Identifier
Mp00283:
Perfect matchings
—non-nesting-exceedence permutation⟶
Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00064: Permutations —reverse⟶ 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] => [3,1,4,2] => [2,4,1,3] => [2,2]
[(1,4),(2,3)] => [3,4,2,1] => [4,1,3,2] => [2,3,1,4] => [2,2]
[(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] => [3,1,4,2,6,5] => [5,6,2,4,1,3] => [2,2,2]
[(1,4),(2,3),(5,6)] => [3,4,2,1,6,5] => [4,1,3,2,6,5] => [5,6,2,3,1,4] => [2,2,2]
[(1,5),(2,3),(4,6)] => [3,5,2,6,1,4] => [5,1,3,2,6,4] => [4,6,2,3,1,5] => [2,2,2]
[(1,6),(2,3),(4,5)] => [3,5,2,6,4,1] => [6,1,3,2,5,4] => [4,5,2,3,1,6] => [2,2,2]
[(1,6),(2,4),(3,5)] => [4,5,6,2,3,1] => [6,1,4,2,5,3] => [3,5,2,4,1,6] => [2,2,2]
[(1,5),(2,4),(3,6)] => [4,5,6,2,1,3] => [5,1,4,2,6,3] => [3,6,2,4,1,5] => [2,2,2]
[(1,4),(2,5),(3,6)] => [4,5,6,1,2,3] => [4,1,5,2,6,3] => [3,6,2,5,1,4] => [2,2,2]
[(1,3),(2,5),(4,6)] => [3,5,1,6,2,4] => [3,1,5,2,6,4] => [4,6,2,5,1,3] => [2,2,2]
[(1,2),(3,5),(4,6)] => [2,1,5,6,3,4] => [2,1,5,3,6,4] => [4,6,3,5,1,2] => [2,2,2]
[(1,2),(3,6),(4,5)] => [2,1,5,6,4,3] => [2,1,6,3,5,4] => [4,5,3,6,1,2] => [2,2,2]
[(1,3),(2,6),(4,5)] => [3,5,1,6,4,2] => [3,1,6,2,5,4] => [4,5,2,6,1,3] => [2,2,2]
[(1,4),(2,6),(3,5)] => [4,5,6,1,3,2] => [4,1,6,2,5,3] => [3,5,2,6,1,4] => [2,2,2]
[(1,5),(2,6),(3,4)] => [4,5,6,3,1,2] => [6,2,5,1,4,3] => [3,4,1,5,2,6] => [2,2,2]
[(1,6),(2,5),(3,4)] => [4,5,6,3,2,1] => [5,2,6,1,4,3] => [3,4,1,6,2,5] => [2,2,2]
[(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] => [3,1,4,2,6,5,8,7] => [7,8,5,6,2,4,1,3] => [2,2,2,2]
[(1,4),(2,3),(5,6),(7,8)] => [3,4,2,1,6,5,8,7] => [4,1,3,2,6,5,8,7] => [7,8,5,6,2,3,1,4] => [2,2,2,2]
[(1,5),(2,3),(4,6),(7,8)] => [3,5,2,6,1,4,8,7] => [5,1,3,2,6,4,8,7] => [7,8,4,6,2,3,1,5] => [2,2,2,2]
[(1,6),(2,3),(4,5),(7,8)] => [3,5,2,6,4,1,8,7] => [6,1,3,2,5,4,8,7] => [7,8,4,5,2,3,1,6] => [2,2,2,2]
[(1,7),(2,3),(4,5),(6,8)] => [3,5,2,7,4,8,1,6] => [7,1,3,2,5,4,8,6] => [6,8,4,5,2,3,1,7] => [2,2,2,2]
[(1,8),(2,3),(4,5),(6,7)] => [3,5,2,7,4,8,6,1] => [8,1,3,2,5,4,7,6] => [6,7,4,5,2,3,1,8] => [2,2,2,2]
[(1,8),(2,4),(3,5),(6,7)] => [4,5,7,2,3,8,6,1] => [8,1,4,2,5,3,7,6] => [6,7,3,5,2,4,1,8] => [2,2,2,2]
[(1,7),(2,4),(3,5),(6,8)] => [4,5,7,2,3,8,1,6] => [7,1,4,2,5,3,8,6] => [6,8,3,5,2,4,1,7] => [2,2,2,2]
[(1,6),(2,4),(3,5),(7,8)] => [4,5,6,2,3,1,8,7] => [6,1,4,2,5,3,8,7] => [7,8,3,5,2,4,1,6] => [2,2,2,2]
[(1,5),(2,4),(3,6),(7,8)] => [4,5,6,2,1,3,8,7] => [5,1,4,2,6,3,8,7] => [7,8,3,6,2,4,1,5] => [2,2,2,2]
[(1,4),(2,5),(3,6),(7,8)] => [4,5,6,1,2,3,8,7] => [4,1,5,2,6,3,8,7] => [7,8,3,6,2,5,1,4] => [2,2,2,2]
[(1,3),(2,5),(4,6),(7,8)] => [3,5,1,6,2,4,8,7] => [3,1,5,2,6,4,8,7] => [7,8,4,6,2,5,1,3] => [2,2,2,2]
[(1,2),(3,5),(4,6),(7,8)] => [2,1,5,6,3,4,8,7] => [2,1,5,3,6,4,8,7] => [7,8,4,6,3,5,1,2] => [2,2,2,2]
[(1,2),(3,6),(4,5),(7,8)] => [2,1,5,6,4,3,8,7] => [2,1,6,3,5,4,8,7] => [7,8,4,5,3,6,1,2] => [2,2,2,2]
[(1,3),(2,6),(4,5),(7,8)] => [3,5,1,6,4,2,8,7] => [3,1,6,2,5,4,8,7] => [7,8,4,5,2,6,1,3] => [2,2,2,2]
[(1,4),(2,6),(3,5),(7,8)] => [4,5,6,1,3,2,8,7] => [4,1,6,2,5,3,8,7] => [7,8,3,5,2,6,1,4] => [2,2,2,2]
[(1,5),(2,6),(3,4),(7,8)] => [4,5,6,3,1,2,8,7] => [6,2,5,1,4,3,8,7] => [7,8,3,4,1,5,2,6] => [2,2,2,2]
[(1,6),(2,5),(3,4),(7,8)] => [4,5,6,3,2,1,8,7] => [5,2,6,1,4,3,8,7] => [7,8,3,4,1,6,2,5] => [2,2,2,2]
[(1,7),(2,5),(3,4),(6,8)] => [4,5,7,3,2,8,1,6] => [5,2,7,1,4,3,8,6] => [6,8,3,4,1,7,2,5] => [2,2,2,2]
[(1,8),(2,5),(3,4),(6,7)] => [4,5,7,3,2,8,6,1] => [5,2,8,1,4,3,7,6] => [6,7,3,4,1,8,2,5] => [2,2,2,2]
[(1,8),(2,6),(3,4),(5,7)] => [4,6,7,3,8,2,5,1] => [6,2,8,1,4,3,7,5] => [5,7,3,4,1,8,2,6] => [2,2,2,2]
[(1,7),(2,6),(3,4),(5,8)] => [4,6,7,3,8,2,1,5] => [6,2,7,1,4,3,8,5] => [5,8,3,4,1,7,2,6] => [2,2,2,2]
[(1,6),(2,7),(3,4),(5,8)] => [4,6,7,3,8,1,2,5] => [7,2,6,1,4,3,8,5] => [5,8,3,4,1,6,2,7] => [2,2,2,2]
[(1,5),(2,7),(3,4),(6,8)] => [4,5,7,3,1,8,2,6] => [7,2,5,1,4,3,8,6] => [6,8,3,4,1,5,2,7] => [2,2,2,2]
[(1,4),(2,7),(3,5),(6,8)] => [4,5,7,1,3,8,2,6] => [4,1,7,2,5,3,8,6] => [6,8,3,5,2,7,1,4] => [2,2,2,2]
[(1,3),(2,7),(4,5),(6,8)] => [3,5,1,7,4,8,2,6] => [3,1,7,2,5,4,8,6] => [6,8,4,5,2,7,1,3] => [2,2,2,2]
[(1,2),(3,7),(4,5),(6,8)] => [2,1,5,7,4,8,3,6] => [2,1,7,3,5,4,8,6] => [6,8,4,5,3,7,1,2] => [2,2,2,2]
[(1,2),(3,8),(4,5),(6,7)] => [2,1,5,7,4,8,6,3] => [2,1,8,3,5,4,7,6] => [6,7,4,5,3,8,1,2] => [2,2,2,2]
[(1,3),(2,8),(4,5),(6,7)] => [3,5,1,7,4,8,6,2] => [3,1,8,2,5,4,7,6] => [6,7,4,5,2,8,1,3] => [2,2,2,2]
[(1,4),(2,8),(3,5),(6,7)] => [4,5,7,1,3,8,6,2] => [4,1,8,2,5,3,7,6] => [6,7,3,5,2,8,1,4] => [2,2,2,2]
[(1,5),(2,8),(3,4),(6,7)] => [4,5,7,3,1,8,6,2] => [8,2,5,1,4,3,7,6] => [6,7,3,4,1,5,2,8] => [2,2,2,2]
[(1,6),(2,8),(3,4),(5,7)] => [4,6,7,3,8,1,5,2] => [8,2,6,1,4,3,7,5] => [5,7,3,4,1,6,2,8] => [2,2,2,2]
[(1,7),(2,8),(3,4),(5,6)] => [4,6,7,3,8,5,1,2] => [7,1,4,3,8,2,6,5] => [5,6,2,8,3,4,1,7] => [2,2,2,2]
[(1,8),(2,7),(3,4),(5,6)] => [4,6,7,3,8,5,2,1] => [8,1,4,3,7,2,6,5] => [5,6,2,7,3,4,1,8] => [2,2,2,2]
[(1,8),(2,7),(3,5),(4,6)] => [5,6,7,8,3,4,2,1] => [8,1,5,3,7,2,6,4] => [4,6,2,7,3,5,1,8] => [2,2,2,2]
[(1,7),(2,8),(3,5),(4,6)] => [5,6,7,8,3,4,1,2] => [7,1,5,3,8,2,6,4] => [4,6,2,8,3,5,1,7] => [2,2,2,2]
[(1,6),(2,8),(3,5),(4,7)] => [5,6,7,8,3,1,4,2] => [8,2,6,1,5,3,7,4] => [4,7,3,5,1,6,2,8] => [2,2,2,2]
[(1,5),(2,8),(3,6),(4,7)] => [5,6,7,8,1,3,4,2] => [5,1,8,2,6,3,7,4] => [4,7,3,6,2,8,1,5] => [2,2,2,2]
[(1,4),(2,8),(3,6),(5,7)] => [4,6,7,1,8,3,5,2] => [4,1,8,2,6,3,7,5] => [5,7,3,6,2,8,1,4] => [2,2,2,2]
[(1,3),(2,8),(4,6),(5,7)] => [3,6,1,7,8,4,5,2] => [3,1,8,2,6,4,7,5] => [5,7,4,6,2,8,1,3] => [2,2,2,2]
[(1,2),(3,8),(4,6),(5,7)] => [2,1,6,7,8,4,5,3] => [2,1,8,3,6,4,7,5] => [5,7,4,6,3,8,1,2] => [2,2,2,2]
[(1,2),(3,7),(4,6),(5,8)] => [2,1,6,7,8,4,3,5] => [2,1,7,3,6,4,8,5] => [5,8,4,6,3,7,1,2] => [2,2,2,2]
[(1,3),(2,7),(4,6),(5,8)] => [3,6,1,7,8,4,2,5] => [3,1,7,2,6,4,8,5] => [5,8,4,6,2,7,1,3] => [2,2,2,2]
[(1,4),(2,7),(3,6),(5,8)] => [4,6,7,1,8,3,2,5] => [4,1,7,2,6,3,8,5] => [5,8,3,6,2,7,1,4] => [2,2,2,2]
[(1,5),(2,7),(3,6),(4,8)] => [5,6,7,8,1,3,2,4] => [5,1,7,2,6,3,8,4] => [4,8,3,6,2,7,1,5] => [2,2,2,2]
[(1,6),(2,7),(3,5),(4,8)] => [5,6,7,8,3,1,2,4] => [7,2,6,1,5,3,8,4] => [4,8,3,5,1,6,2,7] => [2,2,2,2]
[(1,7),(2,6),(3,5),(4,8)] => [5,6,7,8,3,2,1,4] => [6,2,7,1,5,3,8,4] => [4,8,3,5,1,7,2,6] => [2,2,2,2]
[(1,8),(2,6),(3,5),(4,7)] => [5,6,7,8,3,2,4,1] => [6,2,8,1,5,3,7,4] => [4,7,3,5,1,8,2,6] => [2,2,2,2]
[(1,8),(2,5),(3,6),(4,7)] => [5,6,7,8,2,3,4,1] => [8,1,5,2,6,3,7,4] => [4,7,3,6,2,5,1,8] => [2,2,2,2]
[(1,7),(2,5),(3,6),(4,8)] => [5,6,7,8,2,3,1,4] => [7,1,5,2,6,3,8,4] => [4,8,3,6,2,5,1,7] => [2,2,2,2]
[(1,6),(2,5),(3,7),(4,8)] => [5,6,7,8,2,1,3,4] => [6,1,5,2,7,3,8,4] => [4,8,3,7,2,5,1,6] => [2,2,2,2]
[(1,5),(2,6),(3,7),(4,8)] => [5,6,7,8,1,2,3,4] => [5,1,6,2,7,3,8,4] => [4,8,3,7,2,6,1,5] => [2,2,2,2]
[(1,4),(2,6),(3,7),(5,8)] => [4,6,7,1,8,2,3,5] => [4,1,6,2,7,3,8,5] => [5,8,3,7,2,6,1,4] => [2,2,2,2]
[(1,3),(2,6),(4,7),(5,8)] => [3,6,1,7,8,2,4,5] => [3,1,6,2,7,4,8,5] => [5,8,4,7,2,6,1,3] => [2,2,2,2]
[(1,2),(3,6),(4,7),(5,8)] => [2,1,6,7,8,3,4,5] => [2,1,6,3,7,4,8,5] => [5,8,4,7,3,6,1,2] => [2,2,2,2]
[(1,2),(3,5),(4,7),(6,8)] => [2,1,5,7,3,8,4,6] => [2,1,5,3,7,4,8,6] => [6,8,4,7,3,5,1,2] => [2,2,2,2]
[(1,3),(2,5),(4,7),(6,8)] => [3,5,1,7,2,8,4,6] => [3,1,5,2,7,4,8,6] => [6,8,4,7,2,5,1,3] => [2,2,2,2]
[(1,4),(2,5),(3,7),(6,8)] => [4,5,7,1,2,8,3,6] => [4,1,5,2,7,3,8,6] => [6,8,3,7,2,5,1,4] => [2,2,2,2]
[(1,5),(2,4),(3,7),(6,8)] => [4,5,7,2,1,8,3,6] => [5,1,4,2,7,3,8,6] => [6,8,3,7,2,4,1,5] => [2,2,2,2]
[(1,6),(2,4),(3,7),(5,8)] => [4,6,7,2,8,1,3,5] => [6,1,4,2,7,3,8,5] => [5,8,3,7,2,4,1,6] => [2,2,2,2]
[(1,7),(2,4),(3,6),(5,8)] => [4,6,7,2,8,3,1,5] => [7,1,4,2,6,3,8,5] => [5,8,3,6,2,4,1,7] => [2,2,2,2]
[(1,8),(2,4),(3,6),(5,7)] => [4,6,7,2,8,3,5,1] => [8,1,4,2,6,3,7,5] => [5,7,3,6,2,4,1,8] => [2,2,2,2]
[(1,8),(2,3),(4,6),(5,7)] => [3,6,2,7,8,4,5,1] => [8,1,3,2,6,4,7,5] => [5,7,4,6,2,3,1,8] => [2,2,2,2]
[(1,7),(2,3),(4,6),(5,8)] => [3,6,2,7,8,4,1,5] => [7,1,3,2,6,4,8,5] => [5,8,4,6,2,3,1,7] => [2,2,2,2]
[(1,6),(2,3),(4,7),(5,8)] => [3,6,2,7,8,1,4,5] => [6,1,3,2,7,4,8,5] => [5,8,4,7,2,3,1,6] => [2,2,2,2]
[(1,5),(2,3),(4,7),(6,8)] => [3,5,2,7,1,8,4,6] => [5,1,3,2,7,4,8,6] => [6,8,4,7,2,3,1,5] => [2,2,2,2]
[(1,4),(2,3),(5,7),(6,8)] => [3,4,2,1,7,8,5,6] => [4,1,3,2,7,5,8,6] => [6,8,5,7,2,3,1,4] => [2,2,2,2]
[(1,3),(2,4),(5,7),(6,8)] => [3,4,1,2,7,8,5,6] => [3,1,4,2,7,5,8,6] => [6,8,5,7,2,4,1,3] => [2,2,2,2]
[(1,2),(3,4),(5,7),(6,8)] => [2,1,4,3,7,8,5,6] => [2,1,4,3,7,5,8,6] => [6,8,5,7,3,4,1,2] => [2,2,2,2]
[(1,2),(3,4),(5,8),(6,7)] => [2,1,4,3,7,8,6,5] => [2,1,4,3,8,5,7,6] => [6,7,5,8,3,4,1,2] => [2,2,2,2]
[(1,3),(2,4),(5,8),(6,7)] => [3,4,1,2,7,8,6,5] => [3,1,4,2,8,5,7,6] => [6,7,5,8,2,4,1,3] => [2,2,2,2]
[(1,4),(2,3),(5,8),(6,7)] => [3,4,2,1,7,8,6,5] => [4,1,3,2,8,5,7,6] => [6,7,5,8,2,3,1,4] => [2,2,2,2]
[(1,5),(2,3),(4,8),(6,7)] => [3,5,2,7,1,8,6,4] => [5,1,3,2,8,4,7,6] => [6,7,4,8,2,3,1,5] => [2,2,2,2]
[(1,6),(2,3),(4,8),(5,7)] => [3,6,2,7,8,1,5,4] => [6,1,3,2,8,4,7,5] => [5,7,4,8,2,3,1,6] => [2,2,2,2]
[(1,7),(2,3),(4,8),(5,6)] => [3,6,2,7,8,5,1,4] => [8,4,7,1,3,2,6,5] => [5,6,2,3,1,7,4,8] => [2,2,2,2]
[(1,8),(2,3),(4,7),(5,6)] => [3,6,2,7,8,5,4,1] => [7,4,8,1,3,2,6,5] => [5,6,2,3,1,8,4,7] => [2,2,2,2]
[(1,8),(2,4),(3,7),(5,6)] => [4,6,7,2,8,5,3,1] => [7,3,8,1,4,2,6,5] => [5,6,2,4,1,8,3,7] => [2,2,2,2]
[(1,7),(2,4),(3,8),(5,6)] => [4,6,7,2,8,5,1,3] => [8,3,7,1,4,2,6,5] => [5,6,2,4,1,7,3,8] => [2,2,2,2]
[(1,6),(2,4),(3,8),(5,7)] => [4,6,7,2,8,1,5,3] => [6,1,4,2,8,3,7,5] => [5,7,3,8,2,4,1,6] => [2,2,2,2]
[(1,5),(2,4),(3,8),(6,7)] => [4,5,7,2,1,8,6,3] => [5,1,4,2,8,3,7,6] => [6,7,3,8,2,4,1,5] => [2,2,2,2]
[(1,4),(2,5),(3,8),(6,7)] => [4,5,7,1,2,8,6,3] => [4,1,5,2,8,3,7,6] => [6,7,3,8,2,5,1,4] => [2,2,2,2]
>>> Load all 203 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
inverse first fundamental transformation
Description
Let $\sigma = (i_{11}\cdots i_{1k_1})\cdots(i_{\ell 1}\cdots i_{\ell k_\ell})$ be a permutation given by cycle notation such that every cycle starts with its maximal entry, and all cycles are ordered increasingly by these maximal entries.
Maps $\sigma$ to the permutation $[i_{11},\ldots,i_{1k_1},\ldots,i_{\ell 1},\ldots,i_{\ell k_\ell}]$ in one-line notation.
In other words, this map sends the maximal entries of the cycles to the left-to-right maxima, and the sequences between two left-to-right maxima are given by the cycles.
Maps $\sigma$ to the permutation $[i_{11},\ldots,i_{1k_1},\ldots,i_{\ell 1},\ldots,i_{\ell k_\ell}]$ in one-line notation.
In other words, this map sends the maximal entries of the cycles to the left-to-right maxima, and the sequences between two left-to-right maxima are given by the cycles.
Map
reverse
Description
Sends a permutation to its reverse.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.
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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