If $\alpha\in S_{k+l}$, $\alpha=\left(\begin{array}{cccccc}1&\cdots&k&k+1&\cdots&k+l\\l+1&\cdots&l+k&1&\cdots&l\end{array}\right)$, for $k,l\in\mathbb{Z}^+$ then how can I express $\alpha$ as the composition of adjacent transpositions? Indeed I just need the number of adjacent transpositions that lies in this decomposition.
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I think you want the minimal number of adjacent transpositions whose product is $\alpha$. This number is called the length of $\alpha$, which is also the number of inversions of $\alpha$, i.e. pairs of integers $(i, j)$, $i<j$ such that $\alpha(i)>\alpha(j)$. So the number of inversions of $\alpha$ in your question is obviously $lk$.
Alex Fok
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I can't see this "obviously". May you help me? – kurtzdoni Jun 02 '15 at 14:32
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Note that the first $k$ numbers are all greater than the latter $l$ numbers. Then use the definition of inversion I gave in the answer. – Alex Fok Jun 02 '15 at 14:34
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I got it, thanks. – kurtzdoni Jun 02 '15 at 14:54