Identifier
-
Mp00023:
Dyck paths
—to non-crossing permutation⟶
Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000015: Dyck paths ⟶ ℤ (values match St000053The number of valleys of the Dyck path., St001068Number of torsionless simple modules in the corresponding Nakayama algebra., St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra.)
Values
[1,0] => [1] => [1] => [1,0] => 1
[1,0,1,0] => [1,2] => [1,2] => [1,0,1,0] => 2
[1,1,0,0] => [2,1] => [1,2] => [1,0,1,0] => 2
[1,0,1,0,1,0] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0] => 3
[1,0,1,1,0,0] => [1,3,2] => [1,2,3] => [1,0,1,0,1,0] => 3
[1,1,0,0,1,0] => [2,1,3] => [1,2,3] => [1,0,1,0,1,0] => 3
[1,1,0,1,0,0] => [2,3,1] => [1,2,3] => [1,0,1,0,1,0] => 3
[1,1,1,0,0,0] => [3,2,1] => [1,3,2] => [1,0,1,1,0,0] => 2
[1,0,1,0,1,0,1,0] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => 4
[1,0,1,0,1,1,0,0] => [1,2,4,3] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => 4
[1,0,1,1,0,0,1,0] => [1,3,2,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => 4
[1,0,1,1,0,1,0,0] => [1,3,4,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => 4
[1,0,1,1,1,0,0,0] => [1,4,3,2] => [1,2,4,3] => [1,0,1,0,1,1,0,0] => 3
[1,1,0,0,1,0,1,0] => [2,1,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => 4
[1,1,0,0,1,1,0,0] => [2,1,4,3] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => 4
[1,1,0,1,0,0,1,0] => [2,3,1,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => 4
[1,1,0,1,0,1,0,0] => [2,3,4,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => 4
[1,1,0,1,1,0,0,0] => [2,4,3,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0] => 3
[1,1,1,0,0,0,1,0] => [3,2,1,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0] => 3
[1,1,1,0,0,1,0,0] => [3,2,4,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0] => 3
[1,1,1,0,1,0,0,0] => [4,2,3,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0] => 2
[1,1,1,1,0,0,0,0] => [4,3,2,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0] => 2
[1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 5
[1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 5
[1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 5
[1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 5
[1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => 4
[1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 5
[1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 5
[1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 5
[1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 5
[1,0,1,1,0,1,1,0,0,0] => [1,3,5,4,2] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => 4
[1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => 4
[1,0,1,1,1,0,0,1,0,0] => [1,4,3,5,2] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0] => 4
[1,0,1,1,1,0,1,0,0,0] => [1,5,3,4,2] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0] => 3
[1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0] => 3
[1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 5
[1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 5
[1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 5
[1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 5
[1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => 4
[1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 5
[1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 5
[1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 5
[1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 5
[1,1,0,1,0,1,1,0,0,0] => [2,3,5,4,1] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => 4
[1,1,0,1,1,0,0,0,1,0] => [2,4,3,1,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => 4
[1,1,0,1,1,0,0,1,0,0] => [2,4,3,5,1] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0] => 4
[1,1,0,1,1,0,1,0,0,0] => [2,5,3,4,1] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0] => 3
[1,1,0,1,1,1,0,0,0,0] => [2,5,4,3,1] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0] => 3
[1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0] => 4
[1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0] => 4
[1,1,1,0,0,1,0,0,1,0] => [3,2,4,1,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0] => 4
[1,1,1,0,0,1,0,1,0,0] => [3,2,4,5,1] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0] => 4
[1,1,1,0,0,1,1,0,0,0] => [3,2,5,4,1] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0] => 3
[1,1,1,0,1,0,0,0,1,0] => [4,2,3,1,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0] => 3
[1,1,1,0,1,0,0,1,0,0] => [4,2,3,5,1] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0] => 3
[1,1,1,0,1,0,1,0,0,0] => [5,2,3,4,1] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0] => 2
[1,1,1,0,1,1,0,0,0,0] => [5,2,4,3,1] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0] => 2
[1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0] => 3
[1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0] => 3
[1,1,1,1,0,0,1,0,0,0] => [5,3,2,4,1] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0] => 2
[1,1,1,1,0,1,0,0,0,0] => [5,3,4,2,1] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0] => 2
[1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0] => 2
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 6
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 6
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 6
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 6
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,5,4] => [1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0] => 5
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 6
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 6
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,4,5,3,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 6
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 6
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,4,6,5,3] => [1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0] => 5
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,4,3,6] => [1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0] => 5
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,4,6,3] => [1,2,3,5,6,4] => [1,0,1,0,1,0,1,1,0,1,0,0] => 5
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,6,4,5,3] => [1,2,3,6,4,5] => [1,0,1,0,1,0,1,1,1,0,0,0] => 4
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,5,4,3] => [1,2,3,6,4,5] => [1,0,1,0,1,0,1,1,1,0,0,0] => 4
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 6
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 6
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 6
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,3,2,5,6,4] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 6
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,5,4] => [1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0] => 5
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,4,2,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 6
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,4,2,6,5] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 6
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,3,4,5,2,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 6
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 6
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,3,4,6,5,2] => [1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0] => 5
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,3,5,4,2,6] => [1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0] => 5
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,4,6,2] => [1,2,3,5,6,4] => [1,0,1,0,1,0,1,1,0,1,0,0] => 5
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,6,4,5,2] => [1,2,3,6,4,5] => [1,0,1,0,1,0,1,1,1,0,0,0] => 4
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,5,4,2] => [1,2,3,6,4,5] => [1,0,1,0,1,0,1,1,1,0,0,0] => 4
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,3,2,5,6] => [1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0] => 5
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,3,2,6,5] => [1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0] => 5
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,3,5,2,6] => [1,2,4,5,3,6] => [1,0,1,0,1,1,0,1,0,0,1,0] => 5
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,3,5,6,2] => [1,2,4,5,6,3] => [1,0,1,0,1,1,0,1,0,1,0,0] => 5
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,4,3,6,5,2] => [1,2,4,6,3,5] => [1,0,1,0,1,1,0,1,1,0,0,0] => 4
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,5,3,4,2,6] => [1,2,5,3,4,6] => [1,0,1,0,1,1,1,0,0,0,1,0] => 4
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,5,3,4,6,2] => [1,2,5,6,3,4] => [1,0,1,0,1,1,1,0,1,0,0,0] => 4
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,6,3,4,5,2] => [1,2,6,3,4,5] => [1,0,1,0,1,1,1,1,0,0,0,0] => 3
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,6,3,5,4,2] => [1,2,6,3,4,5] => [1,0,1,0,1,1,1,1,0,0,0,0] => 3
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Description
The number of peaks of a Dyck path.
Map
cycle-as-one-line notation
Description
Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.
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..
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.
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.
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