- FindStat

Identifier

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001225: Dyck paths ⟶ ℤ

Values

[.,.] => [1] => [1,0] => 0

[.,[.,.]] => [2,1] => [1,1,0,0] => 0

[[.,.],.] => [1,2] => [1,0,1,0] => 1

[.,[.,[.,.]]] => [3,2,1] => [1,1,1,0,0,0] => 0

[.,[[.,.],.]] => [2,3,1] => [1,1,0,1,0,0] => 2

[[.,.],[.,.]] => [3,1,2] => [1,1,1,0,0,0] => 0

[[.,[.,.]],.] => [2,1,3] => [1,1,0,0,1,0] => 1

[[[.,.],.],.] => [1,2,3] => [1,0,1,0,1,0] => 2

[.,[.,[.,[.,.]]]] => [4,3,2,1] => [1,1,1,1,0,0,0,0] => 0

[.,[.,[[.,.],.]]] => [3,4,2,1] => [1,1,1,0,1,0,0,0] => 3

[.,[[.,.],[.,.]]] => [4,2,3,1] => [1,1,1,1,0,0,0,0] => 0

[.,[[.,[.,.]],.]] => [3,2,4,1] => [1,1,1,0,0,1,0,0] => 2

[.,[[[.,.],.],.]] => [2,3,4,1] => [1,1,0,1,0,1,0,0] => 3

[[.,.],[.,[.,.]]] => [4,3,1,2] => [1,1,1,1,0,0,0,0] => 0

[[.,.],[[.,.],.]] => [3,4,1,2] => [1,1,1,0,1,0,0,0] => 3

[[.,[.,.]],[.,.]] => [4,2,1,3] => [1,1,1,1,0,0,0,0] => 0

[[[.,.],.],[.,.]] => [4,1,2,3] => [1,1,1,1,0,0,0,0] => 0

[[.,[.,[.,.]]],.] => [3,2,1,4] => [1,1,1,0,0,0,1,0] => 1

[[.,[[.,.],.]],.] => [2,3,1,4] => [1,1,0,1,0,0,1,0] => 3

[[[.,.],[.,.]],.] => [3,1,2,4] => [1,1,1,0,0,0,1,0] => 1

[[[.,[.,.]],.],.] => [2,1,3,4] => [1,1,0,0,1,0,1,0] => 2

[[[[.,.],.],.],.] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => 3

[.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0] => 0

[.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0] => 4

[.,[.,[[.,.],[.,.]]]] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0] => 0

[.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0] => 3

[.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0] => 4

[.,[[.,.],[.,[.,.]]]] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0] => 0

[.,[[.,.],[[.,.],.]]] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0] => 4

[.,[[.,[.,.]],[.,.]]] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0] => 0

[.,[[[.,.],.],[.,.]]] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0] => 0

[.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0] => 2

[.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0] => 4

[.,[[[.,.],[.,.]],.]] => [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0] => 2

[.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0] => 3

[.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0] => 4

[[.,.],[.,[.,[.,.]]]] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0] => 0

[[.,.],[.,[[.,.],.]]] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0] => 4

[[.,.],[[.,.],[.,.]]] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0] => 0

[[.,.],[[.,[.,.]],.]] => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0] => 3

[[.,.],[[[.,.],.],.]] => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0] => 4

[[.,[.,.]],[.,[.,.]]] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0] => 0

[[.,[.,.]],[[.,.],.]] => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0] => 4

[[[.,.],.],[.,[.,.]]] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0] => 0

[[[.,.],.],[[.,.],.]] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0] => 4

[[.,[.,[.,.]]],[.,.]] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0] => 0

[[.,[[.,.],.]],[.,.]] => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0] => 0

[[[.,.],[.,.]],[.,.]] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0] => 0

[[[.,[.,.]],.],[.,.]] => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0] => 0

[[[[.,.],.],.],[.,.]] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0] => 0

[[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0] => 1

[[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0] => 4

[[.,[[.,.],[.,.]]],.] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0] => 1

[[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0] => 3

[[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0] => 4

[[[.,.],[.,[.,.]]],.] => [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0] => 1

[[[.,.],[[.,.],.]],.] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0] => 4

[[[.,[.,.]],[.,.]],.] => [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0] => 1

[[[[.,.],.],[.,.]],.] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0] => 1

[[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0] => 2

[[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0] => 4

[[[[.,.],[.,.]],.],.] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0] => 2

[[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0] => 3

[[[[[.,.],.],.],.],.] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 4

[.,[.,[.,[.,[.,[.,.]]]]]] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0] => 0

[.,[.,[.,[.,[[.,.],.]]]]] => [5,6,4,3,2,1] => [1,1,1,1,1,0,1,0,0,0,0,0] => 5

[.,[.,[.,[[.,.],[.,.]]]]] => [6,4,5,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0] => 0

[.,[.,[.,[[.,[.,.]],.]]]] => [5,4,6,3,2,1] => [1,1,1,1,1,0,0,1,0,0,0,0] => 4

[.,[.,[.,[[[.,.],.],.]]]] => [4,5,6,3,2,1] => [1,1,1,1,0,1,0,1,0,0,0,0] => 5

[.,[.,[[.,.],[.,[.,.]]]]] => [6,5,3,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0] => 0

[.,[.,[[.,.],[[.,.],.]]]] => [5,6,3,4,2,1] => [1,1,1,1,1,0,1,0,0,0,0,0] => 5

[.,[.,[[.,[.,.]],[.,.]]]] => [6,4,3,5,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0] => 0

[.,[.,[[[.,.],.],[.,.]]]] => [6,3,4,5,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0] => 0

[.,[.,[[.,[.,[.,.]]],.]]] => [5,4,3,6,2,1] => [1,1,1,1,1,0,0,0,1,0,0,0] => 3

[.,[.,[[.,[[.,.],.]],.]]] => [4,5,3,6,2,1] => [1,1,1,1,0,1,0,0,1,0,0,0] => 5

[.,[.,[[[.,.],[.,.]],.]]] => [5,3,4,6,2,1] => [1,1,1,1,1,0,0,0,1,0,0,0] => 3

[.,[.,[[[.,[.,.]],.],.]]] => [4,3,5,6,2,1] => [1,1,1,1,0,0,1,0,1,0,0,0] => 4

[.,[.,[[[[.,.],.],.],.]]] => [3,4,5,6,2,1] => [1,1,1,0,1,0,1,0,1,0,0,0] => 5

[.,[[.,.],[.,[.,[.,.]]]]] => [6,5,4,2,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0] => 0

[.,[[.,.],[.,[[.,.],.]]]] => [5,6,4,2,3,1] => [1,1,1,1,1,0,1,0,0,0,0,0] => 5

[.,[[.,.],[[.,.],[.,.]]]] => [6,4,5,2,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0] => 0

[.,[[.,.],[[.,[.,.]],.]]] => [5,4,6,2,3,1] => [1,1,1,1,1,0,0,1,0,0,0,0] => 4

[.,[[.,.],[[[.,.],.],.]]] => [4,5,6,2,3,1] => [1,1,1,1,0,1,0,1,0,0,0,0] => 5

[.,[[.,[.,.]],[.,[.,.]]]] => [6,5,3,2,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0] => 0

[.,[[.,[.,.]],[[.,.],.]]] => [5,6,3,2,4,1] => [1,1,1,1,1,0,1,0,0,0,0,0] => 5

[.,[[[.,.],.],[.,[.,.]]]] => [6,5,2,3,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0] => 0

[.,[[[.,.],.],[[.,.],.]]] => [5,6,2,3,4,1] => [1,1,1,1,1,0,1,0,0,0,0,0] => 5

[.,[[.,[.,[.,.]]],[.,.]]] => [6,4,3,2,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0] => 0

[.,[[.,[[.,.],.]],[.,.]]] => [6,3,4,2,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0] => 0

[.,[[[.,.],[.,.]],[.,.]]] => [6,4,2,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0] => 0

[.,[[[.,[.,.]],.],[.,.]]] => [6,3,2,4,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0] => 0

[.,[[[[.,.],.],.],[.,.]]] => [6,2,3,4,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0] => 0

[.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => [1,1,1,1,1,0,0,0,0,1,0,0] => 2

[.,[[.,[.,[[.,.],.]]],.]] => [4,5,3,2,6,1] => [1,1,1,1,0,1,0,0,0,1,0,0] => 5

[.,[[.,[[.,.],[.,.]]],.]] => [5,3,4,2,6,1] => [1,1,1,1,1,0,0,0,0,1,0,0] => 2

[.,[[.,[[.,[.,.]],.]],.]] => [4,3,5,2,6,1] => [1,1,1,1,0,0,1,0,0,1,0,0] => 4

[.,[[.,[[[.,.],.],.]],.]] => [3,4,5,2,6,1] => [1,1,1,0,1,0,1,0,0,1,0,0] => 5

[.,[[[.,.],[.,[.,.]]],.]] => [5,4,2,3,6,1] => [1,1,1,1,1,0,0,0,0,1,0,0] => 2

[.,[[[.,.],[[.,.],.]],.]] => [4,5,2,3,6,1] => [1,1,1,1,0,1,0,0,0,1,0,0] => 5

[.,[[[.,[.,.]],[.,.]],.]] => [5,3,2,4,6,1] => [1,1,1,1,1,0,0,0,0,1,0,0] => 2

[.,[[[[.,.],.],[.,.]],.]] => [5,2,3,4,6,1] => [1,1,1,1,1,0,0,0,0,1,0,0] => 2

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$F_{1} = 1$

$F_{2} = 1 + q$

$F_{3} = 2 + q + 2\ q^{2}$

$F_{4} = 5 + 2\ q + 2\ q^{2} + 5\ q^{3}$

$F_{5} = 14 + 5\ q + 4\ q^{2} + 5\ q^{3} + 14\ q^{4}$

$F_{6} = 42 + 14\ q + 10\ q^{2} + 10\ q^{3} + 14\ q^{4} + 42\ q^{5}$

Description

The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra.

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 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.