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The OEIS entries A019538, A049019, and A133314, relate a refinement of the face polynomials of the permutohedra (A049019) to partition polynomials (A133314) defined by multiplicative inversion of an exponential generating function (or surjections as noted in A049019 and A133314).

For example, with the Taylor series expansion of an analytic function (or formal power series, or e.g.f.)

$$f(x) = a_0 + a_1 x + a_2 \frac{x^2}{2!} + \ldots = \exp[a.x]$$

where $a.^n = a_n$, the series expansion of the reciprocal is formally

\begin{align} \frac{1}{f(x)} &= a_0^{-1} + a_0^{-2} [-a_1] x \\ &+ a_0^{-3} [2a_1^2 - a_2a_0] \frac{x^2}{2!} + a_0^{-4}[-6 a_1^3 + 6 a_1 a_2 a_0 - a_3 a_0^2 ] \frac{x^3}{3!} \\ &+ a_0^{-5} [24 a_1^4 - 36 a_1^2 a_2 a_0 + (8 a_1 a_3 + 6 a_2^2) a_0^2 - a_4 a_0^3] \frac{x^4}{4!}+ \ldots \\ &= \exp[Pt.(a_0,a_1, \ldots )x/a_0] / a_0\; , \end{align}

and the unsigned coefficients of the partition polynomial $Pt_4(a_0,a_1,a_2,a_4)$ for the fourth order term with partitions of the integer four characterize the $P_3$ permutohedron depicted in Wikipedia with 24 vertices (0-D faces), 36 edges (1-D faces), 8 hexagons (2-D faces), 6 tetragons (2-D faces), and 1 3-D permutohedron. Summing coefficients over like dimensions gives A019538 and A090582.

Question I: Is there a bijective mapping between the integer partitions and the different polytopes of the n-D faces of the permutohedra for higher dimensions?

There is a listing of the different polytopes comprising the faces of the 4-D permutohedron, the runcinated 5-cell, in Wikipedia, with such a bijection also.

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  • $\begingroup$ I hope you don't mind me breaking the lines for $1/f$. If you do not like this, you can of course revert the edit (it seemed less than ideal to be several screens wide, originally!) $\endgroup$
    – pjs36
    May 28, 2016 at 18:11
  • $\begingroup$ For the actual question, I'm not familiar with the polynomial approach here, but I would not be surprised. Any face of a permutohedron is a product of smaller permutohedra (but maybe you already knew this). For example, the squares in $P_3$ are $P_1 \times P_1$ (while the hexagons are $P_0 \times P_2 \cong P_2).$ This recursive nature of faces feels quite similar to integer partitions (but I know more about permutohedra than everything else in your post). $\endgroup$
    – pjs36
    May 28, 2016 at 18:23
  • $\begingroup$ @pjs36--This is analogous to the associahedra and compositional inversion, as described in mathoverflow.net/questions/145555/… $\endgroup$ May 28, 2016 at 18:44
  • $\begingroup$ See also "Mirror symmetry for honeycombs" by Benjamin Gammage, David Nadler (arxiv.org/abs/1702.03255) $\endgroup$ Dec 18, 2018 at 16:26

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Rodica Simion in "Convex polytopes and enumeration" affirms the bijection on pages 162 and 163.

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