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Kregnach
Kregnach
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For a given triangulation (combinatorial Type I. or Type II.) of a $2$-sphere, what is the number of unique polygonal covers with $n$ polygons where ($n$ goes from $2$ to $N$)?

Under polygonal cover, I mean, you take a particular triangulations of the sphere with $N$ triangles, then select a few connected triangles and "mark them" as a polygon, then move to the next ones. Obviously, one possibility is having a triangle, plus all the rest, so technically two triangles. The second trivial cover is the other side, where all triangles are marked separately, so we have $n = N.$ But there are exponentially many options (I guess), and I would be interested if they can be enumerated with some close form or not.

For a given triangulation of a $2$-sphere, what is the number of unique polygonal covers with $n$ polygons where ($n$ goes from $2$ to $N$)?

Under polygonal cover, I mean, you take a particular triangulations of the sphere with $N$ triangles, then select a few connected triangles and "mark them" as a polygon, then move to the next ones. Obviously, one possibility is having a triangle, plus all the rest, so technically two triangles. The second trivial cover is the other side, where all triangles are marked separately, so we have $n = N.$ But there are exponentially many options (I guess), and I would be interested if they can be enumerated with some close form or not.

For a given triangulation (combinatorial Type I. or Type II.) of a $2$-sphere, what is the number of unique polygonal covers with $n$ polygons where ($n$ goes from $2$ to $N$)?

Under polygonal cover, I mean, you take a particular triangulations of the sphere with $N$ triangles, then select a few connected triangles and "mark them" as a polygon, then move to the next ones. Obviously, one possibility is having a triangle, plus all the rest, so technically two triangles. The second trivial cover is the other side, where all triangles are marked separately, so we have $n = N.$ But there are exponentially many options (I guess), and I would be interested if they can be enumerated with some close form or not.

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Michael Hardy
Michael Hardy
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Number of polyhedral covers of a triangulation of S^2$S^2$

For a given triangulation of a 2$2$-sphere, what is the number of unique polygonialpolygonal covers with n$n$ polygons where (n$n$ goes from 2$2$ to N$N$)?

Under polygonialpolygonal cover, I mean, you take a particular triangulations of the sphere with N$N$ triangles, then select a few connected triangles and "mark them" as a polygon, then move to the next ones. Obviously, one possibility is having a triangle, plus all the rest, so technically two triangles. The second trivial cover is the other side, where all triangles are marked separately, so we have n = N.$n = N.$ But there are exponentially many options (I guess), and I would be interested if they can be enumerated with some close form or not.

Number of polyhedral covers of a triangulation of S^2

For a given triangulation of a 2-sphere, what is the number of unique polygonial covers with n polygons where (n goes from 2 to N)?

Under polygonial cover, I mean, you take a particular triangulations of the sphere with N triangles, then select a few connected triangles and "mark them" as a polygon, then move to the next ones. Obviously, one possibility is having a triangle, plus all the rest, so technically two triangles. The second trivial cover is the other side, where all triangles are marked separately, so we have n = N. But there are exponentially many options (I guess), and I would be interested if they can be enumerated with some close form or not.

Number of polyhedral covers of a triangulation of $S^2$

For a given triangulation of a $2$-sphere, what is the number of unique polygonal covers with $n$ polygons where ($n$ goes from $2$ to $N$)?

Under polygonal cover, I mean, you take a particular triangulations of the sphere with $N$ triangles, then select a few connected triangles and "mark them" as a polygon, then move to the next ones. Obviously, one possibility is having a triangle, plus all the rest, so technically two triangles. The second trivial cover is the other side, where all triangles are marked separately, so we have $n = N.$ But there are exponentially many options (I guess), and I would be interested if they can be enumerated with some close form or not.

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Kregnach
Kregnach
  • 183
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Number of polyhedral covers of a triangulation of S^2

For a given triangulation of a 2-sphere, what is the number of unique polygonial covers with n polygons where (n goes from 2 to N)?

Under polygonial cover, I mean, you take a particular triangulations of the sphere with N triangles, then select a few connected triangles and "mark them" as a polygon, then move to the next ones. Obviously, one possibility is having a triangle, plus all the rest, so technically two triangles. The second trivial cover is the other side, where all triangles are marked separately, so we have n = N. But there are exponentially many options (I guess), and I would be interested if they can be enumerated with some close form or not.