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- W4288804610 abstract "Partial permutohedra are lattice polytopes which were recently introduced and studied by Heuer and Striker. For positive integers $m$ and $n$, the partial permutohedron $mathcal{P}(m,n)$ is the convex hull of all vectors in ${0,1,ldots,n}^m$ whose nonzero entries are distinct. We study the face lattice, volume and Ehrhart polynomial of $mathcal{P}(m,n)$, and our methods and results include the following. For any $m$ and $n$, we obtain a bijection between the nonempty faces of $mathcal{P}(m,n)$ and certain chains of subsets of ${1,dots,m}$, thereby confirming a conjecture of Heuer and Striker, and we then use this characterization of faces to obtain a closed expression for the $h$-polynomial of $mathcal{P}(m,n)$. For any $m$ and $n$ with $nge m-1$, we use a pyramidal subdivision of $mathcal{P}(m,n)$ to establish a recursive formula for the normalized volume of $mathcal{P}(m,n)$, from which we then obtain closed expressions for this volume. We also use a sculpting process (in which $mathcal{P}(m,n)$ is reached by successively removing certain pieces from a simplex or hypercube) to obtain closed expressions for the Ehrhart polynomial of $mathcal{P}(m,n)$ with arbitrary $m$ and fixed $nle 3$, the volume of $mathcal{P}(m,4)$ with arbitrary $m$, and the Ehrhart polynomial of $mathcal{P}(m,n)$ with fixed $mle4$ and arbitrary $nge m-1$." @default.
- W4288804610 created "2022-07-30" @default.
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- W4288804610 date "2022-07-28" @default.
- W4288804610 modified "2023-09-30" @default.
- W4288804610 title "Partial permutohedra" @default.
- W4288804610 doi "https://doi.org/10.48550/arxiv.2207.14253" @default.
- W4288804610 hasPublicationYear "2022" @default.
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