Identifier
Values
[1,0] => [1] => 0
[1,0,1,0] => [1,2] => 0
[1,1,0,0] => [2,1] => 1
[1,0,1,0,1,0] => [1,2,3] => 0
[1,0,1,1,0,0] => [1,3,2] => 2
[1,1,0,0,1,0] => [2,1,3] => 1
[1,1,0,1,0,0] => [2,3,1] => 3
[1,1,1,0,0,0] => [3,1,2] => 1
[1,0,1,0,1,0,1,0] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0] => [1,2,4,3] => 3
[1,0,1,1,0,0,1,0] => [1,3,2,4] => 2
[1,0,1,1,0,1,0,0] => [1,3,4,2] => 5
[1,0,1,1,1,0,0,0] => [1,4,2,3] => 2
[1,1,0,0,1,0,1,0] => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0] => [2,1,4,3] => 4
[1,1,0,1,0,0,1,0] => [2,3,1,4] => 3
[1,1,0,1,0,1,0,0] => [2,3,4,1] => 6
[1,1,0,1,1,0,0,0] => [2,4,1,3] => 3
[1,1,1,0,0,0,1,0] => [3,1,2,4] => 1
[1,1,1,0,0,1,0,0] => [3,1,4,2] => 4
[1,1,1,0,1,0,0,0] => [3,4,1,2] => 3
[1,1,1,1,0,0,0,0] => [4,1,2,3] => 1
[1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => 4
[1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => 3
[1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => 7
[1,0,1,0,1,1,1,0,0,0] => [1,2,5,3,4] => 3
[1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => 2
[1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => 6
[1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => 5
[1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => 9
[1,0,1,1,0,1,1,0,0,0] => [1,3,5,2,4] => 5
[1,0,1,1,1,0,0,0,1,0] => [1,4,2,3,5] => 2
[1,0,1,1,1,0,0,1,0,0] => [1,4,2,5,3] => 6
[1,0,1,1,1,0,1,0,0,0] => [1,4,5,2,3] => 5
[1,0,1,1,1,1,0,0,0,0] => [1,5,2,3,4] => 2
[1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => 5
[1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => 4
[1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => 8
[1,1,0,0,1,1,1,0,0,0] => [2,1,5,3,4] => 4
[1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => 3
[1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => 7
[1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => 6
[1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => 10
[1,1,0,1,0,1,1,0,0,0] => [2,3,5,1,4] => 6
[1,1,0,1,1,0,0,0,1,0] => [2,4,1,3,5] => 3
[1,1,0,1,1,0,0,1,0,0] => [2,4,1,5,3] => 7
[1,1,0,1,1,0,1,0,0,0] => [2,4,5,1,3] => 6
[1,1,0,1,1,1,0,0,0,0] => [2,5,1,3,4] => 3
[1,1,1,0,0,0,1,0,1,0] => [3,1,2,4,5] => 1
[1,1,1,0,0,0,1,1,0,0] => [3,1,2,5,4] => 5
[1,1,1,0,0,1,0,0,1,0] => [3,1,4,2,5] => 4
[1,1,1,0,0,1,0,1,0,0] => [3,1,4,5,2] => 8
[1,1,1,0,0,1,1,0,0,0] => [3,1,5,2,4] => 4
[1,1,1,0,1,0,0,0,1,0] => [3,4,1,2,5] => 3
[1,1,1,0,1,0,0,1,0,0] => [3,4,1,5,2] => 7
[1,1,1,0,1,0,1,0,0,0] => [3,4,5,1,2] => 6
[1,1,1,0,1,1,0,0,0,0] => [3,5,1,2,4] => 3
[1,1,1,1,0,0,0,0,1,0] => [4,1,2,3,5] => 1
[1,1,1,1,0,0,0,1,0,0] => [4,1,2,5,3] => 5
[1,1,1,1,0,0,1,0,0,0] => [4,1,5,2,3] => 4
[1,1,1,1,0,1,0,0,0,0] => [4,5,1,2,3] => 3
[1,1,1,1,1,0,0,0,0,0] => [5,1,2,3,4] => 1
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => 5
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => 4
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => 9
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,4,5] => 4
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => 3
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => 8
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,4,5,3,6] => 7
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => 12
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,4,6,3,5] => 7
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,3,4,6] => 3
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,3,6,4] => 8
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,5,6,3,4] => 7
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,3,4,5] => 3
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => 2
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => 7
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => 6
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,3,2,5,6,4] => 11
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,4,5] => 6
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,4,2,5,6] => 5
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,4,2,6,5] => 10
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,3,4,5,2,6] => 9
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => 14
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,3,4,6,2,5] => 9
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,3,5,2,4,6] => 5
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,2,6,4] => 10
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,5,6,2,4] => 9
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,2,4,5] => 5
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,2,3,5,6] => 2
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,2,3,6,5] => 7
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,2,5,3,6] => 6
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,2,5,6,3] => 11
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,4,2,6,3,5] => 6
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,4,5,2,3,6] => 5
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,4,5,2,6,3] => 10
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,4,5,6,2,3] => 9
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,4,6,2,3,5] => 5
>>> Load all 196 entries. <<<
[1,0,1,1,1,1,0,0,0,0,1,0] => [1,5,2,3,4,6] => 2
[1,0,1,1,1,1,0,0,0,1,0,0] => [1,5,2,3,6,4] => 7
[1,0,1,1,1,1,0,0,1,0,0,0] => [1,5,2,6,3,4] => 6
[1,0,1,1,1,1,0,1,0,0,0,0] => [1,5,6,2,3,4] => 5
[1,0,1,1,1,1,1,0,0,0,0,0] => [1,6,2,3,4,5] => 2
[1,1,0,0,1,0,1,0,1,0,1,0] => [2,1,3,4,5,6] => 1
[1,1,0,0,1,0,1,0,1,1,0,0] => [2,1,3,4,6,5] => 6
[1,1,0,0,1,0,1,1,0,0,1,0] => [2,1,3,5,4,6] => 5
[1,1,0,0,1,0,1,1,0,1,0,0] => [2,1,3,5,6,4] => 10
[1,1,0,0,1,0,1,1,1,0,0,0] => [2,1,3,6,4,5] => 5
[1,1,0,0,1,1,0,0,1,0,1,0] => [2,1,4,3,5,6] => 4
[1,1,0,0,1,1,0,0,1,1,0,0] => [2,1,4,3,6,5] => 9
[1,1,0,0,1,1,0,1,0,0,1,0] => [2,1,4,5,3,6] => 8
[1,1,0,0,1,1,0,1,0,1,0,0] => [2,1,4,5,6,3] => 13
[1,1,0,0,1,1,0,1,1,0,0,0] => [2,1,4,6,3,5] => 8
[1,1,0,0,1,1,1,0,0,0,1,0] => [2,1,5,3,4,6] => 4
[1,1,0,0,1,1,1,0,0,1,0,0] => [2,1,5,3,6,4] => 9
[1,1,0,0,1,1,1,0,1,0,0,0] => [2,1,5,6,3,4] => 8
[1,1,0,0,1,1,1,1,0,0,0,0] => [2,1,6,3,4,5] => 4
[1,1,0,1,0,0,1,0,1,0,1,0] => [2,3,1,4,5,6] => 3
[1,1,0,1,0,0,1,0,1,1,0,0] => [2,3,1,4,6,5] => 8
[1,1,0,1,0,0,1,1,0,0,1,0] => [2,3,1,5,4,6] => 7
[1,1,0,1,0,0,1,1,0,1,0,0] => [2,3,1,5,6,4] => 12
[1,1,0,1,0,0,1,1,1,0,0,0] => [2,3,1,6,4,5] => 7
[1,1,0,1,0,1,0,0,1,0,1,0] => [2,3,4,1,5,6] => 6
[1,1,0,1,0,1,0,0,1,1,0,0] => [2,3,4,1,6,5] => 11
[1,1,0,1,0,1,0,1,0,0,1,0] => [2,3,4,5,1,6] => 10
[1,1,0,1,0,1,0,1,0,1,0,0] => [2,3,4,5,6,1] => 15
[1,1,0,1,0,1,0,1,1,0,0,0] => [2,3,4,6,1,5] => 10
[1,1,0,1,0,1,1,0,0,0,1,0] => [2,3,5,1,4,6] => 6
[1,1,0,1,0,1,1,0,0,1,0,0] => [2,3,5,1,6,4] => 11
[1,1,0,1,0,1,1,0,1,0,0,0] => [2,3,5,6,1,4] => 10
[1,1,0,1,0,1,1,1,0,0,0,0] => [2,3,6,1,4,5] => 6
[1,1,0,1,1,0,0,0,1,0,1,0] => [2,4,1,3,5,6] => 3
[1,1,0,1,1,0,0,0,1,1,0,0] => [2,4,1,3,6,5] => 8
[1,1,0,1,1,0,0,1,0,0,1,0] => [2,4,1,5,3,6] => 7
[1,1,0,1,1,0,0,1,0,1,0,0] => [2,4,1,5,6,3] => 12
[1,1,0,1,1,0,0,1,1,0,0,0] => [2,4,1,6,3,5] => 7
[1,1,0,1,1,0,1,0,0,0,1,0] => [2,4,5,1,3,6] => 6
[1,1,0,1,1,0,1,0,0,1,0,0] => [2,4,5,1,6,3] => 11
[1,1,0,1,1,0,1,0,1,0,0,0] => [2,4,5,6,1,3] => 10
[1,1,0,1,1,0,1,1,0,0,0,0] => [2,4,6,1,3,5] => 6
[1,1,0,1,1,1,0,0,0,0,1,0] => [2,5,1,3,4,6] => 3
[1,1,0,1,1,1,0,0,0,1,0,0] => [2,5,1,3,6,4] => 8
[1,1,0,1,1,1,0,0,1,0,0,0] => [2,5,1,6,3,4] => 7
[1,1,0,1,1,1,0,1,0,0,0,0] => [2,5,6,1,3,4] => 6
[1,1,0,1,1,1,1,0,0,0,0,0] => [2,6,1,3,4,5] => 3
[1,1,1,0,0,0,1,0,1,0,1,0] => [3,1,2,4,5,6] => 1
[1,1,1,0,0,0,1,0,1,1,0,0] => [3,1,2,4,6,5] => 6
[1,1,1,0,0,0,1,1,0,0,1,0] => [3,1,2,5,4,6] => 5
[1,1,1,0,0,0,1,1,0,1,0,0] => [3,1,2,5,6,4] => 10
[1,1,1,0,0,0,1,1,1,0,0,0] => [3,1,2,6,4,5] => 5
[1,1,1,0,0,1,0,0,1,0,1,0] => [3,1,4,2,5,6] => 4
[1,1,1,0,0,1,0,0,1,1,0,0] => [3,1,4,2,6,5] => 9
[1,1,1,0,0,1,0,1,0,0,1,0] => [3,1,4,5,2,6] => 8
[1,1,1,0,0,1,0,1,0,1,0,0] => [3,1,4,5,6,2] => 13
[1,1,1,0,0,1,0,1,1,0,0,0] => [3,1,4,6,2,5] => 8
[1,1,1,0,0,1,1,0,0,0,1,0] => [3,1,5,2,4,6] => 4
[1,1,1,0,0,1,1,0,0,1,0,0] => [3,1,5,2,6,4] => 9
[1,1,1,0,0,1,1,0,1,0,0,0] => [3,1,5,6,2,4] => 8
[1,1,1,0,0,1,1,1,0,0,0,0] => [3,1,6,2,4,5] => 4
[1,1,1,0,1,0,0,0,1,0,1,0] => [3,4,1,2,5,6] => 3
[1,1,1,0,1,0,0,0,1,1,0,0] => [3,4,1,2,6,5] => 8
[1,1,1,0,1,0,0,1,0,0,1,0] => [3,4,1,5,2,6] => 7
[1,1,1,0,1,0,0,1,0,1,0,0] => [3,4,1,5,6,2] => 12
[1,1,1,0,1,0,0,1,1,0,0,0] => [3,4,1,6,2,5] => 7
[1,1,1,0,1,0,1,0,0,0,1,0] => [3,4,5,1,2,6] => 6
[1,1,1,0,1,0,1,0,0,1,0,0] => [3,4,5,1,6,2] => 11
[1,1,1,0,1,0,1,0,1,0,0,0] => [3,4,5,6,1,2] => 10
[1,1,1,0,1,0,1,1,0,0,0,0] => [3,4,6,1,2,5] => 6
[1,1,1,0,1,1,0,0,0,0,1,0] => [3,5,1,2,4,6] => 3
[1,1,1,0,1,1,0,0,0,1,0,0] => [3,5,1,2,6,4] => 8
[1,1,1,0,1,1,0,0,1,0,0,0] => [3,5,1,6,2,4] => 7
[1,1,1,0,1,1,0,1,0,0,0,0] => [3,5,6,1,2,4] => 6
[1,1,1,0,1,1,1,0,0,0,0,0] => [3,6,1,2,4,5] => 3
[1,1,1,1,0,0,0,0,1,0,1,0] => [4,1,2,3,5,6] => 1
[1,1,1,1,0,0,0,0,1,1,0,0] => [4,1,2,3,6,5] => 6
[1,1,1,1,0,0,0,1,0,0,1,0] => [4,1,2,5,3,6] => 5
[1,1,1,1,0,0,0,1,0,1,0,0] => [4,1,2,5,6,3] => 10
[1,1,1,1,0,0,0,1,1,0,0,0] => [4,1,2,6,3,5] => 5
[1,1,1,1,0,0,1,0,0,0,1,0] => [4,1,5,2,3,6] => 4
[1,1,1,1,0,0,1,0,0,1,0,0] => [4,1,5,2,6,3] => 9
[1,1,1,1,0,0,1,0,1,0,0,0] => [4,1,5,6,2,3] => 8
[1,1,1,1,0,0,1,1,0,0,0,0] => [4,1,6,2,3,5] => 4
[1,1,1,1,0,1,0,0,0,0,1,0] => [4,5,1,2,3,6] => 3
[1,1,1,1,0,1,0,0,0,1,0,0] => [4,5,1,2,6,3] => 8
[1,1,1,1,0,1,0,0,1,0,0,0] => [4,5,1,6,2,3] => 7
[1,1,1,1,0,1,0,1,0,0,0,0] => [4,5,6,1,2,3] => 6
[1,1,1,1,0,1,1,0,0,0,0,0] => [4,6,1,2,3,5] => 3
[1,1,1,1,1,0,0,0,0,0,1,0] => [5,1,2,3,4,6] => 1
[1,1,1,1,1,0,0,0,0,1,0,0] => [5,1,2,3,6,4] => 6
[1,1,1,1,1,0,0,0,1,0,0,0] => [5,1,2,6,3,4] => 5
[1,1,1,1,1,0,0,1,0,0,0,0] => [5,1,6,2,3,4] => 4
[1,1,1,1,1,0,1,0,0,0,0,0] => [5,6,1,2,3,4] => 3
[1,1,1,1,1,1,0,0,0,0,0,0] => [6,1,2,3,4,5] => 1
search for individual values
searching the database for the individual values of this statistic
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searching the database for statistics with the same generating function
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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..
Map
to 321-avoiding permutation (Krattenthaler)
Description
Krattenthaler's bijection to 321-avoiding permutations.
Draw the path of semilength $n$ in an $n\times n$ square matrix, starting at the upper left corner, with right and down steps, and staying below the diagonal. Then the permutation matrix is obtained by placing ones into the cells corresponding to the peaks of the path and placing ones into the remaining columns from left to right, such that the row indices of the cells increase.