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Is there such an immediate proof to the statement or is it simply not true?

Recall that an adjacent transposition is just any transposition $(k\ k+1)$, every permutation $\sigma$ can be written as composition of at least $inv(\sigma)$ adjacent transpositions, where $inv(\sigma)$ is called the inversion number of $\sigma$. We refer to https://dlmf.nist.gov/26.13

Can we also prove that $inv(\sigma)$ is exactly the number of pairs $i,j\in \{1,2,.., n\}$ such that $i<j$ and $\sigma(i)> \sigma(j)$, for $\sigma\in S_n$ ?

NotaChoice
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    I'm confused that the question in the title does not appear in the body text. Anyway, it's false: (12)(23)(12)=(23)(12)(23). – Brauer Suzuki Feb 17 '22 at 17:32
  • thank you, any idea on how to prove that the inversion number correspond to the other definition ? – NotaChoice Feb 17 '22 at 18:11
  • I think it can be proven by (tricky) induction. There is a general statement for Coxeter groups, but I don't have a good reference at hand. – Brauer Suzuki Feb 17 '22 at 18:27
  • On second thought, your question is very reasonable and not on MSE or overflow yet, as far as I can see. It's easy to see that $\sigma$ cannot be written with fewer than inv($\sigma$) adjacent transpositions. For the reverse inequality, one can use the so-called deletion condition. See Humphreys' book on Coxeter groups, Theorem 1.7. I wish, there is a more elementary proof. – Brauer Suzuki Feb 19 '22 at 07:18
  • Thanks for this reference, I might add that I also found the answer of this question useful: https://math.stackexchange.com/questions/3420570/writing-permutations-as-products-of-adjacent-transposition Especially the part : "A Theorem by Tits (explained, e.g., in these notes on Coxeter groups in Section 4.3) states that after writing a permutation as a product of adjacent permutations, this expression can be turned into an expression that is as short as possible by applying only commutativity (where it applies)..." But I wasn't able to find this Tits theorem. – NotaChoice Feb 20 '22 at 00:20
  • I was thinking one can claim that the same transpositions show up in such decompositions, but I did not check that – NotaChoice Feb 20 '22 at 00:23
  • Interesting. The link to the file in this other post has changed. It is now http://arpeg.nl/wp-content/uploads/2016/01/CoxNotes.pdf. Tits' theorem is Theorem 4.3.1 therein. I'm afraid that it is more complicated than the deletion condition. – Brauer Suzuki Feb 20 '22 at 06:41

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