 16, 2020 at 14:59
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    $\begingroup$ Is it by any chance oeis.org/A004123 ? $\endgroup$
    – Gerry Myerson
    Commented Feb 17, 2020 at 22:12
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    $\begingroup$ Yes, thanks for the ref. I do not understand why it could be the same. But I am busy on that week and will look on it later $\endgroup$
    – Arseniy Akopyan
    Commented Feb 21, 2020 at 2:56

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