Identifier
Mp00283:
Perfect matchings
—non-nesting-exceedence permutation⟶
Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
Images
[(1,2)] => [2,1] => [2,1] => 0
[(1,2),(3,4)] => [2,1,4,3] => [3,2,4,1] => 000
[(1,3),(2,4)] => [3,4,1,2] => [1,4,2,3] => 100
[(1,4),(2,3)] => [3,4,2,1] => [4,3,1,2] => 000
[(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [4,3,5,2,6,1] => 00000
[(1,3),(2,4),(5,6)] => [3,4,1,2,6,5] => [2,5,3,4,6,1] => 00000
[(1,4),(2,3),(5,6)] => [3,4,2,1,6,5] => [5,4,2,3,6,1] => 00000
[(1,5),(2,3),(4,6)] => [3,5,2,6,1,4] => [1,5,3,6,2,4] => 10000
[(1,6),(2,3),(4,5)] => [3,5,2,6,4,1] => [6,4,2,5,1,3] => 00000
[(1,6),(2,4),(3,5)] => [4,5,6,2,3,1] => [6,1,5,2,3,4] => 00000
[(1,5),(2,4),(3,6)] => [4,5,6,2,1,3] => [5,1,6,2,3,4] => 00000
[(1,4),(2,5),(3,6)] => [4,5,6,1,2,3] => [1,2,6,3,4,5] => 11000
[(1,3),(2,5),(4,6)] => [3,5,1,6,2,4] => [4,1,3,6,2,5] => 00000
[(1,2),(3,5),(4,6)] => [2,1,5,6,3,4] => [3,2,1,6,4,5] => 00100
[(1,2),(3,6),(4,5)] => [2,1,5,6,4,3] => [4,3,6,5,1,2] => 00000
[(1,3),(2,6),(4,5)] => [3,5,1,6,4,2] => [5,6,2,4,1,3] => 00000
[(1,4),(2,6),(3,5)] => [4,5,6,1,3,2] => [2,6,5,1,3,4] => 00000
[(1,5),(2,6),(3,4)] => [4,5,6,3,1,2] => [1,6,5,2,3,4] => 10000
[(1,6),(2,5),(3,4)] => [4,5,6,3,2,1] => [6,5,4,1,2,3] => 00000
[(1,2),(3,4),(5,6),(7,8)] => [2,1,4,3,6,5,8,7] => [5,4,6,3,7,2,8,1] => 0000000
[(1,3),(2,4),(5,6),(7,8)] => [3,4,1,2,6,5,8,7] => [3,6,4,5,7,2,8,1] => 0000000
[(1,4),(2,3),(5,6),(7,8)] => [3,4,2,1,6,5,8,7] => [6,5,3,4,7,2,8,1] => 0000000
[(1,5),(2,3),(4,6),(7,8)] => [3,5,2,6,1,4,8,7] => [2,6,4,7,3,5,8,1] => 0000000
[(1,6),(2,3),(4,5),(7,8)] => [3,5,2,6,4,1,8,7] => [7,5,3,6,2,4,8,1] => 0000000
[(1,7),(2,3),(4,5),(6,8)] => [3,5,2,7,4,8,1,6] => [1,6,4,7,3,8,2,5] => 1000000
[(1,8),(2,3),(4,5),(6,7)] => [3,5,2,7,4,8,6,1] => [8,5,3,6,2,7,1,4] => 0000000
[(1,8),(2,4),(3,5),(6,7)] => [4,5,7,2,3,8,6,1] => [8,2,6,3,4,7,1,5] => 0000000
[(1,7),(2,4),(3,5),(6,8)] => [4,5,7,2,3,8,1,6] => [1,3,7,4,5,8,2,6] => 1000000
[(1,6),(2,4),(3,5),(7,8)] => [4,5,6,2,3,1,8,7] => [7,2,6,3,4,5,8,1] => 0000000
[(1,5),(2,4),(3,6),(7,8)] => [4,5,6,2,1,3,8,7] => [6,2,7,3,4,5,8,1] => 0000000
[(1,4),(2,5),(3,6),(7,8)] => [4,5,6,1,2,3,8,7] => [2,3,7,4,5,6,8,1] => 0000000
[(1,3),(2,5),(4,6),(7,8)] => [3,5,1,6,2,4,8,7] => [5,2,4,7,3,6,8,1] => 0000000
[(1,2),(3,5),(4,6),(7,8)] => [2,1,5,6,3,4,8,7] => [4,3,2,7,5,6,8,1] => 0000000
[(1,2),(3,6),(4,5),(7,8)] => [2,1,5,6,4,3,8,7] => [5,4,7,6,2,3,8,1] => 0000000
[(1,3),(2,6),(4,5),(7,8)] => [3,5,1,6,4,2,8,7] => [6,7,3,5,2,4,8,1] => 0000000
[(1,4),(2,6),(3,5),(7,8)] => [4,5,6,1,3,2,8,7] => [3,7,6,2,4,5,8,1] => 0000000
[(1,5),(2,6),(3,4),(7,8)] => [4,5,6,3,1,2,8,7] => [2,7,6,3,4,5,8,1] => 0000000
[(1,6),(2,5),(3,4),(7,8)] => [4,5,6,3,2,1,8,7] => [7,6,5,2,3,4,8,1] => 0000000
[(1,7),(2,5),(3,4),(6,8)] => [4,5,7,3,2,8,1,6] => [1,7,6,3,4,8,2,5] => 1000000
[(1,8),(2,5),(3,4),(6,7)] => [4,5,7,3,2,8,6,1] => [8,6,5,2,3,7,1,4] => 0000000
[(1,8),(2,6),(3,4),(5,7)] => [4,6,7,3,8,2,5,1] => [8,1,6,3,7,2,4,5] => 0000000
[(1,7),(2,6),(3,4),(5,8)] => [4,6,7,3,8,2,1,5] => [7,1,6,3,8,2,4,5] => 0000000
[(1,6),(2,7),(3,4),(5,8)] => [4,6,7,3,8,1,2,5] => [1,2,7,4,8,3,5,6] => 1100000
[(1,5),(2,7),(3,4),(6,8)] => [4,5,7,3,1,8,2,6] => [6,1,7,3,4,8,2,5] => 0000000
[(1,4),(2,7),(3,5),(6,8)] => [4,5,7,1,3,8,2,6] => [2,1,7,4,5,8,3,6] => 0100000
[(1,3),(2,7),(4,5),(6,8)] => [3,5,1,7,4,8,2,6] => [5,1,4,7,3,8,2,6] => 0000000
[(1,2),(3,7),(4,5),(6,8)] => [2,1,5,7,4,8,3,6] => [4,3,1,7,5,8,2,6] => 0000000
[(1,2),(3,8),(4,5),(6,7)] => [2,1,5,7,4,8,6,3] => [5,4,8,6,2,7,1,3] => 0000000
[(1,3),(2,8),(4,5),(6,7)] => [3,5,1,7,4,8,6,2] => [6,8,3,5,2,7,1,4] => 0000000
[(1,4),(2,8),(3,5),(6,7)] => [4,5,7,1,3,8,6,2] => [3,8,6,2,4,7,1,5] => 0000000
[(1,5),(2,8),(3,4),(6,7)] => [4,5,7,3,1,8,6,2] => [7,8,5,2,3,6,1,4] => 0000000
[(1,6),(2,8),(3,4),(5,7)] => [4,6,7,3,8,1,5,2] => [2,8,6,3,7,1,4,5] => 0000000
[(1,7),(2,8),(3,4),(5,6)] => [4,6,7,3,8,5,1,2] => [1,8,6,3,7,2,4,5] => 1000000
[(1,8),(2,7),(3,4),(5,6)] => [4,6,7,3,8,5,2,1] => [8,7,5,2,6,1,3,4] => 0000000
[(1,8),(2,7),(3,5),(4,6)] => [5,6,7,8,3,4,2,1] => [8,7,1,6,2,3,4,5] => 0000000
[(1,7),(2,8),(3,5),(4,6)] => [5,6,7,8,3,4,1,2] => [1,8,2,7,3,4,5,6] => 1000000
[(1,6),(2,8),(3,5),(4,7)] => [5,6,7,8,3,1,4,2] => [7,8,1,6,2,3,4,5] => 0000000
[(1,5),(2,8),(3,6),(4,7)] => [5,6,7,8,1,3,4,2] => [2,8,1,7,3,4,5,6] => 0000000
[(1,4),(2,8),(3,6),(5,7)] => [4,6,7,1,8,3,5,2] => [6,8,1,3,7,2,4,5] => 0000000
[(1,3),(2,8),(4,6),(5,7)] => [3,6,1,7,8,4,5,2] => [5,8,2,1,7,3,4,6] => 0000000
[(1,2),(3,8),(4,6),(5,7)] => [2,1,6,7,8,4,5,3] => [4,3,8,1,7,2,5,6] => 0000000
[(1,2),(3,7),(4,6),(5,8)] => [2,1,6,7,8,4,3,5] => [4,3,7,1,8,2,5,6] => 0000000
[(1,3),(2,7),(4,6),(5,8)] => [3,6,1,7,8,4,2,5] => [5,7,2,1,8,3,4,6] => 0000000
[(1,4),(2,7),(3,6),(5,8)] => [4,6,7,1,8,3,2,5] => [6,7,1,3,8,2,4,5] => 0000000
[(1,5),(2,7),(3,6),(4,8)] => [5,6,7,8,1,3,2,4] => [2,7,1,8,3,4,5,6] => 0000000
[(1,6),(2,7),(3,5),(4,8)] => [5,6,7,8,3,1,2,4] => [1,7,2,8,3,4,5,6] => 1000000
[(1,7),(2,6),(3,5),(4,8)] => [5,6,7,8,3,2,1,4] => [7,6,1,8,2,3,4,5] => 0000000
[(1,8),(2,6),(3,5),(4,7)] => [5,6,7,8,3,2,4,1] => [8,6,1,7,2,3,4,5] => 0000000
[(1,8),(2,5),(3,6),(4,7)] => [5,6,7,8,2,3,4,1] => [8,1,2,7,3,4,5,6] => 0000000
[(1,7),(2,5),(3,6),(4,8)] => [5,6,7,8,2,3,1,4] => [7,1,2,8,3,4,5,6] => 0000000
[(1,6),(2,5),(3,7),(4,8)] => [5,6,7,8,2,1,3,4] => [6,1,2,8,3,4,5,7] => 0000000
[(1,5),(2,6),(3,7),(4,8)] => [5,6,7,8,1,2,3,4] => [1,2,3,8,4,5,6,7] => 1110000
[(1,4),(2,6),(3,7),(5,8)] => [4,6,7,1,8,2,3,5] => [5,1,2,4,8,3,6,7] => 0000000
[(1,3),(2,6),(4,7),(5,8)] => [3,6,1,7,8,2,4,5] => [4,1,3,2,8,5,6,7] => 0001000
[(1,2),(3,6),(4,7),(5,8)] => [2,1,6,7,8,3,4,5] => [3,2,1,4,8,5,6,7] => 0011000
[(1,2),(3,5),(4,7),(6,8)] => [2,1,5,7,3,8,4,6] => [4,3,6,1,5,8,2,7] => 0000000
[(1,3),(2,5),(4,7),(6,8)] => [3,5,1,7,2,8,4,6] => [5,6,2,1,4,8,3,7] => 0000000
[(1,4),(2,5),(3,7),(6,8)] => [4,5,7,1,2,8,3,6] => [2,6,1,4,5,8,3,7] => 0000000
[(1,5),(2,4),(3,7),(6,8)] => [4,5,7,2,1,8,3,6] => [6,5,1,3,4,8,2,7] => 0000000
[(1,6),(2,4),(3,7),(5,8)] => [4,6,7,2,8,1,3,5] => [1,6,2,4,8,3,5,7] => 1000000
[(1,7),(2,4),(3,6),(5,8)] => [4,6,7,2,8,3,1,5] => [7,5,1,3,8,2,4,6] => 0000000
[(1,8),(2,4),(3,6),(5,7)] => [4,6,7,2,8,3,5,1] => [8,5,1,3,7,2,4,6] => 0000000
[(1,8),(2,3),(4,6),(5,7)] => [3,6,2,7,8,4,5,1] => [8,4,2,1,7,3,5,6] => 0000000
[(1,7),(2,3),(4,6),(5,8)] => [3,6,2,7,8,4,1,5] => [7,4,2,1,8,3,5,6] => 0000000
[(1,6),(2,3),(4,7),(5,8)] => [3,6,2,7,8,1,4,5] => [1,5,3,2,8,4,6,7] => 1000000
[(1,5),(2,3),(4,7),(6,8)] => [3,5,2,7,1,8,4,6] => [6,4,2,1,5,8,3,7] => 0000000
[(1,4),(2,3),(5,7),(6,8)] => [3,4,2,1,7,8,5,6] => [5,4,2,3,1,8,6,7] => 0000100
[(1,3),(2,4),(5,7),(6,8)] => [3,4,1,2,7,8,5,6] => [2,5,3,4,1,8,6,7] => 0000100
[(1,2),(3,4),(5,7),(6,8)] => [2,1,4,3,7,8,5,6] => [4,3,5,2,1,8,6,7] => 0000100
[(1,2),(3,4),(5,8),(6,7)] => [2,1,4,3,7,8,6,5] => [5,4,6,3,8,7,1,2] => 0000000
[(1,3),(2,4),(5,8),(6,7)] => [3,4,1,2,7,8,6,5] => [3,6,4,5,8,7,1,2] => 0000000
[(1,4),(2,3),(5,8),(6,7)] => [3,4,2,1,7,8,6,5] => [6,5,3,4,8,7,1,2] => 0000000
[(1,5),(2,3),(4,8),(6,7)] => [3,5,2,7,1,8,6,4] => [7,5,3,8,2,6,1,4] => 0000000
[(1,6),(2,3),(4,8),(5,7)] => [3,6,2,7,8,1,5,4] => [2,6,4,8,7,1,3,5] => 0000000
[(1,7),(2,3),(4,8),(5,6)] => [3,6,2,7,8,5,1,4] => [1,6,4,8,7,2,3,5] => 1000000
[(1,8),(2,3),(4,7),(5,6)] => [3,6,2,7,8,5,4,1] => [8,5,3,7,6,1,2,4] => 0000000
[(1,8),(2,4),(3,7),(5,6)] => [4,6,7,2,8,5,3,1] => [8,6,7,2,5,1,3,4] => 0000000
[(1,7),(2,4),(3,8),(5,6)] => [4,6,7,2,8,5,1,3] => [1,7,8,3,6,2,4,5] => 1000000
[(1,6),(2,4),(3,8),(5,7)] => [4,6,7,2,8,1,5,3] => [2,7,8,3,6,1,4,5] => 0000000
[(1,5),(2,4),(3,8),(6,7)] => [4,5,7,2,1,8,6,3] => [7,6,8,2,3,5,1,4] => 0000000
[(1,4),(2,5),(3,8),(6,7)] => [4,5,7,1,2,8,6,3] => [3,7,8,2,4,6,1,5] => 0000000
>>> Load all 126 entries. <<<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
major-index to inversion-number bijection
Description
Return the permutation whose Lehmer code equals the major code of the preimage.
This map sends the major index to the number of inversions.
This map sends the major index to the number of inversions.
Map
connectivity set
Description
The connectivity set of a permutation as a binary word.
According to [2], also known as the global ascent set.
The connectivity set is
C(π)={i∈[n−1]|∀1≤j≤i<k≤n:π(j)<π(k)}.
For n>1 it can also be described as the set of occurrences of the mesh pattern
([1,2],{(0,2),(1,0),(1,1),(2,0),(2,1)})
or equivalently
([1,2],{(0,1),(0,2),(1,1),(1,2),(2,0)}),
see [3].
The permutation is connected, when the connectivity set is empty.
According to [2], also known as the global ascent set.
The connectivity set is
C(π)={i∈[n−1]|∀1≤j≤i<k≤n:π(j)<π(k)}.
For n>1 it can also be described as the set of occurrences of the mesh pattern
([1,2],{(0,2),(1,0),(1,1),(2,0),(2,1)})
or equivalently
([1,2],{(0,1),(0,2),(1,1),(1,2),(2,0)}),
see [3].
The permutation is connected, when the connectivity set is empty.
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