Identifier
-
Mp00017:
Binary trees
—to 312-avoiding permutation⟶
Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001256: Dyck paths ⟶ ℤ
Values
[.,.] => [1] => [1] => [1,0] => 1
[.,[.,.]] => [2,1] => [1,1] => [1,0,1,0] => 1
[[.,.],.] => [1,2] => [2] => [1,1,0,0] => 1
[.,[.,[.,.]]] => [3,2,1] => [1,1,1] => [1,0,1,0,1,0] => 1
[.,[[.,.],.]] => [2,3,1] => [2,1] => [1,1,0,0,1,0] => 1
[[.,.],[.,.]] => [1,3,2] => [2,1] => [1,1,0,0,1,0] => 1
[[.,[.,.]],.] => [2,1,3] => [1,2] => [1,0,1,1,0,0] => 1
[[[.,.],.],.] => [1,2,3] => [3] => [1,1,1,0,0,0] => 1
[.,[.,[.,[.,.]]]] => [4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 1
[.,[.,[[.,.],.]]] => [3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0] => 1
[.,[[.,.],[.,.]]] => [2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0] => 1
[.,[[.,[.,.]],.]] => [3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0] => 1
[.,[[[.,.],.],.]] => [2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0] => 1
[[.,.],[.,[.,.]]] => [1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0] => 1
[[.,.],[[.,.],.]] => [1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0] => 1
[[.,[.,.]],[.,.]] => [2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0] => 1
[[[.,.],.],[.,.]] => [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0] => 1
[[.,[.,[.,.]]],.] => [3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0] => 1
[[.,[[.,.],.]],.] => [2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0] => 1
[[[.,.],[.,.]],.] => [1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0] => 1
[[[.,[.,.]],.],.] => [2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0] => 1
[[[[.,.],.],.],.] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0] => 1
[.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 2
[.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 1
[.,[.,[[.,.],[.,.]]]] => [3,5,4,2,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 1
[.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 1
[.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 1
[.,[[.,.],[.,[.,.]]]] => [2,5,4,3,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 1
[.,[[.,.],[[.,.],.]]] => [2,4,5,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 1
[.,[[.,[.,.]],[.,.]]] => [3,2,5,4,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 1
[.,[[[.,.],.],[.,.]]] => [2,3,5,4,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 1
[.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 1
[.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 1
[.,[[[.,.],[.,.]],.]] => [2,4,3,5,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 1
[.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 1
[.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0] => 1
[[.,.],[.,[.,[.,.]]]] => [1,5,4,3,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 1
[[.,.],[.,[[.,.],.]]] => [1,4,5,3,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 1
[[.,.],[[.,.],[.,.]]] => [1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 1
[[.,.],[[.,[.,.]],.]] => [1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 1
[[.,.],[[[.,.],.],.]] => [1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0] => 1
[[.,[.,.]],[.,[.,.]]] => [2,1,5,4,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 1
[[.,[.,.]],[[.,.],.]] => [2,1,4,5,3] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 1
[[[.,.],.],[.,[.,.]]] => [1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 1
[[[.,.],.],[[.,.],.]] => [1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0] => 1
[[.,[.,[.,.]]],[.,.]] => [3,2,1,5,4] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 1
[[.,[[.,.],.]],[.,.]] => [2,3,1,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 1
[[[.,.],[.,.]],[.,.]] => [1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 1
[[[.,[.,.]],.],[.,.]] => [2,1,3,5,4] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 1
[[[[.,.],.],.],[.,.]] => [1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0] => 1
[[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 1
[[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 1
[[.,[[.,.],[.,.]]],.] => [2,4,3,1,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 1
[[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 1
[[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 1
[[[.,.],[.,[.,.]]],.] => [1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 1
[[[.,.],[[.,.],.]],.] => [1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 1
[[[.,[.,.]],[.,.]],.] => [2,1,4,3,5] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 1
[[[[.,.],.],[.,.]],.] => [1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 1
[[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 1
[[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 1
[[[[.,.],[.,.]],.],.] => [1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 1
[[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 1
[[[[[.,.],.],.],.],.] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0] => 1
[.,[.,[.,[.,[.,[.,.]]]]]] => [6,5,4,3,2,1] => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 3
[.,[.,[.,[.,[[.,.],.]]]]] => [5,6,4,3,2,1] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => 2
[.,[.,[.,[[.,.],[.,.]]]]] => [4,6,5,3,2,1] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => 2
[.,[.,[.,[[.,[.,.]],.]]]] => [5,4,6,3,2,1] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => 1
[.,[.,[.,[[[.,.],.],.]]]] => [4,5,6,3,2,1] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => 1
[.,[.,[[.,.],[.,[.,.]]]]] => [3,6,5,4,2,1] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => 2
[.,[.,[[.,.],[[.,.],.]]]] => [3,5,6,4,2,1] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => 1
[.,[.,[[.,[.,.]],[.,.]]]] => [4,3,6,5,2,1] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => 1
[.,[.,[[[.,.],.],[.,.]]]] => [3,4,6,5,2,1] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => 1
[.,[.,[[.,[.,[.,.]]],.]]] => [5,4,3,6,2,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => 1
[.,[.,[[.,[[.,.],.]],.]]] => [4,5,3,6,2,1] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => 1
[.,[.,[[[.,.],[.,.]],.]]] => [3,5,4,6,2,1] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => 1
[.,[.,[[[.,[.,.]],.],.]]] => [4,3,5,6,2,1] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => 1
[.,[.,[[[[.,.],.],.],.]]] => [3,4,5,6,2,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => 1
[.,[[.,.],[.,[.,[.,.]]]]] => [2,6,5,4,3,1] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => 2
[.,[[.,.],[.,[[.,.],.]]]] => [2,5,6,4,3,1] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => 1
[.,[[.,.],[[.,.],[.,.]]]] => [2,4,6,5,3,1] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => 1
[.,[[.,.],[[.,[.,.]],.]]] => [2,5,4,6,3,1] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => 1
[.,[[.,.],[[[.,.],.],.]]] => [2,4,5,6,3,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => 1
[.,[[.,[.,.]],[.,[.,.]]]] => [3,2,6,5,4,1] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => 1
[.,[[.,[.,.]],[[.,.],.]]] => [3,2,5,6,4,1] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => 1
[.,[[[.,.],.],[.,[.,.]]]] => [2,3,6,5,4,1] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => 1
[.,[[[.,.],.],[[.,.],.]]] => [2,3,5,6,4,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => 1
[.,[[.,[.,[.,.]]],[.,.]]] => [4,3,2,6,5,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => 1
[.,[[.,[[.,.],.]],[.,.]]] => [3,4,2,6,5,1] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => 1
[.,[[[.,.],[.,.]],[.,.]]] => [2,4,3,6,5,1] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => 1
[.,[[[.,[.,.]],.],[.,.]]] => [3,2,4,6,5,1] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => 1
[.,[[[[.,.],.],.],[.,.]]] => [2,3,4,6,5,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => 1
[.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => 1
[.,[[.,[.,[[.,.],.]]],.]] => [4,5,3,2,6,1] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => 1
[.,[[.,[[.,.],[.,.]]],.]] => [3,5,4,2,6,1] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => 1
[.,[[.,[[.,[.,.]],.]],.]] => [4,3,5,2,6,1] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 1
[.,[[.,[[[.,.],.],.]],.]] => [3,4,5,2,6,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 1
[.,[[[.,.],[.,[.,.]]],.]] => [2,5,4,3,6,1] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => 1
[.,[[[.,.],[[.,.],.]],.]] => [2,4,5,3,6,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 1
[.,[[[.,[.,.]],[.,.]],.]] => [3,2,5,4,6,1] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 1
[.,[[[[.,.],.],[.,.]],.]] => [2,3,5,4,6,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 1
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Description
Number of simple reflexive modules that are 2-stable reflexive.
See Definition 3.1. in the reference for the definition of 2-stable reflexive.
See Definition 3.1. in the reference for the definition of 2-stable reflexive.
Map
bounce path
Description
The bounce path determined by an integer composition.
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.
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.
Map
descent composition
Description
The descent composition of a permutation.
The descent composition of a permutation $\pi$ of length $n$ is the integer composition of $n$ whose descent set equals the descent set of $\pi$. The descent set of a permutation $\pi$ is $\{i \mid 1 \leq i < n, \pi(i) > \pi(i+1)\}$. The descent set of a composition $c = (i_1, i_2, \ldots, i_k)$ is the set $\{ i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$.
The descent composition of a permutation $\pi$ of length $n$ is the integer composition of $n$ whose descent set equals the descent set of $\pi$. The descent set of a permutation $\pi$ is $\{i \mid 1 \leq i < n, \pi(i) > \pi(i+1)\}$. The descent set of a composition $c = (i_1, i_2, \ldots, i_k)$ is the set $\{ i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$.
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