**Identifier**

Identifier

Values

[1]
=>
0

[-1]
=>
1

[1,2]
=>
0

[1,-2]
=>
3

[-1,2]
=>
1

[-1,-2]
=>
4

[2,1]
=>
1

[2,-1]
=>
2

[-2,1]
=>
2

[-2,-1]
=>
3

[1,2,3]
=>
0

[1,2,-3]
=>
5

[1,-2,3]
=>
3

[1,-2,-3]
=>
8

[-1,2,3]
=>
1

[-1,2,-3]
=>
6

[-1,-2,3]
=>
4

[-1,-2,-3]
=>
9

[1,3,2]
=>
1

[1,3,-2]
=>
4

[1,-3,2]
=>
4

[1,-3,-2]
=>
7

[-1,3,2]
=>
2

[-1,3,-2]
=>
5

[-1,-3,2]
=>
5

[-1,-3,-2]
=>
8

[2,1,3]
=>
1

[2,1,-3]
=>
6

[2,-1,3]
=>
2

[2,-1,-3]
=>
7

[-2,1,3]
=>
2

[-2,1,-3]
=>
7

[-2,-1,3]
=>
3

[-2,-1,-3]
=>
8

[2,3,1]
=>
2

[2,3,-1]
=>
3

[2,-3,1]
=>
5

[2,-3,-1]
=>
6

[-2,3,1]
=>
3

[-2,3,-1]
=>
4

[-2,-3,1]
=>
6

[-2,-3,-1]
=>
7

[3,1,2]
=>
2

[3,1,-2]
=>
5

[3,-1,2]
=>
3

[3,-1,-2]
=>
6

[-3,1,2]
=>
3

[-3,1,-2]
=>
6

[-3,-1,2]
=>
4

[-3,-1,-2]
=>
7

[3,2,1]
=>
3

[3,2,-1]
=>
4

[3,-2,1]
=>
4

[3,-2,-1]
=>
5

[-3,2,1]
=>
4

[-3,2,-1]
=>
5

[-3,-2,1]
=>
5

[-3,-2,-1]
=>
6

Description

The number of B-inversions of a signed permutation.

The number of B-inversions of a signed permtutation $\sigma$ of length $n$ is

$$\operatorname{inv}_B(\sigma) = \big|\{ 1 \leq i < j \leq n \mid \sigma(i) > \sigma(j) \}\big| + \big|\{ 1 \leq i \leq j \leq n \mid \sigma(-i) > \sigma(j) \}\big|,$$

see [1, Eq. (8.2)]. According to [1, Eq. (8.4)], this is the Coxeter length of $\sigma$.

The number of B-inversions of a signed permtutation $\sigma$ of length $n$ is

$$\operatorname{inv}_B(\sigma) = \big|\{ 1 \leq i < j \leq n \mid \sigma(i) > \sigma(j) \}\big| + \big|\{ 1 \leq i \leq j \leq n \mid \sigma(-i) > \sigma(j) \}\big|,$$

see [1, Eq. (8.2)]. According to [1, Eq. (8.4)], this is the Coxeter length of $\sigma$.

References

Code

def statistic(pi): pi = list(pi) n = len(pi) return sum(1 for i in range(n) for j in range(i+1,n) if pi[i] > pi[j]) + \ sum(1 for i in range(n) for j in range(i ,n) if -pi[i] > pi[j])

Created

Jun 21, 2019 at 14:03 by

**Christian Stump**Updated

Jun 21, 2019 at 14:38 by

**Christian Stump**searching the database

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