Identifier
-
Mp00127:
Permutations
—left-to-right-maxima to Dyck path⟶
Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001896: Signed permutations ⟶ ℤ
Values
[1] => [1,0] => [1] => [1] => 0
[1,2] => [1,0,1,0] => [2,1] => [2,1] => 1
[2,1] => [1,1,0,0] => [1,2] => [1,2] => 0
[1,2,3] => [1,0,1,0,1,0] => [2,1,3] => [2,1,3] => 1
[1,3,2] => [1,0,1,1,0,0] => [2,3,1] => [2,3,1] => 1
[2,1,3] => [1,1,0,0,1,0] => [3,1,2] => [3,1,2] => 1
[2,3,1] => [1,1,0,1,0,0] => [1,3,2] => [1,3,2] => 1
[3,1,2] => [1,1,1,0,0,0] => [1,2,3] => [1,2,3] => 0
[3,2,1] => [1,1,1,0,0,0] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,0,1,0,1,0,1,0] => [2,1,4,3] => [2,1,4,3] => 2
[1,2,4,3] => [1,0,1,0,1,1,0,0] => [2,4,1,3] => [2,4,1,3] => 2
[1,3,2,4] => [1,0,1,1,0,0,1,0] => [2,1,3,4] => [2,1,3,4] => 1
[1,3,4,2] => [1,0,1,1,0,1,0,0] => [2,3,1,4] => [2,3,1,4] => 1
[1,4,2,3] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => [2,3,4,1] => 1
[1,4,3,2] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => [2,3,4,1] => 1
[2,1,3,4] => [1,1,0,0,1,0,1,0] => [3,1,4,2] => [3,1,4,2] => 1
[2,1,4,3] => [1,1,0,0,1,1,0,0] => [3,4,1,2] => [3,4,1,2] => 1
[2,3,1,4] => [1,1,0,1,0,0,1,0] => [3,1,2,4] => [3,1,2,4] => 1
[2,3,4,1] => [1,1,0,1,0,1,0,0] => [1,3,2,4] => [1,3,2,4] => 1
[2,4,1,3] => [1,1,0,1,1,0,0,0] => [1,3,4,2] => [1,3,4,2] => 1
[2,4,3,1] => [1,1,0,1,1,0,0,0] => [1,3,4,2] => [1,3,4,2] => 1
[3,1,2,4] => [1,1,1,0,0,0,1,0] => [4,1,2,3] => [4,1,2,3] => 1
[3,1,4,2] => [1,1,1,0,0,1,0,0] => [1,4,2,3] => [1,4,2,3] => 1
[3,2,1,4] => [1,1,1,0,0,0,1,0] => [4,1,2,3] => [4,1,2,3] => 1
[3,2,4,1] => [1,1,1,0,0,1,0,0] => [1,4,2,3] => [1,4,2,3] => 1
[3,4,1,2] => [1,1,1,0,1,0,0,0] => [1,2,4,3] => [1,2,4,3] => 1
[3,4,2,1] => [1,1,1,0,1,0,0,0] => [1,2,4,3] => [1,2,4,3] => 1
[4,1,2,3] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,3,2] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => [1,2,3,4] => 0
[4,2,1,3] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => [1,2,3,4] => 0
[4,2,3,1] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,1,2] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,2,1] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0] => [1,3,5,2,4] => [1,3,5,2,4] => 2
[2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0] => [1,3,5,2,4] => [1,3,5,2,4] => 2
[2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0] => [1,3,4,2,5] => [1,3,4,2,5] => 1
[2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0] => [1,3,4,2,5] => [1,3,4,2,5] => 1
[2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0] => [1,3,4,5,2] => [1,3,4,5,2] => 1
[2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0] => [1,3,4,5,2] => [1,3,4,5,2] => 1
[2,5,3,1,4] => [1,1,0,1,1,1,0,0,0,0] => [1,3,4,5,2] => [1,3,4,5,2] => 1
[2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0] => [1,3,4,5,2] => [1,3,4,5,2] => 1
[2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0] => [1,3,4,5,2] => [1,3,4,5,2] => 1
[2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0] => [1,3,4,5,2] => [1,3,4,5,2] => 1
[3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0] => [1,4,2,5,3] => [1,4,2,5,3] => 1
[3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0] => [1,4,5,2,3] => [1,4,5,2,3] => 1
[3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0] => [1,4,5,2,3] => [1,4,5,2,3] => 1
[3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0] => [1,4,2,5,3] => [1,4,2,5,3] => 1
[3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0] => [1,4,5,2,3] => [1,4,5,2,3] => 1
[3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0] => [1,4,5,2,3] => [1,4,5,2,3] => 1
[3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0] => [1,4,2,3,5] => [1,4,2,3,5] => 1
[3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0] => [1,4,2,3,5] => [1,4,2,3,5] => 1
[3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0] => [1,2,4,5,3] => [1,2,4,5,3] => 1
[3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0] => [1,2,4,5,3] => [1,2,4,5,3] => 1
[3,5,2,1,4] => [1,1,1,0,1,1,0,0,0,0] => [1,2,4,5,3] => [1,2,4,5,3] => 1
[3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0] => [1,2,4,5,3] => [1,2,4,5,3] => 1
[3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0] => [1,2,4,5,3] => [1,2,4,5,3] => 1
[3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0] => [1,2,4,5,3] => [1,2,4,5,3] => 1
[4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0] => [1,5,2,3,4] => [1,5,2,3,4] => 1
[4,1,3,5,2] => [1,1,1,1,0,0,0,1,0,0] => [1,5,2,3,4] => [1,5,2,3,4] => 1
[4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0] => [1,2,5,3,4] => [1,2,5,3,4] => 1
[4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0] => [1,2,5,3,4] => [1,2,5,3,4] => 1
[4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0] => [1,5,2,3,4] => [1,5,2,3,4] => 1
[4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0] => [1,5,2,3,4] => [1,5,2,3,4] => 1
[4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0] => [1,2,5,3,4] => [1,2,5,3,4] => 1
[4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0] => [1,2,5,3,4] => [1,2,5,3,4] => 1
[4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0] => [1,5,2,3,4] => [1,5,2,3,4] => 1
[4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0] => [1,5,2,3,4] => [1,5,2,3,4] => 1
[4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0] => [1,2,5,3,4] => [1,2,5,3,4] => 1
[4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0] => [1,2,5,3,4] => [1,2,5,3,4] => 1
[4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [1,2,3,4,5] => 0
>>> Load all 104 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 right descents of a signed permutations.
An index is a right descent if it is a left descent of the inverse signed permutation.
An index is a right descent if it is a left descent of the inverse signed 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.
Map
to signed permutation
Description
The signed permutation with all signs positive.
Map
to 321-avoiding permutation
Description
Sends a Dyck path to a 321-avoiding permutation.
This bijection defined in [3, pp. 60] and in [2, Section 3.1].
It is shown in [1] that it sends the number of centered tunnels to the number of fixed points, the number of right tunnels to the number of exceedences, and the semilength plus the height of the middle point to 2 times the length of the longest increasing subsequence.
This bijection defined in [3, pp. 60] and in [2, Section 3.1].
It is shown in [1] that it sends the number of centered tunnels to the number of fixed points, the number of right tunnels to the number of exceedences, and the semilength plus the height of the middle point to 2 times the length of the longest increasing subsequence.
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!