Identifier
Values
=>
[1]=>0
[-1]=>0
[1,2]=>0
[1,-2]=>1
[-1,2]=>1
[-1,-2]=>1
[2,1]=>0
[2,-1]=>0
[-2,1]=>0
[-2,-1]=>0
[1,2,3]=>0
[1,2,-3]=>2
[1,-2,3]=>2
[1,-2,-3]=>3
[-1,2,3]=>2
[-1,2,-3]=>3
[-1,-2,3]=>3
[-1,-2,-3]=>3
[1,3,2]=>1
[1,3,-2]=>1
[1,-3,2]=>2
[1,-3,-2]=>2
[-1,3,2]=>2
[-1,3,-2]=>2
[-1,-3,2]=>2
[-1,-3,-2]=>2
[2,1,3]=>1
[2,1,-3]=>2
[2,-1,3]=>1
[2,-1,-3]=>2
[-2,1,3]=>2
[-2,1,-3]=>2
[-2,-1,3]=>2
[-2,-1,-3]=>2
[2,3,1]=>0
[2,3,-1]=>0
[2,-3,1]=>1
[2,-3,-1]=>1
[-2,3,1]=>1
[-2,3,-1]=>1
[-2,-3,1]=>1
[-2,-3,-1]=>1
[3,1,2]=>0
[3,1,-2]=>1
[3,-1,2]=>1
[3,-1,-2]=>1
[-3,1,2]=>0
[-3,1,-2]=>1
[-3,-1,2]=>1
[-3,-1,-2]=>1
[3,2,1]=>1
[3,2,-1]=>1
[3,-2,1]=>0
[3,-2,-1]=>0
[-3,2,1]=>2
[-3,2,-1]=>2
[-3,-2,1]=>0
[-3,-2,-1]=>0
[1,2,3,4]=>0
[1,2,3,-4]=>3
[1,2,-3,4]=>3
[1,2,-3,-4]=>5
[1,-2,3,4]=>3
[1,-2,3,-4]=>5
[1,-2,-3,4]=>5
[1,-2,-3,-4]=>6
[-1,2,3,4]=>3
[-1,2,3,-4]=>5
[-1,2,-3,4]=>5
[-1,2,-3,-4]=>6
[-1,-2,3,4]=>5
[-1,-2,3,-4]=>6
[-1,-2,-3,4]=>6
[-1,-2,-3,-4]=>6
[1,2,4,3]=>2
[1,2,4,-3]=>2
[1,2,-4,3]=>4
[1,2,-4,-3]=>4
[1,-2,4,3]=>4
[1,-2,4,-3]=>4
[1,-2,-4,3]=>5
[1,-2,-4,-3]=>5
[-1,2,4,3]=>4
[-1,2,4,-3]=>4
[-1,2,-4,3]=>5
[-1,2,-4,-3]=>5
[-1,-2,4,3]=>5
[-1,-2,4,-3]=>5
[-1,-2,-4,3]=>5
[-1,-2,-4,-3]=>5
[1,3,2,4]=>2
[1,3,2,-4]=>4
[1,3,-2,4]=>2
[1,3,-2,-4]=>4
[1,-3,2,4]=>4
[1,-3,2,-4]=>5
[1,-3,-2,4]=>4
[1,-3,-2,-4]=>5
[-1,3,2,4]=>4
[-1,3,2,-4]=>5
[-1,3,-2,4]=>4
[-1,3,-2,-4]=>5
[-1,-3,2,4]=>5
[-1,-3,2,-4]=>5
[-1,-3,-2,4]=>5
[-1,-3,-2,-4]=>5
[1,3,4,2]=>1
[1,3,4,-2]=>1
[1,3,-4,2]=>3
[1,3,-4,-2]=>3
[1,-3,4,2]=>3
[1,-3,4,-2]=>3
[1,-3,-4,2]=>4
[1,-3,-4,-2]=>4
[-1,3,4,2]=>3
[-1,3,4,-2]=>3
[-1,3,-4,2]=>4
[-1,3,-4,-2]=>4
[-1,-3,4,2]=>4
[-1,-3,4,-2]=>4
[-1,-3,-4,2]=>4
[-1,-3,-4,-2]=>4
[1,4,2,3]=>2
[1,4,2,-3]=>3
[1,4,-2,3]=>3
[1,4,-2,-3]=>3
[1,-4,2,3]=>3
[1,-4,2,-3]=>4
[1,-4,-2,3]=>4
[1,-4,-2,-3]=>4
[-1,4,2,3]=>3
[-1,4,2,-3]=>4
[-1,4,-2,3]=>4
[-1,4,-2,-3]=>4
[-1,-4,2,3]=>3
[-1,-4,2,-3]=>4
[-1,-4,-2,3]=>4
[-1,-4,-2,-3]=>4
[1,4,3,2]=>2
[1,4,3,-2]=>2
[1,4,-3,2]=>2
[1,4,-3,-2]=>2
[1,-4,3,2]=>4
[1,-4,3,-2]=>4
[1,-4,-3,2]=>3
[1,-4,-3,-2]=>3
[-1,4,3,2]=>4
[-1,4,3,-2]=>4
[-1,4,-3,2]=>3
[-1,4,-3,-2]=>3
[-1,-4,3,2]=>5
[-1,-4,3,-2]=>5
[-1,-4,-3,2]=>3
[-1,-4,-3,-2]=>3
[2,1,3,4]=>2
[2,1,3,-4]=>4
[2,1,-3,4]=>4
[2,1,-3,-4]=>5
[2,-1,3,4]=>2
[2,-1,3,-4]=>4
[2,-1,-3,4]=>4
[2,-1,-3,-4]=>5
[-2,1,3,4]=>4
[-2,1,3,-4]=>5
[-2,1,-3,4]=>5
[-2,1,-3,-4]=>5
[-2,-1,3,4]=>4
[-2,-1,3,-4]=>5
[-2,-1,-3,4]=>5
[-2,-1,-3,-4]=>5
[2,1,4,3]=>2
[2,1,4,-3]=>3
[2,1,-4,3]=>3
[2,1,-4,-3]=>4
[2,-1,4,3]=>3
[2,-1,4,-3]=>3
[2,-1,-4,3]=>4
[2,-1,-4,-3]=>4
[-2,1,4,3]=>3
[-2,1,4,-3]=>4
[-2,1,-4,3]=>3
[-2,1,-4,-3]=>4
[-2,-1,4,3]=>4
[-2,-1,4,-3]=>4
[-2,-1,-4,3]=>4
[-2,-1,-4,-3]=>4
[2,3,1,4]=>1
[2,3,1,-4]=>3
[2,3,-1,4]=>1
[2,3,-1,-4]=>3
[2,-3,1,4]=>3
[2,-3,1,-4]=>4
[2,-3,-1,4]=>3
[2,-3,-1,-4]=>4
[-2,3,1,4]=>3
[-2,3,1,-4]=>4
[-2,3,-1,4]=>3
[-2,3,-1,-4]=>4
[-2,-3,1,4]=>4
[-2,-3,1,-4]=>4
[-2,-3,-1,4]=>4
[-2,-3,-1,-4]=>4
[2,3,4,1]=>0
[2,3,4,-1]=>0
[2,3,-4,1]=>2
[2,3,-4,-1]=>2
[2,-3,4,1]=>2
[2,-3,4,-1]=>2
[2,-3,-4,1]=>3
[2,-3,-4,-1]=>3
[-2,3,4,1]=>2
[-2,3,4,-1]=>2
[-2,3,-4,1]=>3
[-2,3,-4,-1]=>3
[-2,-3,4,1]=>3
[-2,-3,4,-1]=>3
[-2,-3,-4,1]=>3
[-2,-3,-4,-1]=>3
[2,4,1,3]=>1
[2,4,1,-3]=>2
[2,4,-1,3]=>2
[2,4,-1,-3]=>2
[2,-4,1,3]=>2
[2,-4,1,-3]=>3
[2,-4,-1,3]=>3
[2,-4,-1,-3]=>3
[-2,4,1,3]=>2
[-2,4,1,-3]=>3
[-2,4,-1,3]=>3
[-2,4,-1,-3]=>3
[-2,-4,1,3]=>2
[-2,-4,1,-3]=>3
[-2,-4,-1,3]=>3
[-2,-4,-1,-3]=>3
[2,4,3,1]=>1
[2,4,3,-1]=>1
[2,4,-3,1]=>1
[2,4,-3,-1]=>1
[2,-4,3,1]=>3
[2,-4,3,-1]=>3
[2,-4,-3,1]=>2
[2,-4,-3,-1]=>2
[-2,4,3,1]=>3
[-2,4,3,-1]=>3
[-2,4,-3,1]=>2
[-2,4,-3,-1]=>2
[-2,-4,3,1]=>4
[-2,-4,3,-1]=>4
[-2,-4,-3,1]=>2
[-2,-4,-3,-1]=>2
[3,1,2,4]=>2
[3,1,2,-4]=>3
[3,1,-2,4]=>3
[3,1,-2,-4]=>4
[3,-1,2,4]=>3
[3,-1,2,-4]=>4
[3,-1,-2,4]=>3
[3,-1,-2,-4]=>4
[-3,1,2,4]=>3
[-3,1,2,-4]=>3
[-3,1,-2,4]=>4
[-3,1,-2,-4]=>4
[-3,-1,2,4]=>4
[-3,-1,2,-4]=>4
[-3,-1,-2,4]=>4
[-3,-1,-2,-4]=>4
[3,1,4,2]=>1
[3,1,4,-2]=>2
[3,1,-4,2]=>2
[3,1,-4,-2]=>3
[3,-1,4,2]=>2
[3,-1,4,-2]=>2
[3,-1,-4,2]=>3
[3,-1,-4,-2]=>3
[-3,1,4,2]=>2
[-3,1,4,-2]=>3
[-3,1,-4,2]=>2
[-3,1,-4,-2]=>3
[-3,-1,4,2]=>3
[-3,-1,4,-2]=>3
[-3,-1,-4,2]=>3
[-3,-1,-4,-2]=>3
[3,2,1,4]=>2
[3,2,1,-4]=>4
[3,2,-1,4]=>2
[3,2,-1,-4]=>4
[3,-2,1,4]=>2
[3,-2,1,-4]=>3
[3,-2,-1,4]=>2
[3,-2,-1,-4]=>3
[-3,2,1,4]=>4
[-3,2,1,-4]=>5
[-3,2,-1,4]=>4
[-3,2,-1,-4]=>5
[-3,-2,1,4]=>3
[-3,-2,1,-4]=>3
[-3,-2,-1,4]=>3
[-3,-2,-1,-4]=>3
[3,2,4,1]=>1
[3,2,4,-1]=>1
[3,2,-4,1]=>3
[3,2,-4,-1]=>3
[3,-2,4,1]=>1
[3,-2,4,-1]=>1
[3,-2,-4,1]=>2
[3,-2,-4,-1]=>2
[-3,2,4,1]=>3
[-3,2,4,-1]=>3
[-3,2,-4,1]=>4
[-3,2,-4,-1]=>4
[-3,-2,4,1]=>2
[-3,-2,4,-1]=>2
[-3,-2,-4,1]=>2
[-3,-2,-4,-1]=>2
[3,4,1,2]=>0
[3,4,1,-2]=>1
[3,4,-1,2]=>1
[3,4,-1,-2]=>1
[3,-4,1,2]=>1
[3,-4,1,-2]=>2
[3,-4,-1,2]=>2
[3,-4,-1,-2]=>2
[-3,4,1,2]=>1
[-3,4,1,-2]=>2
[-3,4,-1,2]=>2
[-3,4,-1,-2]=>2
[-3,-4,1,2]=>1
[-3,-4,1,-2]=>2
[-3,-4,-1,2]=>2
[-3,-4,-1,-2]=>2
[3,4,2,1]=>1
[3,4,2,-1]=>2
[3,4,-2,1]=>0
[3,4,-2,-1]=>0
[3,-4,2,1]=>2
[3,-4,2,-1]=>3
[3,-4,-2,1]=>1
[3,-4,-2,-1]=>1
[-3,4,2,1]=>2
[-3,4,2,-1]=>3
[-3,4,-2,1]=>1
[-3,4,-2,-1]=>1
[-3,-4,2,1]=>2
[-3,-4,2,-1]=>3
[-3,-4,-2,1]=>1
[-3,-4,-2,-1]=>1
[4,1,2,3]=>0
[4,1,2,-3]=>2
[4,1,-2,3]=>2
[4,1,-2,-3]=>3
[4,-1,2,3]=>2
[4,-1,2,-3]=>3
[4,-1,-2,3]=>3
[4,-1,-2,-3]=>3
[-4,1,2,3]=>0
[-4,1,2,-3]=>2
[-4,1,-2,3]=>2
[-4,1,-2,-3]=>3
[-4,-1,2,3]=>2
[-4,-1,2,-3]=>3
[-4,-1,-2,3]=>3
[-4,-1,-2,-3]=>3
[4,1,3,2]=>2
[4,1,3,-2]=>3
[4,1,-3,2]=>1
[4,1,-3,-2]=>2
[4,-1,3,2]=>3
[4,-1,3,-2]=>3
[4,-1,-3,2]=>2
[4,-1,-3,-2]=>2
[-4,1,3,2]=>3
[-4,1,3,-2]=>4
[-4,1,-3,2]=>1
[-4,1,-3,-2]=>2
[-4,-1,3,2]=>4
[-4,-1,3,-2]=>4
[-4,-1,-3,2]=>2
[-4,-1,-3,-2]=>2
[4,2,1,3]=>2
[4,2,1,-3]=>3
[4,2,-1,3]=>3
[4,2,-1,-3]=>3
[4,-2,1,3]=>1
[4,-2,1,-3]=>2
[4,-2,-1,3]=>2
[4,-2,-1,-3]=>2
[-4,2,1,3]=>3
[-4,2,1,-3]=>4
[-4,2,-1,3]=>4
[-4,2,-1,-3]=>4
[-4,-2,1,3]=>1
[-4,-2,1,-3]=>2
[-4,-2,-1,3]=>2
[-4,-2,-1,-3]=>2
[4,2,3,1]=>2
[4,2,3,-1]=>2
[4,2,-3,1]=>2
[4,2,-3,-1]=>2
[4,-2,3,1]=>2
[4,-2,3,-1]=>2
[4,-2,-3,1]=>1
[4,-2,-3,-1]=>1
[-4,2,3,1]=>4
[-4,2,3,-1]=>4
[-4,2,-3,1]=>3
[-4,2,-3,-1]=>3
[-4,-2,3,1]=>3
[-4,-2,3,-1]=>3
[-4,-2,-3,1]=>1
[-4,-2,-3,-1]=>1
[4,3,1,2]=>1
[4,3,1,-2]=>2
[4,3,-1,2]=>2
[4,3,-1,-2]=>2
[4,-3,1,2]=>0
[4,-3,1,-2]=>1
[4,-3,-1,2]=>1
[4,-3,-1,-2]=>1
[-4,3,1,2]=>2
[-4,3,1,-2]=>3
[-4,3,-1,2]=>3
[-4,3,-1,-2]=>3
[-4,-3,1,2]=>0
[-4,-3,1,-2]=>1
[-4,-3,-1,2]=>1
[-4,-3,-1,-2]=>1
[4,3,2,1]=>2
[4,3,2,-1]=>3
[4,3,-2,1]=>1
[4,3,-2,-1]=>1
[4,-3,2,1]=>1
[4,-3,2,-1]=>2
[4,-3,-2,1]=>0
[4,-3,-2,-1]=>0
[-4,3,2,1]=>3
[-4,3,2,-1]=>4
[-4,3,-2,1]=>2
[-4,3,-2,-1]=>2
[-4,-3,2,1]=>1
[-4,-3,2,-1]=>2
[-4,-3,-2,1]=>0
[-4,-3,-2,-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 $n\in\mathfrak H_n$ 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, $\operatorname{al}+\operatorname{cr}=(n-\operatorname{wex})(\operatorname{wex}-1+\operatorname{neg})+\binom{\operatorname{neg}{2}$.
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, $\operatorname{al}+\operatorname{cr}=(n-\operatorname{wex})(\operatorname{wex}-1+\operatorname{neg})+\binom{\operatorname{neg}{2}$.
References
[1] Cho, S., Park, K. Permutation statistics and weak Bruhat order in permutation tableaux of type $B$ MathSciNet:3319075
Code
def statistic(pi):
n = len(pi)
A_nest = [(i, j) for i in range(1, n+1) for j in range(1, n+1)
if (-i < -j < -pi(j) < -pi(i))
or (-i < j <= pi(j) < -pi(i))
or (i < j <= pi(j) < pi(i))]
A_EN = [(i, j) for i in range(1, n+1) for j in range(1, n+1)
if (-i < 0 < -pi(i) < pi(j) < j)
or (i <= pi(i) < pi(j) < j)]
A_NE = [(i, j) for i in range(1, n+1) for j in range(1, n+1)
if (pi(i) < i < j <= pi(j))]
return len(A_nest + A_EN + A_NE)
Created
Jul 21, 2022 at 07:29 by Dennis Jahn
Updated
Dec 01, 2022 at 14:23 by Martin Rubey
searching the database
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!