Identifier
Values
[1,0] => [1,1,0,0] => [1,2] => 0
[1,0,1,0] => [1,1,0,1,0,0] => [2,1,3] => 1
[1,1,0,0] => [1,1,1,0,0,0] => [1,2,3] => 0
[1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => [3,2,1,4] => 2
[1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => [2,3,1,4] => 2
[1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => [3,1,2,4] => 2
[1,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => [2,1,3,4] => 1
[1,1,1,0,0,0] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => [4,3,2,1,5] => 3
[1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => [3,4,2,1,5] => 3
[1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => [4,2,3,1,5] => 4
[1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => [3,2,4,1,5] => 3
[1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => [2,3,4,1,5] => 3
[1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => [4,3,1,2,5] => 3
[1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => [3,4,1,2,5] => 4
[1,1,0,1,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => [4,2,1,3,5] => 3
[1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,0,0] => [3,2,1,4,5] => 2
[1,1,0,1,1,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => [2,3,1,4,5] => 2
[1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => [4,1,2,3,5] => 3
[1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => [3,1,2,4,5] => 2
[1,1,1,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => [2,1,3,4,5] => 1
[1,1,1,1,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => [5,4,3,2,1,6] => 4
[1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,1,1,0,0,0] => [4,5,3,2,1,6] => 4
[1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,1,0,0,1,0,0] => [5,3,4,2,1,6] => 5
[1,0,1,0,1,1,0,1,0,0] => [1,1,0,1,0,1,1,0,1,0,0,0] => [4,3,5,2,1,6] => 4
[1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,1,1,0,0,0,0] => [3,4,5,2,1,6] => 4
[1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,1,0,0,1,0,1,0,0] => [5,4,2,3,1,6] => 5
[1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,1,0,0,1,1,0,0,0] => [4,5,2,3,1,6] => 6
[1,0,1,1,0,1,0,0,1,0] => [1,1,0,1,1,0,1,0,0,1,0,0] => [5,3,2,4,1,6] => 5
[1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,1,0,1,0,1,0,0,0] => [4,3,2,5,1,6] => 4
[1,0,1,1,0,1,1,0,0,0] => [1,1,0,1,1,0,1,1,0,0,0,0] => [3,4,2,5,1,6] => 4
[1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => [5,2,3,4,1,6] => 6
[1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,1,0,0,1,0,0,0] => [4,2,3,5,1,6] => 5
[1,0,1,1,1,0,1,0,0,0] => [1,1,0,1,1,1,0,1,0,0,0,0] => [3,2,4,5,1,6] => 4
[1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => [2,3,4,5,1,6] => 4
[1,1,0,0,1,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,1,0,0] => [5,4,3,1,2,6] => 4
[1,1,0,0,1,0,1,1,0,0] => [1,1,1,0,0,1,0,1,1,0,0,0] => [4,5,3,1,2,6] => 4
[1,1,0,0,1,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,1,0,0] => [5,3,4,1,2,6] => 6
[1,1,0,0,1,1,0,1,0,0] => [1,1,1,0,0,1,1,0,1,0,0,0] => [4,3,5,1,2,6] => 5
[1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,0,1,1,1,0,0,0,0] => [3,4,5,1,2,6] => 6
[1,1,0,1,0,0,1,0,1,0] => [1,1,1,0,1,0,0,1,0,1,0,0] => [5,4,2,1,3,6] => 4
[1,1,0,1,0,0,1,1,0,0] => [1,1,1,0,1,0,0,1,1,0,0,0] => [4,5,2,1,3,6] => 5
[1,1,0,1,0,1,0,0,1,0] => [1,1,1,0,1,0,1,0,0,1,0,0] => [5,3,2,1,4,6] => 4
[1,1,0,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,1,0,0,0] => [4,3,2,1,5,6] => 3
[1,1,0,1,0,1,1,0,0,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => [3,4,2,1,5,6] => 3
[1,1,0,1,1,0,0,0,1,0] => [1,1,1,0,1,1,0,0,0,1,0,0] => [5,2,3,1,4,6] => 5
[1,1,0,1,1,0,0,1,0,0] => [1,1,1,0,1,1,0,0,1,0,0,0] => [4,2,3,1,5,6] => 4
[1,1,0,1,1,0,1,0,0,0] => [1,1,1,0,1,1,0,1,0,0,0,0] => [3,2,4,1,5,6] => 3
[1,1,0,1,1,1,0,0,0,0] => [1,1,1,0,1,1,1,0,0,0,0,0] => [2,3,4,1,5,6] => 3
[1,1,1,0,0,0,1,0,1,0] => [1,1,1,1,0,0,0,1,0,1,0,0] => [5,4,1,2,3,6] => 4
[1,1,1,0,0,0,1,1,0,0] => [1,1,1,1,0,0,0,1,1,0,0,0] => [4,5,1,2,3,6] => 6
[1,1,1,0,0,1,0,0,1,0] => [1,1,1,1,0,0,1,0,0,1,0,0] => [5,3,1,2,4,6] => 4
[1,1,1,0,0,1,0,1,0,0] => [1,1,1,1,0,0,1,0,1,0,0,0] => [4,3,1,2,5,6] => 3
[1,1,1,0,0,1,1,0,0,0] => [1,1,1,1,0,0,1,1,0,0,0,0] => [3,4,1,2,5,6] => 4
[1,1,1,0,1,0,0,0,1,0] => [1,1,1,1,0,1,0,0,0,1,0,0] => [5,2,1,3,4,6] => 4
[1,1,1,0,1,0,0,1,0,0] => [1,1,1,1,0,1,0,0,1,0,0,0] => [4,2,1,3,5,6] => 3
[1,1,1,0,1,0,1,0,0,0] => [1,1,1,1,0,1,0,1,0,0,0,0] => [3,2,1,4,5,6] => 2
[1,1,1,0,1,1,0,0,0,0] => [1,1,1,1,0,1,1,0,0,0,0,0] => [2,3,1,4,5,6] => 2
[1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,1,0,0,0,0,1,0,0] => [5,1,2,3,4,6] => 4
[1,1,1,1,0,0,0,1,0,0] => [1,1,1,1,1,0,0,0,1,0,0,0] => [4,1,2,3,5,6] => 3
[1,1,1,1,0,0,1,0,0,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => [3,1,2,4,5,6] => 2
[1,1,1,1,0,1,0,0,0,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => [2,1,3,4,5,6] => 1
[1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,2,3,4,5,6] => 0
[1,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,2,3,4,5,6,7] => 0
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Description
The number of Bruhat lower covers of a permutation.
This is, for a permutation $\pi$, the number of permutations $\tau$ with $\operatorname{inv}(\tau) = \operatorname{inv}(\pi) - 1$ such that $\tau*t = \pi$ for a transposition $t$.
This is also the number of occurrences of the boxed pattern $21$: occurrences of the pattern $21$ such that any entry between the two matched entries is either larger or smaller than both of the matched entries.
Map
to 132-avoiding permutation
Description
Sends a Dyck path to a 132-avoiding permutation.
This bijection is defined in [1, Section 2].
Map
prime Dyck path
Description
Return the Dyck path obtained by adding an initial up and a final down step.