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Vince Matsko
Vince Matsko
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A multipermutahedron is the convex hull of all permutations of a list of numbers. For example, $\Pi(0,1,2)$ generates a regular hexagon, and $\Pi(0,1,1,2)$ generates a cuboctahedron.

In general, given a permutahedron such as $\Pi(0,1,1,2,2,2,3,4,4,4),$ faces are determined by partitions of the ordered list with the property that parts of the partitions of length greater than $1$ contain at least two distinct elements. (See the paper The Combinatorics of Permutation Polytopes by Billera and Sarangarajan.) So $0112|2234|4|4$ determines a face, which is the product of a cuboctahedron and a truncated tetrahedron. Note that $0112|2234|44$ generates the same face, since the $44$ part degenerates to a point, so we essentially make sure there is a one-to-one correspondence between suitable partitions and faces.

I have results for many specific cases, but I have not been able to find any general results (e.g., not just looking at simple polytopes), either for the $f$-vectors or for counting this type of permutationpartition. Any pointers would be greatly appreciated.

A multipermutahedron is the convex hull of all permutations of a list of numbers. For example, $\Pi(0,1,2)$ generates a regular hexagon, and $\Pi(0,1,1,2)$ generates a cuboctahedron.

In general, given a permutahedron such as $\Pi(0,1,1,2,2,2,3,4,4,4),$ faces are determined by partitions of the ordered list with the property that parts of the partitions of length greater than $1$ contain at least two distinct elements. (See the paper The Combinatorics of Permutation Polytopes by Billera and Sarangarajan.) So $0112|2234|4|4$ determines a face, which is the product of a cuboctahedron and a truncated tetrahedron. Note that $0112|2234|44$ generates the same face, since the $44$ part degenerates to a point, so we essentially make sure there is a one-to-one correspondence between suitable partitions and faces.

I have results for many specific cases, but I have not been able to find any general results (e.g., not just looking at simple polytopes), either for the $f$-vectors or for counting this type of permutation. Any pointers would be greatly appreciated.

A multipermutahedron is the convex hull of all permutations of a list of numbers. For example, $\Pi(0,1,2)$ generates a regular hexagon, and $\Pi(0,1,1,2)$ generates a cuboctahedron.

In general, given a permutahedron such as $\Pi(0,1,1,2,2,2,3,4,4,4),$ faces are determined by partitions of the ordered list with the property that parts of the partitions of length greater than $1$ contain at least two distinct elements. (See the paper The Combinatorics of Permutation Polytopes by Billera and Sarangarajan.) So $0112|2234|4|4$ determines a face, which is the product of a cuboctahedron and a truncated tetrahedron. Note that $0112|2234|44$ generates the same face, since the $44$ part degenerates to a point, so we essentially make sure there is a one-to-one correspondence between suitable partitions and faces.

I have results for many specific cases, but I have not been able to find any general results (e.g., not just looking at simple polytopes), either for the $f$-vectors or for counting this type of partition. Any pointers would be greatly appreciated.

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Vince Matsko
Vince Matsko
  • 363
  • 3
  • 7

Counting faces on multipermutahedra/multipermutohedra

A multipermutahedron is the convex hull of all permutations of a list of numbers. For example, $\Pi(0,1,2)$ generates a regular hexagon, and $\Pi(0,1,1,2)$ generates a cuboctahedron.

In general, given a permutahedron such as $\Pi(0,1,1,2,2,2,3,4,4,4),$ faces are determined by partitions of the ordered list with the property that parts of the partitions of length greater than $1$ contain at least two distinct elements. (See the paper The Combinatorics of Permutation Polytopes by Billera and Sarangarajan.) So $0112|2234|4|4$ determines a face, which is the product of a cuboctahedron and a truncated tetrahedron. Note that $0112|2234|44$ generates the same face, since the $44$ part degenerates to a point, so we essentially make sure there is a one-to-one correspondence between suitable partitions and faces.

I have results for many specific cases, but I have not been able to find any general results (e.g., not just looking at simple polytopes), either for the $f$-vectors or for counting this type of permutation. Any pointers would be greatly appreciated.