Identifier
-
Mp00025:
Dyck paths
—to 132-avoiding permutation⟶
Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St001960: Permutations ⟶ ℤ
Values
[1,0,1,0] => [2,1] => [1,2] => [1,2] => 0
[1,1,0,0] => [1,2] => [2,1] => [2,1] => 0
[1,0,1,0,1,0] => [3,2,1] => [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0] => [2,3,1] => [1,3,2] => [3,1,2] => 0
[1,1,0,0,1,0] => [3,1,2] => [2,1,3] => [2,1,3] => 0
[1,1,0,1,0,0] => [2,1,3] => [3,1,2] => [1,3,2] => 1
[1,1,1,0,0,0] => [1,2,3] => [3,2,1] => [3,2,1] => 1
[1,0,1,0,1,0,1,0] => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0] => [3,4,2,1] => [1,2,4,3] => [4,1,2,3] => 0
[1,0,1,1,0,0,1,0] => [4,2,3,1] => [1,3,2,4] => [3,1,2,4] => 0
[1,0,1,1,0,1,0,0] => [3,2,4,1] => [1,4,2,3] => [1,4,2,3] => 1
[1,0,1,1,1,0,0,0] => [2,3,4,1] => [1,4,3,2] => [4,3,1,2] => 1
[1,1,0,0,1,0,1,0] => [4,3,1,2] => [2,1,3,4] => [2,1,3,4] => 0
[1,1,0,0,1,1,0,0] => [3,4,1,2] => [2,1,4,3] => [2,4,1,3] => 0
[1,1,0,1,0,0,1,0] => [4,2,1,3] => [3,1,2,4] => [1,3,2,4] => 1
[1,1,0,1,0,1,0,0] => [3,2,1,4] => [4,1,2,3] => [1,2,4,3] => 1
[1,1,0,1,1,0,0,0] => [2,3,1,4] => [4,1,3,2] => [4,1,3,2] => 1
[1,1,1,0,0,0,1,0] => [4,1,2,3] => [3,2,1,4] => [3,2,1,4] => 1
[1,1,1,0,0,1,0,0] => [3,1,2,4] => [4,2,1,3] => [2,1,4,3] => 1
[1,1,1,0,1,0,0,0] => [2,1,3,4] => [4,3,1,2] => [1,4,3,2] => 2
[1,1,1,1,0,0,0,0] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 2
[1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0] => [4,5,3,2,1] => [1,2,3,5,4] => [5,1,2,3,4] => 0
[1,0,1,0,1,1,0,0,1,0] => [5,3,4,2,1] => [1,2,4,3,5] => [4,1,2,3,5] => 0
[1,0,1,0,1,1,0,1,0,0] => [4,3,5,2,1] => [1,2,5,3,4] => [1,5,2,3,4] => 1
[1,0,1,0,1,1,1,0,0,0] => [3,4,5,2,1] => [1,2,5,4,3] => [5,4,1,2,3] => 1
[1,0,1,1,0,0,1,0,1,0] => [5,4,2,3,1] => [1,3,2,4,5] => [3,1,2,4,5] => 0
[1,0,1,1,0,0,1,1,0,0] => [4,5,2,3,1] => [1,3,2,5,4] => [3,5,1,2,4] => 0
[1,0,1,1,0,1,0,0,1,0] => [5,3,2,4,1] => [1,4,2,3,5] => [1,4,2,3,5] => 1
[1,0,1,1,0,1,0,1,0,0] => [4,3,2,5,1] => [1,5,2,3,4] => [1,2,5,3,4] => 1
[1,0,1,1,0,1,1,0,0,0] => [3,4,2,5,1] => [1,5,2,4,3] => [5,1,4,2,3] => 1
[1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => [1,4,3,2,5] => [4,3,1,2,5] => 1
[1,0,1,1,1,0,0,1,0,0] => [4,2,3,5,1] => [1,5,3,2,4] => [5,1,3,2,4] => 1
[1,0,1,1,1,0,1,0,0,0] => [3,2,4,5,1] => [1,5,4,2,3] => [1,5,4,2,3] => 2
[1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [1,5,4,3,2] => [5,4,3,1,2] => 2
[1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,2] => [2,1,3,4,5] => [2,1,3,4,5] => 0
[1,1,0,0,1,0,1,1,0,0] => [4,5,3,1,2] => [2,1,3,5,4] => [2,5,1,3,4] => 0
[1,1,0,0,1,1,0,0,1,0] => [5,3,4,1,2] => [2,1,4,3,5] => [2,4,1,3,5] => 0
[1,1,0,0,1,1,0,1,0,0] => [4,3,5,1,2] => [2,1,5,3,4] => [5,2,1,3,4] => 1
[1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => [2,1,5,4,3] => [5,2,4,1,3] => 1
[1,1,0,1,0,0,1,0,1,0] => [5,4,2,1,3] => [3,1,2,4,5] => [1,3,2,4,5] => 1
[1,1,0,1,0,0,1,1,0,0] => [4,5,2,1,3] => [3,1,2,5,4] => [1,3,5,2,4] => 1
[1,1,0,1,0,1,0,0,1,0] => [5,3,2,1,4] => [4,1,2,3,5] => [1,2,4,3,5] => 1
[1,1,0,1,0,1,0,1,0,0] => [4,3,2,1,5] => [5,1,2,3,4] => [1,2,3,5,4] => 1
[1,1,0,1,0,1,1,0,0,0] => [3,4,2,1,5] => [5,1,2,4,3] => [5,1,2,4,3] => 1
[1,1,0,1,1,0,0,0,1,0] => [5,2,3,1,4] => [4,1,3,2,5] => [4,1,3,2,5] => 1
[1,1,0,1,1,0,0,1,0,0] => [4,2,3,1,5] => [5,1,3,2,4] => [3,1,2,5,4] => 1
[1,1,0,1,1,0,1,0,0,0] => [3,2,4,1,5] => [5,1,4,2,3] => [1,5,2,4,3] => 2
[1,1,0,1,1,1,0,0,0,0] => [2,3,4,1,5] => [5,1,4,3,2] => [5,4,1,3,2] => 2
[1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3] => [3,2,1,4,5] => [3,2,1,4,5] => 1
[1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => [3,2,1,5,4] => [3,2,5,1,4] => 1
[1,1,1,0,0,1,0,0,1,0] => [5,3,1,2,4] => [4,2,1,3,5] => [2,1,4,3,5] => 1
[1,1,1,0,0,1,0,1,0,0] => [4,3,1,2,5] => [5,2,1,3,4] => [2,1,3,5,4] => 1
[1,1,1,0,0,1,1,0,0,0] => [3,4,1,2,5] => [5,2,1,4,3] => [2,5,1,4,3] => 1
[1,1,1,0,1,0,0,0,1,0] => [5,2,1,3,4] => [4,3,1,2,5] => [1,4,3,2,5] => 2
[1,1,1,0,1,0,0,1,0,0] => [4,2,1,3,5] => [5,3,1,2,4] => [1,3,2,5,4] => 2
[1,1,1,0,1,0,1,0,0,0] => [3,2,1,4,5] => [5,4,1,2,3] => [1,2,5,4,3] => 2
[1,1,1,0,1,1,0,0,0,0] => [2,3,1,4,5] => [5,4,1,3,2] => [5,1,4,3,2] => 2
[1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => [4,3,2,1,5] => [4,3,2,1,5] => 2
[1,1,1,1,0,0,0,1,0,0] => [4,1,2,3,5] => [5,3,2,1,4] => [3,2,1,5,4] => 2
[1,1,1,1,0,0,1,0,0,0] => [3,1,2,4,5] => [5,4,2,1,3] => [2,1,5,4,3] => 2
[1,1,1,1,0,1,0,0,0,0] => [2,1,3,4,5] => [5,4,3,1,2] => [1,5,4,3,2] => 3
[1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => 3
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Description
The number of descents of a permutation minus one if its first entry is not one.
This statistic appears in [1, Theorem 2.3] in a gamma-positivity result, see also [2].
This statistic appears in [1, Theorem 2.3] in a gamma-positivity result, see also [2].
Map
reverse
Description
Sends a permutation to its reverse.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.
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].
Map
inverse Foata bijection
Description
The inverse of Foata's bijection.
See Mp00067Foata bijection.
See Mp00067Foata bijection.
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