Identifier
-
Mp00119:
Dyck paths
—to 321-avoiding permutation (Krattenthaler)⟶
Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001169: Dyck paths ⟶ ℤ (values match St000015The number of peaks of a Dyck path., St000053The number of valleys of the Dyck path., St001068Number of torsionless simple modules in the corresponding Nakayama algebra.)
Values
[1,0] => [1] => [1] => [1,0] => 0
[1,0,1,0] => [1,2] => [1,2] => [1,0,1,0] => 1
[1,1,0,0] => [2,1] => [2,1] => [1,1,0,0] => 0
[1,0,1,0,1,0] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0] => 2
[1,0,1,1,0,0] => [1,3,2] => [1,3,2] => [1,0,1,1,0,0] => 1
[1,1,0,0,1,0] => [2,1,3] => [2,1,3] => [1,1,0,0,1,0] => 1
[1,1,0,1,0,0] => [2,3,1] => [3,2,1] => [1,1,1,0,0,0] => 0
[1,1,1,0,0,0] => [3,1,2] => [3,1,2] => [1,1,1,0,0,0] => 0
[1,0,1,0,1,0,1,0] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => 3
[1,0,1,0,1,1,0,0] => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0] => 2
[1,0,1,1,0,0,1,0] => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0] => 2
[1,0,1,1,0,1,0,0] => [1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0] => 1
[1,0,1,1,1,0,0,0] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0] => 1
[1,1,0,0,1,0,1,0] => [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0] => 2
[1,1,0,0,1,1,0,0] => [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0] => 1
[1,1,0,1,0,0,1,0] => [2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0] => 1
[1,1,0,1,0,1,0,0] => [2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0] => 0
[1,1,0,1,1,0,0,0] => [2,4,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0] => 0
[1,1,1,0,0,0,1,0] => [3,1,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0] => 1
[1,1,1,0,0,1,0,0] => [3,1,4,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0] => 1
[1,1,1,0,1,0,0,0] => [3,4,1,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0] => 0
[1,1,1,1,0,0,0,0] => [4,1,2,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0] => 0
[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] => 4
[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] => 3
[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] => 3
[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] => 2
[1,0,1,0,1,1,1,0,0,0] => [1,2,5,3,4] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0] => 2
[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] => 3
[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] => 2
[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] => 2
[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] => 1
[1,0,1,1,0,1,1,0,0,0] => [1,3,5,2,4] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0] => 1
[1,0,1,1,1,0,0,0,1,0] => [1,4,2,3,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0] => 2
[1,0,1,1,1,0,0,1,0,0] => [1,4,2,5,3] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0] => 2
[1,0,1,1,1,0,1,0,0,0] => [1,4,5,2,3] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0] => 1
[1,0,1,1,1,1,0,0,0,0] => [1,5,2,3,4] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0] => 1
[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] => 3
[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] => 2
[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] => 2
[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] => 1
[1,1,0,0,1,1,1,0,0,0] => [2,1,5,3,4] => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0] => 1
[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] => 2
[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] => 1
[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] => 1
[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] => 0
[1,1,0,1,0,1,1,0,0,0] => [2,3,5,1,4] => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0] => 0
[1,1,0,1,1,0,0,0,1,0] => [2,4,1,3,5] => [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0] => 1
[1,1,0,1,1,0,0,1,0,0] => [2,4,1,5,3] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0] => 1
[1,1,0,1,1,0,1,0,0,0] => [2,4,5,1,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0] => 0
[1,1,0,1,1,1,0,0,0,0] => [2,5,1,3,4] => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0] => 0
[1,1,1,0,0,0,1,0,1,0] => [3,1,2,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0] => 2
[1,1,1,0,0,0,1,1,0,0] => [3,1,2,5,4] => [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0] => 1
[1,1,1,0,0,1,0,0,1,0] => [3,1,4,2,5] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0] => 2
[1,1,1,0,0,1,0,1,0,0] => [3,1,4,5,2] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0] => 1
[1,1,1,0,0,1,1,0,0,0] => [3,1,5,2,4] => [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0] => 1
[1,1,1,0,1,0,0,0,1,0] => [3,4,1,2,5] => [4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0] => 1
[1,1,1,0,1,0,0,1,0,0] => [3,4,1,5,2] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0] => 1
[1,1,1,0,1,0,1,0,0,0] => [3,4,5,1,2] => [5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0] => 0
[1,1,1,0,1,1,0,0,0,0] => [3,5,1,2,4] => [5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0] => 0
[1,1,1,1,0,0,0,0,1,0] => [4,1,2,3,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0] => 1
[1,1,1,1,0,0,0,1,0,0] => [4,1,2,5,3] => [4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0] => 1
[1,1,1,1,0,0,1,0,0,0] => [4,1,5,2,3] => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0] => 1
[1,1,1,1,0,1,0,0,0,0] => [4,5,1,2,3] => [5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0] => 0
[1,1,1,1,1,0,0,0,0,0] => [5,1,2,3,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0] => 0
[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] => 5
[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] => 4
[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] => 4
[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] => 3
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,4,5] => [1,2,3,6,4,5] => [1,0,1,0,1,0,1,1,1,0,0,0] => 3
[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] => 4
[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] => 3
[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] => 3
[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] => 2
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,4,6,3,5] => [1,2,6,4,3,5] => [1,0,1,0,1,1,1,1,0,0,0,0] => 2
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,3,4,6] => [1,2,5,3,4,6] => [1,0,1,0,1,1,1,0,0,0,1,0] => 3
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,3,6,4] => [1,2,5,6,3,4] => [1,0,1,0,1,1,1,0,1,0,0,0] => 3
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,5,6,3,4] => [1,2,6,3,5,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => 2
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,3,4,5] => [1,2,6,3,4,5] => [1,0,1,0,1,1,1,1,0,0,0,0] => 2
[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] => 4
[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] => 3
[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] => 3
[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] => 2
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,4,5] => [1,3,2,6,4,5] => [1,0,1,1,0,0,1,1,1,0,0,0] => 2
[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] => 3
[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] => 2
[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] => 2
[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] => 1
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,3,4,6,2,5] => [1,6,3,4,2,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => 1
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,3,5,2,4,6] => [1,5,3,2,4,6] => [1,0,1,1,1,1,0,0,0,0,1,0] => 2
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,2,6,4] => [1,5,3,6,2,4] => [1,0,1,1,1,1,0,0,1,0,0,0] => 2
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,5,6,2,4] => [1,6,3,2,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0] => 1
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,2,4,5] => [1,6,3,2,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => 1
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,2,3,5,6] => [1,4,2,3,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0] => 3
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,2,3,6,5] => [1,4,2,3,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0] => 2
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,2,5,3,6] => [1,4,5,2,3,6] => [1,0,1,1,1,0,1,0,0,0,1,0] => 3
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,2,5,6,3] => [1,4,6,2,5,3] => [1,0,1,1,1,0,1,1,0,0,0,0] => 2
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,4,2,6,3,5] => [1,4,6,2,3,5] => [1,0,1,1,1,0,1,1,0,0,0,0] => 2
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,4,5,2,3,6] => [1,5,2,4,3,6] => [1,0,1,1,1,1,0,0,0,0,1,0] => 2
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,4,5,2,6,3] => [1,5,6,4,2,3] => [1,0,1,1,1,1,0,1,0,0,0,0] => 2
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,4,5,6,2,3] => [1,6,2,4,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0] => 1
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,4,6,2,3,5] => [1,6,2,4,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => 1
>>> Load all 196 entries. <<<
search for individual values
searching the database for the individual values of this statistic
/
search for generating function
searching the database for statistics with the same generating function
Description
Number of simple modules with projective dimension at least two in 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.
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.
Map
descent views to invisible inversion bottoms
Description
Return a permutation whose multiset of invisible inversion bottoms is the multiset of descent views of the given permutation.
An invisible inversion of a permutation $\sigma$ is a pair $i < j$ such that $i < \sigma(j) < \sigma(i)$. The element $\sigma(j)$ is then an invisible inversion bottom.
A descent view in a permutation $\pi$ is an element $\pi(j)$ such that $\pi(i+1) < \pi(j) < \pi(i)$, and additionally the smallest element in the decreasing run containing $\pi(i)$ is smaller than the smallest element in the decreasing run containing $\pi(j)$.
This map is a bijection $\chi:\mathfrak S_n \to \mathfrak S_n$, such that
An invisible inversion of a permutation $\sigma$ is a pair $i < j$ such that $i < \sigma(j) < \sigma(i)$. The element $\sigma(j)$ is then an invisible inversion bottom.
A descent view in a permutation $\pi$ is an element $\pi(j)$ such that $\pi(i+1) < \pi(j) < \pi(i)$, and additionally the smallest element in the decreasing run containing $\pi(i)$ is smaller than the smallest element in the decreasing run containing $\pi(j)$.
This map is a bijection $\chi:\mathfrak S_n \to \mathfrak S_n$, such that
- the multiset of descent views in $\pi$ is the multiset of invisible inversion bottoms in $\chi(\pi)$,
- the set of left-to-right maxima of $\pi$ is the set of maximal elements in the cycles of $\chi(\pi)$,
- the set of global ascent of $\pi$ is the set of global ascent of $\chi(\pi)$,
- the set of maximal elements in the decreasing runs of $\pi$ is the set of weak deficiency positions of $\chi(\pi)$, and
- the set of minimal elements in the decreasing runs of $\pi$ is the set of weak deficiency values of $\chi(\pi)$.
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