# multi-symmetric functions and multi-partitions

Does sage support manipulating multi-symmetric functions/polynomials and/or multi-partitions? Multi-symmetric functions are like the usual symmetric ones, except the symmetric group acts by permuting "vectors" of variables simultaneously, e.g. for an two vectors $x=(x_1,x_2\dotsc), y=(y_1,y_2,\dotsc)$, $\Sigma_2$, acts by permuting $x,y$. A multi-partition of a $n$-tuple $B=(b_1,\dotsc,b_n)$ of natural numbers is a unordered set of $n$-tuples $A_1,\dotsc,A_l$ with $A_1 + \dotsm + A_l = B$.

I'd like to have a combinatorial class of multi-partitions with similar functionality as partitions, e.g. .first(), .last() methods and iter(). I'd also like to have a class like SymmetricFunctionAlgebra, but with multi-symmetric functions instead. I've had a bit of a poke around and there's some functionality in Maxima (in the Sym) package that might help, but not quite like what I want (that I can find). So, before writing code, I'm asking here.

If the code needs to be written, I'm quite keen to make it my first (hopefully of many) contribution to sage...

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I don't think that what you want is available directly in sage, although there may be some tricks available depending on exactly what you want to do. I don't have time to go into more depth right now, but please post this question to the Google Group: sage-combinat-devel. There are a lot of people (including me!) who read that, who would be interested in getting this kind of functionality into sage.

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