Some considerations on permutation trees (Q799671)

The author develops two algorithms to determine all permutations \(\sigma =\left( \begin{matrix} 1\\ x_ 1\end{matrix} \begin{matrix} 2\\ x_ 2\end{matrix} ...\begin{matrix} n\\ x_ n\end{matrix} \right)\) of the same ''pattern'', which is either the sequence \(\alpha_ 1,...,\alpha_{n-1}\in\{+,-\},\) where \(\alpha_ i=+\Leftrightarrow x_ i<x_{i+1},\) or the triple (A,M,cM) of subsets of \(\{\) 1,...,\(n\}\), whose \(x_ i\in A\Leftrightarrow x_ j=x_ i+1\) has \(j>1\), \(x_ i\in M\Leftrightarrow x_ i<x_{i+1}, x_ i\in cM\Leftrightarrow x_{i-1}<x_ i.\)