Identifier
- St000156: Permutations ⟶ ℤ
Values
[1] => 0
[1,2] => 0
[2,1] => 1
[1,2,3] => 0
[1,3,2] => 2
[2,1,3] => 1
[2,3,1] => 3
[3,1,2] => 1
[3,2,1] => 2
[1,2,3,4] => 0
[1,2,4,3] => 3
[1,3,2,4] => 2
[1,3,4,2] => 5
[1,4,2,3] => 2
[1,4,3,2] => 3
[2,1,3,4] => 1
[2,1,4,3] => 4
[2,3,1,4] => 3
[2,3,4,1] => 6
[2,4,1,3] => 3
[2,4,3,1] => 4
[3,1,2,4] => 1
[3,1,4,2] => 4
[3,2,1,4] => 2
[3,2,4,1] => 5
[3,4,1,2] => 3
[3,4,2,1] => 4
[4,1,2,3] => 1
[4,1,3,2] => 2
[4,2,1,3] => 2
[4,2,3,1] => 3
[4,3,1,2] => 4
[4,3,2,1] => 5
[1,2,3,4,5] => 0
[1,2,3,5,4] => 4
[1,2,4,3,5] => 3
[1,2,4,5,3] => 7
[1,2,5,3,4] => 3
[1,2,5,4,3] => 4
[1,3,2,4,5] => 2
[1,3,2,5,4] => 6
[1,3,4,2,5] => 5
[1,3,4,5,2] => 9
[1,3,5,2,4] => 5
[1,3,5,4,2] => 6
[1,4,2,3,5] => 2
[1,4,2,5,3] => 6
[1,4,3,2,5] => 3
[1,4,3,5,2] => 7
[1,4,5,2,3] => 5
[1,4,5,3,2] => 6
[1,5,2,3,4] => 2
[1,5,2,4,3] => 3
[1,5,3,2,4] => 3
[1,5,3,4,2] => 4
[1,5,4,2,3] => 6
[1,5,4,3,2] => 7
[2,1,3,4,5] => 1
[2,1,3,5,4] => 5
[2,1,4,3,5] => 4
[2,1,4,5,3] => 8
[2,1,5,3,4] => 4
[2,1,5,4,3] => 5
[2,3,1,4,5] => 3
[2,3,1,5,4] => 7
[2,3,4,1,5] => 6
[2,3,4,5,1] => 10
[2,3,5,1,4] => 6
[2,3,5,4,1] => 7
[2,4,1,3,5] => 3
[2,4,1,5,3] => 7
[2,4,3,1,5] => 4
[2,4,3,5,1] => 8
[2,4,5,1,3] => 6
[2,4,5,3,1] => 7
[2,5,1,3,4] => 3
[2,5,1,4,3] => 4
[2,5,3,1,4] => 4
[2,5,3,4,1] => 5
[2,5,4,1,3] => 7
[2,5,4,3,1] => 8
[3,1,2,4,5] => 1
[3,1,2,5,4] => 5
[3,1,4,2,5] => 4
[3,1,4,5,2] => 8
[3,1,5,2,4] => 4
[3,1,5,4,2] => 5
[3,2,1,4,5] => 2
[3,2,1,5,4] => 6
[3,2,4,1,5] => 5
[3,2,4,5,1] => 9
[3,2,5,1,4] => 5
[3,2,5,4,1] => 6
[3,4,1,2,5] => 3
[3,4,1,5,2] => 7
[3,4,2,1,5] => 4
[3,4,2,5,1] => 8
[3,4,5,1,2] => 6
[3,4,5,2,1] => 7
[3,5,1,2,4] => 3
[3,5,1,4,2] => 4
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Description
The Denert index of a permutation.
It is defined as
$$ \begin{align*} den(\sigma) &= \#\{ 1\leq l < k \leq n : \sigma(k) < \sigma(l) \leq k \} \\ &+ \#\{ 1\leq l < k \leq n : \sigma(l) \leq k < \sigma(k) \} \\ &+ \#\{ 1\leq l < k \leq n : k < \sigma(k) < \sigma(l) \} \end{align*} $$
where $n$ is the size of $\sigma$. It was studied by Denert in [1], and it was shown by Foata and Zeilberger in [2] that the bistatistic $(exc,den)$ is Euler-Mahonian. Here, $exc$ is the number of weak exceedences, see St000155The number of exceedances (also excedences) of a permutation..
It is defined as
$$ \begin{align*} den(\sigma) &= \#\{ 1\leq l < k \leq n : \sigma(k) < \sigma(l) \leq k \} \\ &+ \#\{ 1\leq l < k \leq n : \sigma(l) \leq k < \sigma(k) \} \\ &+ \#\{ 1\leq l < k \leq n : k < \sigma(k) < \sigma(l) \} \end{align*} $$
where $n$ is the size of $\sigma$. It was studied by Denert in [1], and it was shown by Foata and Zeilberger in [2] that the bistatistic $(exc,den)$ is Euler-Mahonian. Here, $exc$ is the number of weak exceedences, see St000155The number of exceedances (also excedences) of a permutation..
References
[1] Denert, M. The genus zeta function of hereditary orders in central simple algebras over global fields MathSciNet:0993928
[2] Foata, D., Zeilberger, D. Denert's permutation statistic is indeed Euler-Mahonian MathSciNet:1061147
[3] Triangle of Mahonian numbers T(n,k): coefficients in expansion of Product_i=0..n-1 (1 + x + ... + x^i), where k ranges from 0 to A000217(n-1). OEIS:A008302
[2] Foata, D., Zeilberger, D. Denert's permutation statistic is indeed Euler-Mahonian MathSciNet:1061147
[3] Triangle of Mahonian numbers T(n,k): coefficients in expansion of Product_i=0..n-1 (1 + x + ... + x^i), where k ranges from 0 to A000217(n-1). OEIS:A008302
Code
import itertools
def statistic(pi):
return sum(1 for l, k in itertools.combinations(range(pi.size()), 2)
if (pi[k] < pi[l] <= k+1)
or (pi[l] <= k+1 < pi[k])
or (k+1 < pi[k] < pi[l]))
Created
Jul 24, 2013 at 12:25 by Christian Stump
Updated
Feb 27, 2023 at 12:37 by Martin Rubey
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