> Try the following:
>
> sage: e = SymmetricFunctions(QQ).e() # construct the symmetric functions 
> with the e basis 
> sage: m = SymmetricFunctions(QQ).m() # ditto but with the monomial basis
> sage: m421 = m[4, 2, 1] # create the monomial you care about
> sage: e(m421) # coerce the monomial into the ring with the e basis
> e[3, 2, 1, 1] - 2*e[3, 2, 2] ...
>

That's not obviously the same as the result I specified. I left out that I 
know the number of variables I have available; in the example above, I have 
three variables, so I'm looking specifically for e1=x1+x2+x3, etc. I don't 
see how I get that from the expression e[3,2,1,1] - 2*e[3,2,2] ...

I'm quite new to this topic, hence my lack of clarity. Sorry about this.

john perry

>  

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