Identifier
Values
[.,.] => [1] => [1] => [1,0] => 1
[.,[.,.]] => [2,1] => [1,2] => [1,0,1,0] => 1
[[.,.],.] => [1,2] => [1,2] => [1,0,1,0] => 1
[.,[.,[.,.]]] => [3,2,1] => [1,2,3] => [1,0,1,0,1,0] => 2
[.,[[.,.],.]] => [2,3,1] => [1,2,3] => [1,0,1,0,1,0] => 2
[[.,.],[.,.]] => [1,3,2] => [1,3,2] => [1,0,1,1,0,0] => 2
[[.,[.,.]],.] => [2,1,3] => [1,3,2] => [1,0,1,1,0,0] => 2
[[[.,.],.],.] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0] => 2
[.,[.,[.,[.,.]]]] => [4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => 3
[.,[.,[[.,.],.]]] => [3,4,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => 3
[.,[[.,.],[.,.]]] => [2,4,3,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0] => 3
[.,[[.,[.,.]],.]] => [3,2,4,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0] => 3
[.,[[[.,.],.],.]] => [2,3,4,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => 3
[[.,.],[.,[.,.]]] => [1,4,3,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0] => 2
[[.,.],[[.,.],.]] => [1,3,4,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0] => 2
[[.,[.,.]],[.,.]] => [2,1,4,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0] => 2
[[[.,.],.],[.,.]] => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0] => 3
[[.,[.,[.,.]]],.] => [3,2,1,4] => [1,4,2,3] => [1,0,1,1,1,0,0,0] => 2
[[.,[[.,.],.]],.] => [2,3,1,4] => [1,4,2,3] => [1,0,1,1,1,0,0,0] => 2
[[[.,.],[.,.]],.] => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0] => 3
[[[.,[.,.]],.],.] => [2,1,3,4] => [1,3,4,2] => [1,0,1,1,0,1,0,0] => 2
[[[[.,.],.],.],.] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => 3
[.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 4
[.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 4
[.,[.,[[.,.],[.,.]]]] => [3,5,4,2,1] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => 4
[.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => 4
[.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 4
[.,[[.,.],[.,[.,.]]]] => [2,5,4,3,1] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0] => 3
[.,[[.,.],[[.,.],.]]] => [2,4,5,3,1] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0] => 3
[.,[[.,[.,.]],[.,.]]] => [3,2,5,4,1] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0] => 3
[.,[[[.,.],.],[.,.]]] => [2,3,5,4,1] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => 4
[.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0] => 3
[.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0] => 3
[.,[[[.,.],[.,.]],.]] => [2,4,3,5,1] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => 4
[.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0] => 3
[.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 4
[[.,.],[.,[.,[.,.]]]] => [1,5,4,3,2] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0] => 2
[[.,.],[.,[[.,.],.]]] => [1,4,5,3,2] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0] => 2
[[.,.],[[.,.],[.,.]]] => [1,3,5,4,2] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0] => 3
[[.,.],[[.,[.,.]],.]] => [1,4,3,5,2] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0] => 3
[[.,.],[[[.,.],.],.]] => [1,3,4,5,2] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0] => 3
[[.,[.,.]],[.,[.,.]]] => [2,1,5,4,3] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0] => 2
[[.,[.,.]],[[.,.],.]] => [2,1,4,5,3] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0] => 2
[[[.,.],.],[.,[.,.]]] => [1,2,5,4,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0] => 3
[[[.,.],.],[[.,.],.]] => [1,2,4,5,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0] => 3
[[.,[.,[.,.]]],[.,.]] => [3,2,1,5,4] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0] => 2
[[.,[[.,.],.]],[.,.]] => [2,3,1,5,4] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0] => 2
[[[.,.],[.,.]],[.,.]] => [1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0] => 4
[[[.,[.,.]],.],[.,.]] => [2,1,3,5,4] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0] => 3
[[[[.,.],.],.],[.,.]] => [1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => 4
[[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0] => 2
[[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0] => 2
[[.,[[.,.],[.,.]]],.] => [2,4,3,1,5] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0] => 2
[[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0] => 2
[[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0] => 2
[[[.,.],[.,[.,.]]],.] => [1,4,3,2,5] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0] => 3
[[[.,.],[[.,.],.]],.] => [1,3,4,2,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0] => 3
[[[.,[.,.]],[.,.]],.] => [2,1,4,3,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0] => 3
[[[[.,.],.],[.,.]],.] => [1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => 4
[[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0] => 2
[[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0] => 2
[[[[.,.],[.,.]],.],.] => [1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0] => 4
[[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0] => 3
[[[[[.,.],.],.],.],.] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 4
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Description
The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule.
Map
runsort
Description
The permutation obtained by sorting the increasing runs lexicographically.
Map
left-to-right-maxima to Dyck path
Description
The left-to-right maxima of a permutation as a Dyck path.
Let $(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are $c_1, c_1+c_2, \dots, c_1+\dots+c_k$.
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.
Map
to 312-avoiding permutation
Description
Return a 312-avoiding permutation corresponding to a binary tree.
The linear extensions of a binary tree form an interval of the weak order called the Sylvester class of the tree. This permutation is the minimal element of this Sylvester class.