Identifier
-
Mp00119:
Dyck paths
—to 321-avoiding permutation (Krattenthaler)⟶
Permutations
Mp00159: Permutations —Demazure product with inverse⟶ 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] => [2,1] => [1,1,0,0] => 1
[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] => [1,3,2] => [1,0,1,1,0,0] => 2
[1,1,0,0,1,0] => [2,1,3] => [2,1,3] => [1,1,0,0,1,0] => 2
[1,1,0,1,0,0] => [2,3,1] => [3,2,1] => [1,1,1,0,0,0] => 1
[1,1,1,0,0,0] => [3,1,2] => [3,2,1] => [1,1,1,0,0,0] => 1
[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] => [1,2,4,3] => [1,0,1,0,1,1,0,0] => 3
[1,0,1,1,0,0,1,0] => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0] => 3
[1,0,1,1,0,1,0,0] => [1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0] => 2
[1,0,1,1,1,0,0,0] => [1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0] => 2
[1,1,0,0,1,0,1,0] => [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0] => 3
[1,1,0,0,1,1,0,0] => [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0] => 2
[1,1,0,1,0,0,1,0] => [2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0] => 2
[1,1,0,1,0,1,0,0] => [2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0] => 1
[1,1,0,1,1,0,0,0] => [2,4,1,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0] => 2
[1,1,1,0,0,0,1,0] => [3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0] => 2
[1,1,1,0,0,1,0,0] => [3,1,4,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0] => 1
[1,1,1,0,1,0,0,0] => [3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0] => 1
[1,1,1,1,0,0,0,0] => [4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0] => 1
[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] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => 4
[1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => 4
[1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0] => 3
[1,0,1,0,1,1,1,0,0,0] => [1,2,5,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0] => 3
[1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0] => 4
[1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0] => 3
[1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0] => 3
[1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0] => 2
[1,0,1,1,0,1,1,0,0,0] => [1,3,5,2,4] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0] => 3
[1,0,1,1,1,0,0,0,1,0] => [1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0] => 3
[1,0,1,1,1,0,0,1,0,0] => [1,4,2,5,3] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0] => 2
[1,0,1,1,1,0,1,0,0,0] => [1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0] => 2
[1,0,1,1,1,1,0,0,0,0] => [1,5,2,3,4] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0] => 2
[1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0] => 4
[1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0] => 3
[1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0] => 3
[1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0] => 2
[1,1,0,0,1,1,1,0,0,0] => [2,1,5,3,4] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0] => 2
[1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0] => 3
[1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0] => 2
[1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0] => 2
[1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0] => 1
[1,1,0,1,0,1,1,0,0,0] => [2,3,5,1,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0] => 2
[1,1,0,1,1,0,0,0,1,0] => [2,4,1,3,5] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0] => 3
[1,1,0,1,1,0,0,1,0,0] => [2,4,1,5,3] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0] => 2
[1,1,0,1,1,0,1,0,0,0] => [2,4,5,1,3] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0] => 2
[1,1,0,1,1,1,0,0,0,0] => [2,5,1,3,4] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0] => 2
[1,1,1,0,0,0,1,0,1,0] => [3,1,2,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0] => 3
[1,1,1,0,0,0,1,1,0,0] => [3,1,2,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0] => 2
[1,1,1,0,0,1,0,0,1,0] => [3,1,4,2,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0] => 2
[1,1,1,0,0,1,0,1,0,0] => [3,1,4,5,2] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0] => 1
[1,1,1,0,0,1,1,0,0,0] => [3,1,5,2,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0] => 2
[1,1,1,0,1,0,0,0,1,0] => [3,4,1,2,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0] => 2
[1,1,1,0,1,0,0,1,0,0] => [3,4,1,5,2] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0] => 1
[1,1,1,0,1,0,1,0,0,0] => [3,4,5,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0] => 1
[1,1,1,0,1,1,0,0,0,0] => [3,5,1,2,4] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0] => 2
[1,1,1,1,0,0,0,0,1,0] => [4,1,2,3,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0] => 2
[1,1,1,1,0,0,0,1,0,0] => [4,1,2,5,3] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0] => 1
[1,1,1,1,0,0,1,0,0,0] => [4,1,5,2,3] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0] => 1
[1,1,1,1,0,1,0,0,0,0] => [4,5,1,2,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0] => 1
[1,1,1,1,1,0,0,0,0,0] => [5,1,2,3,4] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0] => 1
[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] => [1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0] => 5
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0] => 5
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => [1,2,3,6,5,4] => [1,0,1,0,1,0,1,1,1,0,0,0] => 4
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,4,5] => [1,2,3,6,5,4] => [1,0,1,0,1,0,1,1,1,0,0,0] => 4
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0] => 5
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => [1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0] => 4
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,4,5,3,6] => [1,2,5,4,3,6] => [1,0,1,0,1,1,1,0,0,0,1,0] => 4
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => [1,2,6,4,5,3] => [1,0,1,0,1,1,1,1,0,0,0,0] => 3
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,4,6,3,5] => [1,2,5,6,3,4] => [1,0,1,0,1,1,1,0,1,0,0,0] => 4
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,3,4,6] => [1,2,5,4,3,6] => [1,0,1,0,1,1,1,0,0,0,1,0] => 4
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,3,6,4] => [1,2,6,4,5,3] => [1,0,1,0,1,1,1,1,0,0,0,0] => 3
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,5,6,3,4] => [1,2,6,5,4,3] => [1,0,1,0,1,1,1,1,0,0,0,0] => 3
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,3,4,5] => [1,2,6,4,5,3] => [1,0,1,0,1,1,1,1,0,0,0,0] => 3
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => [1,3,2,4,5,6] => [1,0,1,1,0,0,1,0,1,0,1,0] => 5
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => [1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0] => 4
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => [1,0,1,1,0,0,1,1,0,0,1,0] => 4
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,3,2,5,6,4] => [1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0] => 3
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,4,5] => [1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0] => 3
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,4,2,5,6] => [1,4,3,2,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0] => 4
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,4,2,6,5] => [1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0] => 3
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,3,4,5,2,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0] => 3
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => [1,6,3,4,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,3,4,6,2,5] => [1,5,3,6,2,4] => [1,0,1,1,1,1,0,0,1,0,0,0] => 3
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,3,5,2,4,6] => [1,4,5,2,3,6] => [1,0,1,1,1,0,1,0,0,0,1,0] => 4
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,2,6,4] => [1,4,6,2,5,3] => [1,0,1,1,1,0,1,1,0,0,0,0] => 3
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,5,6,2,4] => [1,5,6,4,2,3] => [1,0,1,1,1,1,0,1,0,0,0,0] => 3
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,2,4,5] => [1,4,6,2,5,3] => [1,0,1,1,1,0,1,1,0,0,0,0] => 3
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,2,3,5,6] => [1,4,3,2,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0] => 4
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,2,3,6,5] => [1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0] => 3
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,2,5,3,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0] => 3
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,2,5,6,3] => [1,6,3,4,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,4,2,6,3,5] => [1,5,3,6,2,4] => [1,0,1,1,1,1,0,0,1,0,0,0] => 3
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,4,5,2,3,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0] => 3
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,4,5,2,6,3] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,4,5,6,2,3] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,4,6,2,3,5] => [1,5,6,4,2,3] => [1,0,1,1,1,1,0,1,0,0,0,0] => 3
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Description
The number of peaks of a Dyck path.
Map
Demazure product with inverse
Description
This map sends a permutation $\pi$ to $\pi^{-1} \star \pi$ where $\star$ denotes the Demazure product on permutations.
This map is a surjection onto the set of involutions, i.e., the set of permutations $\pi$ for which $\pi = \pi^{-1}$.
This map is a surjection onto the set of involutions, i.e., the set of permutations $\pi$ for which $\pi = \pi^{-1}$.
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.
Map
to 321-avoiding permutation (Krattenthaler)
Description
Krattenthaler's bijection to 321-avoiding permutations.
Draw the path of semilength $n$ in an $n\times n$ square matrix, starting at the upper left corner, with right and down steps, and staying below the diagonal. Then the permutation matrix is obtained by placing ones into the cells corresponding to the peaks of the path and placing ones into the remaining columns from left to right, such that the row indices of the cells increase.
Draw the path of semilength $n$ in an $n\times n$ square matrix, starting at the upper left corner, with right and down steps, and staying below the diagonal. Then the permutation matrix is obtained by placing ones into the cells corresponding to the peaks of the path and placing ones into the remaining columns from left to right, such that the row indices of the cells increase.
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