Identifier
Values
=>
[1]=>[1]=>0 [1,2]=>[1,2]=>0 [2,1]=>[2,1]=>0 [1,2,3]=>[1,2,3]=>0 [1,3,2]=>[1,3,2]=>1 [2,1,3]=>[2,1,3]=>1 [2,3,1]=>[2,3,1]=>0 [3,1,2]=>[3,1,2]=>0 [3,2,1]=>[3,2,1]=>1 [1,2,3,4]=>[1,2,3,4]=>0 [1,2,4,3]=>[1,2,4,3]=>2 [1,3,2,4]=>[1,3,2,4]=>2 [1,3,4,2]=>[1,3,4,2]=>1 [1,4,2,3]=>[1,4,2,3]=>2 [1,4,3,2]=>[1,4,3,2]=>2 [2,1,3,4]=>[2,1,3,4]=>2 [2,1,4,3]=>[2,1,4,3]=>2 [2,3,1,4]=>[2,3,1,4]=>1 [2,3,4,1]=>[2,3,4,1]=>0 [2,4,1,3]=>[2,4,1,3]=>1 [2,4,3,1]=>[2,4,3,1]=>1 [3,1,2,4]=>[3,1,2,4]=>2 [3,1,4,2]=>[3,1,4,2]=>1 [3,2,1,4]=>[3,2,1,4]=>2 [3,2,4,1]=>[3,2,4,1]=>1 [3,4,1,2]=>[3,4,1,2]=>0 [3,4,2,1]=>[3,4,2,1]=>1 [4,1,2,3]=>[4,1,2,3]=>0 [4,1,3,2]=>[4,1,3,2]=>2 [4,2,1,3]=>[4,2,1,3]=>2 [4,2,3,1]=>[4,2,3,1]=>2 [4,3,1,2]=>[4,3,1,2]=>1 [4,3,2,1]=>[4,3,2,1]=>2
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Description
The number of alignments of a signed permutation.
An alignment of a signed permutation $n\in\mathfrak H_n$ is either a nesting alignment, St001866The nesting alignments of a signed permutation., an alignment of type EN, St001867The number of alignments of type EN of a signed permutation., or an alignment of type NE, St001868The number of alignments of type NE of a signed permutation..
Let $\operatorname{al}$ be the number of alignments of $\pi$, let \operatorname{cr} be the number of crossings, St001862The number of crossings of a signed permutation., let \operatorname{wex} be the number of weak excedances, St001863The number of weak excedances of a signed permutation., and let \operatorname{neg} be the number of negative entries, St001429The number of negative entries in a signed permutation.. Then, $\operatorname{al}+\operatorname{cr}=(n-\operatorname{wex})(\operatorname{wex}-1+\operatorname{neg})+\binom{\operatorname{neg}{2}$.
Map
to signed permutation
Description
The signed permutation with all signs positive.