Re: Any efficient way to make complete homogeneous symmetric functions in Mathematica?

• To: mathgroup at smc.vnet.net
• Subject: [mg126831] Re: Any efficient way to make complete homogeneous symmetric functions in Mathematica?
• From: Rex <Aoirex at gmail.com>
• Date: Mon, 11 Jun 2012 00:02:07 -0400 (EDT)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com

```Cool. Thanks, Bob.

On Sun, Jun 10, 2012 at 9:23 AM, Bob Hanlon <hanlonr357 at gmail.com> wrote:

> homogeneousSymmetricPolynomial[k_Integer, var_List] :=
>  Module[{n = Length[var], t},
>   SeriesCoefficient[Times @@ (1/(1 - var*t)),
>    {t, 0, k}]] /;
>  0 <= k <= Length[var]
>
> Table[homogeneousSymmetricPolynomial[k, {x1, x2, x3}], {k, 0, 3}]
>
> {1, x1 + x2 + x3, x1^2 + x1*x2 + x2^2 + x1*x3 + x2*x3 +
>     x3^2, x1^3 + x1^2*x2 + x1*x2^2 + x2^3 + x1^2*x3 +
>     x1*x2*x3 + x2^2*x3 + x1*x3^2 + x2*x3^2 + x3^3}
>
> With[{var = {x1, x2, x3, x4, x5}},
>  Module[{m = Length[var]},
>  Sum[(-1)^i*SymmetricPolynomial[i, var]*
>      homogeneousSymmetricPolynomial[m - i, var],
>     {i, 0, m}] == 0 // Simplify]]
>
> True
>
>
> Bob Hanlon
>
>
> On Sun, Jun 10, 2012 at 2:14 AM, Rex <aoirex at gmail.com> wrote:
> > We do have elementary symmetric functions,
> SymmetricPolynomial[k,{x_1..x_n}]
> > http://reference.wolfram.com/mathematica/ref/SymmetricPolynomial.html
> >
> > But I didn't found complete homogeneous symmetric functions.
> >
> > The induction method to compute h_n from e_i and h_j (j<=n-1) is not
> that efficient.
> > https://en.wikipedia.org/wiki/Complete_homogeneous_symmetric_polynomial
> >
> > Is there any easier way to do this?
> >
>

```

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