Identifier
-
Mp00014:
Binary trees
—to 132-avoiding permutation⟶
Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001206: Dyck paths ⟶ ℤ (values match St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.)
Values
[[.,.],.] => [1,2] => [1,2] => [1,0,1,0] => 2
[[.,[.,.]],.] => [2,1,3] => [2,1,3] => [1,1,0,0,1,0] => 2
[[[.,.],.],.] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0] => 2
[[.,[.,[.,.]]],.] => [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0] => 2
[[.,[[.,.],.]],.] => [2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0] => 2
[[[.,.],[.,.]],.] => [3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0] => 2
[[[.,[.,.]],.],.] => [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0] => 2
[[[[.,.],.],.],.] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => 2
[[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0] => 2
[[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0] => 2
[[.,[[.,.],[.,.]]],.] => [4,2,3,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0] => 2
[[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0] => 2
[[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0] => 2
[[[.,.],[.,[.,.]]],.] => [4,3,1,2,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0] => 2
[[[.,.],[[.,.],.]],.] => [3,4,1,2,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0] => 2
[[[.,[.,.]],[.,.]],.] => [4,2,1,3,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0] => 2
[[[[.,.],.],[.,.]],.] => [4,1,2,3,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0] => 2
[[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0] => 2
[[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0] => 2
[[[[.,.],[.,.]],.],.] => [3,1,2,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0] => 2
[[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0] => 2
[[[[[.,.],.],.],.],.] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 2
[[.,[.,[.,[.,[.,.]]]]],.] => [5,4,3,2,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[[.,[.,[.,[[.,.],.]]]],.] => [4,5,3,2,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[[.,[.,[[.,.],[.,.]]]],.] => [5,3,4,2,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[[.,[.,[[.,[.,.]],.]]],.] => [4,3,5,2,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[[.,[.,[[[.,.],.],.]]],.] => [3,4,5,2,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[[.,[[.,.],[.,[.,.]]]],.] => [5,4,2,3,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[[.,[[.,.],[[.,.],.]]],.] => [4,5,2,3,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[[.,[[.,[.,.]],[.,.]]],.] => [5,3,2,4,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[[.,[[[.,.],.],[.,.]]],.] => [5,2,3,4,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[[.,[[.,[.,[.,.]]],.]],.] => [4,3,2,5,1,6] => [5,3,2,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[[.,[[.,[[.,.],.]],.]],.] => [3,4,2,5,1,6] => [5,3,2,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[[.,[[[.,.],[.,.]],.]],.] => [4,2,3,5,1,6] => [5,3,2,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[[.,[[[.,[.,.]],.],.]],.] => [3,2,4,5,1,6] => [5,2,3,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[[.,[[[[.,.],.],.],.]],.] => [2,3,4,5,1,6] => [5,2,3,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[[[.,.],[.,[.,[.,.]]]],.] => [5,4,3,1,2,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[[[.,.],[.,[[.,.],.]]],.] => [4,5,3,1,2,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[[[.,.],[[.,.],[.,.]]],.] => [5,3,4,1,2,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[[[.,.],[[.,[.,.]],.]],.] => [4,3,5,1,2,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[[[.,.],[[[.,.],.],.]],.] => [3,4,5,1,2,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[[[.,[.,.]],[.,[.,.]]],.] => [5,4,2,1,3,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[[[.,[.,.]],[[.,.],.]],.] => [4,5,2,1,3,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[[[[.,.],.],[.,[.,.]]],.] => [5,4,1,2,3,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[[[[.,.],.],[[.,.],.]],.] => [4,5,1,2,3,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[[[.,[.,[.,.]]],[.,.]],.] => [5,3,2,1,4,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[[[.,[[.,.],.]],[.,.]],.] => [5,2,3,1,4,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[[[[.,.],[.,.]],[.,.]],.] => [5,3,1,2,4,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[[[[.,[.,.]],.],[.,.]],.] => [5,2,1,3,4,6] => [5,3,2,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[[[[[.,.],.],.],[.,.]],.] => [5,1,2,3,4,6] => [5,2,3,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[[[.,[.,[.,[.,.]]]],.],.] => [4,3,2,1,5,6] => [4,3,2,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0] => 2
[[[.,[.,[[.,.],.]]],.],.] => [3,4,2,1,5,6] => [4,3,2,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0] => 2
[[[.,[[.,.],[.,.]]],.],.] => [4,2,3,1,5,6] => [4,3,2,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0] => 2
[[[.,[[.,[.,.]],.]],.],.] => [3,2,4,1,5,6] => [4,2,3,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0] => 2
[[[.,[[[.,.],.],.]],.],.] => [2,3,4,1,5,6] => [4,2,3,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0] => 2
[[[[.,.],[.,[.,.]]],.],.] => [4,3,1,2,5,6] => [4,3,2,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0] => 2
[[[[.,.],[[.,.],.]],.],.] => [3,4,1,2,5,6] => [4,3,2,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0] => 2
[[[[.,[.,.]],[.,.]],.],.] => [4,2,1,3,5,6] => [4,3,2,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0] => 2
[[[[[.,.],.],[.,.]],.],.] => [4,1,2,3,5,6] => [4,2,3,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0] => 2
[[[[.,[.,[.,.]]],.],.],.] => [3,2,1,4,5,6] => [3,2,1,4,5,6] => [1,1,1,0,0,0,1,0,1,0,1,0] => 2
[[[[.,[[.,.],.]],.],.],.] => [2,3,1,4,5,6] => [3,2,1,4,5,6] => [1,1,1,0,0,0,1,0,1,0,1,0] => 2
[[[[[.,.],[.,.]],.],.],.] => [3,1,2,4,5,6] => [3,2,1,4,5,6] => [1,1,1,0,0,0,1,0,1,0,1,0] => 2
[[[[[.,[.,.]],.],.],.],.] => [2,1,3,4,5,6] => [2,1,3,4,5,6] => [1,1,0,0,1,0,1,0,1,0,1,0] => 2
[[[[[[.,.],.],.],.],.],.] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 2
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Description
The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$.
Map
Demazure product with inverse
Description
This map sends a permutation $\pi$ to $\pi^{-1} \star \pi$ where $\star$ denotes the Demazure product on permutations.
This map is a surjection onto the set of involutions, i.e., the set of permutations $\pi$ for which $\pi = \pi^{-1}$.
This map is a surjection onto the set of involutions, i.e., the set of permutations $\pi$ for which $\pi = \pi^{-1}$.
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.
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 132-avoiding permutation
Description
Return a 132-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 maximal element of the Sylvester class.
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 maximal element of the Sylvester class.
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