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## Generate all unique polynomials [closed]

To clarify: by unique I mean disregarding permutation of variables. Say I have n variables, and want to generate all "unique" polynomials of degree 2 and 1 (powers are excluded). For example, for n = 3, x1 x2 and x3 are the variables.

x1x2 or x1x2 + x2x3 or x1x3 + x1 or x3 are all example polynomials.

But, x1x2 + x3 can be considered the same polynomial as x1x3 + x2, since variables are simply swapped. I'd like to generate only the "unique" polynomials, (in this case, x1x2 + x3, x1x3 + x2, x2x3 + x1 would all be considered the same under permutation of variables).

I think the polya enumeration theorem is applicable here, but i'm not quite sure how to use it.

Thanks for the help!

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With coefficients in what? If your coefficients are real numbers, for example, then x_1+x_2, x_1+2x_2, x_1+3x_2, x_1+4x_2, ... is an infinite sequence of inequivalent polynomials. – David Speyer May 19 2011 at 16:08
I assume (a) that the coefficients are in $\{0, 1\}$, and (b) this is a homework problem. – Igor Rivin May 19 2011 at 17:51
Please read the faq. Routine applications of Polya enumeration are not within the ambit of this website. – Gerry Myerson May 20 2011 at 0:27

## closed as too localized by Igor Rivin, Pete L. Clark, Harry Gindi, Gerry Myerson, Dan PetersenMay 20 2011 at 3:37

A similar thing is done in this paper: http://arxiv.org/pdf/1104.0589 (I am one of the authors), where one consider homogeneous polynomials, which are unique w.r.t permutation of variables, and translation. These corresponds one-to-one with multigraphs. The translation-invariance can be removed by a change of variables, reducing the number of variables by 1, and adding different homogeneous parts should give arbitrary polynomials.

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