Identifier
Values
[1,0] => [2,1] => [2,1] => 1
[1,0,1,0] => [3,1,2] => [1,3,2] => 1
[1,1,0,0] => [2,3,1] => [3,2,1] => 1
[1,0,1,0,1,0] => [4,1,2,3] => [1,4,2,3] => 2
[1,0,1,1,0,0] => [3,1,4,2] => [1,3,4,2] => 1
[1,1,0,0,1,0] => [2,4,1,3] => [4,2,1,3] => 1
[1,1,0,1,0,0] => [4,3,1,2] => [3,4,1,2] => 2
[1,1,1,0,0,0] => [2,3,4,1] => [3,2,4,1] => 2
[1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [1,5,2,3,4] => 3
[1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [1,4,2,5,3] => 2
[1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [1,3,5,2,4] => 2
[1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [1,5,4,2,3] => 2
[1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [1,3,4,5,2] => 1
[1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [5,2,1,3,4] => 1
[1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [4,2,1,5,3] => 2
[1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [3,5,1,2,4] => 3
[1,1,0,1,0,1,0,0] => [5,4,1,2,3] => [4,5,1,2,3] => 3
[1,1,0,1,1,0,0,0] => [4,3,1,5,2] => [3,4,1,5,2] => 2
[1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [3,2,5,1,4] => 3
[1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [5,2,4,1,3] => 3
[1,1,1,0,1,0,0,0] => [5,3,4,1,2] => [3,5,4,1,2] => 2
[1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [3,2,4,5,1] => 2
[1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [1,6,2,3,4,5] => 4
[1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [1,5,2,3,6,4] => 3
[1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [1,4,2,6,3,5] => 3
[1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => [1,6,2,5,3,4] => 3
[1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [1,4,2,5,6,3] => 2
[1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [1,3,6,2,4,5] => 3
[1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [1,3,5,2,6,4] => 2
[1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => [1,6,4,2,3,5] => 2
[1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => [1,6,5,2,3,4] => 3
[1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [1,5,4,2,6,3] => 2
[1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [1,3,4,6,2,5] => 2
[1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [1,3,6,5,2,4] => 2
[1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => [1,6,4,5,2,3] => 3
[1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [1,3,4,5,6,2] => 1
[1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [6,2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [5,2,1,3,6,4] => 2
[1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [4,2,1,6,3,5] => 3
[1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [6,2,1,5,3,4] => 3
[1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [4,2,1,5,6,3] => 2
[1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => [3,6,1,2,4,5] => 4
[1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => [3,5,1,2,6,4] => 3
[1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => [4,6,1,2,3,5] => 4
[1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => [6,5,1,2,3,4] => 4
[1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => [4,5,1,2,6,3] => 3
[1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => [3,4,1,6,2,5] => 3
[1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => [3,6,1,5,2,4] => 3
[1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => [4,6,1,5,2,3] => 3
[1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => [3,4,1,5,6,2] => 2
[1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [3,2,6,1,4,5] => 4
[1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [3,2,5,1,6,4] => 3
[1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [6,2,4,1,3,5] => 3
[1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [6,2,5,1,3,4] => 4
[1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => [5,2,4,1,6,3] => 3
[1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => [3,6,4,1,2,5] => 2
[1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => [3,6,5,1,2,4] => 3
[1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => [5,6,4,1,2,3] => 3
[1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => [3,5,4,1,6,2] => 2
[1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [3,2,4,6,1,5] => 3
[1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => [3,2,6,5,1,4] => 3
[1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => [6,2,4,5,1,3] => 4
[1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => [3,6,4,5,1,2] => 3
[1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [3,2,4,5,6,1] => 2
[1,0,1,0,1,0,1,0,1,0,1,0] => [7,1,2,3,4,5,6] => [1,7,2,3,4,5,6] => 5
[1,0,1,0,1,0,1,0,1,1,0,0] => [6,1,2,3,4,7,5] => [1,6,2,3,4,7,5] => 4
[1,0,1,1,0,0,1,0,1,0,1,0] => [3,1,7,2,4,5,6] => [1,3,7,2,4,5,6] => 4
[1,0,1,1,0,1,0,1,0,1,0,0] => [6,1,7,2,3,4,5] => [1,6,7,2,3,4,5] => 4
[1,1,0,0,1,0,1,0,1,0,1,0] => [2,7,1,3,4,5,6] => [7,2,1,3,4,5,6] => 1
[1,1,0,1,0,1,0,0,1,1,0,0] => [6,4,1,2,3,7,5] => [4,6,1,2,3,7,5] => 4
[1,1,0,1,0,1,0,1,0,0,1,0] => [5,7,1,2,3,4,6] => [7,5,1,2,3,4,6] => 4
[1,1,0,1,0,1,0,1,0,1,0,0] => [7,6,1,2,3,4,5] => [6,7,1,2,3,4,5] => 5
[1,1,1,0,1,0,1,0,1,0,0,0] => [6,7,5,1,2,3,4] => [7,6,5,1,2,3,4] => 5
[1,1,1,1,0,1,0,1,0,0,0,0] => [7,6,4,5,1,2,3] => [6,7,4,5,1,2,3] => 4
[1,1,1,1,1,1,0,0,0,0,0,0] => [2,3,4,5,6,7,1] => [3,2,4,5,6,7,1] => 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [8,1,2,3,4,5,6,7] => [1,8,2,3,4,5,6,7] => 6
[1,0,1,0,1,1,0,0,1,1,0,0,1,0] => [4,1,2,6,3,8,5,7] => [1,4,2,6,3,8,5,7] => 4
[1,0,1,1,0,1,0,1,0,1,0,0,1,0] => [6,1,8,2,3,4,5,7] => [1,6,8,2,3,4,5,7] => 5
[1,1,1,1,0,1,0,1,0,1,0,0,0,0] => [7,8,6,5,1,2,3,4] => [8,7,6,5,1,2,3,4] => 5
[1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,1] => [3,2,4,5,6,7,8,1] => 2
[] => [1] => [1] => 0
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
click to show known generating functions       
Description
The lec statistic, the sum of the inversion numbers of the hook factors of a permutation.
For a permutation $\sigma = p \tau_{1} \tau_{2} \cdots \tau_{k}$ in its hook factorization, [1] defines $$ \textrm{lec} \, \sigma = \sum_{1 \leq i \leq k} \textrm{inv} \, \tau_{i} \, ,$$ where $\textrm{inv} \, \tau_{i}$ is the number of inversions of $\tau_{i}$.
Map
Alexandersson Kebede
Description
Sends a permutation to a permutation and it preserves the set of right-to-left minima.
Take a permutation $\pi$ of length $n$. The mapping looks for a smallest odd integer $i\in[n-1]$ such that swapping the entries $\pi(i)$ and $\pi(i+1)$ preserves the set of right-to-left minima. Otherwise, $\pi$ will be a fixed element of the mapping. Note that the map changes the sign of all non-fixed elements.
There are exactly $\binom{\lfloor n/2 \rfloor}{k-\lceil n/2 \rceil}$ elements in $S_n$ fixed under this map, with exactly $k$ right-to-left minima, see Lemma 35 in [1].
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.