Identifier
-
Mp00025:
Dyck paths
—to 132-avoiding permutation⟶
Permutations
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
St000078: Permutations ⟶ ℤ
Values
[1,0] => [1] => [1] => 1
[1,0,1,0] => [2,1] => [2,1] => 1
[1,1,0,0] => [1,2] => [1,2] => 1
[1,0,1,0,1,0] => [3,2,1] => [2,3,1] => 1
[1,0,1,1,0,0] => [2,3,1] => [3,2,1] => 1
[1,1,0,0,1,0] => [3,1,2] => [1,3,2] => 2
[1,1,0,1,0,0] => [2,1,3] => [2,1,3] => 1
[1,1,1,0,0,0] => [1,2,3] => [1,2,3] => 1
[1,0,1,0,1,0,1,0] => [4,3,2,1] => [3,4,2,1] => 1
[1,0,1,0,1,1,0,0] => [3,4,2,1] => [4,3,2,1] => 1
[1,0,1,1,0,0,1,0] => [4,2,3,1] => [2,4,3,1] => 2
[1,0,1,1,0,1,0,0] => [3,2,4,1] => [2,3,4,1] => 1
[1,0,1,1,1,0,0,0] => [2,3,4,1] => [3,2,4,1] => 1
[1,1,0,0,1,0,1,0] => [4,3,1,2] => [3,4,1,2] => 1
[1,1,0,0,1,1,0,0] => [3,4,1,2] => [4,3,1,2] => 1
[1,1,0,1,0,0,1,0] => [4,2,1,3] => [2,4,1,3] => 2
[1,1,0,1,0,1,0,0] => [3,2,1,4] => [2,3,1,4] => 1
[1,1,0,1,1,0,0,0] => [2,3,1,4] => [3,2,1,4] => 1
[1,1,1,0,0,0,1,0] => [4,1,2,3] => [1,4,2,3] => 3
[1,1,1,0,0,1,0,0] => [3,1,2,4] => [1,3,2,4] => 2
[1,1,1,0,1,0,0,0] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,1,1,0,0,0,0] => [1,2,3,4] => [1,2,3,4] => 1
[1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => [4,5,3,2,1] => 1
[1,0,1,0,1,0,1,1,0,0] => [4,5,3,2,1] => [5,4,3,2,1] => 1
[1,0,1,0,1,1,0,0,1,0] => [5,3,4,2,1] => [3,5,4,2,1] => 2
[1,0,1,0,1,1,0,1,0,0] => [4,3,5,2,1] => [3,4,5,2,1] => 1
[1,0,1,0,1,1,1,0,0,0] => [3,4,5,2,1] => [4,3,5,2,1] => 1
[1,0,1,1,0,0,1,0,1,0] => [5,4,2,3,1] => [4,5,2,3,1] => 1
[1,0,1,1,0,0,1,1,0,0] => [4,5,2,3,1] => [5,4,2,3,1] => 1
[1,0,1,1,0,1,0,0,1,0] => [5,3,2,4,1] => [3,5,2,4,1] => 2
[1,0,1,1,0,1,0,1,0,0] => [4,3,2,5,1] => [3,4,2,5,1] => 1
[1,0,1,1,0,1,1,0,0,0] => [3,4,2,5,1] => [4,3,2,5,1] => 1
[1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => [2,5,3,4,1] => 3
[1,0,1,1,1,0,0,1,0,0] => [4,2,3,5,1] => [2,4,3,5,1] => 2
[1,0,1,1,1,0,1,0,0,0] => [3,2,4,5,1] => [2,3,4,5,1] => 1
[1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [3,2,4,5,1] => 1
[1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,2] => [4,5,3,1,2] => 1
[1,1,0,0,1,0,1,1,0,0] => [4,5,3,1,2] => [5,4,3,1,2] => 1
[1,1,0,0,1,1,0,0,1,0] => [5,3,4,1,2] => [3,5,4,1,2] => 2
[1,1,0,0,1,1,0,1,0,0] => [4,3,5,1,2] => [3,4,5,1,2] => 1
[1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => [4,3,5,1,2] => 1
[1,1,0,1,0,0,1,0,1,0] => [5,4,2,1,3] => [4,5,2,1,3] => 1
[1,1,0,1,0,0,1,1,0,0] => [4,5,2,1,3] => [5,4,2,1,3] => 1
[1,1,0,1,0,1,0,0,1,0] => [5,3,2,1,4] => [3,5,2,1,4] => 2
[1,1,0,1,0,1,0,1,0,0] => [4,3,2,1,5] => [3,4,2,1,5] => 1
[1,1,0,1,0,1,1,0,0,0] => [3,4,2,1,5] => [4,3,2,1,5] => 1
[1,1,0,1,1,0,0,0,1,0] => [5,2,3,1,4] => [2,5,3,1,4] => 3
[1,1,0,1,1,0,0,1,0,0] => [4,2,3,1,5] => [2,4,3,1,5] => 2
[1,1,0,1,1,0,1,0,0,0] => [3,2,4,1,5] => [2,3,4,1,5] => 1
[1,1,0,1,1,1,0,0,0,0] => [2,3,4,1,5] => [3,2,4,1,5] => 1
[1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3] => [4,5,1,2,3] => 1
[1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => [5,4,1,2,3] => 1
[1,1,1,0,0,1,0,0,1,0] => [5,3,1,2,4] => [3,5,1,2,4] => 2
[1,1,1,0,0,1,0,1,0,0] => [4,3,1,2,5] => [3,4,1,2,5] => 1
[1,1,1,0,0,1,1,0,0,0] => [3,4,1,2,5] => [4,3,1,2,5] => 1
[1,1,1,0,1,0,0,0,1,0] => [5,2,1,3,4] => [2,5,1,3,4] => 3
[1,1,1,0,1,0,0,1,0,0] => [4,2,1,3,5] => [2,4,1,3,5] => 2
[1,1,1,0,1,0,1,0,0,0] => [3,2,1,4,5] => [2,3,1,4,5] => 1
[1,1,1,0,1,1,0,0,0,0] => [2,3,1,4,5] => [3,2,1,4,5] => 1
[1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => [1,5,2,3,4] => 4
[1,1,1,1,0,0,0,1,0,0] => [4,1,2,3,5] => [1,4,2,3,5] => 3
[1,1,1,1,0,0,1,0,0,0] => [3,1,2,4,5] => [1,3,2,4,5] => 2
[1,1,1,1,0,1,0,0,0,0] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,0,1,0,1,0,1,0,1,0,1,0] => [6,5,4,3,2,1] => [5,6,4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,1,0,0] => [5,6,4,3,2,1] => [6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,1,0,0,1,0] => [6,4,5,3,2,1] => [4,6,5,3,2,1] => 2
[1,0,1,0,1,0,1,1,0,1,0,0] => [5,4,6,3,2,1] => [4,5,6,3,2,1] => 1
[1,0,1,0,1,0,1,1,1,0,0,0] => [4,5,6,3,2,1] => [5,4,6,3,2,1] => 1
[1,0,1,0,1,1,0,0,1,0,1,0] => [6,5,3,4,2,1] => [5,6,3,4,2,1] => 1
[1,0,1,0,1,1,0,0,1,1,0,0] => [5,6,3,4,2,1] => [6,5,3,4,2,1] => 1
[1,0,1,0,1,1,0,1,0,0,1,0] => [6,4,3,5,2,1] => [4,6,3,5,2,1] => 2
[1,0,1,0,1,1,0,1,0,1,0,0] => [5,4,3,6,2,1] => [4,5,3,6,2,1] => 1
[1,0,1,0,1,1,0,1,1,0,0,0] => [4,5,3,6,2,1] => [5,4,3,6,2,1] => 1
[1,0,1,0,1,1,1,0,0,0,1,0] => [6,3,4,5,2,1] => [3,6,4,5,2,1] => 3
[1,0,1,0,1,1,1,0,0,1,0,0] => [5,3,4,6,2,1] => [3,5,4,6,2,1] => 2
[1,0,1,0,1,1,1,0,1,0,0,0] => [4,3,5,6,2,1] => [3,4,5,6,2,1] => 1
[1,0,1,0,1,1,1,1,0,0,0,0] => [3,4,5,6,2,1] => [4,3,5,6,2,1] => 1
[1,0,1,1,0,0,1,0,1,0,1,0] => [6,5,4,2,3,1] => [5,6,4,2,3,1] => 1
[1,0,1,1,0,0,1,0,1,1,0,0] => [5,6,4,2,3,1] => [6,5,4,2,3,1] => 1
[1,0,1,1,0,0,1,1,0,0,1,0] => [6,4,5,2,3,1] => [4,6,5,2,3,1] => 2
[1,0,1,1,0,0,1,1,0,1,0,0] => [5,4,6,2,3,1] => [4,5,6,2,3,1] => 1
[1,0,1,1,0,0,1,1,1,0,0,0] => [4,5,6,2,3,1] => [5,4,6,2,3,1] => 1
[1,0,1,1,0,1,0,0,1,0,1,0] => [6,5,3,2,4,1] => [5,6,3,2,4,1] => 1
[1,0,1,1,0,1,0,0,1,1,0,0] => [5,6,3,2,4,1] => [6,5,3,2,4,1] => 1
[1,0,1,1,0,1,0,1,0,0,1,0] => [6,4,3,2,5,1] => [4,6,3,2,5,1] => 2
[1,0,1,1,0,1,0,1,0,1,0,0] => [5,4,3,2,6,1] => [4,5,3,2,6,1] => 1
[1,0,1,1,0,1,0,1,1,0,0,0] => [4,5,3,2,6,1] => [5,4,3,2,6,1] => 1
[1,0,1,1,0,1,1,0,0,0,1,0] => [6,3,4,2,5,1] => [3,6,4,2,5,1] => 3
[1,0,1,1,0,1,1,0,0,1,0,0] => [5,3,4,2,6,1] => [3,5,4,2,6,1] => 2
[1,0,1,1,0,1,1,0,1,0,0,0] => [4,3,5,2,6,1] => [3,4,5,2,6,1] => 1
[1,0,1,1,0,1,1,1,0,0,0,0] => [3,4,5,2,6,1] => [4,3,5,2,6,1] => 1
[1,0,1,1,1,0,0,0,1,0,1,0] => [6,5,2,3,4,1] => [5,6,2,3,4,1] => 1
[1,0,1,1,1,0,0,0,1,1,0,0] => [5,6,2,3,4,1] => [6,5,2,3,4,1] => 1
[1,0,1,1,1,0,0,1,0,0,1,0] => [6,4,2,3,5,1] => [4,6,2,3,5,1] => 2
[1,0,1,1,1,0,0,1,0,1,0,0] => [5,4,2,3,6,1] => [4,5,2,3,6,1] => 1
[1,0,1,1,1,0,0,1,1,0,0,0] => [4,5,2,3,6,1] => [5,4,2,3,6,1] => 1
[1,0,1,1,1,0,1,0,0,0,1,0] => [6,3,2,4,5,1] => [3,6,2,4,5,1] => 3
[1,0,1,1,1,0,1,0,0,1,0,0] => [5,3,2,4,6,1] => [3,5,2,4,6,1] => 2
[1,0,1,1,1,0,1,0,1,0,0,0] => [4,3,2,5,6,1] => [3,4,2,5,6,1] => 1
[1,0,1,1,1,0,1,1,0,0,0,0] => [3,4,2,5,6,1] => [4,3,2,5,6,1] => 1
>>> Load all 293 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 alternating sign matrices whose left key is the permutation.
The left key of an alternating sign matrix was defined by Lascoux
in [2] and is obtained by successively removing all the `-1`'s until what remains is a permutation matrix. This notion corresponds to the notion of left key for semistandard tableaux.
The left key of an alternating sign matrix was defined by Lascoux
in [2] and is obtained by successively removing all the `-1`'s until what remains is a permutation matrix. This notion corresponds to the notion of left key for semistandard tableaux.
Map
Alexandersson Kebede
Description
Sends a permutation to a permutation and it preserves the set of right-to-left minima.
Take a permutation $\pi$ of length $n$. The mapping looks for a smallest odd integer $i\in[n-1]$ such that swapping the entries $\pi(i)$ and $\pi(i+1)$ preserves the set of right-to-left minima. Otherwise, $\pi$ will be a fixed element of the mapping. Note that the map changes the sign of all non-fixed elements.
There are exactly $\binom{\lfloor n/2 \rfloor}{k-\lceil n/2 \rceil}$ elements in $S_n$ fixed under this map, with exactly $k$ right-to-left minima, see Lemma 35 in [1].
Take a permutation $\pi$ of length $n$. The mapping looks for a smallest odd integer $i\in[n-1]$ such that swapping the entries $\pi(i)$ and $\pi(i+1)$ preserves the set of right-to-left minima. Otherwise, $\pi$ will be a fixed element of the mapping. Note that the map changes the sign of all non-fixed elements.
There are exactly $\binom{\lfloor n/2 \rfloor}{k-\lceil n/2 \rceil}$ elements in $S_n$ fixed under this map, with exactly $k$ right-to-left minima, see Lemma 35 in [1].
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!