Identifier
Values
[1,0] => [2,1] => [2,1] => [2,1] => 0
[1,0,1,0] => [3,1,2] => [2,3,1] => [3,2,1] => 0
[1,1,0,0] => [2,3,1] => [3,1,2] => [1,3,2] => 0
[1,0,1,0,1,0] => [4,1,2,3] => [2,3,4,1] => [3,2,4,1] => 0
[1,0,1,1,0,0] => [3,1,4,2] => [2,4,1,3] => [4,2,1,3] => 0
[1,1,0,0,1,0] => [2,4,1,3] => [3,1,4,2] => [1,3,4,2] => 0
[1,1,0,1,0,0] => [4,3,1,2] => [3,4,2,1] => [4,3,2,1] => 1
[1,1,1,0,0,0] => [2,3,4,1] => [4,1,2,3] => [1,4,2,3] => 0
[1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [2,3,4,5,1] => [3,2,4,5,1] => 0
[1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [2,3,5,1,4] => [3,2,5,1,4] => 0
[1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [2,4,1,5,3] => [4,2,1,5,3] => 0
[1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [2,4,5,3,1] => [4,2,5,3,1] => 1
[1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [2,5,1,3,4] => [5,2,1,3,4] => 0
[1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [3,1,4,5,2] => [1,3,4,5,2] => 0
[1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [3,1,5,2,4] => [1,3,5,2,4] => 0
[1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [3,4,2,5,1] => [4,3,2,5,1] => 1
[1,1,0,1,0,1,0,0] => [5,4,1,2,3] => [3,4,5,2,1] => [4,3,5,2,1] => 1
[1,1,0,1,1,0,0,0] => [4,3,1,5,2] => [3,5,2,1,4] => [5,3,2,1,4] => 1
[1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [4,1,2,5,3] => [1,4,2,5,3] => 0
[1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [4,1,5,3,2] => [1,4,5,3,2] => 1
[1,1,1,0,1,0,0,0] => [5,3,4,1,2] => [4,5,2,3,1] => [5,4,2,3,1] => 2
[1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [5,1,2,3,4] => [1,5,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [2,3,4,5,6,1] => [3,2,4,5,6,1] => 0
[1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [2,3,4,6,1,5] => [3,2,4,6,1,5] => 0
[1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [2,3,5,1,6,4] => [3,2,5,1,6,4] => 0
[1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => [2,3,5,6,4,1] => [3,2,5,6,4,1] => 1
[1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [2,3,6,1,4,5] => [3,2,6,1,4,5] => 0
[1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [2,4,1,5,6,3] => [4,2,1,5,6,3] => 0
[1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [2,4,1,6,3,5] => [4,2,1,6,3,5] => 0
[1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => [2,4,5,3,6,1] => [4,2,5,3,6,1] => 1
[1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => [2,4,5,6,3,1] => [4,2,5,6,3,1] => 1
[1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [2,4,6,3,1,5] => [4,2,6,3,1,5] => 1
[1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [2,5,1,3,6,4] => [5,2,1,3,6,4] => 0
[1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [2,5,1,6,4,3] => [5,2,1,6,4,3] => 1
[1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => [2,5,6,3,4,1] => [5,2,6,3,4,1] => 2
[1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [2,6,1,3,4,5] => [6,2,1,3,4,5] => 0
[1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [3,1,4,5,6,2] => [1,3,4,5,6,2] => 0
[1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [3,1,4,6,2,5] => [1,3,4,6,2,5] => 0
[1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [3,1,5,2,6,4] => [1,3,5,2,6,4] => 0
[1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [3,1,5,6,4,2] => [1,3,5,6,4,2] => 1
[1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [3,1,6,2,4,5] => [1,3,6,2,4,5] => 0
[1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => [3,4,2,5,6,1] => [4,3,2,5,6,1] => 1
[1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => [3,4,2,6,1,5] => [4,3,2,6,1,5] => 1
[1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => [3,4,5,2,6,1] => [4,3,5,2,6,1] => 1
[1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => [3,4,5,6,1,2] => [4,3,5,6,1,2] => 0
[1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => [3,4,6,2,1,5] => [4,3,6,2,1,5] => 1
[1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => [3,5,2,1,6,4] => [5,3,2,1,6,4] => 1
[1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => [3,5,2,6,4,1] => [5,3,2,6,4,1] => 2
[1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => [3,5,6,2,4,1] => [5,3,6,2,4,1] => 2
[1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => [3,6,2,1,4,5] => [6,3,2,1,4,5] => 1
[1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [4,1,2,5,6,3] => [1,4,2,5,6,3] => 0
[1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [4,1,2,6,3,5] => [1,4,2,6,3,5] => 0
[1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [4,1,5,3,6,2] => [1,4,5,3,6,2] => 1
[1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [4,1,5,6,3,2] => [1,4,5,6,3,2] => 1
[1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => [4,1,6,3,2,5] => [1,4,6,3,2,5] => 1
[1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => [4,5,2,3,6,1] => [5,4,2,3,6,1] => 2
[1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => [4,5,2,6,3,1] => [5,4,2,6,3,1] => 2
[1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => [4,5,6,3,2,1] => [5,4,6,3,2,1] => 2
[1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => [4,6,2,3,1,5] => [6,4,2,3,1,5] => 2
[1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [5,1,2,3,6,4] => [1,5,2,3,6,4] => 0
[1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => [5,1,2,6,4,3] => [1,5,2,6,4,3] => 1
[1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => [5,1,6,3,4,2] => [1,5,6,3,4,2] => 2
[1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => [5,6,2,3,4,1] => [6,5,2,3,4,1] => 3
[1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [6,1,2,3,4,5] => [1,6,2,3,4,5] => 0
[1,1,0,0,1,0,1,0,1,0,1,0] => [2,7,1,3,4,5,6] => [3,1,4,5,6,7,2] => [1,3,4,5,6,7,2] => 0
[1,1,0,0,1,0,1,0,1,1,0,0] => [2,6,1,3,4,7,5] => [3,1,4,5,7,2,6] => [1,3,4,5,7,2,6] => 0
[1,1,0,0,1,0,1,1,0,0,1,0] => [2,5,1,3,7,4,6] => [3,1,4,6,2,7,5] => [1,3,4,6,2,7,5] => 0
[1,1,0,0,1,0,1,1,0,1,0,0] => [2,7,1,3,6,4,5] => [3,1,4,6,7,5,2] => [1,3,4,6,7,5,2] => 1
[1,1,0,0,1,0,1,1,1,0,0,0] => [2,5,1,3,6,7,4] => [3,1,4,7,2,5,6] => [1,3,4,7,2,5,6] => 0
[1,1,0,0,1,1,0,0,1,0,1,0] => [2,4,1,7,3,5,6] => [3,1,5,2,6,7,4] => [1,3,5,2,6,7,4] => 0
[1,1,0,0,1,1,0,0,1,1,0,0] => [2,4,1,6,3,7,5] => [3,1,5,2,7,4,6] => [1,3,5,2,7,4,6] => 0
[1,1,0,0,1,1,0,1,0,0,1,0] => [2,7,1,5,3,4,6] => [3,1,5,6,4,7,2] => [1,3,5,6,4,7,2] => 1
[1,1,0,0,1,1,0,1,0,1,0,0] => [2,7,1,6,3,4,5] => [3,1,5,6,7,4,2] => [1,3,5,6,7,4,2] => 1
[1,1,0,0,1,1,0,1,1,0,0,0] => [2,6,1,5,3,7,4] => [3,1,5,7,4,2,6] => [1,3,5,7,4,2,6] => 1
[1,1,0,0,1,1,1,0,0,0,1,0] => [2,4,1,5,7,3,6] => [3,1,6,2,4,7,5] => [1,3,6,2,4,7,5] => 0
[1,1,0,0,1,1,1,0,0,1,0,0] => [2,4,1,7,6,3,5] => [3,1,6,2,7,5,4] => [1,3,6,2,7,5,4] => 1
[1,1,0,0,1,1,1,0,1,0,0,0] => [2,7,1,5,6,3,4] => [3,1,6,7,4,5,2] => [1,3,6,7,4,5,2] => 2
[1,1,0,0,1,1,1,1,0,0,0,0] => [2,4,1,5,6,7,3] => [3,1,7,2,4,5,6] => [1,3,7,2,4,5,6] => 0
[1,1,1,0,0,0,1,0,1,0,1,0] => [2,3,7,1,4,5,6] => [4,1,2,5,6,7,3] => [1,4,2,5,6,7,3] => 0
[1,1,1,0,0,0,1,0,1,1,0,0] => [2,3,6,1,4,7,5] => [4,1,2,5,7,3,6] => [1,4,2,5,7,3,6] => 0
[1,1,1,0,0,0,1,1,0,0,1,0] => [2,3,5,1,7,4,6] => [4,1,2,6,3,7,5] => [1,4,2,6,3,7,5] => 0
[1,1,1,0,0,0,1,1,0,1,0,0] => [2,3,7,1,6,4,5] => [4,1,2,6,7,5,3] => [1,4,2,6,7,5,3] => 1
[1,1,1,0,0,0,1,1,1,0,0,0] => [2,3,5,1,6,7,4] => [4,1,2,7,3,5,6] => [1,4,2,7,3,5,6] => 0
[1,1,1,0,0,1,0,0,1,0,1,0] => [2,7,4,1,3,5,6] => [4,1,5,3,6,7,2] => [1,4,5,3,6,7,2] => 1
[1,1,1,0,0,1,0,0,1,1,0,0] => [2,6,4,1,3,7,5] => [4,1,5,3,7,2,6] => [1,4,5,3,7,2,6] => 1
[1,1,1,0,0,1,0,1,0,0,1,0] => [2,7,5,1,3,4,6] => [4,1,5,6,3,7,2] => [1,4,5,6,3,7,2] => 1
[1,1,1,0,0,1,0,1,0,1,0,0] => [2,6,7,1,3,4,5] => [4,1,5,6,7,2,3] => [1,4,5,6,7,2,3] => 0
[1,1,1,0,0,1,0,1,1,0,0,0] => [2,6,5,1,3,7,4] => [4,1,5,7,3,2,6] => [1,4,5,7,3,2,6] => 1
[1,1,1,0,0,1,1,0,0,0,1,0] => [2,5,4,1,7,3,6] => [4,1,6,3,2,7,5] => [1,4,6,3,2,7,5] => 1
[1,1,1,0,0,1,1,0,0,1,0,0] => [2,7,4,1,6,3,5] => [4,1,6,3,7,5,2] => [1,4,6,3,7,5,2] => 2
[] => [1] => [1] => [1] => 0
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Description
The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$.
Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j < j$ and there exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also St000213The number of weak exceedances (also weak excedences) of a permutation. and St000119The number of occurrences of the pattern 321 in a permutation..
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
inverse
Description
Sends a permutation to its inverse.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.