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We all know that the symmetric group $S_n$ could be viewed as a Coxeter group <$t_1,t_2,..,t_{n-1}|(t_i)^2=1 $>, where $t_i$ refers to transposition $(i,i+1)$. We define the length of a sequence of $\{t_1,t_2,..,t_{n-1}\}$ to be the number of bits in the sequence. For example, sequence $t_1t_2t_3t_4t_1$ has length 5 and sequence $t_1t_1t_2$ has length 3. Now suppose we have a sequence of length $k$, is there any theorem telling us how many different permutations are there among all these $(n-1)^k$ sequences?

Jean Marie
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YL C
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    Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. – José Carlos Santos Aug 08 '23 at 09:53
  • I have "latexified" your question. As José Carlos santos said, you have to learn the basics of Latex/Mathjax for writing mathematical expressions on this site, and on many others : expressions must be enclosed between dollar signs, indices are obtained with underscores, etc... – Jean Marie Aug 08 '23 at 10:07
  • Not sure there is a clearcut result, but have you attempted to write a "brute force" program examining all cases allowing to build a conjecture for some given $n,k$, for example $n=5,k=3$ $n=5,k=6$ and $n=5,k=9$ ? – Jean Marie Aug 08 '23 at 10:13
  • A side reference here about the $(i,i+1)$ being generators of $S_n$. – Jean Marie Aug 08 '23 at 10:20
  • @JeanMarie My apology for not using latex form and thank you very much for latexifying my question:) This is my first time using stack exchange. I haven't written a program on this question since I would like to know at first if anyone has already solved this question. Now it seems it would be interesting to start with some programming on this question:) – YL C Aug 08 '23 at 11:09
  • An interesting Edit in this question – Jean Marie Aug 08 '23 at 14:29
  • Connected : https://math.stackexchange.com/a/4278716/305862 – Jean Marie Aug 09 '23 at 21:40
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    @JeanMarie It is really an interesting theorem about the Coxeter length and the number of inversions :). However, I think there is still a gap between the Coxeter length of a permutation and my question of the constant length permutation since in my question, $k$ is a constant length and the Coxeter length of the permutation could be smaller than $k$. – YL C Aug 10 '23 at 05:56
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    well, I think I got the solution for any $n \ge 3$, $k=2, 3$. The next step will be $k\ge 4$. :) – YL C Aug 10 '23 at 06:36
  • Well, if k is large enough, then the sequence will cover $\frac{n!}{2}$ of all the permutations(half the symmetric group). The proof is simple, as the number of odd permutations equals to the number of even permutations. – YL C Aug 22 '23 at 05:24

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