Identifier
-
Mp00025:
Dyck paths
—to 132-avoiding permutation⟶
Permutations
Mp00325: Permutations —ones to leading⟶ Permutations
St000367: Permutations ⟶ ℤ
Values
[1,0] => [1] => [1] => 0
[1,0,1,0] => [2,1] => [2,1] => 0
[1,1,0,0] => [1,2] => [1,2] => 0
[1,0,1,0,1,0] => [3,2,1] => [3,2,1] => 1
[1,0,1,1,0,0] => [2,3,1] => [2,1,3] => 0
[1,1,0,0,1,0] => [3,1,2] => [3,1,2] => 0
[1,1,0,1,0,0] => [2,1,3] => [1,3,2] => 0
[1,1,1,0,0,0] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0] => [4,3,2,1] => [4,3,1,2] => 1
[1,0,1,0,1,1,0,0] => [3,4,2,1] => [3,2,4,1] => 1
[1,0,1,1,0,0,1,0] => [4,2,3,1] => [4,3,2,1] => 3
[1,0,1,1,0,1,0,0] => [3,2,4,1] => [2,1,3,4] => 0
[1,0,1,1,1,0,0,0] => [2,3,4,1] => [2,1,4,3] => 0
[1,1,0,0,1,0,1,0] => [4,3,1,2] => [4,2,1,3] => 1
[1,1,0,0,1,1,0,0] => [3,4,1,2] => [3,1,4,2] => 0
[1,1,0,1,0,0,1,0] => [4,2,1,3] => [4,2,3,1] => 0
[1,1,0,1,0,1,0,0] => [3,2,1,4] => [1,4,3,2] => 1
[1,1,0,1,1,0,0,0] => [2,3,1,4] => [1,4,2,3] => 0
[1,1,1,0,0,0,1,0] => [4,1,2,3] => [4,1,3,2] => 0
[1,1,1,0,0,1,0,0] => [3,1,2,4] => [1,3,4,2] => 0
[1,1,1,0,1,0,0,0] => [2,1,3,4] => [1,3,2,4] => 0
[1,1,1,1,0,0,0,0] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => [5,4,1,2,3] => 1
[1,0,1,0,1,0,1,1,0,0] => [4,5,3,2,1] => [4,3,5,1,2] => 1
[1,0,1,0,1,1,0,0,1,0] => [5,3,4,2,1] => [5,4,2,1,3] => 3
[1,0,1,0,1,1,0,1,0,0] => [4,3,5,2,1] => [3,2,4,5,1] => 1
[1,0,1,0,1,1,1,0,0,0] => [3,4,5,2,1] => [3,2,5,4,1] => 2
[1,0,1,1,0,0,1,0,1,0] => [5,4,2,3,1] => [5,4,1,3,2] => 1
[1,0,1,1,0,0,1,1,0,0] => [4,5,2,3,1] => [4,3,5,2,1] => 4
[1,0,1,1,0,1,0,0,1,0] => [5,3,2,4,1] => [5,4,3,1,2] => 3
[1,0,1,1,0,1,0,1,0,0] => [4,3,2,5,1] => [2,1,3,4,5] => 0
[1,0,1,1,0,1,1,0,0,0] => [3,4,2,5,1] => [2,1,4,3,5] => 0
[1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => [5,4,3,2,1] => 6
[1,0,1,1,1,0,0,1,0,0] => [4,2,3,5,1] => [2,1,3,5,4] => 0
[1,0,1,1,1,0,1,0,0,0] => [3,2,4,5,1] => [2,1,5,3,4] => 0
[1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [2,1,5,4,3] => 1
[1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,2] => [5,3,1,2,4] => 1
[1,1,0,0,1,0,1,1,0,0] => [4,5,3,1,2] => [4,2,5,1,3] => 1
[1,1,0,0,1,1,0,0,1,0] => [5,3,4,1,2] => [5,3,2,1,4] => 4
[1,1,0,0,1,1,0,1,0,0] => [4,3,5,1,2] => [3,1,4,5,2] => 0
[1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => [3,1,5,4,2] => 1
[1,1,0,1,0,0,1,0,1,0] => [5,4,2,1,3] => [5,3,1,4,2] => 1
[1,1,0,1,0,0,1,1,0,0] => [4,5,2,1,3] => [4,2,5,3,1] => 1
[1,1,0,1,0,1,0,0,1,0] => [5,3,2,1,4] => [5,3,4,2,1] => 3
[1,1,0,1,0,1,0,1,0,0] => [4,3,2,1,5] => [1,5,4,3,2] => 3
[1,1,0,1,0,1,1,0,0,0] => [3,4,2,1,5] => [1,5,4,2,3] => 1
[1,1,0,1,1,0,0,0,1,0] => [5,2,3,1,4] => [5,3,4,1,2] => 0
[1,1,0,1,1,0,0,1,0,0] => [4,2,3,1,5] => [1,5,3,4,2] => 0
[1,1,0,1,1,0,1,0,0,0] => [3,2,4,1,5] => [1,5,3,2,4] => 1
[1,1,0,1,1,1,0,0,0,0] => [2,3,4,1,5] => [1,5,2,3,4] => 0
[1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3] => [5,2,1,4,3] => 1
[1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => [4,1,5,3,2] => 1
[1,1,1,0,0,1,0,0,1,0] => [5,3,1,2,4] => [5,2,4,3,1] => 2
[1,1,1,0,0,1,0,1,0,0] => [4,3,1,2,5] => [1,4,5,3,2] => 2
[1,1,1,0,0,1,1,0,0,0] => [3,4,1,2,5] => [1,4,5,2,3] => 0
[1,1,1,0,1,0,0,0,1,0] => [5,2,1,3,4] => [5,2,4,1,3] => 0
[1,1,1,0,1,0,0,1,0,0] => [4,2,1,3,5] => [1,4,3,5,2] => 1
[1,1,1,0,1,0,1,0,0,0] => [3,2,1,4,5] => [1,4,3,2,5] => 2
[1,1,1,0,1,1,0,0,0,0] => [2,3,1,4,5] => [1,4,2,3,5] => 0
[1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => [5,1,4,2,3] => 0
[1,1,1,1,0,0,0,1,0,0] => [4,1,2,3,5] => [1,3,4,5,2] => 0
[1,1,1,1,0,0,1,0,0,0] => [3,1,2,4,5] => [1,3,4,2,5] => 0
[1,1,1,1,0,1,0,0,0,0] => [2,1,3,4,5] => [1,3,2,4,5] => 0
[1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0,1,0,1,0] => [6,5,4,3,2,1] => [6,5,1,2,3,4] => 1
[1,0,1,0,1,0,1,0,1,1,0,0] => [5,6,4,3,2,1] => [5,4,6,1,2,3] => 1
[1,0,1,0,1,0,1,1,0,0,1,0] => [6,4,5,3,2,1] => [6,5,2,1,3,4] => 3
[1,0,1,0,1,0,1,1,0,1,0,0] => [5,4,6,3,2,1] => [4,3,5,6,1,2] => 1
[1,0,1,0,1,0,1,1,1,0,0,0] => [4,5,6,3,2,1] => [4,3,6,5,1,2] => 2
[1,0,1,0,1,1,0,0,1,0,1,0] => [6,5,3,4,2,1] => [6,5,1,3,2,4] => 1
[1,0,1,0,1,1,0,0,1,1,0,0] => [5,6,3,4,2,1] => [5,4,6,2,1,3] => 4
[1,0,1,0,1,1,0,1,0,0,1,0] => [6,4,3,5,2,1] => [6,5,3,1,2,4] => 3
[1,0,1,0,1,1,0,1,0,1,0,0] => [5,4,3,6,2,1] => [3,2,4,5,6,1] => 1
[1,0,1,0,1,1,0,1,1,0,0,0] => [4,5,3,6,2,1] => [3,2,5,4,6,1] => 2
[1,0,1,0,1,1,1,0,0,0,1,0] => [6,3,4,5,2,1] => [6,5,3,2,1,4] => 7
[1,0,1,0,1,1,1,0,0,1,0,0] => [5,3,4,6,2,1] => [3,2,4,6,5,1] => 2
[1,0,1,0,1,1,1,0,1,0,0,0] => [4,3,5,6,2,1] => [3,2,6,4,5,1] => 1
[1,0,1,0,1,1,1,1,0,0,0,0] => [3,4,5,6,2,1] => [3,2,6,5,4,1] => 4
[1,0,1,1,0,0,1,0,1,0,1,0] => [6,5,4,2,3,1] => [6,5,1,2,4,3] => 1
[1,0,1,1,0,0,1,0,1,1,0,0] => [5,6,4,2,3,1] => [5,4,6,1,3,2] => 1
[1,0,1,1,0,0,1,1,0,0,1,0] => [6,4,5,2,3,1] => [6,5,2,1,4,3] => 3
[1,0,1,1,0,0,1,1,0,1,0,0] => [5,4,6,2,3,1] => [4,3,5,6,2,1] => 5
[1,0,1,1,0,0,1,1,1,0,0,0] => [4,5,6,2,3,1] => [4,3,6,5,2,1] => 6
[1,0,1,1,0,1,0,0,1,0,1,0] => [6,5,3,2,4,1] => [6,5,1,4,2,3] => 1
[1,0,1,1,0,1,0,0,1,1,0,0] => [5,6,3,2,4,1] => [5,4,6,3,1,2] => 4
[1,0,1,1,0,1,0,1,0,0,1,0] => [6,4,3,2,5,1] => [6,5,4,1,2,3] => 3
[1,0,1,1,0,1,0,1,0,1,0,0] => [5,4,3,2,6,1] => [2,1,3,4,5,6] => 0
[1,0,1,1,0,1,0,1,1,0,0,0] => [4,5,3,2,6,1] => [2,1,4,3,5,6] => 0
[1,0,1,1,0,1,1,0,0,0,1,0] => [6,3,4,2,5,1] => [6,5,4,2,1,3] => 6
[1,0,1,1,0,1,1,0,0,1,0,0] => [5,3,4,2,6,1] => [2,1,3,5,4,6] => 0
[1,0,1,1,0,1,1,0,1,0,0,0] => [4,3,5,2,6,1] => [2,1,5,3,4,6] => 0
[1,0,1,1,0,1,1,1,0,0,0,0] => [3,4,5,2,6,1] => [2,1,5,4,3,6] => 2
[1,0,1,1,1,0,0,0,1,0,1,0] => [6,5,2,3,4,1] => [6,5,1,4,3,2] => 4
[1,0,1,1,1,0,0,0,1,1,0,0] => [5,6,2,3,4,1] => [5,4,6,3,2,1] => 8
[1,0,1,1,1,0,0,1,0,0,1,0] => [6,4,2,3,5,1] => [6,5,4,1,3,2] => 3
[1,0,1,1,1,0,0,1,0,1,0,0] => [5,4,2,3,6,1] => [2,1,3,4,6,5] => 0
[1,0,1,1,1,0,0,1,1,0,0,0] => [4,5,2,3,6,1] => [2,1,4,3,6,5] => 0
[1,0,1,1,1,0,1,0,0,0,1,0] => [6,3,2,4,5,1] => [6,5,4,3,1,2] => 6
[1,0,1,1,1,0,1,0,0,1,0,0] => [5,3,2,4,6,1] => [2,1,3,6,4,5] => 0
[1,0,1,1,1,0,1,0,1,0,0,0] => [4,3,2,5,6,1] => [2,1,6,3,4,5] => 0
[1,0,1,1,1,0,1,1,0,0,0,0] => [3,4,2,5,6,1] => [2,1,6,4,3,5] => 1
>>> Load all 216 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 simsun double descents of a permutation.
The restriction of a permutation $\pi$ to $[k] = \{1,\ldots,k\}$ is given in one-line notation by the subword of $\pi$ of letters in $[k]$.
A simsun double descent of a permutation $\pi$ is a double descent of any restriction of $\pi$ to $[1,\ldots,k]$ for some $k$. (Note here that the same double descent can appear in multiple restrictions!)
The restriction of a permutation $\pi$ to $[k] = \{1,\ldots,k\}$ is given in one-line notation by the subword of $\pi$ of letters in $[k]$.
A simsun double descent of a permutation $\pi$ is a double descent of any restriction of $\pi$ to $[1,\ldots,k]$ for some $k$. (Note here that the same double descent can appear in multiple restrictions!)
Map
ones to leading
Description
The unique permutation obtained by applying the Foata-Riordan map to obtain a Prüfer code, then prepending zero and cyclically shifting.
Let $c_1,\dots, c_{n-1}$ be the Prüfer code obtained via the Foata-Riordan map described in [1, eq (1.2)] and let $c_0 = 0$.
This map returns the a unique permutation $q_1,\dots, q_n$ such that $q_i - c_{i-1}$ is constant modulo $n+1$.
This map is Mp00299ones to leading restricted to permutations.
Let $c_1,\dots, c_{n-1}$ be the Prüfer code obtained via the Foata-Riordan map described in [1, eq (1.2)] and let $c_0 = 0$.
This map returns the a unique permutation $q_1,\dots, q_n$ such that $q_i - c_{i-1}$ is constant modulo $n+1$.
This map is Mp00299ones to leading restricted to permutations.
Map
to 132-avoiding permutation
Description
Sends a Dyck path to a 132-avoiding permutation.
This bijection is defined in [1, Section 2].
This bijection is defined in [1, Section 2].
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!