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Partitioned Permutohedron

A partitioned permutohedron is the intersection of a permutohedron with selected half-spaces x_i<=x_(i+1). More precisely, if P_n is the convex hull of the permutations of a vector in R^n with distinct coordinates and K subset= {1,...,n-1}, then

P_n(K)=P_n intersection intersection _(i in K){x:x_i<=x_(i+1)}.

These are simple polytopes of dimension n-1 (Horiguchi et al. 2024). For K=emptyset, the original permutohedron is recovered. For n=2, choosing K={1} retains half of the original line segment.

Let f_i count the i-dimensional faces, including f_(n-1)=1, and put

h(t)=sum_(i=0)^(n-1)f_i(t-1)^i.

Horiguchi et al. (2026) prove that the gamma vector of this polynomial has only nonnegative entries. For the ordinary hexagonal permutohedron, h(t)=1+4t+t^2 and the gamma vector is (1,2).

See also

Gamma Vector, Permutohedron

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References

Horiguchi, T.; Masuda, M.; Sato, T.; Shareshian, J.; and Song, J. "Gamma Vectors of Partitioned Permutohedra." Electron. J. Combin. 33, P3.61, 2026. https://doi.org/10.37236/14561.Horiguchi, T.; Masuda, M.; Shareshian, J.; and Song, J. "Toric Orbifolds Associated with Partitioned Weight Polytopes in Classical Types." Selecta Math. 30, 84, 2024.

Cite this as:

Weisstein, Eric W. "Partitioned Permutohedron." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PartitionedPermutohedron.html

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