Identifier
Values
=>
[1]=>[1]=>[1]=>0 [1,2]=>[1,2]=>[1,2]=>0 [2,1]=>[1,2]=>[1,2]=>0 [1,2,3]=>[1,2,3]=>[1,2,3]=>0 [1,3,2]=>[1,3,2]=>[1,3,2]=>0 [2,1,3]=>[1,3,2]=>[1,3,2]=>0 [2,3,1]=>[1,2,3]=>[1,2,3]=>0 [3,1,2]=>[1,2,3]=>[1,2,3]=>0 [3,2,1]=>[1,2,3]=>[1,2,3]=>0 [1,2,3,4]=>[1,2,3,4]=>[1,2,3,4]=>0 [1,2,4,3]=>[1,2,4,3]=>[1,2,4,3]=>0 [1,3,2,4]=>[1,3,2,4]=>[1,3,2,4]=>0 [1,3,4,2]=>[1,3,4,2]=>[1,3,4,2]=>1 [1,4,2,3]=>[1,4,2,3]=>[1,4,2,3]=>0 [1,4,3,2]=>[1,4,2,3]=>[1,4,2,3]=>0 [2,1,3,4]=>[1,3,4,2]=>[1,3,4,2]=>1 [2,1,4,3]=>[1,4,2,3]=>[1,4,2,3]=>0 [2,3,1,4]=>[1,4,2,3]=>[1,4,2,3]=>0 [2,3,4,1]=>[1,2,3,4]=>[1,2,3,4]=>0 [2,4,1,3]=>[1,3,2,4]=>[1,3,2,4]=>0 [2,4,3,1]=>[1,2,4,3]=>[1,2,4,3]=>0 [3,1,2,4]=>[1,2,4,3]=>[1,2,4,3]=>0 [3,1,4,2]=>[1,4,2,3]=>[1,4,2,3]=>0 [3,2,1,4]=>[1,4,2,3]=>[1,4,2,3]=>0 [3,2,4,1]=>[1,2,4,3]=>[1,2,4,3]=>0 [3,4,1,2]=>[1,2,3,4]=>[1,2,3,4]=>0 [3,4,2,1]=>[1,2,3,4]=>[1,2,3,4]=>0 [4,1,2,3]=>[1,2,3,4]=>[1,2,3,4]=>0 [4,1,3,2]=>[1,3,2,4]=>[1,3,2,4]=>0 [4,2,1,3]=>[1,3,2,4]=>[1,3,2,4]=>0 [4,2,3,1]=>[1,2,3,4]=>[1,2,3,4]=>0 [4,3,1,2]=>[1,2,3,4]=>[1,2,3,4]=>0 [4,3,2,1]=>[1,2,3,4]=>[1,2,3,4]=>0 [1,2,3,4,5]=>[1,2,3,4,5]=>[1,2,3,4,5]=>0 [1,2,3,5,4]=>[1,2,3,5,4]=>[1,2,3,5,4]=>0 [1,2,4,3,5]=>[1,2,4,3,5]=>[1,2,4,3,5]=>0 [1,2,4,5,3]=>[1,2,4,5,3]=>[1,2,4,5,3]=>1 [1,2,5,3,4]=>[1,2,5,3,4]=>[1,2,5,3,4]=>0 [1,2,5,4,3]=>[1,2,5,3,4]=>[1,2,5,3,4]=>0 [1,3,2,4,5]=>[1,3,2,4,5]=>[1,3,2,4,5]=>0 [1,3,2,5,4]=>[1,3,2,5,4]=>[1,3,2,5,4]=>0 [1,3,4,2,5]=>[1,3,4,2,5]=>[1,3,4,2,5]=>1 [1,3,4,5,2]=>[1,3,4,5,2]=>[1,3,4,5,2]=>2 [1,3,5,2,4]=>[1,3,5,2,4]=>[1,3,5,2,4]=>1 [1,3,5,4,2]=>[1,3,5,2,4]=>[1,3,5,2,4]=>1 [1,4,2,3,5]=>[1,4,2,3,5]=>[1,4,2,3,5]=>0 [1,4,2,5,3]=>[1,4,2,5,3]=>[1,4,2,5,3]=>1 [1,4,3,2,5]=>[1,4,2,5,3]=>[1,4,2,5,3]=>1 [1,4,3,5,2]=>[1,4,2,3,5]=>[1,4,2,3,5]=>0 [1,4,5,2,3]=>[1,4,5,2,3]=>[1,4,5,2,3]=>1 [1,4,5,3,2]=>[1,4,5,2,3]=>[1,4,5,2,3]=>1 [1,5,2,3,4]=>[1,5,2,3,4]=>[1,5,2,3,4]=>0 [1,5,2,4,3]=>[1,5,2,4,3]=>[1,5,2,4,3]=>0 [1,5,3,2,4]=>[1,5,2,4,3]=>[1,5,2,4,3]=>0 [1,5,3,4,2]=>[1,5,2,3,4]=>[1,5,2,3,4]=>0 [1,5,4,2,3]=>[1,5,2,3,4]=>[1,5,2,3,4]=>0 [1,5,4,3,2]=>[1,5,2,3,4]=>[1,5,2,3,4]=>0 [2,1,3,4,5]=>[1,3,4,5,2]=>[1,3,4,5,2]=>2 [2,1,3,5,4]=>[1,3,5,2,4]=>[1,3,5,2,4]=>1 [2,1,4,3,5]=>[1,4,2,3,5]=>[1,4,2,3,5]=>0 [2,1,4,5,3]=>[1,4,5,2,3]=>[1,4,5,2,3]=>1 [2,1,5,3,4]=>[1,5,2,3,4]=>[1,5,2,3,4]=>0 [2,1,5,4,3]=>[1,5,2,3,4]=>[1,5,2,3,4]=>0 [2,3,1,4,5]=>[1,4,5,2,3]=>[1,4,5,2,3]=>1 [2,3,1,5,4]=>[1,5,2,3,4]=>[1,5,2,3,4]=>0 [2,3,4,1,5]=>[1,5,2,3,4]=>[1,5,2,3,4]=>0 [2,3,4,5,1]=>[1,2,3,4,5]=>[1,2,3,4,5]=>0 [2,3,5,1,4]=>[1,4,2,3,5]=>[1,4,2,3,5]=>0 [2,3,5,4,1]=>[1,2,3,5,4]=>[1,2,3,5,4]=>0 [2,4,1,3,5]=>[1,3,5,2,4]=>[1,3,5,2,4]=>1 [2,4,1,5,3]=>[1,5,2,4,3]=>[1,5,2,4,3]=>0 [2,4,3,1,5]=>[1,5,2,4,3]=>[1,5,2,4,3]=>0 [2,4,3,5,1]=>[1,2,4,3,5]=>[1,2,4,3,5]=>0 [2,4,5,1,3]=>[1,3,2,4,5]=>[1,3,2,4,5]=>0 [2,4,5,3,1]=>[1,2,4,5,3]=>[1,2,4,5,3]=>1 [2,5,1,3,4]=>[1,3,4,2,5]=>[1,3,4,2,5]=>1 [2,5,1,4,3]=>[1,4,2,5,3]=>[1,4,2,5,3]=>1 [2,5,3,1,4]=>[1,4,2,5,3]=>[1,4,2,5,3]=>1 [2,5,3,4,1]=>[1,2,5,3,4]=>[1,2,5,3,4]=>0 [2,5,4,1,3]=>[1,3,2,5,4]=>[1,3,2,5,4]=>0 [2,5,4,3,1]=>[1,2,5,3,4]=>[1,2,5,3,4]=>0 [3,1,2,4,5]=>[1,2,4,5,3]=>[1,2,4,5,3]=>1 [3,1,2,5,4]=>[1,2,5,3,4]=>[1,2,5,3,4]=>0 [3,1,4,2,5]=>[1,4,2,5,3]=>[1,4,2,5,3]=>1 [3,1,4,5,2]=>[1,4,5,2,3]=>[1,4,5,2,3]=>1 [3,1,5,2,4]=>[1,5,2,4,3]=>[1,5,2,4,3]=>0 [3,1,5,4,2]=>[1,5,2,3,4]=>[1,5,2,3,4]=>0 [3,2,1,4,5]=>[1,4,5,2,3]=>[1,4,5,2,3]=>1 [3,2,1,5,4]=>[1,5,2,3,4]=>[1,5,2,3,4]=>0 [3,2,4,1,5]=>[1,5,2,4,3]=>[1,5,2,4,3]=>0 [3,2,4,5,1]=>[1,2,4,5,3]=>[1,2,4,5,3]=>1 [3,2,5,1,4]=>[1,4,2,5,3]=>[1,4,2,5,3]=>1 [3,2,5,4,1]=>[1,2,5,3,4]=>[1,2,5,3,4]=>0 [3,4,1,2,5]=>[1,2,5,3,4]=>[1,2,5,3,4]=>0 [3,4,1,5,2]=>[1,5,2,3,4]=>[1,5,2,3,4]=>0 [3,4,2,1,5]=>[1,5,2,3,4]=>[1,5,2,3,4]=>0 [3,4,2,5,1]=>[1,2,5,3,4]=>[1,2,5,3,4]=>0 [3,4,5,1,2]=>[1,2,3,4,5]=>[1,2,3,4,5]=>0 [3,4,5,2,1]=>[1,2,3,4,5]=>[1,2,3,4,5]=>0 [3,5,1,2,4]=>[1,2,4,3,5]=>[1,2,4,3,5]=>0 [3,5,1,4,2]=>[1,4,2,3,5]=>[1,4,2,3,5]=>0 [3,5,2,1,4]=>[1,4,2,3,5]=>[1,4,2,3,5]=>0 [3,5,2,4,1]=>[1,2,4,3,5]=>[1,2,4,3,5]=>0 [3,5,4,1,2]=>[1,2,3,5,4]=>[1,2,3,5,4]=>0 [3,5,4,2,1]=>[1,2,3,5,4]=>[1,2,3,5,4]=>0 [4,1,2,3,5]=>[1,2,3,5,4]=>[1,2,3,5,4]=>0 [4,1,2,5,3]=>[1,2,5,3,4]=>[1,2,5,3,4]=>0 [4,1,3,2,5]=>[1,3,2,5,4]=>[1,3,2,5,4]=>0 [4,1,3,5,2]=>[1,3,5,2,4]=>[1,3,5,2,4]=>1 [4,1,5,2,3]=>[1,5,2,3,4]=>[1,5,2,3,4]=>0 [4,1,5,3,2]=>[1,5,2,3,4]=>[1,5,2,3,4]=>0 [4,2,1,3,5]=>[1,3,5,2,4]=>[1,3,5,2,4]=>1 [4,2,1,5,3]=>[1,5,2,3,4]=>[1,5,2,3,4]=>0 [4,2,3,1,5]=>[1,5,2,3,4]=>[1,5,2,3,4]=>0 [4,2,3,5,1]=>[1,2,3,5,4]=>[1,2,3,5,4]=>0 [4,2,5,1,3]=>[1,3,2,5,4]=>[1,3,2,5,4]=>0 [4,2,5,3,1]=>[1,2,5,3,4]=>[1,2,5,3,4]=>0 [4,3,1,2,5]=>[1,2,5,3,4]=>[1,2,5,3,4]=>0 [4,3,1,5,2]=>[1,5,2,3,4]=>[1,5,2,3,4]=>0 [4,3,2,1,5]=>[1,5,2,3,4]=>[1,5,2,3,4]=>0 [4,3,2,5,1]=>[1,2,5,3,4]=>[1,2,5,3,4]=>0 [4,3,5,1,2]=>[1,2,3,5,4]=>[1,2,3,5,4]=>0 [4,3,5,2,1]=>[1,2,3,5,4]=>[1,2,3,5,4]=>0 [4,5,1,2,3]=>[1,2,3,4,5]=>[1,2,3,4,5]=>0 [4,5,1,3,2]=>[1,3,2,4,5]=>[1,3,2,4,5]=>0 [4,5,2,1,3]=>[1,3,2,4,5]=>[1,3,2,4,5]=>0 [4,5,2,3,1]=>[1,2,3,4,5]=>[1,2,3,4,5]=>0 [4,5,3,1,2]=>[1,2,3,4,5]=>[1,2,3,4,5]=>0 [4,5,3,2,1]=>[1,2,3,4,5]=>[1,2,3,4,5]=>0 [5,1,2,3,4]=>[1,2,3,4,5]=>[1,2,3,4,5]=>0 [5,1,2,4,3]=>[1,2,4,3,5]=>[1,2,4,3,5]=>0 [5,1,3,2,4]=>[1,3,2,4,5]=>[1,3,2,4,5]=>0 [5,1,3,4,2]=>[1,3,4,2,5]=>[1,3,4,2,5]=>1 [5,1,4,2,3]=>[1,4,2,3,5]=>[1,4,2,3,5]=>0 [5,1,4,3,2]=>[1,4,2,3,5]=>[1,4,2,3,5]=>0 [5,2,1,3,4]=>[1,3,4,2,5]=>[1,3,4,2,5]=>1 [5,2,1,4,3]=>[1,4,2,3,5]=>[1,4,2,3,5]=>0 [5,2,3,1,4]=>[1,4,2,3,5]=>[1,4,2,3,5]=>0 [5,2,3,4,1]=>[1,2,3,4,5]=>[1,2,3,4,5]=>0 [5,2,4,1,3]=>[1,3,2,4,5]=>[1,3,2,4,5]=>0 [5,2,4,3,1]=>[1,2,4,3,5]=>[1,2,4,3,5]=>0 [5,3,1,2,4]=>[1,2,4,3,5]=>[1,2,4,3,5]=>0 [5,3,1,4,2]=>[1,4,2,3,5]=>[1,4,2,3,5]=>0 [5,3,2,1,4]=>[1,4,2,3,5]=>[1,4,2,3,5]=>0 [5,3,2,4,1]=>[1,2,4,3,5]=>[1,2,4,3,5]=>0 [5,3,4,1,2]=>[1,2,3,4,5]=>[1,2,3,4,5]=>0 [5,3,4,2,1]=>[1,2,3,4,5]=>[1,2,3,4,5]=>0 [5,4,1,2,3]=>[1,2,3,4,5]=>[1,2,3,4,5]=>0 [5,4,1,3,2]=>[1,3,2,4,5]=>[1,3,2,4,5]=>0 [5,4,2,1,3]=>[1,3,2,4,5]=>[1,3,2,4,5]=>0 [5,4,2,3,1]=>[1,2,3,4,5]=>[1,2,3,4,5]=>0 [5,4,3,1,2]=>[1,2,3,4,5]=>[1,2,3,4,5]=>0 [5,4,3,2,1]=>[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
runsort
Description
The permutation obtained by sorting the increasing runs lexicographically.