Thanks indeed ! you did not made errors: I could check for some values of $h$ and $n$ and it works. Before I was trying other modulus based ordering, but not this. Does the class of permutations given by the $\sigma^{-1}$ above, have a standard naming ?

@LSpice: I mean that C is an off-diagonal matrix, but where $n-1$ ones are replaced by zeros. We can express the position of the $1/0$ by a permutation of a vector with $n$ zeros and $h-n$ ones. Concerning your 2nd comment, it looks very interesting: could you pls. explicitate in an answer ? many thanks

@RichardStanley: you may be interested to know that such polynomials have a defined structure, as reported in my answer.

@Wolfgang: you are right in your supposition, there is a scheme for that, as reported in my answer

@becko; is everyhing (enough) clear now?

@becko: I added some notes to better clarify the meaning of matrix $\bf E$, and to show that inversion of (1) just involves simple manipulations around the same set of basic matrices.