Identifier
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] => 0
[1,0,1,1,0,0] => [2,3,1] => [2,3,1] => 1
[1,1,0,0,1,0] => [3,1,2] => [3,1,2] => 0
[1,1,0,1,0,0] => [2,1,3] => [2,1,3] => 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] => [4,3,2,1] => 0
[1,0,1,0,1,1,0,0] => [3,4,2,1] => [3,4,2,1] => 1
[1,0,1,1,0,0,1,0] => [4,2,3,1] => [4,2,3,1] => 1
[1,0,1,1,0,1,0,0] => [3,2,4,1] => [3,2,4,1] => 1
[1,0,1,1,1,0,0,0] => [2,3,4,1] => [2,3,4,1] => 2
[1,1,0,0,1,0,1,0] => [4,3,1,2] => [4,3,1,2] => 0
[1,1,0,0,1,1,0,0] => [3,4,1,2] => [3,4,1,2] => 1
[1,1,0,1,0,0,1,0] => [4,2,1,3] => [4,2,1,3] => 0
[1,1,0,1,0,1,0,0] => [3,2,1,4] => [3,2,1,4] => 0
[1,1,0,1,1,0,0,0] => [2,3,1,4] => [2,3,1,4] => 1
[1,1,1,0,0,0,1,0] => [4,1,2,3] => [4,1,2,3] => 0
[1,1,1,0,0,1,0,0] => [3,1,2,4] => [3,1,2,4] => 0
[1,1,1,0,1,0,0,0] => [2,1,3,4] => [2,1,3,4] => 0
[1,1,1,1,0,0,0,0] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [1,2,3,4,5] => 0
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Description
The number of occurrences of a type-B 231 pattern in a signed permutation.
For a signed permutation $\pi\in\mathfrak H_n$, a triple $-n \leq i < j < k\leq n$ is an occurrence of the type-B $231$ pattern, if $1 \leq j < k$, $\pi(i) < \pi(j)$ and $\pi(i)$ is one larger than $\pi(k)$, i.e., $\pi(i) = \pi(k) + 1$ if $\pi(k) \neq -1$ and $\pi(i) = 1$ otherwise.
Map
to signed permutation
Description
The signed permutation with all signs positive.
Map
to 132-avoiding permutation
Description
Sends a Dyck path to a 132-avoiding permutation.
This bijection is defined in [1, Section 2].