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.)

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

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.

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?


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

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