Identifier
-
Mp00025:
Dyck paths
—to 132-avoiding permutation⟶
Permutations
Mp00069: Permutations —complement⟶ Permutations
St000709: Permutations ⟶ ℤ
Values
[1,0,1,0] => [2,1] => [1,2] => 0
[1,1,0,0] => [1,2] => [2,1] => 0
[1,0,1,0,1,0] => [3,2,1] => [1,2,3] => 0
[1,0,1,1,0,0] => [2,3,1] => [2,1,3] => 0
[1,1,0,0,1,0] => [3,1,2] => [1,3,2] => 0
[1,1,0,1,0,0] => [2,1,3] => [2,3,1] => 0
[1,1,1,0,0,0] => [1,2,3] => [3,2,1] => 0
[1,0,1,0,1,0,1,0] => [4,3,2,1] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0] => [3,4,2,1] => [2,1,3,4] => 0
[1,0,1,1,0,0,1,0] => [4,2,3,1] => [1,3,2,4] => 0
[1,0,1,1,0,1,0,0] => [3,2,4,1] => [2,3,1,4] => 0
[1,0,1,1,1,0,0,0] => [2,3,4,1] => [3,2,1,4] => 0
[1,1,0,0,1,0,1,0] => [4,3,1,2] => [1,2,4,3] => 0
[1,1,0,0,1,1,0,0] => [3,4,1,2] => [2,1,4,3] => 0
[1,1,0,1,0,0,1,0] => [4,2,1,3] => [1,3,4,2] => 0
[1,1,0,1,0,1,0,0] => [3,2,1,4] => [2,3,4,1] => 0
[1,1,0,1,1,0,0,0] => [2,3,1,4] => [3,2,4,1] => 0
[1,1,1,0,0,0,1,0] => [4,1,2,3] => [1,4,3,2] => 1
[1,1,1,0,0,1,0,0] => [3,1,2,4] => [2,4,3,1] => 0
[1,1,1,0,1,0,0,0] => [2,1,3,4] => [3,4,2,1] => 0
[1,1,1,1,0,0,0,0] => [1,2,3,4] => [4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0] => [4,5,3,2,1] => [2,1,3,4,5] => 0
[1,0,1,0,1,1,0,0,1,0] => [5,3,4,2,1] => [1,3,2,4,5] => 0
[1,0,1,0,1,1,0,1,0,0] => [4,3,5,2,1] => [2,3,1,4,5] => 0
[1,0,1,0,1,1,1,0,0,0] => [3,4,5,2,1] => [3,2,1,4,5] => 0
[1,0,1,1,0,0,1,0,1,0] => [5,4,2,3,1] => [1,2,4,3,5] => 0
[1,0,1,1,0,0,1,1,0,0] => [4,5,2,3,1] => [2,1,4,3,5] => 0
[1,0,1,1,0,1,0,0,1,0] => [5,3,2,4,1] => [1,3,4,2,5] => 0
[1,0,1,1,0,1,0,1,0,0] => [4,3,2,5,1] => [2,3,4,1,5] => 0
[1,0,1,1,0,1,1,0,0,0] => [3,4,2,5,1] => [3,2,4,1,5] => 0
[1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => [1,4,3,2,5] => 1
[1,0,1,1,1,0,0,1,0,0] => [4,2,3,5,1] => [2,4,3,1,5] => 0
[1,0,1,1,1,0,1,0,0,0] => [3,2,4,5,1] => [3,4,2,1,5] => 0
[1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [4,3,2,1,5] => 0
[1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,2] => [1,2,3,5,4] => 0
[1,1,0,0,1,0,1,1,0,0] => [4,5,3,1,2] => [2,1,3,5,4] => 0
[1,1,0,0,1,1,0,0,1,0] => [5,3,4,1,2] => [1,3,2,5,4] => 0
[1,1,0,0,1,1,0,1,0,0] => [4,3,5,1,2] => [2,3,1,5,4] => 0
[1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => [3,2,1,5,4] => 0
[1,1,0,1,0,0,1,0,1,0] => [5,4,2,1,3] => [1,2,4,5,3] => 0
[1,1,0,1,0,0,1,1,0,0] => [4,5,2,1,3] => [2,1,4,5,3] => 0
[1,1,0,1,0,1,0,0,1,0] => [5,3,2,1,4] => [1,3,4,5,2] => 0
[1,1,0,1,0,1,0,1,0,0] => [4,3,2,1,5] => [2,3,4,5,1] => 0
[1,1,0,1,0,1,1,0,0,0] => [3,4,2,1,5] => [3,2,4,5,1] => 0
[1,1,0,1,1,0,0,0,1,0] => [5,2,3,1,4] => [1,4,3,5,2] => 1
[1,1,0,1,1,0,0,1,0,0] => [4,2,3,1,5] => [2,4,3,5,1] => 0
[1,1,0,1,1,0,1,0,0,0] => [3,2,4,1,5] => [3,4,2,5,1] => 0
[1,1,0,1,1,1,0,0,0,0] => [2,3,4,1,5] => [4,3,2,5,1] => 0
[1,1,1,0,0,0,1,0,1,0] => [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] => [2,1,5,4,3] => 1
[1,1,1,0,0,1,0,0,1,0] => [5,3,1,2,4] => [1,3,5,4,2] => 0
[1,1,1,0,0,1,0,1,0,0] => [4,3,1,2,5] => [2,3,5,4,1] => 0
[1,1,1,0,0,1,1,0,0,0] => [3,4,1,2,5] => [3,2,5,4,1] => 0
[1,1,1,0,1,0,0,0,1,0] => [5,2,1,3,4] => [1,4,5,3,2] => 1
[1,1,1,0,1,0,0,1,0,0] => [4,2,1,3,5] => [2,4,5,3,1] => 0
[1,1,1,0,1,0,1,0,0,0] => [3,2,1,4,5] => [3,4,5,2,1] => 0
[1,1,1,0,1,1,0,0,0,0] => [2,3,1,4,5] => [4,3,5,2,1] => 0
[1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => [1,5,4,3,2] => 3
[1,1,1,1,0,0,0,1,0,0] => [4,1,2,3,5] => [2,5,4,3,1] => 1
[1,1,1,1,0,0,1,0,0,0] => [3,1,2,4,5] => [3,5,4,2,1] => 0
[1,1,1,1,0,1,0,0,0,0] => [2,1,3,4,5] => [4,5,3,2,1] => 0
[1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0,1,0] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,0,1,0,1,1,0,0] => [5,6,4,3,2,1] => [2,1,3,4,5,6] => 0
[1,0,1,0,1,0,1,1,0,0,1,0] => [6,4,5,3,2,1] => [1,3,2,4,5,6] => 0
[1,0,1,0,1,0,1,1,0,1,0,0] => [5,4,6,3,2,1] => [2,3,1,4,5,6] => 0
[1,0,1,0,1,0,1,1,1,0,0,0] => [4,5,6,3,2,1] => [3,2,1,4,5,6] => 0
[1,0,1,0,1,1,0,0,1,0,1,0] => [6,5,3,4,2,1] => [1,2,4,3,5,6] => 0
[1,0,1,0,1,1,0,0,1,1,0,0] => [5,6,3,4,2,1] => [2,1,4,3,5,6] => 0
[1,0,1,0,1,1,0,1,0,0,1,0] => [6,4,3,5,2,1] => [1,3,4,2,5,6] => 0
[1,0,1,0,1,1,0,1,0,1,0,0] => [5,4,3,6,2,1] => [2,3,4,1,5,6] => 0
[1,0,1,0,1,1,0,1,1,0,0,0] => [4,5,3,6,2,1] => [3,2,4,1,5,6] => 0
[1,0,1,0,1,1,1,0,0,0,1,0] => [6,3,4,5,2,1] => [1,4,3,2,5,6] => 1
[1,0,1,0,1,1,1,0,0,1,0,0] => [5,3,4,6,2,1] => [2,4,3,1,5,6] => 0
[1,0,1,0,1,1,1,0,1,0,0,0] => [4,3,5,6,2,1] => [3,4,2,1,5,6] => 0
[1,0,1,0,1,1,1,1,0,0,0,0] => [3,4,5,6,2,1] => [4,3,2,1,5,6] => 0
[1,0,1,1,0,0,1,0,1,0,1,0] => [6,5,4,2,3,1] => [1,2,3,5,4,6] => 0
[1,0,1,1,0,0,1,0,1,1,0,0] => [5,6,4,2,3,1] => [2,1,3,5,4,6] => 0
[1,0,1,1,0,0,1,1,0,0,1,0] => [6,4,5,2,3,1] => [1,3,2,5,4,6] => 0
[1,0,1,1,0,0,1,1,0,1,0,0] => [5,4,6,2,3,1] => [2,3,1,5,4,6] => 0
[1,0,1,1,0,0,1,1,1,0,0,0] => [4,5,6,2,3,1] => [3,2,1,5,4,6] => 0
[1,0,1,1,0,1,0,0,1,0,1,0] => [6,5,3,2,4,1] => [1,2,4,5,3,6] => 0
[1,0,1,1,0,1,0,0,1,1,0,0] => [5,6,3,2,4,1] => [2,1,4,5,3,6] => 0
[1,0,1,1,0,1,0,1,0,0,1,0] => [6,4,3,2,5,1] => [1,3,4,5,2,6] => 0
[1,0,1,1,0,1,0,1,0,1,0,0] => [5,4,3,2,6,1] => [2,3,4,5,1,6] => 0
[1,0,1,1,0,1,0,1,1,0,0,0] => [4,5,3,2,6,1] => [3,2,4,5,1,6] => 0
[1,0,1,1,0,1,1,0,0,0,1,0] => [6,3,4,2,5,1] => [1,4,3,5,2,6] => 1
[1,0,1,1,0,1,1,0,0,1,0,0] => [5,3,4,2,6,1] => [2,4,3,5,1,6] => 0
[1,0,1,1,0,1,1,0,1,0,0,0] => [4,3,5,2,6,1] => [3,4,2,5,1,6] => 0
[1,0,1,1,0,1,1,1,0,0,0,0] => [3,4,5,2,6,1] => [4,3,2,5,1,6] => 0
[1,0,1,1,1,0,0,0,1,0,1,0] => [6,5,2,3,4,1] => [1,2,5,4,3,6] => 1
[1,0,1,1,1,0,0,0,1,1,0,0] => [5,6,2,3,4,1] => [2,1,5,4,3,6] => 1
[1,0,1,1,1,0,0,1,0,0,1,0] => [6,4,2,3,5,1] => [1,3,5,4,2,6] => 0
[1,0,1,1,1,0,0,1,0,1,0,0] => [5,4,2,3,6,1] => [2,3,5,4,1,6] => 0
[1,0,1,1,1,0,0,1,1,0,0,0] => [4,5,2,3,6,1] => [3,2,5,4,1,6] => 0
[1,0,1,1,1,0,1,0,0,0,1,0] => [6,3,2,4,5,1] => [1,4,5,3,2,6] => 1
[1,0,1,1,1,0,1,0,0,1,0,0] => [5,3,2,4,6,1] => [2,4,5,3,1,6] => 0
[1,0,1,1,1,0,1,0,1,0,0,0] => [4,3,2,5,6,1] => [3,4,5,2,1,6] => 0
[1,0,1,1,1,0,1,1,0,0,0,0] => [3,4,2,5,6,1] => [4,3,5,2,1,6] => 0
[1,0,1,1,1,1,0,0,0,0,1,0] => [6,2,3,4,5,1] => [1,5,4,3,2,6] => 3
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Description
The number of occurrences of 14-2-3 or 14-3-2.
The number of permutations avoiding both of these patterns is the case $k=2$ of the third item in Corollary 34 of [1].
The number of permutations avoiding both of these patterns is the case $k=2$ of the third item in Corollary 34 of [1].
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 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].
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