Identifier
Mp00058: Perfect matchings to permutationPermutations
Mp00235: Permutations descent views to invisible inversion bottomsPermutations
Mp00252: Permutations restrictionPermutations
Images
[(1,2)] => [2,1] => [2,1] => [1]
[(1,2),(3,4)] => [2,1,4,3] => [2,1,4,3] => [2,1,3]
[(1,3),(2,4)] => [3,4,1,2] => [4,1,3,2] => [1,3,2]
[(1,4),(2,3)] => [4,3,2,1] => [2,3,4,1] => [2,3,1]
[(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [2,1,4,3,6,5] => [2,1,4,3,5]
[(1,3),(2,4),(5,6)] => [3,4,1,2,6,5] => [4,1,3,2,6,5] => [4,1,3,2,5]
[(1,4),(2,3),(5,6)] => [4,3,2,1,6,5] => [2,3,4,1,6,5] => [2,3,4,1,5]
[(1,5),(2,3),(4,6)] => [5,3,2,6,1,4] => [6,3,5,1,2,4] => [3,5,1,2,4]
[(1,6),(2,3),(4,5)] => [6,3,2,5,4,1] => [4,3,5,6,1,2] => [4,3,5,1,2]
[(1,6),(2,4),(3,5)] => [6,4,5,2,3,1] => [3,5,2,6,4,1] => [3,5,2,4,1]
[(1,5),(2,4),(3,6)] => [5,4,6,2,1,3] => [2,6,1,5,4,3] => [2,1,5,4,3]
[(1,4),(2,5),(3,6)] => [4,5,6,1,2,3] => [6,1,2,4,5,3] => [1,2,4,5,3]
[(1,3),(2,5),(4,6)] => [3,5,1,6,2,4] => [5,6,3,2,1,4] => [5,3,2,1,4]
[(1,2),(3,5),(4,6)] => [2,1,5,6,3,4] => [2,1,6,3,5,4] => [2,1,3,5,4]
[(1,2),(3,6),(4,5)] => [2,1,6,5,4,3] => [2,1,4,5,6,3] => [2,1,4,5,3]
[(1,3),(2,6),(4,5)] => [3,6,1,5,4,2] => [6,4,3,5,1,2] => [4,3,5,1,2]
[(1,4),(2,6),(3,5)] => [4,6,5,1,3,2] => [5,3,1,4,6,2] => [5,3,1,4,2]
[(1,5),(2,6),(3,4)] => [5,6,4,3,1,2] => [3,1,4,6,5,2] => [3,1,4,5,2]
[(1,6),(2,5),(3,4)] => [6,5,4,3,2,1] => [2,3,4,5,6,1] => [2,3,4,5,1]
[(1,2),(3,4),(5,6),(7,8)] => [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,7]
[(1,3),(2,4),(5,6),(7,8)] => [3,4,1,2,6,5,8,7] => [4,1,3,2,6,5,8,7] => [4,1,3,2,6,5,7]
[(1,4),(2,3),(5,6),(7,8)] => [4,3,2,1,6,5,8,7] => [2,3,4,1,6,5,8,7] => [2,3,4,1,6,5,7]
[(1,5),(2,3),(4,6),(7,8)] => [5,3,2,6,1,4,8,7] => [6,3,5,1,2,4,8,7] => [6,3,5,1,2,4,7]
[(1,6),(2,3),(4,5),(7,8)] => [6,3,2,5,4,1,8,7] => [4,3,5,6,1,2,8,7] => [4,3,5,6,1,2,7]
[(1,7),(2,3),(4,5),(6,8)] => [7,3,2,5,4,8,1,6] => [8,3,7,5,2,1,4,6] => [3,7,5,2,1,4,6]
[(1,8),(2,3),(4,5),(6,7)] => [8,3,2,5,4,7,6,1] => [6,3,7,5,1,8,4,2] => [6,3,7,5,1,4,2]
[(1,8),(2,4),(3,5),(6,7)] => [8,4,5,2,3,7,6,1] => [6,5,2,7,1,8,3,4] => [6,5,2,7,1,3,4]
[(1,7),(2,4),(3,5),(6,8)] => [7,4,5,2,3,8,1,6] => [8,5,2,7,4,1,3,6] => [5,2,7,4,1,3,6]
[(1,6),(2,4),(3,5),(7,8)] => [6,4,5,2,3,1,8,7] => [3,5,2,6,4,1,8,7] => [3,5,2,6,4,1,7]
[(1,5),(2,4),(3,6),(7,8)] => [5,4,6,2,1,3,8,7] => [2,6,1,5,4,3,8,7] => [2,6,1,5,4,3,7]
[(1,4),(2,5),(3,6),(7,8)] => [4,5,6,1,2,3,8,7] => [6,1,2,4,5,3,8,7] => [6,1,2,4,5,3,7]
[(1,3),(2,5),(4,6),(7,8)] => [3,5,1,6,2,4,8,7] => [5,6,3,2,1,4,8,7] => [5,6,3,2,1,4,7]
[(1,2),(3,5),(4,6),(7,8)] => [2,1,5,6,3,4,8,7] => [2,1,6,3,5,4,8,7] => [2,1,6,3,5,4,7]
[(1,2),(3,6),(4,5),(7,8)] => [2,1,6,5,4,3,8,7] => [2,1,4,5,6,3,8,7] => [2,1,4,5,6,3,7]
[(1,3),(2,6),(4,5),(7,8)] => [3,6,1,5,4,2,8,7] => [6,4,3,5,1,2,8,7] => [6,4,3,5,1,2,7]
[(1,4),(2,6),(3,5),(7,8)] => [4,6,5,1,3,2,8,7] => [5,3,1,4,6,2,8,7] => [5,3,1,4,6,2,7]
[(1,5),(2,6),(3,4),(7,8)] => [5,6,4,3,1,2,8,7] => [3,1,4,6,5,2,8,7] => [3,1,4,6,5,2,7]
[(1,6),(2,5),(3,4),(7,8)] => [6,5,4,3,2,1,8,7] => [2,3,4,5,6,1,8,7] => [2,3,4,5,6,1,7]
[(1,7),(2,5),(3,4),(6,8)] => [7,5,4,3,2,8,1,6] => [8,3,4,5,7,1,2,6] => [3,4,5,7,1,2,6]
[(1,8),(2,5),(3,4),(6,7)] => [8,5,4,3,2,7,6,1] => [6,3,4,5,7,8,1,2] => [6,3,4,5,7,1,2]
[(1,8),(2,6),(3,4),(5,7)] => [8,6,4,3,7,2,5,1] => [5,7,4,6,2,8,3,1] => [5,7,4,6,2,3,1]
[(1,7),(2,6),(3,4),(5,8)] => [7,6,4,3,8,2,1,5] => [2,8,4,6,1,7,3,5] => [2,4,6,1,7,3,5]
[(1,6),(2,7),(3,4),(5,8)] => [6,7,4,3,8,1,2,5] => [8,1,4,7,2,6,3,5] => [1,4,7,2,6,3,5]
[(1,5),(2,7),(3,4),(6,8)] => [5,7,4,3,1,8,2,6] => [3,4,7,8,5,2,1,6] => [3,4,7,5,2,1,6]
[(1,4),(2,7),(3,5),(6,8)] => [4,7,5,1,3,8,2,6] => [5,7,2,4,8,1,3,6] => [5,7,2,4,1,3,6]
[(1,3),(2,7),(4,5),(6,8)] => [3,7,1,5,4,8,2,6] => [7,8,3,5,1,2,4,6] => [7,3,5,1,2,4,6]
[(1,2),(3,7),(4,5),(6,8)] => [2,1,7,5,4,8,3,6] => [2,1,8,5,7,3,4,6] => [2,1,5,7,3,4,6]
[(1,2),(3,8),(4,5),(6,7)] => [2,1,8,5,4,7,6,3] => [2,1,6,5,7,8,3,4] => [2,1,6,5,7,3,4]
[(1,3),(2,8),(4,5),(6,7)] => [3,8,1,5,4,7,6,2] => [8,6,3,5,1,7,4,2] => [6,3,5,1,7,4,2]
[(1,4),(2,8),(3,5),(6,7)] => [4,8,5,1,3,7,6,2] => [5,8,2,4,6,7,3,1] => [5,2,4,6,7,3,1]
[(1,5),(2,8),(3,4),(6,7)] => [5,8,4,3,1,7,6,2] => [3,4,8,6,5,7,1,2] => [3,4,6,5,7,1,2]
[(1,6),(2,8),(3,4),(5,7)] => [6,8,4,3,7,1,5,2] => [7,5,4,8,1,6,3,2] => [7,5,4,1,6,3,2]
[(1,7),(2,8),(3,4),(5,6)] => [7,8,4,3,6,5,1,2] => [5,1,4,6,8,2,7,3] => [5,1,4,6,2,7,3]
[(1,8),(2,7),(3,4),(5,6)] => [8,7,4,3,6,5,2,1] => [2,5,4,6,7,1,8,3] => [2,5,4,6,7,1,3]
[(1,8),(2,7),(3,5),(4,6)] => [8,7,5,6,3,4,2,1] => [2,4,6,3,7,5,8,1] => [2,4,6,3,7,5,1]
[(1,7),(2,8),(3,5),(4,6)] => [7,8,5,6,3,4,1,2] => [4,1,6,3,8,5,7,2] => [4,1,6,3,5,7,2]
[(1,6),(2,8),(3,5),(4,7)] => [6,8,5,7,3,1,4,2] => [3,7,4,2,8,6,5,1] => [3,7,4,2,6,5,1]
[(1,5),(2,8),(3,6),(4,7)] => [5,8,6,7,1,3,4,2] => [7,4,1,3,5,8,6,2] => [7,4,1,3,5,6,2]
[(1,4),(2,8),(3,6),(5,7)] => [4,8,6,1,7,3,5,2] => [6,5,8,4,1,7,3,2] => [6,5,4,1,7,3,2]
[(1,3),(2,8),(4,6),(5,7)] => [3,8,1,6,7,4,5,2] => [8,5,3,7,4,1,6,2] => [5,3,7,4,1,6,2]
[(1,2),(3,8),(4,6),(5,7)] => [2,1,8,6,7,4,5,3] => [2,1,5,7,4,8,6,3] => [2,1,5,7,4,6,3]
[(1,2),(3,7),(4,6),(5,8)] => [2,1,7,6,8,4,3,5] => [2,1,4,8,3,7,6,5] => [2,1,4,3,7,6,5]
[(1,3),(2,7),(4,6),(5,8)] => [3,7,1,6,8,4,2,5] => [7,4,3,8,2,1,6,5] => [7,4,3,2,1,6,5]
[(1,4),(2,7),(3,6),(5,8)] => [4,7,6,1,8,3,2,5] => [6,3,7,4,1,8,2,5] => [6,3,7,4,1,2,5]
[(1,5),(2,7),(3,6),(4,8)] => [5,7,6,8,1,3,2,4] => [8,3,1,2,5,7,6,4] => [3,1,2,5,7,6,4]
[(1,6),(2,7),(3,5),(4,8)] => [6,7,5,8,3,1,2,4] => [3,1,8,2,7,6,5,4] => [3,1,2,7,6,5,4]
[(1,7),(2,6),(3,5),(4,8)] => [7,6,5,8,3,2,1,4] => [2,3,8,1,6,7,5,4] => [2,3,1,6,7,5,4]
[(1,8),(2,6),(3,5),(4,7)] => [8,6,5,7,3,2,4,1] => [4,3,7,2,6,8,5,1] => [4,3,7,2,6,5,1]
[(1,8),(2,5),(3,6),(4,7)] => [8,5,6,7,2,3,4,1] => [4,7,2,3,8,5,6,1] => [4,7,2,3,5,6,1]
[(1,7),(2,5),(3,6),(4,8)] => [7,5,6,8,2,3,1,4] => [3,8,2,1,7,5,6,4] => [3,2,1,7,5,6,4]
[(1,6),(2,5),(3,7),(4,8)] => [6,5,7,8,2,1,3,4] => [2,8,1,3,6,5,7,4] => [2,1,3,6,5,7,4]
[(1,5),(2,6),(3,7),(4,8)] => [5,6,7,8,1,2,3,4] => [8,1,2,3,5,6,7,4] => [1,2,3,5,6,7,4]
[(1,4),(2,6),(3,7),(5,8)] => [4,6,7,1,8,2,3,5] => [7,8,2,4,3,6,1,5] => [7,2,4,3,6,1,5]
[(1,3),(2,6),(4,7),(5,8)] => [3,6,1,7,8,2,4,5] => [6,8,3,2,4,1,7,5] => [6,3,2,4,1,7,5]
[(1,2),(3,6),(4,7),(5,8)] => [2,1,6,7,8,3,4,5] => [2,1,8,3,4,6,7,5] => [2,1,3,4,6,7,5]
[(1,2),(3,5),(4,7),(6,8)] => [2,1,5,7,3,8,4,6] => [2,1,7,8,5,4,3,6] => [2,1,7,5,4,3,6]
[(1,3),(2,5),(4,7),(6,8)] => [3,5,1,7,2,8,4,6] => [5,7,3,8,1,4,2,6] => [5,7,3,1,4,2,6]
[(1,4),(2,5),(3,7),(6,8)] => [4,5,7,1,2,8,3,6] => [7,1,8,4,5,3,2,6] => [7,1,4,5,3,2,6]
[(1,5),(2,4),(3,7),(6,8)] => [5,4,7,2,1,8,3,6] => [2,7,8,5,4,3,1,6] => [2,7,5,4,3,1,6]
[(1,6),(2,4),(3,7),(5,8)] => [6,4,7,2,8,1,3,5] => [8,7,1,6,3,4,2,5] => [7,1,6,3,4,2,5]
[(1,7),(2,4),(3,6),(5,8)] => [7,4,6,2,8,3,1,5] => [3,8,6,7,2,4,1,5] => [3,6,7,2,4,1,5]
[(1,8),(2,4),(3,6),(5,7)] => [8,4,6,2,7,3,5,1] => [5,6,7,8,3,4,2,1] => [5,6,7,3,4,2,1]
[(1,8),(2,3),(4,6),(5,7)] => [8,3,2,6,7,4,5,1] => [5,3,8,7,4,2,6,1] => [5,3,7,4,2,6,1]
[(1,7),(2,3),(4,6),(5,8)] => [7,3,2,6,8,4,1,5] => [4,3,8,7,2,1,6,5] => [4,3,7,2,1,6,5]
[(1,6),(2,3),(4,7),(5,8)] => [6,3,2,7,8,1,4,5] => [8,3,6,1,4,2,7,5] => [3,6,1,4,2,7,5]
[(1,5),(2,3),(4,7),(6,8)] => [5,3,2,7,1,8,4,6] => [7,3,5,8,2,4,1,6] => [7,3,5,2,4,1,6]
[(1,4),(2,3),(5,7),(6,8)] => [4,3,2,1,7,8,5,6] => [2,3,4,1,8,5,7,6] => [2,3,4,1,5,7,6]
[(1,3),(2,4),(5,7),(6,8)] => [3,4,1,2,7,8,5,6] => [4,1,3,2,8,5,7,6] => [4,1,3,2,5,7,6]
[(1,2),(3,4),(5,7),(6,8)] => [2,1,4,3,7,8,5,6] => [2,1,4,3,8,5,7,6] => [2,1,4,3,5,7,6]
[(1,2),(3,4),(5,8),(6,7)] => [2,1,4,3,8,7,6,5] => [2,1,4,3,6,7,8,5] => [2,1,4,3,6,7,5]
[(1,3),(2,4),(5,8),(6,7)] => [3,4,1,2,8,7,6,5] => [4,1,3,2,6,7,8,5] => [4,1,3,2,6,7,5]
[(1,4),(2,3),(5,8),(6,7)] => [4,3,2,1,8,7,6,5] => [2,3,4,1,6,7,8,5] => [2,3,4,1,6,7,5]
[(1,5),(2,3),(4,8),(6,7)] => [5,3,2,8,1,7,6,4] => [8,3,5,6,2,7,1,4] => [3,5,6,2,7,1,4]
[(1,6),(2,3),(4,8),(5,7)] => [6,3,2,8,7,1,5,4] => [7,3,6,5,1,2,8,4] => [7,3,6,5,1,2,4]
[(1,7),(2,3),(4,8),(5,6)] => [7,3,2,8,6,5,1,4] => [5,3,6,1,8,7,2,4] => [5,3,6,1,7,2,4]
[(1,8),(2,3),(4,7),(5,6)] => [8,3,2,7,6,5,4,1] => [4,3,5,6,7,8,1,2] => [4,3,5,6,7,1,2]
[(1,8),(2,4),(3,7),(5,6)] => [8,4,7,2,6,5,3,1] => [3,5,6,7,8,4,2,1] => [3,5,6,7,4,2,1]
[(1,7),(2,4),(3,8),(5,6)] => [7,4,8,2,6,5,1,3] => [5,6,2,8,7,4,1,3] => [5,6,2,7,4,1,3]
[(1,6),(2,4),(3,8),(5,7)] => [6,4,8,2,7,1,5,3] => [7,8,5,6,1,4,2,3] => [7,5,6,1,4,2,3]
[(1,5),(2,4),(3,8),(6,7)] => [5,4,8,2,1,7,6,3] => [2,8,6,5,4,7,1,3] => [2,6,5,4,7,1,3]
[(1,4),(2,5),(3,8),(6,7)] => [4,5,8,1,2,7,6,3] => [8,1,6,4,5,7,2,3] => [1,6,4,5,7,2,3]
>>> Load all 131 entries. <<<
[(1,3),(2,5),(4,8),(6,7)] => [3,5,1,8,2,7,6,4] => [5,8,3,6,1,7,2,4] => [5,3,6,1,7,2,4]
[(1,2),(3,5),(4,8),(6,7)] => [2,1,5,8,3,7,6,4] => [2,1,8,6,5,7,3,4] => [2,1,6,5,7,3,4]
[(1,2),(3,6),(4,8),(5,7)] => [2,1,6,8,7,3,5,4] => [2,1,7,5,3,6,8,4] => [2,1,7,5,3,6,4]
[(1,3),(2,6),(4,8),(5,7)] => [3,6,1,8,7,2,5,4] => [6,7,3,5,2,1,8,4] => [6,7,3,5,2,1,4]
[(1,4),(2,6),(3,8),(5,7)] => [4,6,8,1,7,2,5,3] => [8,7,5,4,2,6,1,3] => [7,5,4,2,6,1,3]
[(1,5),(2,6),(3,8),(4,7)] => [5,6,8,7,1,2,4,3] => [7,1,4,2,5,6,8,3] => [7,1,4,2,5,6,3]
[(1,6),(2,5),(3,8),(4,7)] => [6,5,8,7,2,1,4,3] => [2,7,4,1,6,5,8,3] => [2,7,4,1,6,5,3]
[(1,7),(2,5),(3,8),(4,6)] => [7,5,8,6,2,4,1,3] => [4,6,1,2,8,7,3,5] => [4,6,1,2,7,3,5]
[(1,8),(2,5),(3,7),(4,6)] => [8,5,7,6,2,4,3,1] => [3,4,6,1,7,8,2,5] => [3,4,6,1,7,2,5]
[(1,8),(2,6),(3,7),(4,5)] => [8,6,7,5,4,2,3,1] => [3,4,2,5,7,8,6,1] => [3,4,2,5,7,6,1]
[(1,7),(2,6),(3,8),(4,5)] => [7,6,8,5,4,2,1,3] => [2,4,1,5,8,7,6,3] => [2,4,1,5,7,6,3]
[(1,6),(2,7),(3,8),(4,5)] => [6,7,8,5,4,1,2,3] => [4,1,2,5,8,6,7,3] => [4,1,2,5,6,7,3]
[(1,5),(2,7),(3,8),(4,6)] => [5,7,8,6,1,4,2,3] => [6,4,2,1,5,8,7,3] => [6,4,2,1,5,7,3]
[(1,4),(2,7),(3,8),(5,6)] => [4,7,8,1,6,5,2,3] => [8,5,2,4,6,1,7,3] => [5,2,4,6,1,7,3]
[(1,3),(2,7),(4,8),(5,6)] => [3,7,1,8,6,5,2,4] => [7,5,3,2,6,8,1,4] => [7,5,3,2,6,1,4]
[(1,2),(3,7),(4,8),(5,6)] => [2,1,7,8,6,5,3,4] => [2,1,5,3,6,8,7,4] => [2,1,5,3,6,7,4]
[(1,2),(3,8),(4,7),(5,6)] => [2,1,8,7,6,5,4,3] => [2,1,4,5,6,7,8,3] => [2,1,4,5,6,7,3]
[(1,3),(2,8),(4,7),(5,6)] => [3,8,1,7,6,5,4,2] => [8,4,3,5,6,7,1,2] => [4,3,5,6,7,1,2]
[(1,4),(2,8),(3,7),(5,6)] => [4,8,7,1,6,5,3,2] => [7,3,5,4,6,1,8,2] => [7,3,5,4,6,1,2]
[(1,5),(2,8),(3,7),(4,6)] => [5,8,7,6,1,4,3,2] => [6,3,4,1,5,7,8,2] => [6,3,4,1,5,7,2]
[(1,6),(2,8),(3,7),(4,5)] => [6,8,7,5,4,1,3,2] => [4,3,1,5,7,6,8,2] => [4,3,1,5,7,6,2]
[(1,7),(2,8),(3,6),(4,5)] => [7,8,6,5,4,3,1,2] => [3,1,4,5,6,8,7,2] => [3,1,4,5,6,7,2]
[(1,8),(2,7),(3,6),(4,5)] => [8,7,6,5,4,3,2,1] => [2,3,4,5,6,7,8,1] => [2,3,4,5,6,7,1]
[(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [2,1,4,3,6,5,8,7,10,9] => [2,1,4,3,6,5,8,7,9]
[(1,6),(2,4),(3,5),(7,8),(9,10)] => [6,4,5,2,3,1,8,7,10,9] => [3,5,2,6,4,1,8,7,10,9] => [3,5,2,6,4,1,8,7,9]
[(1,2),(3,8),(4,6),(5,7),(9,10)] => [2,1,8,6,7,4,5,3,10,9] => [2,1,5,7,4,8,6,3,10,9] => [2,1,5,7,4,8,6,3,9]
[(1,10),(2,8),(3,9),(4,6),(5,7)] => [10,8,9,6,7,4,5,2,3,1] => [3,5,2,7,4,9,6,10,8,1] => [3,5,2,7,4,9,6,8,1]
[(1,2),(3,4),(5,10),(6,8),(7,9)] => [2,1,4,3,10,8,9,6,7,5] => [2,1,4,3,7,9,6,10,8,5] => [2,1,4,3,7,9,6,8,5]
[(1,10),(2,9),(3,8),(4,7),(5,6)] => [10,9,8,7,6,5,4,3,2,1] => [2,3,4,5,6,7,8,9,10,1] => [2,3,4,5,6,7,8,9,1]
[(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,1,4,3,6,5,8,7,10,9,11]
Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.
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.
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 maxima 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 weak deficiency positions of $\chi(\pi)$, and
  • the set of minimal elements in the decreasing runs of $\pi$ is the set of weak deficiency values of $\chi(\pi)$.
Map
restriction
Description
The permutation obtained by removing the largest letter.
This map is undefined for the empty permutation.