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Fixing some typos
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ulfarsson committed Jun 17, 2019
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Expand Up @@ -8,17 +8,17 @@ Tilescope is an algorithm for finding and proving structures of permutation clas
When it succeeds it returns a combinatorial specification for the class. In many
cases the specification only uses Cartesian products and disjoint unions, and this
allows us to turn the specification into a system of equations with a single variable.
Some notable examples of that
Some notable examples include:

- Several 2x4 classes enumerated by the Schröder numbers: [Av(1342,1432)](http://combopal.ru.is/tree?treeId=5cf9b19d8d535eb769ca50fb), [Av(1342,3142)](http://combopal.ru.is/tree?treeId=5cf9b19d8d535eb769ca50f8), [Av(2413,3142)](http://combopal.ru.is/tree?treeId=5cf9b19d8d535eb769ca50f9)
- All but one of the 2x4 classes equinumerous to the smooth permutations: [Av(1324,2143)](http://combopal.ru.is/tree?treeId=5cf9b19a8d535eb769ca4ebc)), [Av(1324,2413)](http://combopal.ru.is/tree?treeId=5cf9b19d8d535eb769ca50f5), [Av(1342,2431)](http://combopal.ru.is/tree?treeId=5cf9b19d8d535eb769ca50fa), [Av(1342,3241)](http://combopal.ru.is/tree?treeId=5cf9b19c8d535eb769ca4fc0)
- All but one of the 2x4 classes equinumerous to the smooth permutations: [Av(1324,2143)](http://combopal.ru.is/tree?treeId=5cf9b19a8d535eb769ca4ebc), [Av(1324,2413)](http://combopal.ru.is/tree?treeId=5cf9b19d8d535eb769ca50f5), [Av(1342,2431)](http://combopal.ru.is/tree?treeId=5cf9b19d8d535eb769ca50fa), [Av(1342,3241)](http://combopal.ru.is/tree?treeId=5cf9b19c8d535eb769ca4fc0)
- Permutations drawn on an X and a diamond: [Av(2143,2413,3142,3412)](http://combopal.ru.is/tree?treeId=5cf9b19d8d535eb769ca500f)

In other cases more complicated operators are needed, producing a system with
a catalytic variable. The system can still be solved using the kernel method
or other approaches
or other approaches:

- The remainder of the 2x4 classes enumerated by the Schröder numbers ([Av(1234,1243)](https://permpal.ru.is/perms/av/1234_1243/), [Av(1243,1324)](https://permpal.ru.is/perms/av/1243_1324/), [Av(1243,1342)](https://permpal.ru.is/perms/av/1243_1342/), [Av(1243,2143)](https://permpal.ru.is/perms/av/1243_2143/), [Av(1324,1342)](https://permpal.ru.is/perms/av/1324_1342/), [Av(1342,1423)](https://permpal.ru.is/perms/av/1342_1423/), [Av(1342,2341)](https://permpal.ru.is/perms/av/1342_2341/))
- The remainder of the 2x4 classes enumerated by the Schröder numbers: [Av(1234,1243)](https://permpal.ru.is/perms/av/1234_1243/), [Av(1243,1324)](https://permpal.ru.is/perms/av/1243_1324/), [Av(1243,1342)](https://permpal.ru.is/perms/av/1243_1342/), [Av(1243,2143)](https://permpal.ru.is/perms/av/1243_2143/), [Av(1324,1342)](https://permpal.ru.is/perms/av/1324_1342/), [Av(1342,1423)](https://permpal.ru.is/perms/av/1342_1423/), [Av(1342,2341)](https://permpal.ru.is/perms/av/1342_2341/)
- The remaining 2x4 class equinumerous to the smooth permutations: [Av(1342,2314)](https://permpal.ru.is/perms/av/1342_2314/)

Finally, there are cases when we get systems with several catalytic variables.
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