Identifier
Mp00113:
Perfect matchings
—reverse⟶
Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Images
[(1,2)] => [(1,2)] => [2,1] => [2,1] => [1,2]
[(1,2),(3,4)] => [(1,2),(3,4)] => [2,1,4,3] => [2,1,4,3] => [3,4,1,2]
[(1,3),(2,4)] => [(1,3),(2,4)] => [3,4,1,2] => [2,4,1,3] => [3,1,4,2]
[(1,4),(2,3)] => [(1,4),(2,3)] => [4,3,2,1] => [4,3,2,1] => [1,2,3,4]
[(1,2),(3,4),(5,6)] => [(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [2,1,4,3,6,5] => [5,6,3,4,1,2]
[(1,3),(2,4),(5,6)] => [(1,2),(3,5),(4,6)] => [2,1,5,6,3,4] => [2,1,4,6,3,5] => [5,6,3,1,4,2]
[(1,4),(2,3),(5,6)] => [(1,2),(3,6),(4,5)] => [2,1,6,5,4,3] => [2,1,6,5,4,3] => [5,6,1,2,3,4]
[(1,5),(2,3),(4,6)] => [(1,3),(2,6),(4,5)] => [3,6,1,5,4,2] => [5,4,2,6,1,3] => [2,3,5,1,6,4]
[(1,6),(2,3),(4,5)] => [(1,6),(2,3),(4,5)] => [6,3,2,5,4,1] => [5,4,1,6,3,2] => [2,3,6,1,4,5]
[(1,6),(2,4),(3,5)] => [(1,6),(2,4),(3,5)] => [6,4,5,2,3,1] => [3,1,5,2,6,4] => [4,6,2,5,1,3]
[(1,5),(2,4),(3,6)] => [(1,4),(2,6),(3,5)] => [4,6,5,1,3,2] => [3,2,6,5,1,4] => [4,5,1,2,6,3]
[(1,4),(2,5),(3,6)] => [(1,4),(2,5),(3,6)] => [4,5,6,1,2,3] => [2,3,6,1,4,5] => [5,4,1,6,3,2]
[(1,3),(2,5),(4,6)] => [(1,3),(2,5),(4,6)] => [3,5,1,6,2,4] => [4,6,2,5,1,3] => [3,1,5,2,6,4]
[(1,2),(3,5),(4,6)] => [(1,3),(2,4),(5,6)] => [3,4,1,2,6,5] => [2,4,1,3,6,5] => [5,3,6,4,1,2]
[(1,2),(3,6),(4,5)] => [(1,4),(2,3),(5,6)] => [4,3,2,1,6,5] => [4,3,2,1,6,5] => [3,4,5,6,1,2]
[(1,3),(2,6),(4,5)] => [(1,5),(2,3),(4,6)] => [5,3,2,6,1,4] => [4,6,1,5,3,2] => [3,1,6,2,4,5]
[(1,4),(2,6),(3,5)] => [(1,5),(2,4),(3,6)] => [5,4,6,2,1,3] => [3,6,2,1,5,4] => [4,1,5,6,2,3]
[(1,5),(2,6),(3,4)] => [(1,5),(2,6),(3,4)] => [5,6,4,3,1,2] => [2,6,4,3,1,5] => [5,1,3,4,6,2]
[(1,6),(2,5),(3,4)] => [(1,6),(2,5),(3,4)] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6]
[(1,2),(3,4),(5,6),(7,8)] => [(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]
[(1,3),(2,4),(5,6),(7,8)] => [(1,2),(3,4),(5,7),(6,8)] => [2,1,4,3,7,8,5,6] => [2,1,4,3,6,8,5,7] => [7,8,5,6,3,1,4,2]
[(1,4),(2,3),(5,6),(7,8)] => [(1,2),(3,4),(5,8),(6,7)] => [2,1,4,3,8,7,6,5] => [2,1,4,3,8,7,6,5] => [7,8,5,6,1,2,3,4]
[(1,5),(2,3),(4,6),(7,8)] => [(1,2),(3,5),(4,8),(6,7)] => [2,1,5,8,3,7,6,4] => [2,1,7,6,4,8,3,5] => [7,8,2,3,5,1,6,4]
[(1,6),(2,3),(4,5),(7,8)] => [(1,2),(3,8),(4,5),(6,7)] => [2,1,8,5,4,7,6,3] => [2,1,7,6,3,8,5,4] => [7,8,2,3,6,1,4,5]
[(1,7),(2,3),(4,5),(6,8)] => [(1,3),(2,8),(4,5),(6,7)] => [3,8,1,5,4,7,6,2] => [7,6,2,5,4,8,1,3] => [2,3,7,4,5,1,8,6]
[(1,8),(2,3),(4,5),(6,7)] => [(1,8),(2,3),(4,5),(6,7)] => [8,3,2,5,4,7,6,1] => [7,6,1,5,4,8,3,2] => [2,3,8,4,5,1,6,7]
[(1,8),(2,4),(3,5),(6,7)] => [(1,8),(2,3),(4,6),(5,7)] => [8,3,2,6,7,4,5,1] => [5,1,7,4,6,8,3,2] => [4,8,2,5,3,1,6,7]
[(1,7),(2,4),(3,5),(6,8)] => [(1,3),(2,8),(4,6),(5,7)] => [3,8,1,6,7,4,5,2] => [5,2,7,4,6,8,1,3] => [4,7,2,5,3,1,8,6]
[(1,6),(2,4),(3,5),(7,8)] => [(1,2),(3,8),(4,6),(5,7)] => [2,1,8,6,7,4,5,3] => [2,1,5,3,7,4,8,6] => [7,8,4,6,2,5,1,3]
[(1,5),(2,4),(3,6),(7,8)] => [(1,2),(3,6),(4,8),(5,7)] => [2,1,6,8,7,3,5,4] => [2,1,5,4,8,7,3,6] => [7,8,4,5,1,2,6,3]
[(1,4),(2,5),(3,6),(7,8)] => [(1,2),(3,6),(4,7),(5,8)] => [2,1,6,7,8,3,4,5] => [2,1,4,5,8,3,6,7] => [7,8,5,4,1,6,3,2]
[(1,3),(2,5),(4,6),(7,8)] => [(1,2),(3,5),(4,7),(6,8)] => [2,1,5,7,3,8,4,6] => [2,1,6,8,4,7,3,5] => [7,8,3,1,5,2,6,4]
[(1,2),(3,5),(4,6),(7,8)] => [(1,2),(3,5),(4,6),(7,8)] => [2,1,5,6,3,4,8,7] => [2,1,4,6,3,5,8,7] => [7,8,5,3,6,4,1,2]
[(1,2),(3,6),(4,5),(7,8)] => [(1,2),(3,6),(4,5),(7,8)] => [2,1,6,5,4,3,8,7] => [2,1,6,5,4,3,8,7] => [7,8,3,4,5,6,1,2]
[(1,3),(2,6),(4,5),(7,8)] => [(1,2),(3,7),(4,5),(6,8)] => [2,1,7,5,4,8,3,6] => [2,1,6,8,3,7,5,4] => [7,8,3,1,6,2,4,5]
[(1,4),(2,6),(3,5),(7,8)] => [(1,2),(3,7),(4,6),(5,8)] => [2,1,7,6,8,4,3,5] => [2,1,5,8,4,3,7,6] => [7,8,4,1,5,6,2,3]
[(1,5),(2,6),(3,4),(7,8)] => [(1,2),(3,7),(4,8),(5,6)] => [2,1,7,8,6,5,3,4] => [2,1,4,8,6,5,3,7] => [7,8,5,1,3,4,6,2]
[(1,6),(2,5),(3,4),(7,8)] => [(1,2),(3,8),(4,7),(5,6)] => [2,1,8,7,6,5,4,3] => [2,1,8,7,6,5,4,3] => [7,8,1,2,3,4,5,6]
[(1,7),(2,5),(3,4),(6,8)] => [(1,3),(2,8),(4,7),(5,6)] => [3,8,1,7,6,5,4,2] => [7,6,5,4,2,8,1,3] => [2,3,4,5,7,1,8,6]
[(1,8),(2,5),(3,4),(6,7)] => [(1,8),(2,3),(4,7),(5,6)] => [8,3,2,7,6,5,4,1] => [7,6,5,4,1,8,3,2] => [2,3,4,5,8,1,6,7]
[(1,8),(2,6),(3,4),(5,7)] => [(1,8),(2,4),(3,7),(5,6)] => [8,4,7,2,6,5,3,1] => [6,5,3,1,7,2,8,4] => [3,4,6,8,2,7,1,5]
[(1,7),(2,6),(3,4),(5,8)] => [(1,4),(2,8),(3,7),(5,6)] => [4,8,7,1,6,5,3,2] => [6,5,3,2,8,7,1,4] => [3,4,6,7,1,2,8,5]
[(1,6),(2,7),(3,4),(5,8)] => [(1,4),(2,7),(3,8),(5,6)] => [4,7,8,1,6,5,2,3] => [3,6,5,2,8,1,4,7] => [6,3,4,7,1,8,5,2]
[(1,5),(2,7),(3,4),(6,8)] => [(1,3),(2,7),(4,8),(5,6)] => [3,7,1,8,6,5,2,4] => [4,8,6,5,2,7,1,3] => [5,1,3,4,7,2,8,6]
[(1,4),(2,7),(3,5),(6,8)] => [(1,3),(2,7),(4,6),(5,8)] => [3,7,1,6,8,4,2,5] => [5,8,4,2,6,7,1,3] => [4,1,5,7,3,2,8,6]
[(1,3),(2,7),(4,5),(6,8)] => [(1,3),(2,7),(4,5),(6,8)] => [3,7,1,5,4,8,2,6] => [6,8,2,5,4,7,1,3] => [3,1,7,4,5,2,8,6]
[(1,2),(3,7),(4,5),(6,8)] => [(1,3),(2,6),(4,5),(7,8)] => [3,6,1,5,4,2,8,7] => [5,4,2,6,1,3,8,7] => [4,5,7,3,8,6,1,2]
[(1,2),(3,8),(4,5),(6,7)] => [(1,6),(2,3),(4,5),(7,8)] => [6,3,2,5,4,1,8,7] => [5,4,1,6,3,2,8,7] => [4,5,8,3,6,7,1,2]
[(1,3),(2,8),(4,5),(6,7)] => [(1,7),(2,3),(4,5),(6,8)] => [7,3,2,5,4,8,1,6] => [6,8,1,5,4,7,3,2] => [3,1,8,4,5,2,6,7]
[(1,4),(2,8),(3,5),(6,7)] => [(1,7),(2,3),(4,6),(5,8)] => [7,3,2,6,8,4,1,5] => [5,8,4,1,6,7,3,2] => [4,1,5,8,3,2,6,7]
[(1,5),(2,8),(3,4),(6,7)] => [(1,7),(2,3),(4,8),(5,6)] => [7,3,2,8,6,5,1,4] => [4,8,6,5,1,7,3,2] => [5,1,3,4,8,2,6,7]
[(1,6),(2,8),(3,4),(5,7)] => [(1,7),(2,4),(3,8),(5,6)] => [7,4,8,2,6,5,1,3] => [3,6,5,1,8,2,7,4] => [6,3,4,8,1,7,2,5]
[(1,7),(2,8),(3,4),(5,6)] => [(1,7),(2,8),(3,4),(5,6)] => [7,8,4,3,6,5,1,2] => [2,6,5,1,8,4,3,7] => [7,3,4,8,1,5,6,2]
[(1,8),(2,7),(3,4),(5,6)] => [(1,8),(2,7),(3,4),(5,6)] => [8,7,4,3,6,5,2,1] => [6,5,2,1,8,7,4,3] => [3,4,7,8,1,2,5,6]
[(1,8),(2,7),(3,5),(4,6)] => [(1,8),(2,7),(3,5),(4,6)] => [8,7,5,6,3,4,2,1] => [4,2,1,6,3,8,7,5] => [5,7,8,3,6,1,2,4]
[(1,7),(2,8),(3,5),(4,6)] => [(1,7),(2,8),(3,5),(4,6)] => [7,8,5,6,3,4,1,2] => [2,4,1,6,3,8,5,7] => [7,5,8,3,6,1,4,2]
[(1,6),(2,8),(3,5),(4,7)] => [(1,7),(2,5),(3,8),(4,6)] => [7,5,8,6,2,4,1,3] => [3,4,1,8,6,2,7,5] => [6,5,8,1,3,7,2,4]
[(1,5),(2,8),(3,6),(4,7)] => [(1,7),(2,5),(3,6),(4,8)] => [7,5,6,8,2,3,1,4] => [3,1,4,8,2,6,7,5] => [6,8,5,1,7,3,2,4]
[(1,4),(2,8),(3,6),(5,7)] => [(1,7),(2,4),(3,6),(5,8)] => [7,4,6,2,8,3,1,5] => [5,8,3,1,6,2,7,4] => [4,1,6,8,3,7,2,5]
[(1,3),(2,8),(4,6),(5,7)] => [(1,7),(2,4),(3,5),(6,8)] => [7,4,5,2,3,8,1,6] => [6,8,1,3,5,2,7,4] => [3,1,8,6,4,7,2,5]
[(1,2),(3,8),(4,6),(5,7)] => [(1,6),(2,4),(3,5),(7,8)] => [6,4,5,2,3,1,8,7] => [3,1,5,2,6,4,8,7] => [6,8,4,7,3,5,1,2]
[(1,2),(3,7),(4,6),(5,8)] => [(1,4),(2,6),(3,5),(7,8)] => [4,6,5,1,3,2,8,7] => [3,2,6,5,1,4,8,7] => [6,7,3,4,8,5,1,2]
[(1,3),(2,7),(4,6),(5,8)] => [(1,4),(2,7),(3,5),(6,8)] => [4,7,5,1,3,8,2,6] => [6,8,2,3,7,5,1,4] => [3,1,7,6,2,4,8,5]
[(1,4),(2,7),(3,6),(5,8)] => [(1,4),(2,7),(3,6),(5,8)] => [4,7,6,1,8,3,2,5] => [5,8,3,2,7,6,1,4] => [4,1,6,7,2,3,8,5]
[(1,5),(2,7),(3,6),(4,8)] => [(1,5),(2,7),(3,6),(4,8)] => [5,7,6,8,1,3,2,4] => [3,2,4,8,1,5,7,6] => [6,7,5,1,8,4,2,3]
[(1,6),(2,7),(3,5),(4,8)] => [(1,5),(2,7),(3,8),(4,6)] => [5,7,8,6,1,4,2,3] => [3,4,2,8,6,1,5,7] => [6,5,7,1,3,8,4,2]
[(1,7),(2,6),(3,5),(4,8)] => [(1,5),(2,8),(3,7),(4,6)] => [5,8,7,6,1,4,3,2] => [4,3,2,8,7,6,1,5] => [5,6,7,1,2,3,8,4]
[(1,8),(2,6),(3,5),(4,7)] => [(1,8),(2,5),(3,7),(4,6)] => [8,5,7,6,2,4,3,1] => [4,3,1,7,6,2,8,5] => [5,6,8,2,3,7,1,4]
[(1,8),(2,5),(3,6),(4,7)] => [(1,8),(2,5),(3,6),(4,7)] => [8,5,6,7,2,3,4,1] => [4,1,3,7,2,6,8,5] => [5,8,6,2,7,3,1,4]
[(1,7),(2,5),(3,6),(4,8)] => [(1,5),(2,8),(3,6),(4,7)] => [5,8,6,7,1,3,4,2] => [4,2,3,7,1,5,8,6] => [5,7,6,2,8,4,1,3]
[(1,6),(2,5),(3,7),(4,8)] => [(1,5),(2,6),(3,8),(4,7)] => [5,6,8,7,1,2,4,3] => [2,4,3,8,7,1,5,6] => [7,5,6,1,2,8,4,3]
[(1,5),(2,6),(3,7),(4,8)] => [(1,5),(2,6),(3,7),(4,8)] => [5,6,7,8,1,2,3,4] => [2,3,4,8,1,5,6,7] => [7,6,5,1,8,4,3,2]
[(1,4),(2,6),(3,7),(5,8)] => [(1,4),(2,6),(3,7),(5,8)] => [4,6,7,1,8,2,3,5] => [3,5,8,2,7,1,4,6] => [6,4,1,7,2,8,5,3]
[(1,3),(2,6),(4,7),(5,8)] => [(1,4),(2,5),(3,7),(6,8)] => [4,5,7,1,2,8,3,6] => [2,6,8,3,7,1,4,5] => [7,3,1,6,2,8,5,4]
[(1,2),(3,6),(4,7),(5,8)] => [(1,4),(2,5),(3,6),(7,8)] => [4,5,6,1,2,3,8,7] => [2,3,6,1,4,5,8,7] => [7,6,3,8,5,4,1,2]
[(1,2),(3,5),(4,7),(6,8)] => [(1,3),(2,5),(4,6),(7,8)] => [3,5,1,6,2,4,8,7] => [4,6,2,5,1,3,8,7] => [5,3,7,4,8,6,1,2]
[(1,3),(2,5),(4,7),(6,8)] => [(1,3),(2,5),(4,7),(6,8)] => [3,5,1,7,2,8,4,6] => [6,8,4,7,2,5,1,3] => [3,1,5,2,7,4,8,6]
[(1,4),(2,5),(3,7),(6,8)] => [(1,3),(2,6),(4,7),(5,8)] => [3,6,1,7,8,2,4,5] => [4,5,8,2,6,1,3,7] => [5,4,1,7,3,8,6,2]
[(1,5),(2,4),(3,7),(6,8)] => [(1,3),(2,6),(4,8),(5,7)] => [3,6,1,8,7,2,5,4] => [5,4,8,7,2,6,1,3] => [4,5,1,2,7,3,8,6]
[(1,6),(2,4),(3,7),(5,8)] => [(1,4),(2,6),(3,8),(5,7)] => [4,6,8,1,7,2,5,3] => [5,3,7,2,8,1,4,6] => [4,6,2,7,1,8,5,3]
[(1,7),(2,4),(3,6),(5,8)] => [(1,4),(2,8),(3,6),(5,7)] => [4,8,6,1,7,3,5,2] => [5,2,7,3,8,6,1,4] => [4,7,2,6,1,3,8,5]
[(1,8),(2,4),(3,6),(5,7)] => [(1,8),(2,4),(3,6),(5,7)] => [8,4,6,2,7,3,5,1] => [5,1,7,3,6,2,8,4] => [4,8,2,6,3,7,1,5]
[(1,8),(2,3),(4,6),(5,7)] => [(1,8),(2,4),(3,5),(6,7)] => [8,4,5,2,3,7,6,1] => [7,6,1,3,5,2,8,4] => [2,3,8,6,4,7,1,5]
[(1,7),(2,3),(4,6),(5,8)] => [(1,4),(2,8),(3,5),(6,7)] => [4,8,5,1,3,7,6,2] => [7,6,2,3,8,5,1,4] => [2,3,7,6,1,4,8,5]
[(1,6),(2,3),(4,7),(5,8)] => [(1,4),(2,5),(3,8),(6,7)] => [4,5,8,1,2,7,6,3] => [2,7,6,3,8,1,4,5] => [7,2,3,6,1,8,5,4]
[(1,5),(2,3),(4,7),(6,8)] => [(1,3),(2,5),(4,8),(6,7)] => [3,5,1,8,2,7,6,4] => [7,6,4,8,2,5,1,3] => [2,3,5,1,7,4,8,6]
[(1,4),(2,3),(5,7),(6,8)] => [(1,3),(2,4),(5,8),(6,7)] => [3,4,1,2,8,7,6,5] => [2,4,1,3,8,7,6,5] => [7,5,8,6,1,2,3,4]
[(1,3),(2,4),(5,7),(6,8)] => [(1,3),(2,4),(5,7),(6,8)] => [3,4,1,2,7,8,5,6] => [2,4,1,3,6,8,5,7] => [7,5,8,6,3,1,4,2]
[(1,2),(3,4),(5,7),(6,8)] => [(1,3),(2,4),(5,6),(7,8)] => [3,4,1,2,6,5,8,7] => [2,4,1,3,6,5,8,7] => [7,5,8,6,3,4,1,2]
[(1,2),(3,4),(5,8),(6,7)] => [(1,4),(2,3),(5,6),(7,8)] => [4,3,2,1,6,5,8,7] => [4,3,2,1,6,5,8,7] => [5,6,7,8,3,4,1,2]
[(1,3),(2,4),(5,8),(6,7)] => [(1,4),(2,3),(5,7),(6,8)] => [4,3,2,1,7,8,5,6] => [4,3,2,1,6,8,5,7] => [5,6,7,8,3,1,4,2]
[(1,4),(2,3),(5,8),(6,7)] => [(1,4),(2,3),(5,8),(6,7)] => [4,3,2,1,8,7,6,5] => [4,3,2,1,8,7,6,5] => [5,6,7,8,1,2,3,4]
[(1,5),(2,3),(4,8),(6,7)] => [(1,5),(2,3),(4,8),(6,7)] => [5,3,2,8,1,7,6,4] => [7,6,4,8,1,5,3,2] => [2,3,5,1,8,4,6,7]
[(1,6),(2,3),(4,8),(5,7)] => [(1,5),(2,4),(3,8),(6,7)] => [5,4,8,2,1,7,6,3] => [7,6,3,8,2,1,5,4] => [2,3,6,1,7,8,4,5]
[(1,7),(2,3),(4,8),(5,6)] => [(1,5),(2,8),(3,4),(6,7)] => [5,8,4,3,1,7,6,2] => [7,6,2,8,4,3,1,5] => [2,3,7,1,5,6,8,4]
[(1,8),(2,3),(4,7),(5,6)] => [(1,8),(2,5),(3,4),(6,7)] => [8,5,4,3,2,7,6,1] => [7,6,1,8,5,4,3,2] => [2,3,8,1,4,5,6,7]
[(1,8),(2,4),(3,7),(5,6)] => [(1,8),(2,6),(3,4),(5,7)] => [8,6,4,3,7,2,5,1] => [5,1,7,2,8,6,4,3] => [4,8,2,7,1,3,5,6]
[(1,7),(2,4),(3,8),(5,6)] => [(1,6),(2,8),(3,4),(5,7)] => [6,8,4,3,7,1,5,2] => [5,2,7,1,8,4,3,6] => [4,7,2,8,1,5,6,3]
[(1,6),(2,4),(3,8),(5,7)] => [(1,6),(2,4),(3,8),(5,7)] => [6,4,8,2,7,1,5,3] => [5,3,7,1,8,2,6,4] => [4,6,2,8,1,7,3,5]
[(1,5),(2,4),(3,8),(6,7)] => [(1,6),(2,3),(4,8),(5,7)] => [6,3,2,8,7,1,5,4] => [5,4,8,7,1,6,3,2] => [4,5,1,2,8,3,6,7]
[(1,4),(2,5),(3,8),(6,7)] => [(1,6),(2,3),(4,7),(5,8)] => [6,3,2,7,8,1,4,5] => [4,5,8,1,6,3,2,7] => [5,4,1,8,3,6,7,2]
>>> Load all 180 entries. <<<Map
reverse
Description
The reverse of a perfect matching of $\{1,2,...,n\}$.
This is the perfect matching obtained by replacing $i$ by $n+1-i$.
This is the perfect matching obtained by replacing $i$ by $n+1-i$.
Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.
Map
invert Laguerre heap
Description
The permutation obtained by inverting the corresponding Laguerre heap, according to Viennot.
Let $\pi$ be a permutation. Following Viennot [1], we associate to $\pi$ a heap of pieces, by considering each decreasing run $(\pi_i, \pi_{i+1}, \dots, \pi_j)$ of $\pi$ as one piece, beginning with the left most run. Two pieces commute if and only if the minimal element of one piece is larger than the maximal element of the other piece.
This map yields the permutation corresponding to the heap obtained by reversing the reading direction of the heap.
Equivalently, this is the permutation obtained by flipping the noncrossing arc diagram of Reading [2] vertically.
By definition, this map preserves the set of decreasing runs.
Let $\pi$ be a permutation. Following Viennot [1], we associate to $\pi$ a heap of pieces, by considering each decreasing run $(\pi_i, \pi_{i+1}, \dots, \pi_j)$ of $\pi$ as one piece, beginning with the left most run. Two pieces commute if and only if the minimal element of one piece is larger than the maximal element of the other piece.
This map yields the permutation corresponding to the heap obtained by reversing the reading direction of the heap.
Equivalently, this is the permutation obtained by flipping the noncrossing arc diagram of Reading [2] vertically.
By definition, this map preserves the set of decreasing runs.
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 $\tau(i) = n+1-\sigma(i)$
searching the database
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