Identifier
Mp00283:
Perfect matchings
—non-nesting-exceedence permutation⟶
Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00239: Permutations —Corteel⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Images
[(1,2)] => [2,1] => [2,1] => [1,2] => [2]
[(1,2),(3,4)] => [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => [2,2]
[(1,3),(2,4)] => [3,4,1,2] => [4,3,2,1] => [1,2,3,4] => [4]
[(1,4),(2,3)] => [3,4,2,1] => [4,3,1,2] => [1,2,4,3] => [3,1]
[(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [2,1,4,3,6,5] => [5,6,3,4,1,2] => [2,2,2]
[(1,3),(2,4),(5,6)] => [3,4,1,2,6,5] => [4,3,2,1,6,5] => [3,4,5,6,1,2] => [4,2]
[(1,4),(2,3),(5,6)] => [3,4,2,1,6,5] => [4,3,1,2,6,5] => [3,4,6,5,1,2] => [3,1,2]
[(1,5),(2,3),(4,6)] => [3,5,2,6,1,4] => [6,3,1,5,4,2] => [1,4,6,2,3,5] => [3,3]
[(1,6),(2,3),(4,5)] => [3,5,2,6,4,1] => [6,3,1,5,2,4] => [1,4,6,2,5,3] => [3,2,1]
[(1,6),(2,4),(3,5)] => [4,5,6,2,3,1] => [6,5,4,2,1,3] => [1,2,3,5,6,4] => [5,1]
[(1,5),(2,4),(3,6)] => [4,5,6,2,1,3] => [6,5,4,2,3,1] => [1,2,3,5,4,6] => [4,2]
[(1,4),(2,5),(3,6)] => [4,5,6,1,2,3] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => [6]
[(1,3),(2,5),(4,6)] => [3,5,1,6,2,4] => [6,3,2,5,4,1] => [1,4,5,2,3,6] => [3,3]
[(1,2),(3,5),(4,6)] => [2,1,5,6,3,4] => [2,1,6,5,4,3] => [5,6,1,2,3,4] => [2,4]
[(1,2),(3,6),(4,5)] => [2,1,5,6,4,3] => [2,1,6,5,3,4] => [5,6,1,2,4,3] => [2,3,1]
[(1,3),(2,6),(4,5)] => [3,5,1,6,4,2] => [6,3,2,5,1,4] => [1,4,5,2,6,3] => [3,2,1]
[(1,4),(2,6),(3,5)] => [4,5,6,1,3,2] => [6,5,4,3,1,2] => [1,2,3,4,6,5] => [5,1]
[(1,5),(2,6),(3,4)] => [4,5,6,3,1,2] => [6,5,4,1,3,2] => [1,2,3,6,4,5] => [4,2]
[(1,6),(2,5),(3,4)] => [4,5,6,3,2,1] => [6,5,4,1,2,3] => [1,2,3,6,5,4] => [4,1,1]
[(1,2),(3,4),(5,6),(7,8)] => [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => [7,8,5,6,3,4,1,2] => [2,2,2,2]
[(1,3),(2,4),(5,6),(7,8)] => [3,4,1,2,6,5,8,7] => [4,3,2,1,6,5,8,7] => [5,6,7,8,3,4,1,2] => [4,2,2]
[(1,4),(2,3),(5,6),(7,8)] => [3,4,2,1,6,5,8,7] => [4,3,1,2,6,5,8,7] => [5,6,8,7,3,4,1,2] => [3,1,2,2]
[(1,5),(2,3),(4,6),(7,8)] => [3,5,2,6,1,4,8,7] => [6,3,1,5,4,2,8,7] => [3,6,8,4,5,7,1,2] => [3,3,2]
[(1,6),(2,3),(4,5),(7,8)] => [3,5,2,6,4,1,8,7] => [6,3,1,5,2,4,8,7] => [3,6,8,4,7,5,1,2] => [3,2,1,2]
[(1,7),(2,3),(4,5),(6,8)] => [3,5,2,7,4,8,1,6] => [8,3,1,5,2,7,6,4] => [1,6,8,4,7,2,3,5] => [3,2,3]
[(1,8),(2,3),(4,5),(6,7)] => [3,5,2,7,4,8,6,1] => [8,3,1,5,2,7,4,6] => [1,6,8,4,7,2,5,3] => [3,2,2,1]
[(1,8),(2,4),(3,5),(6,7)] => [4,5,7,2,3,8,6,1] => [8,5,4,2,1,7,3,6] => [1,4,5,7,8,2,6,3] => [5,2,1]
[(1,7),(2,4),(3,5),(6,8)] => [4,5,7,2,3,8,1,6] => [8,5,4,2,1,7,6,3] => [1,4,5,7,8,2,3,6] => [5,3]
[(1,6),(2,4),(3,5),(7,8)] => [4,5,6,2,3,1,8,7] => [6,5,4,2,1,3,8,7] => [3,4,5,7,8,6,1,2] => [5,1,2]
[(1,5),(2,4),(3,6),(7,8)] => [4,5,6,2,1,3,8,7] => [6,5,4,2,3,1,8,7] => [3,4,5,7,6,8,1,2] => [4,2,2]
[(1,4),(2,5),(3,6),(7,8)] => [4,5,6,1,2,3,8,7] => [6,5,4,3,2,1,8,7] => [3,4,5,6,7,8,1,2] => [6,2]
[(1,3),(2,5),(4,6),(7,8)] => [3,5,1,6,2,4,8,7] => [6,3,2,5,4,1,8,7] => [3,6,7,4,5,8,1,2] => [3,3,2]
[(1,2),(3,5),(4,6),(7,8)] => [2,1,5,6,3,4,8,7] => [2,1,6,5,4,3,8,7] => [7,8,3,4,5,6,1,2] => [2,4,2]
[(1,2),(3,6),(4,5),(7,8)] => [2,1,5,6,4,3,8,7] => [2,1,6,5,3,4,8,7] => [7,8,3,4,6,5,1,2] => [2,3,1,2]
[(1,3),(2,6),(4,5),(7,8)] => [3,5,1,6,4,2,8,7] => [6,3,2,5,1,4,8,7] => [3,6,7,4,8,5,1,2] => [3,2,1,2]
[(1,4),(2,6),(3,5),(7,8)] => [4,5,6,1,3,2,8,7] => [6,5,4,3,1,2,8,7] => [3,4,5,6,8,7,1,2] => [5,1,2]
[(1,5),(2,6),(3,4),(7,8)] => [4,5,6,3,1,2,8,7] => [6,5,4,1,3,2,8,7] => [3,4,5,8,6,7,1,2] => [4,2,2]
[(1,6),(2,5),(3,4),(7,8)] => [4,5,6,3,2,1,8,7] => [6,5,4,1,2,3,8,7] => [3,4,5,8,7,6,1,2] => [4,1,1,2]
[(1,7),(2,5),(3,4),(6,8)] => [4,5,7,3,2,8,1,6] => [8,5,4,1,2,7,6,3] => [1,4,5,8,7,2,3,6] => [4,1,3]
[(1,8),(2,5),(3,4),(6,7)] => [4,5,7,3,2,8,6,1] => [8,5,4,1,2,7,3,6] => [1,4,5,8,7,2,6,3] => [4,1,2,1]
[(1,8),(2,6),(3,4),(5,7)] => [4,6,7,3,8,2,5,1] => [8,7,4,1,6,3,2,5] => [1,2,5,8,3,6,7,4] => [4,3,1]
[(1,7),(2,6),(3,4),(5,8)] => [4,6,7,3,8,2,1,5] => [8,7,4,1,6,3,5,2] => [1,2,5,8,3,6,4,7] => [4,2,2]
[(1,6),(2,7),(3,4),(5,8)] => [4,6,7,3,8,1,2,5] => [8,7,4,1,6,5,3,2] => [1,2,5,8,3,4,6,7] => [4,4]
[(1,5),(2,7),(3,4),(6,8)] => [4,5,7,3,1,8,2,6] => [8,5,4,1,3,7,6,2] => [1,4,5,8,6,2,3,7] => [4,1,3]
[(1,4),(2,7),(3,5),(6,8)] => [4,5,7,1,3,8,2,6] => [8,5,4,3,1,7,6,2] => [1,4,5,6,8,2,3,7] => [5,3]
[(1,3),(2,7),(4,5),(6,8)] => [3,5,1,7,4,8,2,6] => [8,3,2,5,1,7,6,4] => [1,6,7,4,8,2,3,5] => [3,2,3]
[(1,2),(3,7),(4,5),(6,8)] => [2,1,5,7,4,8,3,6] => [2,1,8,5,3,7,6,4] => [7,8,1,4,6,2,3,5] => [2,3,3]
[(1,2),(3,8),(4,5),(6,7)] => [2,1,5,7,4,8,6,3] => [2,1,8,5,3,7,4,6] => [7,8,1,4,6,2,5,3] => [2,3,2,1]
[(1,3),(2,8),(4,5),(6,7)] => [3,5,1,7,4,8,6,2] => [8,3,2,5,1,7,4,6] => [1,6,7,4,8,2,5,3] => [3,2,2,1]
[(1,4),(2,8),(3,5),(6,7)] => [4,5,7,1,3,8,6,2] => [8,5,4,3,1,7,2,6] => [1,4,5,6,8,2,7,3] => [5,2,1]
[(1,5),(2,8),(3,4),(6,7)] => [4,5,7,3,1,8,6,2] => [8,5,4,1,3,7,2,6] => [1,4,5,8,6,2,7,3] => [4,1,2,1]
[(1,6),(2,8),(3,4),(5,7)] => [4,6,7,3,8,1,5,2] => [8,7,4,1,6,5,2,3] => [1,2,5,8,3,4,7,6] => [4,3,1]
[(1,7),(2,8),(3,4),(5,6)] => [4,6,7,3,8,5,1,2] => [8,7,4,1,6,2,5,3] => [1,2,5,8,3,7,4,6] => [4,2,2]
[(1,8),(2,7),(3,4),(5,6)] => [4,6,7,3,8,5,2,1] => [8,7,4,1,6,2,3,5] => [1,2,5,8,3,7,6,4] => [4,2,1,1]
[(1,8),(2,7),(3,5),(4,6)] => [5,6,7,8,3,4,2,1] => [8,7,6,5,2,1,3,4] => [1,2,3,4,7,8,6,5] => [6,1,1]
[(1,7),(2,8),(3,5),(4,6)] => [5,6,7,8,3,4,1,2] => [8,7,6,5,2,1,4,3] => [1,2,3,4,7,8,5,6] => [6,2]
[(1,6),(2,8),(3,5),(4,7)] => [5,6,7,8,3,1,4,2] => [8,7,6,5,2,4,1,3] => [1,2,3,4,7,5,8,6] => [5,2,1]
[(1,5),(2,8),(3,6),(4,7)] => [5,6,7,8,1,3,4,2] => [8,7,6,5,4,2,1,3] => [1,2,3,4,5,7,8,6] => [7,1]
[(1,4),(2,8),(3,6),(5,7)] => [4,6,7,1,8,3,5,2] => [8,7,4,3,6,2,1,5] => [1,2,5,6,3,7,8,4] => [4,3,1]
[(1,3),(2,8),(4,6),(5,7)] => [3,6,1,7,8,4,5,2] => [8,3,2,7,6,4,1,5] => [1,6,7,2,3,5,8,4] => [3,4,1]
[(1,2),(3,8),(4,6),(5,7)] => [2,1,6,7,8,4,5,3] => [2,1,8,7,6,4,3,5] => [7,8,1,2,3,5,6,4] => [2,5,1]
[(1,2),(3,7),(4,6),(5,8)] => [2,1,6,7,8,4,3,5] => [2,1,8,7,6,4,5,3] => [7,8,1,2,3,5,4,6] => [2,4,2]
[(1,3),(2,7),(4,6),(5,8)] => [3,6,1,7,8,4,2,5] => [8,3,2,7,6,4,5,1] => [1,6,7,2,3,5,4,8] => [3,3,2]
[(1,4),(2,7),(3,6),(5,8)] => [4,6,7,1,8,3,2,5] => [8,7,4,3,6,2,5,1] => [1,2,5,6,3,7,4,8] => [4,2,2]
[(1,5),(2,7),(3,6),(4,8)] => [5,6,7,8,1,3,2,4] => [8,7,6,5,4,2,3,1] => [1,2,3,4,5,7,6,8] => [6,2]
[(1,6),(2,7),(3,5),(4,8)] => [5,6,7,8,3,1,2,4] => [8,7,6,5,2,4,3,1] => [1,2,3,4,7,5,6,8] => [5,3]
[(1,7),(2,6),(3,5),(4,8)] => [5,6,7,8,3,2,1,4] => [8,7,6,5,2,3,4,1] => [1,2,3,4,7,6,5,8] => [5,1,2]
[(1,8),(2,6),(3,5),(4,7)] => [5,6,7,8,3,2,4,1] => [8,7,6,5,2,3,1,4] => [1,2,3,4,7,6,8,5] => [5,2,1]
[(1,8),(2,5),(3,6),(4,7)] => [5,6,7,8,2,3,4,1] => [8,7,6,5,3,2,1,4] => [1,2,3,4,6,7,8,5] => [7,1]
[(1,7),(2,5),(3,6),(4,8)] => [5,6,7,8,2,3,1,4] => [8,7,6,5,3,2,4,1] => [1,2,3,4,6,7,5,8] => [6,2]
[(1,6),(2,5),(3,7),(4,8)] => [5,6,7,8,2,1,3,4] => [8,7,6,5,3,4,2,1] => [1,2,3,4,6,5,7,8] => [5,3]
[(1,5),(2,6),(3,7),(4,8)] => [5,6,7,8,1,2,3,4] => [8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => [8]
[(1,4),(2,6),(3,7),(5,8)] => [4,6,7,1,8,2,3,5] => [8,7,4,3,6,5,2,1] => [1,2,5,6,3,4,7,8] => [4,4]
[(1,3),(2,6),(4,7),(5,8)] => [3,6,1,7,8,2,4,5] => [8,3,2,7,6,5,4,1] => [1,6,7,2,3,4,5,8] => [3,5]
[(1,2),(3,6),(4,7),(5,8)] => [2,1,6,7,8,3,4,5] => [2,1,8,7,6,5,4,3] => [7,8,1,2,3,4,5,6] => [2,6]
[(1,2),(3,5),(4,7),(6,8)] => [2,1,5,7,3,8,4,6] => [2,1,8,5,4,7,6,3] => [7,8,1,4,5,2,3,6] => [2,3,3]
[(1,3),(2,5),(4,7),(6,8)] => [3,5,1,7,2,8,4,6] => [8,3,2,5,4,7,6,1] => [1,6,7,4,5,2,3,8] => [3,2,3]
[(1,4),(2,5),(3,7),(6,8)] => [4,5,7,1,2,8,3,6] => [8,5,4,3,2,7,6,1] => [1,4,5,6,7,2,3,8] => [5,3]
[(1,5),(2,4),(3,7),(6,8)] => [4,5,7,2,1,8,3,6] => [8,5,4,2,3,7,6,1] => [1,4,5,7,6,2,3,8] => [4,1,3]
[(1,6),(2,4),(3,7),(5,8)] => [4,6,7,2,8,1,3,5] => [8,7,4,2,6,5,3,1] => [1,2,5,7,3,4,6,8] => [4,4]
[(1,7),(2,4),(3,6),(5,8)] => [4,6,7,2,8,3,1,5] => [8,7,4,2,6,3,5,1] => [1,2,5,7,3,6,4,8] => [4,2,2]
[(1,8),(2,4),(3,6),(5,7)] => [4,6,7,2,8,3,5,1] => [8,7,4,2,6,3,1,5] => [1,2,5,7,3,6,8,4] => [4,3,1]
[(1,8),(2,3),(4,6),(5,7)] => [3,6,2,7,8,4,5,1] => [8,3,1,7,6,4,2,5] => [1,6,8,2,3,5,7,4] => [3,4,1]
[(1,7),(2,3),(4,6),(5,8)] => [3,6,2,7,8,4,1,5] => [8,3,1,7,6,4,5,2] => [1,6,8,2,3,5,4,7] => [3,3,2]
[(1,6),(2,3),(4,7),(5,8)] => [3,6,2,7,8,1,4,5] => [8,3,1,7,6,5,4,2] => [1,6,8,2,3,4,5,7] => [3,5]
[(1,5),(2,3),(4,7),(6,8)] => [3,5,2,7,1,8,4,6] => [8,3,1,5,4,7,6,2] => [1,6,8,4,5,2,3,7] => [3,2,3]
[(1,4),(2,3),(5,7),(6,8)] => [3,4,2,1,7,8,5,6] => [4,3,1,2,8,7,6,5] => [5,6,8,7,1,2,3,4] => [3,1,4]
[(1,3),(2,4),(5,7),(6,8)] => [3,4,1,2,7,8,5,6] => [4,3,2,1,8,7,6,5] => [5,6,7,8,1,2,3,4] => [4,4]
[(1,2),(3,4),(5,7),(6,8)] => [2,1,4,3,7,8,5,6] => [2,1,4,3,8,7,6,5] => [7,8,5,6,1,2,3,4] => [2,2,4]
[(1,2),(3,4),(5,8),(6,7)] => [2,1,4,3,7,8,6,5] => [2,1,4,3,8,7,5,6] => [7,8,5,6,1,2,4,3] => [2,2,3,1]
[(1,3),(2,4),(5,8),(6,7)] => [3,4,1,2,7,8,6,5] => [4,3,2,1,8,7,5,6] => [5,6,7,8,1,2,4,3] => [4,3,1]
[(1,4),(2,3),(5,8),(6,7)] => [3,4,2,1,7,8,6,5] => [4,3,1,2,8,7,5,6] => [5,6,8,7,1,2,4,3] => [3,1,3,1]
[(1,5),(2,3),(4,8),(6,7)] => [3,5,2,7,1,8,6,4] => [8,3,1,5,4,7,2,6] => [1,6,8,4,5,2,7,3] => [3,2,2,1]
[(1,6),(2,3),(4,8),(5,7)] => [3,6,2,7,8,1,5,4] => [8,3,1,7,6,5,2,4] => [1,6,8,2,3,4,7,5] => [3,4,1]
[(1,7),(2,3),(4,8),(5,6)] => [3,6,2,7,8,5,1,4] => [8,3,1,7,6,2,5,4] => [1,6,8,2,3,7,4,5] => [3,3,2]
[(1,8),(2,3),(4,7),(5,6)] => [3,6,2,7,8,5,4,1] => [8,3,1,7,6,2,4,5] => [1,6,8,2,3,7,5,4] => [3,3,1,1]
[(1,8),(2,4),(3,7),(5,6)] => [4,6,7,2,8,5,3,1] => [8,7,4,2,6,1,3,5] => [1,2,5,7,3,8,6,4] => [4,2,1,1]
[(1,7),(2,4),(3,8),(5,6)] => [4,6,7,2,8,5,1,3] => [8,7,4,2,6,1,5,3] => [1,2,5,7,3,8,4,6] => [4,2,2]
[(1,6),(2,4),(3,8),(5,7)] => [4,6,7,2,8,1,5,3] => [8,7,4,2,6,5,1,3] => [1,2,5,7,3,4,8,6] => [4,3,1]
[(1,5),(2,4),(3,8),(6,7)] => [4,5,7,2,1,8,6,3] => [8,5,4,2,3,7,1,6] => [1,4,5,7,6,2,8,3] => [4,1,2,1]
[(1,4),(2,5),(3,8),(6,7)] => [4,5,7,1,2,8,6,3] => [8,5,4,3,2,7,1,6] => [1,4,5,6,7,2,8,3] => [5,2,1]
>>> Load all 311 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
Corteel
Description
Corteel's map interchanging the number of crossings and the number of nestings of a permutation.
This involution creates a labelled bicoloured Motzkin path, using the Foata-Zeilberger map. In the corresponding bump diagram, each label records the number of arcs nesting the given arc. Then each label is replaced by its complement, and the inverse of the Foata-Zeilberger map is applied.
This involution creates a labelled bicoloured Motzkin path, using the Foata-Zeilberger map. In the corresponding bump diagram, each label records the number of arcs nesting the given arc. Then each label is replaced by its complement, and the inverse of the Foata-Zeilberger map is applied.
Map
complement
Description
Sents a permutation to its complement.
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$
Map
descent composition
Description
The descent composition of a permutation.
The descent composition of a permutation $\pi$ of length $n$ is the integer composition of $n$ whose descent set equals the descent set of $\pi$. The descent set of a permutation $\pi$ is $\{i \mid 1 \leq i < n, \pi(i) > \pi(i+1)\}$. The descent set of a composition $c = (i_1, i_2, \ldots, i_k)$ is the set $\{ i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$.
The descent composition of a permutation $\pi$ of length $n$ is the integer composition of $n$ whose descent set equals the descent set of $\pi$. The descent set of a permutation $\pi$ is $\{i \mid 1 \leq i < n, \pi(i) > \pi(i+1)\}$. The descent set of a composition $c = (i_1, i_2, \ldots, i_k)$ is the set $\{ i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$.
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