- FindStat

Identifier

Mp00277: Permutations —catalanization⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000032: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[2,4,1,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0] => 14

[2,4,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0] => 7

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

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

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

[3,2,4,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0] => 7

[3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0] => 14

[3,4,2,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0] => 9

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

[4,1,3,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0] => 7

[4,2,1,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0] => 7

[4,2,3,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0] => 9

[4,3,1,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0] => 9

[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0] => 14

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

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

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

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

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

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

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

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

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

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

[1,3,5,2,4] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0] => 14

[1,3,5,4,2] => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0] => 7

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

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

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

[1,4,3,5,2] => [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0] => 7

[1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0] => 14

[1,4,5,3,2] => [1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0] => 9

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

[1,5,2,4,3] => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0] => 7

[1,5,3,2,4] => [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0] => 7

[1,5,3,4,2] => [1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0] => 9

[1,5,4,2,3] => [1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0] => 9

[1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0] => 14

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

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

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

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

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

[2,1,5,4,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0] => 10

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

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

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

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

[2,3,5,1,4] => [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0] => 19

[2,3,5,4,1] => [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0] => 9

[2,4,1,3,5] => [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0] => 14

[2,4,1,5,3] => [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0] => 19

[2,4,3,1,5] => [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0] => 7

[2,4,3,5,1] => [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0] => 9

[2,4,5,1,3] => [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0] => 23

[2,4,5,3,1] => [2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0] => 12

[2,5,1,3,4] => [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0] => 19

[2,5,1,4,3] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0] => 42

[2,5,3,1,4] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0] => 42

[2,5,3,4,1] => [2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0] => 12

[2,5,4,1,3] => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0] => 14

[2,5,4,3,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0] => 19

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

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

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

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

[3,1,5,2,4] => [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0] => 19

[3,1,5,4,2] => [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0] => 9

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

[3,2,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0] => 10

[3,2,4,1,5] => [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0] => 7

[3,2,4,5,1] => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0] => 9

[3,2,5,1,4] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0] => 42

[3,2,5,4,1] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0] => 16

[3,4,1,2,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0] => 14

[3,4,1,5,2] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0] => 19

[3,4,2,1,5] => [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0] => 9

[3,4,2,5,1] => [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0] => 12

[3,4,5,1,2] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0] => 23

[3,4,5,2,1] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0] => 14

[3,5,1,2,4] => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0] => 23

[3,5,1,4,2] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0] => 42

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Description

The number of elements smaller than the given Dyck path in the Tamari Order.

Map

catalanization

Description

The catalanization of a permutation.
For a permutation $\sigma$, this is the product of the reflections corresponding to the inversions of $\sigma$ in lex-order.
A permutation is $231$-avoiding if and only if it is a fixpoint of this map. Also, for every permutation there exists an index $k$ such that the $k$-fold application of this map is $231$-avoiding.

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.