The OEIS entries [A019538](http://oeis.org/A019538), [A049019](http://oeis.org/A049019), and [A133314](http://oeis.org/A133314), relate a refinement of the face polynomials of the [permutohedra](http://en.wikipedia.org/wiki/Permutohedron) (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.)


$$f(x) = a\_0 + a\_1 x + a\_2 \frac{x^2}{2!} + ... = \exp[a.x]$$


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


$$\frac{1}{f(x)}= a\_0^{-1} + a\_0^{-2} [-a\_1] x + a\_0^{-3} [2a\_1^2 - a\_2a\_0] \frac{x^2}{2!} + a\_0^{-4}[-6 a\_1^3 + 6 a\_1 a\_2 a\_0 - a\_3 a\_0^2 ] \frac{x^3}{3!} + a\_0^{-5} [24 a\_1^4 - 36 a\_1^2 a\_2 a\_0 + (8 a\_1 a\_3 + 6 a\_2^2) a\_0^2 - a\_4 a\_0^3] \frac{x^4}{4!}+ ... = \exp[Pt.(a\_0,a\_1,..)x]\; ,$$


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](http://oeis.org/A090582). 


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



Relating face polytopes of permutohedra to integer partitions
Rodica Simion in "Convex polytopes and enumeration" affirms the bijection on pages 162 and 163.


