Identifier
Mp00058:
Perfect matchings
—to permutation⟶
Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00237: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00237: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Images
[(1,2)] => [2,1] => [2,1] => [2,1] => [2,1]
[(1,2),(3,4)] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3]
[(1,3),(2,4)] => [3,4,1,2] => [3,4,1,2] => [3,1,4,2] => [3,4,1,2]
[(1,4),(2,3)] => [4,3,2,1] => [4,3,2,1] => [3,2,4,1] => [4,3,2,1]
[(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [2,1,6,5,4,3] => [2,1,5,4,6,3] => [2,1,6,5,4,3]
[(1,3),(2,4),(5,6)] => [3,4,1,2,6,5] => [3,6,1,5,4,2] => [3,1,5,4,6,2] => [3,6,1,5,4,2]
[(1,4),(2,3),(5,6)] => [4,3,2,1,6,5] => [4,3,2,1,6,5] => [3,2,4,1,6,5] => [4,3,2,1,6,5]
[(1,5),(2,3),(4,6)] => [5,3,2,6,1,4] => [5,3,2,6,1,4] => [3,2,5,1,6,4] => [5,3,2,6,1,4]
[(1,6),(2,3),(4,5)] => [6,3,2,5,4,1] => [6,3,2,5,4,1] => [3,2,5,4,6,1] => [6,3,2,5,4,1]
[(1,6),(2,4),(3,5)] => [6,4,5,2,3,1] => [6,4,5,2,3,1] => [4,2,5,3,6,1] => [6,4,5,2,3,1]
[(1,5),(2,4),(3,6)] => [5,4,6,2,1,3] => [5,4,6,2,1,3] => [4,2,5,1,6,3] => [5,4,6,2,1,3]
[(1,4),(2,5),(3,6)] => [4,5,6,1,2,3] => [4,6,5,1,3,2] => [4,1,5,3,6,2] => [4,6,5,1,3,2]
[(1,3),(2,5),(4,6)] => [3,5,1,6,2,4] => [3,6,1,5,4,2] => [3,1,5,4,6,2] => [3,6,1,5,4,2]
[(1,2),(3,5),(4,6)] => [2,1,5,6,3,4] => [2,1,6,5,4,3] => [2,1,5,4,6,3] => [2,1,6,5,4,3]
[(1,2),(3,6),(4,5)] => [2,1,6,5,4,3] => [2,1,6,5,4,3] => [2,1,5,4,6,3] => [2,1,6,5,4,3]
[(1,3),(2,6),(4,5)] => [3,6,1,5,4,2] => [3,6,1,5,4,2] => [3,1,5,4,6,2] => [3,6,1,5,4,2]
[(1,4),(2,6),(3,5)] => [4,6,5,1,3,2] => [4,6,5,1,3,2] => [4,1,5,3,6,2] => [4,6,5,1,3,2]
[(1,5),(2,6),(3,4)] => [5,6,4,3,1,2] => [5,6,4,3,1,2] => [4,3,5,1,6,2] => [5,6,4,3,1,2]
[(1,6),(2,5),(3,4)] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => [4,3,5,2,6,1] => [6,5,4,3,2,1]
[(1,2),(3,4),(5,6),(7,8)] => [2,1,4,3,6,5,8,7] => [2,1,8,7,6,5,4,3] => [2,1,6,5,7,4,8,3] => [2,1,8,7,6,5,4,3]
[(1,3),(2,4),(5,6),(7,8)] => [3,4,1,2,6,5,8,7] => [3,8,1,7,6,5,4,2] => [3,1,6,5,7,4,8,2] => [3,8,1,7,6,5,4,2]
[(1,4),(2,3),(5,6),(7,8)] => [4,3,2,1,6,5,8,7] => [4,3,2,1,8,7,6,5] => [3,2,4,1,7,6,8,5] => [4,3,2,1,8,7,6,5]
[(1,5),(2,3),(4,6),(7,8)] => [5,3,2,6,1,4,8,7] => [5,3,2,8,1,7,6,4] => [3,2,5,1,7,6,8,4] => [5,3,2,8,1,7,6,4]
[(1,6),(2,3),(4,5),(7,8)] => [6,3,2,5,4,1,8,7] => [6,3,2,8,7,1,5,4] => [3,2,6,1,7,5,8,4] => [6,3,2,8,7,1,5,4]
[(1,7),(2,3),(4,5),(6,8)] => [7,3,2,5,4,8,1,6] => [7,3,2,8,6,5,1,4] => [3,2,6,5,7,1,8,4] => [7,3,2,8,6,5,1,4]
[(1,8),(2,3),(4,5),(6,7)] => [8,3,2,5,4,7,6,1] => [8,3,2,7,6,5,4,1] => [3,2,6,5,7,4,8,1] => [8,3,2,7,6,5,4,1]
[(1,8),(2,4),(3,5),(6,7)] => [8,4,5,2,3,7,6,1] => [8,4,7,2,6,5,3,1] => [4,2,6,5,7,3,8,1] => [8,4,7,2,6,5,3,1]
[(1,7),(2,4),(3,5),(6,8)] => [7,4,5,2,3,8,1,6] => [7,4,8,2,6,5,1,3] => [4,2,6,5,7,1,8,3] => [7,4,8,2,6,5,1,3]
[(1,6),(2,4),(3,5),(7,8)] => [6,4,5,2,3,1,8,7] => [6,4,8,2,7,1,5,3] => [4,2,6,1,7,5,8,3] => [6,4,8,2,7,1,5,3]
[(1,5),(2,4),(3,6),(7,8)] => [5,4,6,2,1,3,8,7] => [5,4,8,2,1,7,6,3] => [4,2,5,1,7,6,8,3] => [5,4,8,2,1,7,6,3]
[(1,4),(2,5),(3,6),(7,8)] => [4,5,6,1,2,3,8,7] => [4,8,7,1,6,5,3,2] => [4,1,6,5,7,3,8,2] => [4,8,7,1,6,5,3,2]
[(1,3),(2,5),(4,6),(7,8)] => [3,5,1,6,2,4,8,7] => [3,8,1,7,6,5,4,2] => [3,1,6,5,7,4,8,2] => [3,8,1,7,6,5,4,2]
[(1,2),(3,5),(4,6),(7,8)] => [2,1,5,6,3,4,8,7] => [2,1,8,7,6,5,4,3] => [2,1,6,5,7,4,8,3] => [2,1,8,7,6,5,4,3]
[(1,2),(3,6),(4,5),(7,8)] => [2,1,6,5,4,3,8,7] => [2,1,8,7,6,5,4,3] => [2,1,6,5,7,4,8,3] => [2,1,8,7,6,5,4,3]
[(1,3),(2,6),(4,5),(7,8)] => [3,6,1,5,4,2,8,7] => [3,8,1,7,6,5,4,2] => [3,1,6,5,7,4,8,2] => [3,8,1,7,6,5,4,2]
[(1,4),(2,6),(3,5),(7,8)] => [4,6,5,1,3,2,8,7] => [4,8,7,1,6,5,3,2] => [4,1,6,5,7,3,8,2] => [4,8,7,1,6,5,3,2]
[(1,5),(2,6),(3,4),(7,8)] => [5,6,4,3,1,2,8,7] => [5,8,4,3,1,7,6,2] => [4,3,5,1,7,6,8,2] => [5,8,4,3,1,7,6,2]
[(1,6),(2,5),(3,4),(7,8)] => [6,5,4,3,2,1,8,7] => [6,5,4,3,2,1,8,7] => [4,3,5,2,6,1,8,7] => [6,5,4,3,2,1,8,7]
[(1,7),(2,5),(3,4),(6,8)] => [7,5,4,3,2,8,1,6] => [7,5,4,3,2,8,1,6] => [4,3,5,2,7,1,8,6] => [7,5,4,3,2,8,1,6]
[(1,8),(2,5),(3,4),(6,7)] => [8,5,4,3,2,7,6,1] => [8,5,4,3,2,7,6,1] => [4,3,5,2,7,6,8,1] => [8,5,4,3,2,7,6,1]
[(1,8),(2,6),(3,4),(5,7)] => [8,6,4,3,7,2,5,1] => [8,6,4,3,7,2,5,1] => [4,3,6,2,7,5,8,1] => [8,6,4,3,7,2,5,1]
[(1,7),(2,6),(3,4),(5,8)] => [7,6,4,3,8,2,1,5] => [7,6,4,3,8,2,1,5] => [4,3,6,2,7,1,8,5] => [7,6,4,3,8,2,1,5]
[(1,6),(2,7),(3,4),(5,8)] => [6,7,4,3,8,1,2,5] => [6,8,4,3,7,1,5,2] => [4,3,6,1,7,5,8,2] => [6,8,4,3,7,1,5,2]
[(1,5),(2,7),(3,4),(6,8)] => [5,7,4,3,1,8,2,6] => [5,8,4,3,1,7,6,2] => [4,3,5,1,7,6,8,2] => [5,8,4,3,1,7,6,2]
[(1,4),(2,7),(3,5),(6,8)] => [4,7,5,1,3,8,2,6] => [4,8,7,1,6,5,3,2] => [4,1,6,5,7,3,8,2] => [4,8,7,1,6,5,3,2]
[(1,3),(2,7),(4,5),(6,8)] => [3,7,1,5,4,8,2,6] => [3,8,1,7,6,5,4,2] => [3,1,6,5,7,4,8,2] => [3,8,1,7,6,5,4,2]
[(1,2),(3,7),(4,5),(6,8)] => [2,1,7,5,4,8,3,6] => [2,1,8,7,6,5,4,3] => [2,1,6,5,7,4,8,3] => [2,1,8,7,6,5,4,3]
[(1,2),(3,8),(4,5),(6,7)] => [2,1,8,5,4,7,6,3] => [2,1,8,7,6,5,4,3] => [2,1,6,5,7,4,8,3] => [2,1,8,7,6,5,4,3]
[(1,3),(2,8),(4,5),(6,7)] => [3,8,1,5,4,7,6,2] => [3,8,1,7,6,5,4,2] => [3,1,6,5,7,4,8,2] => [3,8,1,7,6,5,4,2]
[(1,4),(2,8),(3,5),(6,7)] => [4,8,5,1,3,7,6,2] => [4,8,7,1,6,5,3,2] => [4,1,6,5,7,3,8,2] => [4,8,7,1,6,5,3,2]
[(1,5),(2,8),(3,4),(6,7)] => [5,8,4,3,1,7,6,2] => [5,8,4,3,1,7,6,2] => [4,3,5,1,7,6,8,2] => [5,8,4,3,1,7,6,2]
[(1,6),(2,8),(3,4),(5,7)] => [6,8,4,3,7,1,5,2] => [6,8,4,3,7,1,5,2] => [4,3,6,1,7,5,8,2] => [6,8,4,3,7,1,5,2]
[(1,7),(2,8),(3,4),(5,6)] => [7,8,4,3,6,5,1,2] => [7,8,4,3,6,5,1,2] => [4,3,6,5,7,1,8,2] => [7,8,4,3,6,5,1,2]
[(1,8),(2,7),(3,4),(5,6)] => [8,7,4,3,6,5,2,1] => [8,7,4,3,6,5,2,1] => [4,3,6,5,7,2,8,1] => [8,7,4,3,6,5,2,1]
[(1,8),(2,7),(3,5),(4,6)] => [8,7,5,6,3,4,2,1] => [8,7,5,6,3,4,2,1] => [5,3,6,4,7,2,8,1] => [8,7,5,6,3,4,2,1]
[(1,7),(2,8),(3,5),(4,6)] => [7,8,5,6,3,4,1,2] => [7,8,5,6,3,4,1,2] => [5,3,6,4,7,1,8,2] => [7,8,5,6,3,4,1,2]
[(1,6),(2,8),(3,5),(4,7)] => [6,8,5,7,3,1,4,2] => [6,8,5,7,3,1,4,2] => [5,3,6,1,7,4,8,2] => [6,8,5,7,3,1,4,2]
[(1,5),(2,8),(3,6),(4,7)] => [5,8,6,7,1,3,4,2] => [5,8,7,6,1,4,3,2] => [5,1,6,4,7,3,8,2] => [5,8,7,6,1,4,3,2]
[(1,4),(2,8),(3,6),(5,7)] => [4,8,6,1,7,3,5,2] => [4,8,7,1,6,5,3,2] => [4,1,6,5,7,3,8,2] => [4,8,7,1,6,5,3,2]
[(1,3),(2,8),(4,6),(5,7)] => [3,8,1,6,7,4,5,2] => [3,8,1,7,6,5,4,2] => [3,1,6,5,7,4,8,2] => [3,8,1,7,6,5,4,2]
[(1,2),(3,8),(4,6),(5,7)] => [2,1,8,6,7,4,5,3] => [2,1,8,7,6,5,4,3] => [2,1,6,5,7,4,8,3] => [2,1,8,7,6,5,4,3]
[(1,2),(3,7),(4,6),(5,8)] => [2,1,7,6,8,4,3,5] => [2,1,8,7,6,5,4,3] => [2,1,6,5,7,4,8,3] => [2,1,8,7,6,5,4,3]
[(1,3),(2,7),(4,6),(5,8)] => [3,7,1,6,8,4,2,5] => [3,8,1,7,6,5,4,2] => [3,1,6,5,7,4,8,2] => [3,8,1,7,6,5,4,2]
[(1,4),(2,7),(3,6),(5,8)] => [4,7,6,1,8,3,2,5] => [4,8,7,1,6,5,3,2] => [4,1,6,5,7,3,8,2] => [4,8,7,1,6,5,3,2]
[(1,5),(2,7),(3,6),(4,8)] => [5,7,6,8,1,3,2,4] => [5,8,7,6,1,4,3,2] => [5,1,6,4,7,3,8,2] => [5,8,7,6,1,4,3,2]
[(1,6),(2,7),(3,5),(4,8)] => [6,7,5,8,3,1,2,4] => [6,8,5,7,3,1,4,2] => [5,3,6,1,7,4,8,2] => [6,8,5,7,3,1,4,2]
[(1,7),(2,6),(3,5),(4,8)] => [7,6,5,8,3,2,1,4] => [7,6,5,8,3,2,1,4] => [5,3,6,2,7,1,8,4] => [7,6,5,8,3,2,1,4]
[(1,8),(2,6),(3,5),(4,7)] => [8,6,5,7,3,2,4,1] => [8,6,5,7,3,2,4,1] => [5,3,6,2,7,4,8,1] => [8,6,5,7,3,2,4,1]
[(1,8),(2,5),(3,6),(4,7)] => [8,5,6,7,2,3,4,1] => [8,5,7,6,2,4,3,1] => [5,2,6,4,7,3,8,1] => [8,5,7,6,2,4,3,1]
[(1,7),(2,5),(3,6),(4,8)] => [7,5,6,8,2,3,1,4] => [7,5,8,6,2,4,1,3] => [5,2,6,4,7,1,8,3] => [7,5,8,6,2,4,1,3]
[(1,6),(2,5),(3,7),(4,8)] => [6,5,7,8,2,1,3,4] => [6,5,8,7,2,1,4,3] => [5,2,6,1,7,4,8,3] => [6,5,8,7,2,1,4,3]
[(1,5),(2,6),(3,7),(4,8)] => [5,6,7,8,1,2,3,4] => [5,8,7,6,1,4,3,2] => [5,1,6,4,7,3,8,2] => [5,8,7,6,1,4,3,2]
[(1,4),(2,6),(3,7),(5,8)] => [4,6,7,1,8,2,3,5] => [4,8,7,1,6,5,3,2] => [4,1,6,5,7,3,8,2] => [4,8,7,1,6,5,3,2]
[(1,3),(2,6),(4,7),(5,8)] => [3,6,1,7,8,2,4,5] => [3,8,1,7,6,5,4,2] => [3,1,6,5,7,4,8,2] => [3,8,1,7,6,5,4,2]
[(1,2),(3,6),(4,7),(5,8)] => [2,1,6,7,8,3,4,5] => [2,1,8,7,6,5,4,3] => [2,1,6,5,7,4,8,3] => [2,1,8,7,6,5,4,3]
[(1,2),(3,5),(4,7),(6,8)] => [2,1,5,7,3,8,4,6] => [2,1,8,7,6,5,4,3] => [2,1,6,5,7,4,8,3] => [2,1,8,7,6,5,4,3]
[(1,3),(2,5),(4,7),(6,8)] => [3,5,1,7,2,8,4,6] => [3,8,1,7,6,5,4,2] => [3,1,6,5,7,4,8,2] => [3,8,1,7,6,5,4,2]
[(1,4),(2,5),(3,7),(6,8)] => [4,5,7,1,2,8,3,6] => [4,8,7,1,6,5,3,2] => [4,1,6,5,7,3,8,2] => [4,8,7,1,6,5,3,2]
[(1,5),(2,4),(3,7),(6,8)] => [5,4,7,2,1,8,3,6] => [5,4,8,2,1,7,6,3] => [4,2,5,1,7,6,8,3] => [5,4,8,2,1,7,6,3]
[(1,6),(2,4),(3,7),(5,8)] => [6,4,7,2,8,1,3,5] => [6,4,8,2,7,1,5,3] => [4,2,6,1,7,5,8,3] => [6,4,8,2,7,1,5,3]
[(1,7),(2,4),(3,6),(5,8)] => [7,4,6,2,8,3,1,5] => [7,4,8,2,6,5,1,3] => [4,2,6,5,7,1,8,3] => [7,4,8,2,6,5,1,3]
[(1,8),(2,4),(3,6),(5,7)] => [8,4,6,2,7,3,5,1] => [8,4,7,2,6,5,3,1] => [4,2,6,5,7,3,8,1] => [8,4,7,2,6,5,3,1]
[(1,8),(2,3),(4,6),(5,7)] => [8,3,2,6,7,4,5,1] => [8,3,2,7,6,5,4,1] => [3,2,6,5,7,4,8,1] => [8,3,2,7,6,5,4,1]
[(1,7),(2,3),(4,6),(5,8)] => [7,3,2,6,8,4,1,5] => [7,3,2,8,6,5,1,4] => [3,2,6,5,7,1,8,4] => [7,3,2,8,6,5,1,4]
[(1,6),(2,3),(4,7),(5,8)] => [6,3,2,7,8,1,4,5] => [6,3,2,8,7,1,5,4] => [3,2,6,1,7,5,8,4] => [6,3,2,8,7,1,5,4]
[(1,5),(2,3),(4,7),(6,8)] => [5,3,2,7,1,8,4,6] => [5,3,2,8,1,7,6,4] => [3,2,5,1,7,6,8,4] => [5,3,2,8,1,7,6,4]
[(1,4),(2,3),(5,7),(6,8)] => [4,3,2,1,7,8,5,6] => [4,3,2,1,8,7,6,5] => [3,2,4,1,7,6,8,5] => [4,3,2,1,8,7,6,5]
[(1,3),(2,4),(5,7),(6,8)] => [3,4,1,2,7,8,5,6] => [3,8,1,7,6,5,4,2] => [3,1,6,5,7,4,8,2] => [3,8,1,7,6,5,4,2]
[(1,2),(3,4),(5,7),(6,8)] => [2,1,4,3,7,8,5,6] => [2,1,8,7,6,5,4,3] => [2,1,6,5,7,4,8,3] => [2,1,8,7,6,5,4,3]
[(1,2),(3,4),(5,8),(6,7)] => [2,1,4,3,8,7,6,5] => [2,1,8,7,6,5,4,3] => [2,1,6,5,7,4,8,3] => [2,1,8,7,6,5,4,3]
[(1,3),(2,4),(5,8),(6,7)] => [3,4,1,2,8,7,6,5] => [3,8,1,7,6,5,4,2] => [3,1,6,5,7,4,8,2] => [3,8,1,7,6,5,4,2]
[(1,4),(2,3),(5,8),(6,7)] => [4,3,2,1,8,7,6,5] => [4,3,2,1,8,7,6,5] => [3,2,4,1,7,6,8,5] => [4,3,2,1,8,7,6,5]
[(1,5),(2,3),(4,8),(6,7)] => [5,3,2,8,1,7,6,4] => [5,3,2,8,1,7,6,4] => [3,2,5,1,7,6,8,4] => [5,3,2,8,1,7,6,4]
[(1,6),(2,3),(4,8),(5,7)] => [6,3,2,8,7,1,5,4] => [6,3,2,8,7,1,5,4] => [3,2,6,1,7,5,8,4] => [6,3,2,8,7,1,5,4]
[(1,7),(2,3),(4,8),(5,6)] => [7,3,2,8,6,5,1,4] => [7,3,2,8,6,5,1,4] => [3,2,6,5,7,1,8,4] => [7,3,2,8,6,5,1,4]
[(1,8),(2,3),(4,7),(5,6)] => [8,3,2,7,6,5,4,1] => [8,3,2,7,6,5,4,1] => [3,2,6,5,7,4,8,1] => [8,3,2,7,6,5,4,1]
[(1,8),(2,4),(3,7),(5,6)] => [8,4,7,2,6,5,3,1] => [8,4,7,2,6,5,3,1] => [4,2,6,5,7,3,8,1] => [8,4,7,2,6,5,3,1]
[(1,7),(2,4),(3,8),(5,6)] => [7,4,8,2,6,5,1,3] => [7,4,8,2,6,5,1,3] => [4,2,6,5,7,1,8,3] => [7,4,8,2,6,5,1,3]
[(1,6),(2,4),(3,8),(5,7)] => [6,4,8,2,7,1,5,3] => [6,4,8,2,7,1,5,3] => [4,2,6,1,7,5,8,3] => [6,4,8,2,7,1,5,3]
[(1,5),(2,4),(3,8),(6,7)] => [5,4,8,2,1,7,6,3] => [5,4,8,2,1,7,6,3] => [4,2,5,1,7,6,8,3] => [5,4,8,2,1,7,6,3]
[(1,4),(2,5),(3,8),(6,7)] => [4,5,8,1,2,7,6,3] => [4,8,7,1,6,5,3,2] => [4,1,6,5,7,3,8,2] => [4,8,7,1,6,5,3,2]
>>> Load all 127 entries. <<<Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.
Map
Simion-Schmidt map
Description
The Simion-Schmidt map sends any permutation to a $123$-avoiding permutation.
Details can be found in [1].
In particular, this is a bijection between $132$-avoiding permutations and $123$-avoiding permutations, see [1, Proposition 19].
Details can be found in [1].
In particular, this is a bijection between $132$-avoiding permutations and $123$-avoiding permutations, see [1, Proposition 19].
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
descent views to invisible inversion bottoms
Description
Return a permutation whose multiset of invisible inversion bottoms is the multiset of descent views of the given permutation.
This map is similar to Mp00235descent views to invisible inversion bottoms, but different beginning with permutations of six elements.
An invisible inversion of a permutation $\sigma$ is a pair $i < j$ such that $i < \sigma(j) < \sigma(i)$. The element $\sigma(j)$ is then an invisible inversion bottom.
A descent view in a permutation $\pi$ is an element $\pi(j)$ such that $\pi(i+1) < \pi(j) < \pi(i)$, and additionally the smallest element in the decreasing run containing $\pi(i)$ is smaller than the smallest element in the decreasing run containing $\pi(j)$.
This map is a bijection $\chi:\mathfrak S_n \to \mathfrak S_n$, such that
This map is similar to Mp00235descent views to invisible inversion bottoms, but different beginning with permutations of six elements.
An invisible inversion of a permutation $\sigma$ is a pair $i < j$ such that $i < \sigma(j) < \sigma(i)$. The element $\sigma(j)$ is then an invisible inversion bottom.
A descent view in a permutation $\pi$ is an element $\pi(j)$ such that $\pi(i+1) < \pi(j) < \pi(i)$, and additionally the smallest element in the decreasing run containing $\pi(i)$ is smaller than the smallest element in the decreasing run containing $\pi(j)$.
This map is a bijection $\chi:\mathfrak S_n \to \mathfrak S_n$, such that
- the multiset of descent views in $\pi$ is the multiset of invisible inversion bottoms in $\chi(\pi)$,
- the set of left-to-right maximima of $\pi$ is the set of maximal elements in the cycles of $\chi(\pi)$,
- the set of global ascent of $\pi$ is the set of global ascent of $\chi(\pi)$,
- the set of maximal elements in the decreasing runs of $\pi$ is the set of deficiency positions of $\chi(\pi)$, and
- the set of minimal elements in the decreasing runs of $\pi$ is the set of deficiency values of $\chi(\pi)$.
searching the database
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