- FindStat

Identifier

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000021: Permutations ⟶ ℤ (values match St000325The width of the tree associated to a permutation., St000470The number of runs in a permutation.)

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of descents of a permutation.
This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].

Map

to non-crossing permutation

Description

Sends a Dyck path $D$ with valley at positions $\{(i_1,j_1),\ldots,(i_k,j_k)\}$ to the unique non-crossing permutation $\pi$ having descents $\{i_1,\ldots,i_k\}$ and whose inverse has descents $\{j_1,\ldots,j_k\}$.
It sends the area St000012The area of a Dyck path. to the number of inversions St000018The number of inversions of a permutation. and the major index St000027The major index of a Dyck path. to $n(n-1)$ minus the sum of the major index St000004The major index of a permutation. and the inverse major index St000305The inverse major index of a permutation..

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.