Identifier
-
Mp00024:
Dyck paths
—to 321-avoiding permutation⟶
Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St001682: Permutations ⟶ ℤ
Values
[1,0] => [1] => [1] => [1] => 0
[1,0,1,0] => [2,1] => [2,1] => [1,2] => 0
[1,1,0,0] => [1,2] => [1,2] => [2,1] => 0
[1,0,1,0,1,0] => [2,1,3] => [2,1,3] => [2,3,1] => 0
[1,0,1,1,0,0] => [2,3,1] => [3,2,1] => [1,2,3] => 1
[1,1,0,0,1,0] => [3,1,2] => [3,1,2] => [1,3,2] => 0
[1,1,0,1,0,0] => [1,3,2] => [1,3,2] => [3,1,2] => 0
[1,1,1,0,0,0] => [1,2,3] => [1,2,3] => [3,2,1] => 0
[1,0,1,0,1,0,1,0] => [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 0
[1,0,1,0,1,1,0,0] => [2,4,1,3] => [4,2,1,3] => [1,3,4,2] => 1
[1,0,1,1,0,0,1,0] => [2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 0
[1,0,1,1,0,1,0,0] => [2,3,1,4] => [3,2,1,4] => [2,3,4,1] => 1
[1,0,1,1,1,0,0,0] => [2,3,4,1] => [4,2,3,1] => [1,3,2,4] => 1
[1,1,0,0,1,0,1,0] => [3,1,4,2] => [4,3,1,2] => [1,2,4,3] => 1
[1,1,0,0,1,1,0,0] => [3,4,1,2] => [4,1,3,2] => [1,4,2,3] => 1
[1,1,0,1,0,0,1,0] => [3,1,2,4] => [3,1,2,4] => [2,4,3,1] => 0
[1,1,0,1,0,1,0,0] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 0
[1,1,0,1,1,0,0,0] => [1,3,4,2] => [1,4,3,2] => [4,1,2,3] => 1
[1,1,1,0,0,0,1,0] => [4,1,2,3] => [4,1,2,3] => [1,4,3,2] => 0
[1,1,1,0,0,1,0,0] => [1,4,2,3] => [1,4,2,3] => [4,1,3,2] => 0
[1,1,1,0,1,0,0,0] => [1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 0
[1,1,1,1,0,0,0,0] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0] => [2,1,4,3,5] => [2,1,4,3,5] => [4,5,2,3,1] => 0
[1,0,1,0,1,0,1,1,0,0] => [2,4,1,3,5] => [4,2,1,3,5] => [2,4,5,3,1] => 1
[1,0,1,0,1,1,0,0,1,0] => [2,1,4,5,3] => [2,1,5,4,3] => [4,5,1,2,3] => 1
[1,0,1,0,1,1,0,1,0,0] => [2,4,1,5,3] => [5,2,4,1,3] => [1,4,2,5,3] => 1
[1,0,1,0,1,1,1,0,0,0] => [2,4,5,1,3] => [5,2,1,4,3] => [1,4,5,2,3] => 1
[1,0,1,1,0,0,1,0,1,0] => [2,1,5,3,4] => [2,1,5,3,4] => [4,5,1,3,2] => 0
[1,0,1,1,0,0,1,1,0,0] => [2,5,1,3,4] => [5,2,1,3,4] => [1,4,5,3,2] => 1
[1,0,1,1,0,1,0,0,1,0] => [2,1,3,5,4] => [2,1,3,5,4] => [4,5,3,1,2] => 0
[1,0,1,1,0,1,0,1,0,0] => [2,3,1,5,4] => [3,2,1,5,4] => [3,4,5,1,2] => 1
[1,0,1,1,0,1,1,0,0,0] => [2,3,5,1,4] => [5,2,3,1,4] => [1,4,3,5,2] => 1
[1,0,1,1,1,0,0,0,1,0] => [2,1,3,4,5] => [2,1,3,4,5] => [4,5,3,2,1] => 0
[1,0,1,1,1,0,0,1,0,0] => [2,3,1,4,5] => [3,2,1,4,5] => [3,4,5,2,1] => 1
[1,0,1,1,1,0,1,0,0,0] => [2,3,4,1,5] => [4,2,3,1,5] => [2,4,3,5,1] => 1
[1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [5,2,3,4,1] => [1,4,3,2,5] => 1
[1,1,0,0,1,0,1,0,1,0] => [3,1,4,2,5] => [4,3,1,2,5] => [2,3,5,4,1] => 1
[1,1,0,0,1,0,1,1,0,0] => [3,4,1,2,5] => [4,1,3,2,5] => [2,5,3,4,1] => 1
[1,1,0,0,1,1,0,0,1,0] => [3,1,4,5,2] => [5,3,1,4,2] => [1,3,5,2,4] => 1
[1,1,0,0,1,1,0,1,0,0] => [3,4,1,5,2] => [5,4,3,1,2] => [1,2,3,5,4] => 2
[1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => [5,1,3,4,2] => [1,5,3,2,4] => 1
[1,1,0,1,0,0,1,0,1,0] => [3,1,5,2,4] => [5,3,1,2,4] => [1,3,5,4,2] => 1
[1,1,0,1,0,0,1,1,0,0] => [3,5,1,2,4] => [5,1,3,2,4] => [1,5,3,4,2] => 1
[1,1,0,1,0,1,0,0,1,0] => [3,1,2,5,4] => [3,1,2,5,4] => [3,5,4,1,2] => 0
[1,1,0,1,0,1,0,1,0,0] => [1,3,2,5,4] => [1,3,2,5,4] => [5,3,4,1,2] => 0
[1,1,0,1,0,1,1,0,0,0] => [1,3,5,2,4] => [1,5,3,2,4] => [5,1,3,4,2] => 1
[1,1,0,1,1,0,0,0,1,0] => [3,1,2,4,5] => [3,1,2,4,5] => [3,5,4,2,1] => 0
[1,1,0,1,1,0,0,1,0,0] => [1,3,2,4,5] => [1,3,2,4,5] => [5,3,4,2,1] => 0
[1,1,0,1,1,0,1,0,0,0] => [1,3,4,2,5] => [1,4,3,2,5] => [5,2,3,4,1] => 1
[1,1,0,1,1,1,0,0,0,0] => [1,3,4,5,2] => [1,5,3,4,2] => [5,1,3,2,4] => 1
[1,1,1,0,0,0,1,0,1,0] => [4,1,5,2,3] => [5,4,1,2,3] => [1,2,5,4,3] => 1
[1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => [5,1,2,4,3] => [1,5,4,2,3] => 1
[1,1,1,0,0,1,0,0,1,0] => [4,1,2,5,3] => [5,1,4,2,3] => [1,5,2,4,3] => 1
[1,1,1,0,0,1,0,1,0,0] => [1,4,2,5,3] => [1,5,4,2,3] => [5,1,2,4,3] => 1
[1,1,1,0,0,1,1,0,0,0] => [1,4,5,2,3] => [1,5,2,4,3] => [5,1,4,2,3] => 1
[1,1,1,0,1,0,0,0,1,0] => [4,1,2,3,5] => [4,1,2,3,5] => [2,5,4,3,1] => 0
[1,1,1,0,1,0,0,1,0,0] => [1,4,2,3,5] => [1,4,2,3,5] => [5,2,4,3,1] => 0
[1,1,1,0,1,0,1,0,0,0] => [1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 0
[1,1,1,0,1,1,0,0,0,0] => [1,2,4,5,3] => [1,2,5,4,3] => [5,4,1,2,3] => 1
[1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => [5,1,2,3,4] => [1,5,4,3,2] => 0
[1,1,1,1,0,0,0,1,0,0] => [1,5,2,3,4] => [1,5,2,3,4] => [5,1,4,3,2] => 0
[1,1,1,1,0,0,1,0,0,0] => [1,2,5,3,4] => [1,2,5,3,4] => [5,4,1,3,2] => 0
[1,1,1,1,0,1,0,0,0,0] => [1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => 0
[1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0,1,0] => [2,1,4,3,6,5] => [2,1,4,3,6,5] => [5,6,3,4,1,2] => 0
[1,0,1,0,1,0,1,0,1,1,0,0] => [2,4,1,3,6,5] => [4,2,1,3,6,5] => [3,5,6,4,1,2] => 1
[1,0,1,0,1,0,1,1,0,0,1,0] => [2,1,4,6,3,5] => [2,1,6,4,3,5] => [5,6,1,3,4,2] => 1
[1,0,1,0,1,0,1,1,0,1,0,0] => [2,4,1,6,3,5] => [6,2,4,1,3,5] => [1,5,3,6,4,2] => 1
[1,0,1,0,1,0,1,1,1,0,0,0] => [2,4,6,1,3,5] => [6,2,1,4,3,5] => [1,5,6,3,4,2] => 1
[1,0,1,0,1,1,0,0,1,0,1,0] => [2,1,4,3,5,6] => [2,1,4,3,5,6] => [5,6,3,4,2,1] => 0
[1,0,1,0,1,1,0,0,1,1,0,0] => [2,4,1,3,5,6] => [4,2,1,3,5,6] => [3,5,6,4,2,1] => 1
[1,0,1,0,1,1,0,1,0,0,1,0] => [2,1,4,5,3,6] => [2,1,5,4,3,6] => [5,6,2,3,4,1] => 1
[1,0,1,0,1,1,0,1,0,1,0,0] => [2,4,1,5,3,6] => [5,2,4,1,3,6] => [2,5,3,6,4,1] => 1
[1,0,1,0,1,1,0,1,1,0,0,0] => [2,4,5,1,3,6] => [5,2,1,4,3,6] => [2,5,6,3,4,1] => 1
[1,0,1,0,1,1,1,0,0,0,1,0] => [2,1,4,5,6,3] => [2,1,6,4,5,3] => [5,6,1,3,2,4] => 1
[1,0,1,0,1,1,1,0,0,1,0,0] => [2,4,1,5,6,3] => [6,2,4,1,5,3] => [1,5,3,6,2,4] => 1
[1,0,1,0,1,1,1,0,1,0,0,0] => [2,4,5,1,6,3] => [6,2,5,4,1,3] => [1,5,2,3,6,4] => 2
[1,0,1,0,1,1,1,1,0,0,0,0] => [2,4,5,6,1,3] => [6,2,1,4,5,3] => [1,5,6,3,2,4] => 1
[1,0,1,1,0,0,1,0,1,0,1,0] => [2,1,5,3,6,4] => [2,1,6,5,3,4] => [5,6,1,2,4,3] => 1
[1,0,1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [6,2,1,5,3,4] => [1,5,6,2,4,3] => 1
[1,0,1,1,0,0,1,1,0,0,1,0] => [2,1,5,6,3,4] => [2,1,6,3,5,4] => [5,6,1,4,2,3] => 1
[1,0,1,1,0,0,1,1,0,1,0,0] => [2,5,1,6,3,4] => [6,2,5,1,3,4] => [1,5,2,6,4,3] => 1
[1,0,1,1,0,0,1,1,1,0,0,0] => [2,5,6,1,3,4] => [6,2,1,3,5,4] => [1,5,6,4,2,3] => 1
[1,0,1,1,0,1,0,0,1,0,1,0] => [2,1,5,3,4,6] => [2,1,5,3,4,6] => [5,6,2,4,3,1] => 0
[1,0,1,1,0,1,0,0,1,1,0,0] => [2,5,1,3,4,6] => [5,2,1,3,4,6] => [2,5,6,4,3,1] => 1
[1,0,1,1,0,1,0,1,0,0,1,0] => [2,1,3,5,4,6] => [2,1,3,5,4,6] => [5,6,4,2,3,1] => 0
[1,0,1,1,0,1,0,1,0,1,0,0] => [2,3,1,5,4,6] => [3,2,1,5,4,6] => [4,5,6,2,3,1] => 1
[1,0,1,1,0,1,0,1,1,0,0,0] => [2,3,5,1,4,6] => [5,2,3,1,4,6] => [2,5,4,6,3,1] => 1
[1,0,1,1,0,1,1,0,0,0,1,0] => [2,1,3,5,6,4] => [2,1,3,6,5,4] => [5,6,4,1,2,3] => 1
[1,0,1,1,0,1,1,0,0,1,0,0] => [2,3,1,5,6,4] => [3,2,1,6,5,4] => [4,5,6,1,2,3] => 2
[1,0,1,1,0,1,1,0,1,0,0,0] => [2,3,5,1,6,4] => [6,2,3,5,1,4] => [1,5,4,2,6,3] => 1
[1,0,1,1,0,1,1,1,0,0,0,0] => [2,3,5,6,1,4] => [6,2,3,1,5,4] => [1,5,4,6,2,3] => 1
[1,0,1,1,1,0,0,0,1,0,1,0] => [2,1,6,3,4,5] => [2,1,6,3,4,5] => [5,6,1,4,3,2] => 0
[1,0,1,1,1,0,0,0,1,1,0,0] => [2,6,1,3,4,5] => [6,2,1,3,4,5] => [1,5,6,4,3,2] => 1
[1,0,1,1,1,0,0,1,0,0,1,0] => [2,1,3,6,4,5] => [2,1,3,6,4,5] => [5,6,4,1,3,2] => 0
[1,0,1,1,1,0,0,1,0,1,0,0] => [2,3,1,6,4,5] => [3,2,1,6,4,5] => [4,5,6,1,3,2] => 1
[1,0,1,1,1,0,0,1,1,0,0,0] => [2,3,6,1,4,5] => [6,2,3,1,4,5] => [1,5,4,6,3,2] => 1
[1,0,1,1,1,0,1,0,0,0,1,0] => [2,1,3,4,6,5] => [2,1,3,4,6,5] => [5,6,4,3,1,2] => 0
[1,0,1,1,1,0,1,0,0,1,0,0] => [2,3,1,4,6,5] => [3,2,1,4,6,5] => [4,5,6,3,1,2] => 1
[1,0,1,1,1,0,1,0,1,0,0,0] => [2,3,4,1,6,5] => [4,2,3,1,6,5] => [3,5,4,6,1,2] => 1
[1,0,1,1,1,0,1,1,0,0,0,0] => [2,3,4,6,1,5] => [6,2,3,4,1,5] => [1,5,4,3,6,2] => 1
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Description
The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation.
Map
complement
Description
Sents a permutation to its complement.
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$
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.
Map
Clarke-Steingrimsson-Zeng
Description
The Clarke-Steingrimsson-Zeng map sending descents to excedances.
This is the map $\Phi$ in [1, sec.3]. In particular, it satisfies
$$ (des, Dbot, Ddif, Res)\pi = (exc, Ebot, Edif, Ine)\Phi(\pi), $$
where
This is the map $\Phi$ in [1, sec.3]. In particular, it satisfies
$$ (des, Dbot, Ddif, Res)\pi = (exc, Ebot, Edif, Ine)\Phi(\pi), $$
where
- $des$ is the number of descents, St000021The number of descents of a permutation.,
- $exc$ is the number of (strict) excedances, St000155The number of exceedances (also excedences) of a permutation.,
- $Dbot$ is the sum of the descent bottoms, St000154The sum of the descent bottoms of a permutation.,
- $Ebot$ is the sum of the excedance bottoms,
- $Ddif$ is the sum of the descent differences, St000030The sum of the descent differences of a permutations.,
- $Edif$ is the sum of the excedance differences (or depth), St000029The depth of a permutation.,
- $Res$ is the sum of the (right) embracing numbers,
- $Ine$ is the sum of the side numbers.
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