Processing math: 100%

Identifier
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
Images
[(1,2)] => [2,1] => [2,1] => 1
[(1,2),(3,4)] => [2,1,4,3] => [2,1,4,3] => 101
[(1,3),(2,4)] => [3,4,1,2] => [4,1,3,2] => 110
[(1,4),(2,3)] => [3,4,2,1] => [4,2,3,1] => 110
[(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [2,1,4,3,6,5] => 10101
[(1,3),(2,4),(5,6)] => [3,4,1,2,6,5] => [4,1,3,2,6,5] => 11001
[(1,4),(2,3),(5,6)] => [3,4,2,1,6,5] => [4,2,3,1,6,5] => 11001
[(1,5),(2,3),(4,6)] => [3,5,2,6,1,4] => [6,2,3,1,5,4] => 11010
[(1,6),(2,3),(4,5)] => [3,5,2,6,4,1] => [6,4,5,2,3,1] => 11010
[(1,6),(2,4),(3,5)] => [4,5,6,2,3,1] => [6,2,5,3,4,1] => 11100
[(1,5),(2,4),(3,6)] => [4,5,6,2,1,3] => [6,2,5,1,4,3] => 11100
[(1,4),(2,5),(3,6)] => [4,5,6,1,2,3] => [6,1,5,2,4,3] => 11100
[(1,3),(2,5),(4,6)] => [3,5,1,6,2,4] => [6,1,3,2,5,4] => 11010
[(1,2),(3,5),(4,6)] => [2,1,5,6,3,4] => [2,1,6,3,5,4] => 10110
[(1,2),(3,6),(4,5)] => [2,1,5,6,4,3] => [2,1,6,4,5,3] => 10110
[(1,3),(2,6),(4,5)] => [3,5,1,6,4,2] => [6,4,5,1,3,2] => 11010
[(1,4),(2,6),(3,5)] => [4,5,6,1,3,2] => [6,1,5,3,4,2] => 11100
[(1,5),(2,6),(3,4)] => [4,5,6,3,1,2] => [6,3,5,1,4,2] => 11100
[(1,6),(2,5),(3,4)] => [4,5,6,3,2,1] => [6,3,5,2,4,1] => 11100
[(1,2),(3,4),(5,6),(7,8)] => [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => 1010101
[(1,3),(2,4),(5,6),(7,8)] => [3,4,1,2,6,5,8,7] => [4,1,3,2,6,5,8,7] => 1100101
[(1,4),(2,3),(5,6),(7,8)] => [3,4,2,1,6,5,8,7] => [4,2,3,1,6,5,8,7] => 1100101
[(1,5),(2,3),(4,6),(7,8)] => [3,5,2,6,1,4,8,7] => [6,2,3,1,5,4,8,7] => 1101001
[(1,6),(2,3),(4,5),(7,8)] => [3,5,2,6,4,1,8,7] => [6,4,5,2,3,1,8,7] => 1101001
[(1,7),(2,3),(4,5),(6,8)] => [3,5,2,7,4,8,1,6] => [8,4,5,2,3,1,7,6] => 1101010
[(1,8),(2,3),(4,5),(6,7)] => [3,5,2,7,4,8,6,1] => [8,6,7,4,5,2,3,1] => 1101010
[(1,8),(2,4),(3,5),(6,7)] => [4,5,7,2,3,8,6,1] => [8,6,7,2,5,3,4,1] => 1110010
[(1,7),(2,4),(3,5),(6,8)] => [4,5,7,2,3,8,1,6] => [8,2,5,3,4,1,7,6] => 1110010
[(1,6),(2,4),(3,5),(7,8)] => [4,5,6,2,3,1,8,7] => [6,2,5,3,4,1,8,7] => 1110001
[(1,5),(2,4),(3,6),(7,8)] => [4,5,6,2,1,3,8,7] => [6,2,5,1,4,3,8,7] => 1110001
[(1,4),(2,5),(3,6),(7,8)] => [4,5,6,1,2,3,8,7] => [6,1,5,2,4,3,8,7] => 1110001
[(1,3),(2,5),(4,6),(7,8)] => [3,5,1,6,2,4,8,7] => [6,1,3,2,5,4,8,7] => 1101001
[(1,2),(3,5),(4,6),(7,8)] => [2,1,5,6,3,4,8,7] => [2,1,6,3,5,4,8,7] => 1011001
[(1,2),(3,6),(4,5),(7,8)] => [2,1,5,6,4,3,8,7] => [2,1,6,4,5,3,8,7] => 1011001
[(1,3),(2,6),(4,5),(7,8)] => [3,5,1,6,4,2,8,7] => [6,4,5,1,3,2,8,7] => 1101001
[(1,4),(2,6),(3,5),(7,8)] => [4,5,6,1,3,2,8,7] => [6,1,5,3,4,2,8,7] => 1110001
[(1,5),(2,6),(3,4),(7,8)] => [4,5,6,3,1,2,8,7] => [6,3,5,1,4,2,8,7] => 1110001
[(1,6),(2,5),(3,4),(7,8)] => [4,5,6,3,2,1,8,7] => [6,3,5,2,4,1,8,7] => 1110001
[(1,7),(2,5),(3,4),(6,8)] => [4,5,7,3,2,8,1,6] => [8,3,5,2,4,1,7,6] => 1110010
[(1,8),(2,5),(3,4),(6,7)] => [4,5,7,3,2,8,6,1] => [8,6,7,3,5,2,4,1] => 1110010
[(1,8),(2,6),(3,4),(5,7)] => [4,6,7,3,8,2,5,1] => [8,3,7,5,6,2,4,1] => 1110100
[(1,7),(2,6),(3,4),(5,8)] => [4,6,7,3,8,2,1,5] => [8,3,7,2,4,1,6,5] => 1110100
[(1,6),(2,7),(3,4),(5,8)] => [4,6,7,3,8,1,2,5] => [8,3,7,1,4,2,6,5] => 1110100
[(1,5),(2,7),(3,4),(6,8)] => [4,5,7,3,1,8,2,6] => [8,3,5,1,4,2,7,6] => 1110010
[(1,4),(2,7),(3,5),(6,8)] => [4,5,7,1,3,8,2,6] => [8,1,5,3,4,2,7,6] => 1110010
[(1,3),(2,7),(4,5),(6,8)] => [3,5,1,7,4,8,2,6] => [8,4,5,1,3,2,7,6] => 1101010
[(1,2),(3,7),(4,5),(6,8)] => [2,1,5,7,4,8,3,6] => [2,1,8,4,5,3,7,6] => 1011010
[(1,2),(3,8),(4,5),(6,7)] => [2,1,5,7,4,8,6,3] => [2,1,8,6,7,4,5,3] => 1011010
[(1,3),(2,8),(4,5),(6,7)] => [3,5,1,7,4,8,6,2] => [8,6,7,4,5,1,3,2] => 1101010
[(1,4),(2,8),(3,5),(6,7)] => [4,5,7,1,3,8,6,2] => [8,6,7,1,5,3,4,2] => 1110010
[(1,5),(2,8),(3,4),(6,7)] => [4,5,7,3,1,8,6,2] => [8,6,7,3,5,1,4,2] => 1110010
[(1,6),(2,8),(3,4),(5,7)] => [4,6,7,3,8,1,5,2] => [8,3,7,5,6,1,4,2] => 1110100
[(1,7),(2,8),(3,4),(5,6)] => [4,6,7,3,8,5,1,2] => [8,5,7,3,6,1,4,2] => 1110100
[(1,8),(2,7),(3,4),(5,6)] => [4,6,7,3,8,5,2,1] => [8,5,7,3,6,2,4,1] => 1110100
[(1,8),(2,7),(3,5),(4,6)] => [5,6,7,8,3,4,2,1] => [8,3,7,4,6,2,5,1] => 1111000
[(1,7),(2,8),(3,5),(4,6)] => [5,6,7,8,3,4,1,2] => [8,3,7,4,6,1,5,2] => 1111000
[(1,6),(2,8),(3,5),(4,7)] => [5,6,7,8,3,1,4,2] => [8,3,7,1,6,4,5,2] => 1111000
[(1,5),(2,8),(3,6),(4,7)] => [5,6,7,8,1,3,4,2] => [8,1,7,3,6,4,5,2] => 1111000
[(1,4),(2,8),(3,6),(5,7)] => [4,6,7,1,8,3,5,2] => [8,1,7,5,6,3,4,2] => 1110100
[(1,3),(2,8),(4,6),(5,7)] => [3,6,1,7,8,4,5,2] => [8,4,7,5,6,1,3,2] => 1101100
[(1,2),(3,8),(4,6),(5,7)] => [2,1,6,7,8,4,5,3] => [2,1,8,4,7,5,6,3] => 1011100
[(1,2),(3,7),(4,6),(5,8)] => [2,1,6,7,8,4,3,5] => [2,1,8,4,7,3,6,5] => 1011100
[(1,3),(2,7),(4,6),(5,8)] => [3,6,1,7,8,4,2,5] => [8,4,7,1,3,2,6,5] => 1101100
[(1,4),(2,7),(3,6),(5,8)] => [4,6,7,1,8,3,2,5] => [8,1,7,3,4,2,6,5] => 1110100
[(1,5),(2,7),(3,6),(4,8)] => [5,6,7,8,1,3,2,4] => [8,1,7,3,6,2,5,4] => 1111000
[(1,6),(2,7),(3,5),(4,8)] => [5,6,7,8,3,1,2,4] => [8,3,7,1,6,2,5,4] => 1111000
[(1,7),(2,6),(3,5),(4,8)] => [5,6,7,8,3,2,1,4] => [8,3,7,2,6,1,5,4] => 1111000
[(1,8),(2,6),(3,5),(4,7)] => [5,6,7,8,3,2,4,1] => [8,3,7,2,6,4,5,1] => 1111000
[(1,8),(2,5),(3,6),(4,7)] => [5,6,7,8,2,3,4,1] => [8,2,7,3,6,4,5,1] => 1111000
[(1,7),(2,5),(3,6),(4,8)] => [5,6,7,8,2,3,1,4] => [8,2,7,3,6,1,5,4] => 1111000
[(1,6),(2,5),(3,7),(4,8)] => [5,6,7,8,2,1,3,4] => [8,2,7,1,6,3,5,4] => 1111000
[(1,5),(2,6),(3,7),(4,8)] => [5,6,7,8,1,2,3,4] => [8,1,7,2,6,3,5,4] => 1111000
[(1,4),(2,6),(3,7),(5,8)] => [4,6,7,1,8,2,3,5] => [8,1,7,2,4,3,6,5] => 1110100
[(1,3),(2,6),(4,7),(5,8)] => [3,6,1,7,8,2,4,5] => [8,1,3,2,7,4,6,5] => 1101100
[(1,2),(3,6),(4,7),(5,8)] => [2,1,6,7,8,3,4,5] => [2,1,8,3,7,4,6,5] => 1011100
[(1,2),(3,5),(4,7),(6,8)] => [2,1,5,7,3,8,4,6] => [2,1,8,3,5,4,7,6] => 1011010
[(1,3),(2,5),(4,7),(6,8)] => [3,5,1,7,2,8,4,6] => [8,1,3,2,5,4,7,6] => 1101010
[(1,4),(2,5),(3,7),(6,8)] => [4,5,7,1,2,8,3,6] => [8,1,5,2,4,3,7,6] => 1110010
[(1,5),(2,4),(3,7),(6,8)] => [4,5,7,2,1,8,3,6] => [8,2,5,1,4,3,7,6] => 1110010
[(1,6),(2,4),(3,7),(5,8)] => [4,6,7,2,8,1,3,5] => [8,2,7,1,4,3,6,5] => 1110100
[(1,7),(2,4),(3,6),(5,8)] => [4,6,7,2,8,3,1,5] => [8,2,7,3,4,1,6,5] => 1110100
[(1,8),(2,4),(3,6),(5,7)] => [4,6,7,2,8,3,5,1] => [8,2,7,5,6,3,4,1] => 1110100
[(1,8),(2,3),(4,6),(5,7)] => [3,6,2,7,8,4,5,1] => [8,4,7,5,6,2,3,1] => 1101100
[(1,7),(2,3),(4,6),(5,8)] => [3,6,2,7,8,4,1,5] => [8,4,7,2,3,1,6,5] => 1101100
[(1,6),(2,3),(4,7),(5,8)] => [3,6,2,7,8,1,4,5] => [8,2,3,1,7,4,6,5] => 1101100
[(1,5),(2,3),(4,7),(6,8)] => [3,5,2,7,1,8,4,6] => [8,2,3,1,5,4,7,6] => 1101010
[(1,4),(2,3),(5,7),(6,8)] => [3,4,2,1,7,8,5,6] => [4,2,3,1,8,5,7,6] => 1100110
[(1,3),(2,4),(5,7),(6,8)] => [3,4,1,2,7,8,5,6] => [4,1,3,2,8,5,7,6] => 1100110
[(1,2),(3,4),(5,7),(6,8)] => [2,1,4,3,7,8,5,6] => [2,1,4,3,8,5,7,6] => 1010110
[(1,2),(3,4),(5,8),(6,7)] => [2,1,4,3,7,8,6,5] => [2,1,4,3,8,6,7,5] => 1010110
[(1,3),(2,4),(5,8),(6,7)] => [3,4,1,2,7,8,6,5] => [4,1,3,2,8,6,7,5] => 1100110
[(1,4),(2,3),(5,8),(6,7)] => [3,4,2,1,7,8,6,5] => [4,2,3,1,8,6,7,5] => 1100110
[(1,5),(2,3),(4,8),(6,7)] => [3,5,2,7,1,8,6,4] => [8,6,7,2,3,1,5,4] => 1101010
[(1,6),(2,3),(4,8),(5,7)] => [3,6,2,7,8,1,5,4] => [8,2,3,1,7,5,6,4] => 1101100
[(1,7),(2,3),(4,8),(5,6)] => [3,6,2,7,8,5,1,4] => [8,5,7,2,3,1,6,4] => 1101100
[(1,8),(2,3),(4,7),(5,6)] => [3,6,2,7,8,5,4,1] => [8,5,7,4,6,2,3,1] => 1101100
[(1,8),(2,4),(3,7),(5,6)] => [4,6,7,2,8,5,3,1] => [8,5,7,2,6,3,4,1] => 1110100
[(1,7),(2,4),(3,8),(5,6)] => [4,6,7,2,8,5,1,3] => [8,5,7,2,6,1,4,3] => 1110100
[(1,6),(2,4),(3,8),(5,7)] => [4,6,7,2,8,1,5,3] => [8,2,7,5,6,1,4,3] => 1110100
[(1,5),(2,4),(3,8),(6,7)] => [4,5,7,2,1,8,6,3] => [8,6,7,2,5,1,4,3] => 1110010
[(1,4),(2,5),(3,8),(6,7)] => [4,5,7,1,2,8,6,3] => [8,6,7,1,5,2,4,3] => 1110010
>>> Load all 150 entries. <<<
[(1,3),(2,5),(4,8),(6,7)] => [3,5,1,7,2,8,6,4] => [8,6,7,1,3,2,5,4] => 1101010
[(1,2),(3,5),(4,8),(6,7)] => [2,1,5,7,3,8,6,4] => [2,1,8,6,7,3,5,4] => 1011010
[(1,2),(3,6),(4,8),(5,7)] => [2,1,6,7,8,3,5,4] => [2,1,8,3,7,5,6,4] => 1011100
[(1,3),(2,6),(4,8),(5,7)] => [3,6,1,7,8,2,5,4] => [8,1,3,2,7,5,6,4] => 1101100
[(1,4),(2,6),(3,8),(5,7)] => [4,6,7,1,8,2,5,3] => [8,1,7,5,6,2,4,3] => 1110100
[(1,5),(2,6),(3,8),(4,7)] => [5,6,7,8,1,2,4,3] => [8,1,7,2,6,4,5,3] => 1111000
[(1,6),(2,5),(3,8),(4,7)] => [5,6,7,8,2,1,4,3] => [8,2,7,1,6,4,5,3] => 1111000
[(1,7),(2,5),(3,8),(4,6)] => [5,6,7,8,2,4,1,3] => [8,2,7,4,6,1,5,3] => 1111000
[(1,8),(2,5),(3,7),(4,6)] => [5,6,7,8,2,4,3,1] => [8,2,7,4,6,3,5,1] => 1111000
[(1,8),(2,6),(3,7),(4,5)] => [5,6,7,8,4,2,3,1] => [8,4,7,2,6,3,5,1] => 1111000
[(1,7),(2,6),(3,8),(4,5)] => [5,6,7,8,4,2,1,3] => [8,4,7,2,6,1,5,3] => 1111000
[(1,6),(2,7),(3,8),(4,5)] => [5,6,7,8,4,1,2,3] => [8,4,7,1,6,2,5,3] => 1111000
[(1,5),(2,7),(3,8),(4,6)] => [5,6,7,8,1,4,2,3] => [8,1,7,4,6,2,5,3] => 1111000
[(1,4),(2,7),(3,8),(5,6)] => [4,6,7,1,8,5,2,3] => [8,5,7,1,6,2,4,3] => 1110100
[(1,3),(2,7),(4,8),(5,6)] => [3,6,1,7,8,5,2,4] => [8,5,7,1,3,2,6,4] => 1101100
[(1,2),(3,7),(4,8),(5,6)] => [2,1,6,7,8,5,3,4] => [2,1,8,5,7,3,6,4] => 1011100
[(1,2),(3,8),(4,7),(5,6)] => [2,1,6,7,8,5,4,3] => [2,1,8,5,7,4,6,3] => 1011100
[(1,3),(2,8),(4,7),(5,6)] => [3,6,1,7,8,5,4,2] => [8,5,7,4,6,1,3,2] => 1101100
[(1,4),(2,8),(3,7),(5,6)] => [4,6,7,1,8,5,3,2] => [8,5,7,1,6,3,4,2] => 1110100
[(1,5),(2,8),(3,7),(4,6)] => [5,6,7,8,1,4,3,2] => [8,1,7,4,6,3,5,2] => 1111000
[(1,6),(2,8),(3,7),(4,5)] => [5,6,7,8,4,1,3,2] => [8,4,7,1,6,3,5,2] => 1111000
[(1,7),(2,8),(3,6),(4,5)] => [5,6,7,8,4,3,1,2] => [8,4,7,3,6,1,5,2] => 1111000
[(1,8),(2,7),(3,6),(4,5)] => [5,6,7,8,4,3,2,1] => [8,4,7,3,6,2,5,1] => 1111000
[(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] => 101010101
[(1,3),(2,4),(5,6),(7,8),(9,10)] => [3,4,1,2,6,5,8,7,10,9] => [4,1,3,2,6,5,8,7,10,9] => 110010101
[(1,10),(2,3),(4,5),(6,7),(8,9)] => [3,5,2,7,4,9,6,10,8,1] => [10,8,9,6,7,4,5,2,3,1] => 110101010
[(1,2),(3,5),(4,6),(7,8),(9,10)] => [2,1,5,6,3,4,8,7,10,9] => [2,1,6,3,5,4,8,7,10,9] => 101100101
[(1,3),(2,4),(5,7),(6,8),(9,10)] => [3,4,1,2,7,8,5,6,10,9] => [4,1,3,2,8,5,7,6,10,9] => 110011001
[(1,2),(3,4),(5,7),(6,8),(9,10)] => [2,1,4,3,7,8,5,6,10,9] => [2,1,4,3,8,5,7,6,10,9] => 101011001
[(1,6),(2,7),(3,8),(4,9),(5,10)] => [6,7,8,9,10,1,2,3,4,5] => [10,1,9,2,8,3,7,4,6,5] => 111110000
[(1,3),(2,5),(4,7),(6,9),(8,10)] => [3,5,1,7,2,9,4,10,6,8] => [10,1,3,2,5,4,7,6,9,8] => 110101010
[(1,2),(3,5),(4,6),(7,9),(8,10)] => [2,1,5,6,3,4,9,10,7,8] => [2,1,6,3,5,4,10,7,9,8] => 101100110
[(1,3),(2,4),(5,6),(7,9),(8,10)] => [3,4,1,2,6,5,9,10,7,8] => [4,1,3,2,6,5,10,7,9,8] => 110010110
[(1,2),(3,4),(5,6),(7,9),(8,10)] => [2,1,4,3,6,5,9,10,7,8] => [2,1,4,3,6,5,10,7,9,8] => 101010110
[(1,10),(2,9),(3,8),(4,7),(5,6)] => [6,7,8,9,10,5,4,3,2,1] => [10,5,9,4,8,3,7,2,6,1] => 111110000
[(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] => 10101010101
[(1,2),(3,4),(5,6),(7,8),(9,11),(10,12)] => [2,1,4,3,6,5,8,7,11,12,9,10] => [2,1,4,3,6,5,8,7,12,9,11,10] => 10101010110
[(1,2),(3,4),(5,6),(7,9),(8,10),(11,12)] => [2,1,4,3,6,5,9,10,7,8,12,11] => [2,1,4,3,6,5,10,7,9,8,12,11] => 10101011001
[(1,2),(3,4),(5,7),(6,8),(9,10),(11,12)] => [2,1,4,3,7,8,5,6,10,9,12,11] => [2,1,4,3,8,5,7,6,10,9,12,11] => 10101100101
[(1,2),(3,4),(5,7),(6,8),(9,11),(10,12)] => [2,1,4,3,7,8,5,6,11,12,9,10] => [2,1,4,3,8,5,7,6,12,9,11,10] => 10101100110
[(1,2),(3,5),(4,6),(7,8),(9,10),(11,12)] => [2,1,5,6,3,4,8,7,10,9,12,11] => [2,1,6,3,5,4,8,7,10,9,12,11] => 10110010101
[(1,2),(3,5),(4,6),(7,8),(9,11),(10,12)] => [2,1,5,6,3,4,8,7,11,12,9,10] => [2,1,6,3,5,4,8,7,12,9,11,10] => 10110010110
[(1,2),(3,5),(4,6),(7,9),(8,10),(11,12)] => [2,1,5,6,3,4,9,10,7,8,12,11] => [2,1,6,3,5,4,10,7,9,8,12,11] => 10110011001
[(1,3),(2,4),(5,6),(7,8),(9,10),(11,12)] => [3,4,1,2,6,5,8,7,10,9,12,11] => [4,1,3,2,6,5,8,7,10,9,12,11] => 11001010101
[(1,3),(2,4),(5,6),(7,8),(9,11),(10,12)] => [3,4,1,2,6,5,8,7,11,12,9,10] => [4,1,3,2,6,5,8,7,12,9,11,10] => 11001010110
[(1,3),(2,4),(5,6),(7,9),(8,10),(11,12)] => [3,4,1,2,6,5,9,10,7,8,12,11] => [4,1,3,2,6,5,10,7,9,8,12,11] => 11001011001
[(1,3),(2,4),(5,7),(6,8),(9,10),(11,12)] => [3,4,1,2,7,8,5,6,10,9,12,11] => [4,1,3,2,8,5,7,6,10,9,12,11] => 11001100101
[(1,3),(2,4),(5,7),(6,8),(9,11),(10,12)] => [3,4,1,2,7,8,5,6,11,12,9,10] => [4,1,3,2,8,5,7,6,12,9,11,10] => 11001100110
[(1,3),(2,5),(4,7),(6,9),(8,11),(10,12)] => [3,5,1,7,2,9,4,11,6,12,8,10] => [12,1,3,2,5,4,7,6,9,8,11,10] => 11010101010
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.
Map
Clarke-Steingrimsson-Zeng inverse
Description
The inverse of the Clarke-Steingrimsson-Zeng map, sending excedances to descents.
This is the inverse of the map Φ in [1, sec.3].
Map
descent bottoms
Description
The descent bottoms of a permutation as a binary word.