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Due to the fundamental theorem of symmetric polynomials every symmetric polynomial can be expressed as a polynomial in the elementary symmetric polynomials (here an example).

I know from here and here Gauss's algorithm to get that expression.

However, I would like to know if there exist other algorithms not involving expanding the polynomial and processing the monomials to find the highest in lex order, for example like for this symmetric polynomial i am especially interested in:

$$P(x_1,\ldots,x_n)=\prod_{1 \le i \lt j \le n}{\left(1+x_{i}x_{j}\right)}$$

which requires $\mathcal{O}(2^{n \choose 2})$ time to be expanded.

Fabius Wiesner
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  • You asked about this polynomial in a recent question. What exactly is the problem you are trying to solve using this polynomial? – Somos Oct 09 '21 at 22:15
  • I don't have an exact problem, just many curiosities. Mainly I wanted to know if a closed formula might be obtained for the two sequences described here where a connection with the polynomial above is explained. There there is another polynomial but I will add an explanation there that this one gives the same result. – Fabius Wiesner Oct 10 '21 at 06:02
  • Also I noticed that the number of monomials of the above polynomials expressed in the elementary symmetric polynomials (then with these as variables) is A004251 at least for 4,11,31, and I wanted to test more terms. Rethinking about it, I admit that anyway the number of terms of the symmetric reduction is growing exponentially, so another algorithm won't help so much I think, but it would be nice to know about alternatives to the Gauss's method anyway. – Fabius Wiesner Oct 10 '21 at 06:13

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