Processing math: 100%

Identifier
Values
[1,0] => [2,1] => [2,1] => 0
[1,0,1,0] => [3,1,2] => [3,1,2] => 0
[1,1,0,0] => [2,3,1] => [2,3,1] => 0
[1,0,1,0,1,0] => [4,1,2,3] => [4,1,2,3] => 0
[1,0,1,1,0,0] => [3,1,4,2] => [3,1,4,2] => 1
[1,1,0,0,1,0] => [2,4,1,3] => [2,4,1,3] => 1
[1,1,0,1,0,0] => [4,3,1,2] => [4,3,1,2] => 1
[1,1,1,0,0,0] => [2,3,4,1] => [2,3,4,1] => 0
[] => [1] => [1] => 0
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
Description
The number of alignments of a signed permutation.
An alignment of a signed permutation n∈Hn 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 al be the number of alignments of π, 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.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.