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The monomial symmetric polynomials are defined see Wikipedia.

For an arbitrary partition $\lambda$ with $n$ parts I'm trying to find the following values: $$m_{\lambda}(\omega_0,\dotsc,\omega_{n-1})$$ where $\left\lbrace \omega_{0}=1,\omega_{1},\dotsc, \omega_{n-1} \right\rbrace $ is the set of $n$-th roots of unity.

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    $\begingroup$ Essentially a repeat of mathoverflow.net/questions/67600. $\endgroup$
    – Richard Stanley
    Commented Jul 24 at 15:19
  • $\begingroup$ Is there any particular paper for special cases ? $\endgroup$
    – wkmath
    Commented Jul 24 at 20:33
  • $\begingroup$ You can find small values using Sage or SF (Stembridge's Maple package) to compute the plethysm $h_\lambda[p_n]$. $\endgroup$
    – Richard Stanley
    Commented Jul 26 at 13:42
  • $\begingroup$ Here is a simple demo in Mathematica $\endgroup$
    – 138 Aspen
    Commented Aug 2 at 2:00
  • $\begingroup$ Here's another way to think about it, a bit different from the solution linked to above: Note that at this specialization $e_1 = e_2 = \dots = e_{n-1} = 0$ and $e_n = (-1)^{n-1}$. So if you expand $m_\lambda$ in the elementary symmetric polynomial basis, then every term vanishes under this specialization except for the $e_{n^k} = e_n^k$ ones. So up to a sign you are just looking for the coefficient of $e_{n^k}$ in the elementary expansion. $\endgroup$
    – Nate
    Commented Aug 2 at 19:06

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