Identifier
- St000339: Permutations ⟶ ℤ
Values
[1] => 0
[1,2] => 0
[2,1] => 1
[1,2,3] => 0
[1,3,2] => 1
[2,1,3] => 3
[2,3,1] => 2
[3,1,2] => 1
[3,2,1] => 2
[1,2,3,4] => 0
[1,2,4,3] => 1
[1,3,2,4] => 3
[1,3,4,2] => 2
[1,4,2,3] => 1
[1,4,3,2] => 2
[2,1,3,4] => 5
[2,1,4,3] => 4
[2,3,1,4] => 5
[2,3,4,1] => 3
[2,4,1,3] => 2
[2,4,3,1] => 4
[3,1,2,4] => 4
[3,1,4,2] => 4
[3,2,1,4] => 4
[3,2,4,1] => 3
[3,4,1,2] => 2
[3,4,2,1] => 5
[4,1,2,3] => 1
[4,1,3,2] => 3
[4,2,1,3] => 2
[4,2,3,1] => 3
[4,3,1,2] => 3
[4,3,2,1] => 6
[1,2,3,4,5] => 0
[1,2,3,5,4] => 1
[1,2,4,3,5] => 3
[1,2,4,5,3] => 2
[1,2,5,3,4] => 1
[1,2,5,4,3] => 2
[1,3,2,4,5] => 5
[1,3,2,5,4] => 4
[1,3,4,2,5] => 5
[1,3,4,5,2] => 3
[1,3,5,2,4] => 2
[1,3,5,4,2] => 4
[1,4,2,3,5] => 4
[1,4,2,5,3] => 4
[1,4,3,2,5] => 4
[1,4,3,5,2] => 3
[1,4,5,2,3] => 2
[1,4,5,3,2] => 5
[1,5,2,3,4] => 1
[1,5,2,4,3] => 3
[1,5,3,2,4] => 2
[1,5,3,4,2] => 3
[1,5,4,2,3] => 3
[1,5,4,3,2] => 6
[2,1,3,4,5] => 7
[2,1,3,5,4] => 6
[2,1,4,3,5] => 8
[2,1,4,5,3] => 5
[2,1,5,3,4] => 4
[2,1,5,4,3] => 7
[2,3,1,4,5] => 8
[2,3,1,5,4] => 6
[2,3,4,1,5] => 7
[2,3,4,5,1] => 4
[2,3,5,1,4] => 3
[2,3,5,4,1] => 6
[2,4,1,3,5] => 6
[2,4,1,5,3] => 6
[2,4,3,1,5] => 7
[2,4,3,5,1] => 5
[2,4,5,1,3] => 3
[2,4,5,3,1] => 7
[2,5,1,3,4] => 2
[2,5,1,4,3] => 5
[2,5,3,1,4] => 4
[2,5,3,4,1] => 6
[2,5,4,1,3] => 5
[2,5,4,3,1] => 9
[3,1,2,4,5] => 7
[3,1,2,5,4] => 5
[3,1,4,2,5] => 8
[3,1,4,5,2] => 5
[3,1,5,2,4] => 4
[3,1,5,4,2] => 7
[3,2,1,4,5] => 6
[3,2,1,5,4] => 5
[3,2,4,1,5] => 6
[3,2,4,5,1] => 4
[3,2,5,1,4] => 3
[3,2,5,4,1] => 5
[3,4,1,2,5] => 6
[3,4,1,5,2] => 6
[3,4,2,1,5] => 9
[3,4,2,5,1] => 6
[3,4,5,1,2] => 3
[3,4,5,2,1] => 7
[3,5,1,2,4] => 2
[3,5,1,4,2] => 5
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Description
The maf index of a permutation.
Let $\sigma$ be a permutation with fixed point set $\operatorname{FIX}(\sigma)$, and let $\operatorname{Der}(\sigma)$ be the derangement obtained from $\sigma$ by removing the fixed points.
Then
$$\operatorname{maf}(\sigma) = \sum_{i \in \operatorname{FIX}(\sigma)} i - \binom{|\operatorname{FIX}(\sigma)|+1}{2} + \operatorname{maj}(\operatorname{Der}(\sigma)),$$
where $\operatorname{maj}(\operatorname{Der}(\sigma))$ is the major index of the derangement of $\sigma$.
Let $\sigma$ be a permutation with fixed point set $\operatorname{FIX}(\sigma)$, and let $\operatorname{Der}(\sigma)$ be the derangement obtained from $\sigma$ by removing the fixed points.
Then
$$\operatorname{maf}(\sigma) = \sum_{i \in \operatorname{FIX}(\sigma)} i - \binom{|\operatorname{FIX}(\sigma)|+1}{2} + \operatorname{maj}(\operatorname{Der}(\sigma)),$$
where $\operatorname{maj}(\operatorname{Der}(\sigma))$ is the major index of the derangement of $\sigma$.
References
[1] Foata, D., Han, G.-N. Fix-Mahonian Calculus, II: further statistics arXiv:math/0703101
[2] Han, G.-N., Xin, G. Permutations with Extremal number of Fixed Points arXiv:0706.1738
[2] Han, G.-N., Xin, G. Permutations with Extremal number of Fixed Points arXiv:0706.1738
Code
def statistic(x):
f = len(x.fixed_points())
return sum(x.fixed_points()) - ((f+1)*f)//2 + Der(x).major_index()
def Der(x):
return Word([i for i in sz(x) if i != 0]).standard_permutation()
#Returns the word (as a list) where fixed points of x are changed to zero (see [2])
def sz(x):
l = list(x)
for i in x.fixed_points():
l[i-1] = 0
return l
Created
Dec 18, 2015 at 13:45 by Joseph Bernstein
Updated
Dec 13, 2024 at 13:30 by Nupur Jain
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