Identifier
-
Mp00283:
Perfect matchings
—non-nesting-exceedence permutation⟶
Permutations
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
St000648: Permutations ⟶ ℤ
Values
[(1,2)] => [2,1] => [2,1] => 0
[(1,2),(3,4)] => [2,1,4,3] => [2,1,4,3] => 0
[(1,3),(2,4)] => [3,4,1,2] => [4,3,1,2] => 0
[(1,4),(2,3)] => [3,4,2,1] => [4,3,2,1] => 0
[(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [2,1,4,3,6,5] => 0
[(1,3),(2,4),(5,6)] => [3,4,1,2,6,5] => [4,3,1,2,6,5] => 0
[(1,4),(2,3),(5,6)] => [3,4,2,1,6,5] => [4,3,2,1,6,5] => 0
[(1,5),(2,3),(4,6)] => [3,5,2,6,1,4] => [5,3,2,6,1,4] => 1
[(1,6),(2,3),(4,5)] => [3,5,2,6,4,1] => [5,3,2,6,4,1] => 1
[(1,6),(2,4),(3,5)] => [4,5,6,2,3,1] => [5,4,6,2,3,1] => 1
[(1,5),(2,4),(3,6)] => [4,5,6,2,1,3] => [5,4,6,2,1,3] => 1
[(1,4),(2,5),(3,6)] => [4,5,6,1,2,3] => [5,4,6,1,2,3] => 1
[(1,3),(2,5),(4,6)] => [3,5,1,6,2,4] => [5,3,1,6,2,4] => 1
[(1,2),(3,5),(4,6)] => [2,1,5,6,3,4] => [2,1,6,5,3,4] => 0
[(1,2),(3,6),(4,5)] => [2,1,5,6,4,3] => [2,1,6,5,4,3] => 0
[(1,3),(2,6),(4,5)] => [3,5,1,6,4,2] => [5,3,1,6,4,2] => 1
[(1,4),(2,6),(3,5)] => [4,5,6,1,3,2] => [5,4,6,1,3,2] => 1
[(1,5),(2,6),(3,4)] => [4,5,6,3,1,2] => [5,4,6,3,1,2] => 1
[(1,6),(2,5),(3,4)] => [4,5,6,3,2,1] => [5,4,6,3,2,1] => 1
[(1,2),(3,4),(5,6),(7,8)] => [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => 0
[(1,4),(2,3),(5,6),(7,8)] => [3,4,2,1,6,5,8,7] => [4,3,2,1,6,5,8,7] => 0
[(1,5),(2,3),(4,6),(7,8)] => [3,5,2,6,1,4,8,7] => [5,3,2,6,1,4,8,7] => 1
[(1,5),(2,4),(3,6),(7,8)] => [4,5,6,2,1,3,8,7] => [5,4,6,2,1,3,8,7] => 1
[(1,2),(3,6),(4,5),(7,8)] => [2,1,5,6,4,3,8,7] => [2,1,6,5,4,3,8,7] => 0
[(1,5),(2,7),(3,4),(6,8)] => [4,5,7,3,1,8,2,6] => [5,4,7,3,1,8,2,6] => 2
[(1,4),(2,7),(3,5),(6,8)] => [4,5,7,1,3,8,2,6] => [5,4,7,1,3,8,2,6] => 2
[(1,2),(3,7),(4,5),(6,8)] => [2,1,5,7,4,8,3,6] => [2,1,7,5,4,8,3,6] => 1
[(1,2),(3,7),(4,6),(5,8)] => [2,1,6,7,8,4,3,5] => [2,1,7,6,8,4,3,5] => 1
[(1,6),(2,5),(3,7),(4,8)] => [5,6,7,8,2,1,3,4] => [6,5,7,8,2,1,3,4] => 0
[(1,5),(2,4),(3,7),(6,8)] => [4,5,7,2,1,8,3,6] => [5,4,7,2,1,8,3,6] => 2
[(1,6),(2,4),(3,7),(5,8)] => [4,6,7,2,8,1,3,5] => [6,4,7,2,8,1,3,5] => 1
[(1,6),(2,3),(4,7),(5,8)] => [3,6,2,7,8,1,4,5] => [6,3,2,7,8,1,4,5] => 0
[(1,5),(2,3),(4,7),(6,8)] => [3,5,2,7,1,8,4,6] => [5,3,2,7,1,8,4,6] => 1
[(1,4),(2,3),(5,7),(6,8)] => [3,4,2,1,7,8,5,6] => [4,3,2,1,7,8,5,6] => 2
[(1,2),(3,4),(5,8),(6,7)] => [2,1,4,3,7,8,6,5] => [2,1,4,3,8,7,6,5] => 0
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Description
The number of 2-excedences of a permutation.
This is the number of positions $1\leq i\leq n$ such that $\sigma(i)=i+2$.
This is the number of positions $1\leq i\leq n$ such that $\sigma(i)=i+2$.
Map
non-nesting-exceedence permutation
Description
The fixed-point-free permutation with deficiencies given by the perfect matching, no alignments and no inversions between exceedences.
Put differently, the exceedences form the unique non-nesting perfect matching whose openers coincide with those of the given perfect matching.
Put differently, the exceedences form the unique non-nesting perfect matching whose openers coincide with those of the given perfect matching.
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].
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].
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