Your data matches 1 statistic following compositions of up to 3 maps.
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Matching statistic: St000677
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Mapping
St000677: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 0
[2,1] => 1
[1,2,3] => 0
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 1
[1,2,3,4] => 1
[1,2,4,3] => 2
[1,3,2,4] => 2
[1,3,4,2] => 1
[1,4,2,3] => 1
[1,4,3,2] => 2
[2,1,3,4] => 2
[2,1,4,3] => 3
[2,3,1,4] => 1
[2,3,4,1] => 2
[2,4,1,3] => 0
[2,4,3,1] => 3
[3,1,2,4] => 1
[3,1,4,2] => 4
[3,2,1,4] => 2
[3,2,4,1] => 3
[3,4,1,2] => 1
[3,4,2,1] => 2
[4,1,2,3] => 2
[4,1,3,2] => 3
[4,2,1,3] => 3
[4,2,3,1] => 2
[4,3,1,2] => 2
[4,3,2,1] => 3
[1,2,3,4,5] => 1
[1,2,3,5,4] => 2
[1,2,4,3,5] => 2
[1,2,4,5,3] => 1
[1,2,5,3,4] => 1
[1,2,5,4,3] => 2
[1,3,2,4,5] => 2
[1,3,2,5,4] => 3
[1,3,4,2,5] => 1
[1,3,4,5,2] => 2
[1,3,5,2,4] => 0
[1,3,5,4,2] => 3
[1,4,2,3,5] => 1
[1,4,2,5,3] => 4
[1,4,3,2,5] => 2
[1,4,3,5,2] => 3
[1,4,5,2,3] => 1
[1,4,5,3,2] => 2
Description
The standardized bi-alternating inversion number of a permutation. The bi-alternating inversion number $i(\pi)$ is $\sum_{1\leq y < x\leq n} (-1)^{y+x} \mathrm{sgn}(\pi(x)-\pi(y))$, see Section 3.1 of [1]. It takes values in between $-\lfloor\frac n2\rfloor^2$ and $\lfloor\frac n2\rfloor^2$, and $i(\pi)+\lfloor\frac n2\rfloor^2$ is even, so we standardize it to $(i(\pi)+\lfloor\frac n2\rfloor^2)/2$. For odd $n$ the bi-alternating inversion is invariant under rotation, that is, conjugation with the long cycle.