Identifier
-
Mp00119:
Dyck paths
—to 321-avoiding permutation (Krattenthaler)⟶
Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000099: Permutations ⟶ ℤ (values match St000023The number of inner peaks of a permutation.)
Values
[1,0] => [1] => [1] => 1
[1,0,1,0] => [1,2] => [1,2] => 1
[1,1,0,0] => [2,1] => [2,1] => 1
[1,0,1,0,1,0] => [1,2,3] => [1,2,3] => 1
[1,0,1,1,0,0] => [1,3,2] => [1,3,2] => 2
[1,1,0,0,1,0] => [2,1,3] => [2,1,3] => 1
[1,1,0,1,0,0] => [2,3,1] => [3,2,1] => 1
[1,1,1,0,0,0] => [3,1,2] => [2,3,1] => 2
[1,0,1,0,1,0,1,0] => [1,2,3,4] => [1,2,3,4] => 1
[1,0,1,0,1,1,0,0] => [1,2,4,3] => [1,2,4,3] => 2
[1,0,1,1,0,0,1,0] => [1,3,2,4] => [1,3,2,4] => 2
[1,0,1,1,0,1,0,0] => [1,3,4,2] => [1,4,3,2] => 2
[1,0,1,1,1,0,0,0] => [1,4,2,3] => [1,3,4,2] => 2
[1,1,0,0,1,0,1,0] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0] => [2,1,4,3] => [2,1,4,3] => 2
[1,1,0,1,0,0,1,0] => [2,3,1,4] => [3,2,1,4] => 1
[1,1,0,1,0,1,0,0] => [2,3,4,1] => [4,2,3,1] => 2
[1,1,0,1,1,0,0,0] => [2,4,1,3] => [3,2,4,1] => 2
[1,1,1,0,0,0,1,0] => [3,1,2,4] => [2,3,1,4] => 2
[1,1,1,0,0,1,0,0] => [3,1,4,2] => [3,4,1,2] => 2
[1,1,1,0,1,0,0,0] => [3,4,1,2] => [2,4,3,1] => 2
[1,1,1,1,0,0,0,0] => [4,1,2,3] => [2,3,4,1] => 2
[1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => [1,2,3,5,4] => 2
[1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => [1,2,4,3,5] => 2
[1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => [1,2,5,4,3] => 2
[1,0,1,0,1,1,1,0,0,0] => [1,2,5,3,4] => [1,2,4,5,3] => 2
[1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => [1,3,2,4,5] => 2
[1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => [1,3,2,5,4] => 3
[1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => [1,4,3,2,5] => 2
[1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => [1,5,3,4,2] => 3
[1,0,1,1,0,1,1,0,0,0] => [1,3,5,2,4] => [1,4,3,5,2] => 3
[1,0,1,1,1,0,0,0,1,0] => [1,4,2,3,5] => [1,3,4,2,5] => 2
[1,0,1,1,1,0,0,1,0,0] => [1,4,2,5,3] => [1,4,5,2,3] => 2
[1,0,1,1,1,0,1,0,0,0] => [1,4,5,2,3] => [1,3,5,4,2] => 2
[1,0,1,1,1,1,0,0,0,0] => [1,5,2,3,4] => [1,3,4,5,2] => 2
[1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => [2,1,3,5,4] => 2
[1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => [2,1,4,3,5] => 2
[1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => [2,1,5,4,3] => 2
[1,1,0,0,1,1,1,0,0,0] => [2,1,5,3,4] => [2,1,4,5,3] => 2
[1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => [3,2,1,4,5] => 1
[1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => [3,2,1,5,4] => 2
[1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => [4,2,3,1,5] => 2
[1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => [5,2,3,4,1] => 2
[1,1,0,1,0,1,1,0,0,0] => [2,3,5,1,4] => [4,2,3,5,1] => 2
[1,1,0,1,1,0,0,0,1,0] => [2,4,1,3,5] => [3,2,4,1,5] => 2
[1,1,0,1,1,0,0,1,0,0] => [2,4,1,5,3] => [4,2,5,1,3] => 2
[1,1,0,1,1,0,1,0,0,0] => [2,4,5,1,3] => [3,2,5,4,1] => 2
[1,1,0,1,1,1,0,0,0,0] => [2,5,1,3,4] => [3,2,4,5,1] => 2
[1,1,1,0,0,0,1,0,1,0] => [3,1,2,4,5] => [2,3,1,4,5] => 2
[1,1,1,0,0,0,1,1,0,0] => [3,1,2,5,4] => [2,3,1,5,4] => 3
[1,1,1,0,0,1,0,0,1,0] => [3,1,4,2,5] => [3,4,1,2,5] => 2
[1,1,1,0,0,1,0,1,0,0] => [3,1,4,5,2] => [3,5,1,4,2] => 3
[1,1,1,0,0,1,1,0,0,0] => [3,1,5,2,4] => [3,4,1,5,2] => 3
[1,1,1,0,1,0,0,0,1,0] => [3,4,1,2,5] => [2,4,3,1,5] => 2
[1,1,1,0,1,0,0,1,0,0] => [3,4,1,5,2] => [4,5,3,1,2] => 2
[1,1,1,0,1,0,1,0,0,0] => [3,4,5,1,2] => [2,5,3,4,1] => 3
[1,1,1,0,1,1,0,0,0,0] => [3,5,1,2,4] => [2,4,3,5,1] => 3
[1,1,1,1,0,0,0,0,1,0] => [4,1,2,3,5] => [2,3,4,1,5] => 2
[1,1,1,1,0,0,0,1,0,0] => [4,1,2,5,3] => [2,4,5,1,3] => 2
[1,1,1,1,0,0,1,0,0,0] => [4,1,5,2,3] => [4,3,5,1,2] => 2
[1,1,1,1,0,1,0,0,0,0] => [4,5,1,2,3] => [2,3,5,4,1] => 2
[1,1,1,1,1,0,0,0,0,0] => [5,1,2,3,4] => [2,3,4,5,1] => 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
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => 2
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => 2
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => [1,2,3,6,5,4] => 2
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,4,5] => [1,2,3,5,6,4] => 2
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => 2
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => [1,2,4,3,6,5] => 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] => 2
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => [1,2,6,4,5,3] => 3
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,4,6,3,5] => [1,2,5,4,6,3] => 3
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,3,4,6] => [1,2,4,5,3,6] => 2
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,3,6,4] => [1,2,5,6,3,4] => 2
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,5,6,3,4] => [1,2,4,6,5,3] => 2
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,3,4,5] => [1,2,4,5,6,3] => 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] => 2
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => [1,3,2,4,6,5] => 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] => 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] => 3
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,4,5] => [1,3,2,5,6,4] => 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] => 2
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,4,2,6,5] => [1,4,3,2,6,5] => 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] => 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] => 3
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,3,4,6,2,5] => [1,5,3,4,6,2] => 3
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,3,5,2,4,6] => [1,4,3,5,2,6] => 3
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,2,6,4] => [1,5,3,6,2,4] => 3
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,5,6,2,4] => [1,4,3,6,5,2] => 3
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,2,4,5] => [1,4,3,5,6,2] => 3
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,2,3,5,6] => [1,3,4,2,5,6] => 2
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,2,3,6,5] => [1,3,4,2,6,5] => 3
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,2,5,3,6] => [1,4,5,2,3,6] => 2
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,2,5,6,3] => [1,4,6,2,5,3] => 3
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,4,2,6,3,5] => [1,4,5,2,6,3] => 3
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,4,5,2,3,6] => [1,3,5,4,2,6] => 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] => 2
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,4,5,6,2,3] => [1,3,6,4,5,2] => 3
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,4,6,2,3,5] => [1,3,5,4,6,2] => 3
>>> 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
The number of valleys of a permutation, including the boundary.
The number of valleys excluding the boundary is St000353The number of inner valleys of a permutation..
The number of valleys excluding the boundary is St000353The number of inner valleys of a 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
first fundamental transformation
Description
Return the permutation whose cycles are the subsequences between successive left to right maxima.
searching the database
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!