How to calculate number of possible lattice polygons given restriction?
I am trying to determine the number of non-self-intersection lattice polygons with perimeter $n$ I can construct given that a certain point, as well as all of its orthogonal neighbors, must lie on its boundary. The polygon need not be convex.
An example of what I mean is given in the attached image. The perimeter of the lattice polygon is $n=10$, while we require that the red point as well as its four orthogonal neighbors (not including diagonal neighbors) lie on the boundary.
As an added question, how would I calculate the number of lattice walks of length $n$ if I relaxed the condition that they need be non-self-intersecting? An example of a walk with length $n=11$ that satisfies the above criteria is given below.
Unfortunately, my strength in combinatorics is lackluster, so I'm unsure how to answer this question.
combinatorics
asked 2 days ago
Derek AdamsDerek Adams
515413
add a comment |
I am trying to determine the number of non-self-intersection lattice polygons with perimeter $n$ I can construct given that a certain point, as well as all of its orthogonal neighbors, must lie on its boundary. The polygon need not be convex.
An example of what I mean is given in the attached image. The perimeter of the lattice polygon is $n=10$, while we require that the red point as well as its four orthogonal neighbors (not including diagonal neighbors) lie on the boundary.
As an added question, how would I calculate the number of lattice walks of length $n$ if I relaxed the condition that they need be non-self-intersecting? An example of a walk with length $n=11$ that satisfies the above criteria is given below.
Unfortunately, my strength in combinatorics is lackluster, so I'm unsure how to answer this question.
combinatorics
asked 2 days ago
Derek AdamsDerek Adams
515413
add a comment |
I am trying to determine the number of non-self-intersection lattice polygons with perimeter $n$ I can construct given that a certain point, as well as all of its orthogonal neighbors, must lie on its boundary. The polygon need not be convex.
An example of what I mean is given in the attached image. The perimeter of the lattice polygon is $n=10$, while we require that the red point as well as its four orthogonal neighbors (not including diagonal neighbors) lie on the boundary.
As an added question, how would I calculate the number of lattice walks of length $n$ if I relaxed the condition that they need be non-self-intersecting? An example of a walk with length $n=11$ that satisfies the above criteria is given below.
Unfortunately, my strength in combinatorics is lackluster, so I'm unsure how to answer this question.
combinatorics
asked 2 days ago
Derek AdamsDerek Adams
515413
I am trying to determine the number of non-self-intersection lattice polygons with perimeter $n$ I can construct given that a certain point, as well as all of its orthogonal neighbors, must lie on its boundary. The polygon need not be convex.
An example of what I mean is given in the attached image. The perimeter of the lattice polygon is $n=10$, while we require that the red point as well as its four orthogonal neighbors (not including diagonal neighbors) lie on the boundary.
As an added question, how would I calculate the number of lattice walks of length $n$ if I relaxed the condition that they need be non-self-intersecting? An example of a walk with length $n=11$ that satisfies the above criteria is given below.
Unfortunately, my strength in combinatorics is lackluster, so I'm unsure how to answer this question.
combinatorics
combinatorics
asked 2 days ago
Derek AdamsDerek Adams
515413
asked 2 days ago
Derek AdamsDerek Adams
515413
asked 2 days ago
Derek AdamsDerek Adams
515413
asked 2 days ago
Derek AdamsDerek Adams
515413
asked 2 days ago
Derek AdamsDerek Adams
515413
515413
add a comment |
add a comment |
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