Identifier
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
Images
[(1,2)] => [2,1] => [1] =>
[(1,2),(3,4)] => [2,1,4,3] => [2,1,3] => 01
[(1,3),(2,4)] => [3,4,1,2] => [3,1,2] => 00
[(1,4),(2,3)] => [4,3,2,1] => [3,2,1] => 00
[(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [2,1,4,3,5] => 0101
[(1,3),(2,4),(5,6)] => [3,4,1,2,6,5] => [3,4,1,2,5] => 0001
[(1,4),(2,3),(5,6)] => [4,3,2,1,6,5] => [4,3,2,1,5] => 0001
[(1,5),(2,3),(4,6)] => [5,3,2,6,1,4] => [5,3,2,1,4] => 0000
[(1,6),(2,3),(4,5)] => [6,3,2,5,4,1] => [3,2,5,4,1] => 0000
[(1,6),(2,4),(3,5)] => [6,4,5,2,3,1] => [4,5,2,3,1] => 0000
[(1,5),(2,4),(3,6)] => [5,4,6,2,1,3] => [5,4,2,1,3] => 0000
[(1,4),(2,5),(3,6)] => [4,5,6,1,2,3] => [4,5,1,2,3] => 0000
[(1,3),(2,5),(4,6)] => [3,5,1,6,2,4] => [3,5,1,2,4] => 0000
[(1,2),(3,5),(4,6)] => [2,1,5,6,3,4] => [2,1,5,3,4] => 0100
[(1,2),(3,6),(4,5)] => [2,1,6,5,4,3] => [2,1,5,4,3] => 0100
[(1,3),(2,6),(4,5)] => [3,6,1,5,4,2] => [3,1,5,4,2] => 0000
[(1,4),(2,6),(3,5)] => [4,6,5,1,3,2] => [4,5,1,3,2] => 0000
[(1,5),(2,6),(3,4)] => [5,6,4,3,1,2] => [5,4,3,1,2] => 0000
[(1,6),(2,5),(3,4)] => [6,5,4,3,2,1] => [5,4,3,2,1] => 0000
[(1,2),(3,4),(5,6),(7,8)] => [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,7] => 010101
[(1,3),(2,4),(5,6),(7,8)] => [3,4,1,2,6,5,8,7] => [3,4,1,2,6,5,7] => 000101
[(1,4),(2,3),(5,6),(7,8)] => [4,3,2,1,6,5,8,7] => [4,3,2,1,6,5,7] => 000101
[(1,5),(2,3),(4,6),(7,8)] => [5,3,2,6,1,4,8,7] => [5,3,2,6,1,4,7] => 000001
[(1,6),(2,3),(4,5),(7,8)] => [6,3,2,5,4,1,8,7] => [6,3,2,5,4,1,7] => 000001
[(1,7),(2,3),(4,5),(6,8)] => [7,3,2,5,4,8,1,6] => [7,3,2,5,4,1,6] => 000000
[(1,8),(2,3),(4,5),(6,7)] => [8,3,2,5,4,7,6,1] => [3,2,5,4,7,6,1] => 000000
[(1,8),(2,4),(3,5),(6,7)] => [8,4,5,2,3,7,6,1] => [4,5,2,3,7,6,1] => 000000
[(1,7),(2,4),(3,5),(6,8)] => [7,4,5,2,3,8,1,6] => [7,4,5,2,3,1,6] => 000000
[(1,6),(2,4),(3,5),(7,8)] => [6,4,5,2,3,1,8,7] => [6,4,5,2,3,1,7] => 000001
[(1,5),(2,4),(3,6),(7,8)] => [5,4,6,2,1,3,8,7] => [5,4,6,2,1,3,7] => 000001
[(1,4),(2,5),(3,6),(7,8)] => [4,5,6,1,2,3,8,7] => [4,5,6,1,2,3,7] => 000001
[(1,3),(2,5),(4,6),(7,8)] => [3,5,1,6,2,4,8,7] => [3,5,1,6,2,4,7] => 000001
[(1,2),(3,5),(4,6),(7,8)] => [2,1,5,6,3,4,8,7] => [2,1,5,6,3,4,7] => 010001
[(1,2),(3,6),(4,5),(7,8)] => [2,1,6,5,4,3,8,7] => [2,1,6,5,4,3,7] => 010001
[(1,3),(2,6),(4,5),(7,8)] => [3,6,1,5,4,2,8,7] => [3,6,1,5,4,2,7] => 000001
[(1,4),(2,6),(3,5),(7,8)] => [4,6,5,1,3,2,8,7] => [4,6,5,1,3,2,7] => 000001
[(1,5),(2,6),(3,4),(7,8)] => [5,6,4,3,1,2,8,7] => [5,6,4,3,1,2,7] => 000001
[(1,6),(2,5),(3,4),(7,8)] => [6,5,4,3,2,1,8,7] => [6,5,4,3,2,1,7] => 000001
[(1,7),(2,5),(3,4),(6,8)] => [7,5,4,3,2,8,1,6] => [7,5,4,3,2,1,6] => 000000
[(1,8),(2,5),(3,4),(6,7)] => [8,5,4,3,2,7,6,1] => [5,4,3,2,7,6,1] => 000000
[(1,8),(2,6),(3,4),(5,7)] => [8,6,4,3,7,2,5,1] => [6,4,3,7,2,5,1] => 000000
[(1,7),(2,6),(3,4),(5,8)] => [7,6,4,3,8,2,1,5] => [7,6,4,3,2,1,5] => 000000
[(1,6),(2,7),(3,4),(5,8)] => [6,7,4,3,8,1,2,5] => [6,7,4,3,1,2,5] => 000000
[(1,5),(2,7),(3,4),(6,8)] => [5,7,4,3,1,8,2,6] => [5,7,4,3,1,2,6] => 000000
[(1,4),(2,7),(3,5),(6,8)] => [4,7,5,1,3,8,2,6] => [4,7,5,1,3,2,6] => 000000
[(1,3),(2,7),(4,5),(6,8)] => [3,7,1,5,4,8,2,6] => [3,7,1,5,4,2,6] => 000000
[(1,2),(3,7),(4,5),(6,8)] => [2,1,7,5,4,8,3,6] => [2,1,7,5,4,3,6] => 010000
[(1,2),(3,8),(4,5),(6,7)] => [2,1,8,5,4,7,6,3] => [2,1,5,4,7,6,3] => 010000
[(1,3),(2,8),(4,5),(6,7)] => [3,8,1,5,4,7,6,2] => [3,1,5,4,7,6,2] => 000000
[(1,4),(2,8),(3,5),(6,7)] => [4,8,5,1,3,7,6,2] => [4,5,1,3,7,6,2] => 000000
[(1,5),(2,8),(3,4),(6,7)] => [5,8,4,3,1,7,6,2] => [5,4,3,1,7,6,2] => 000000
[(1,6),(2,8),(3,4),(5,7)] => [6,8,4,3,7,1,5,2] => [6,4,3,7,1,5,2] => 000000
[(1,7),(2,8),(3,4),(5,6)] => [7,8,4,3,6,5,1,2] => [7,4,3,6,5,1,2] => 000000
[(1,8),(2,7),(3,4),(5,6)] => [8,7,4,3,6,5,2,1] => [7,4,3,6,5,2,1] => 000000
[(1,8),(2,7),(3,5),(4,6)] => [8,7,5,6,3,4,2,1] => [7,5,6,3,4,2,1] => 000000
[(1,7),(2,8),(3,5),(4,6)] => [7,8,5,6,3,4,1,2] => [7,5,6,3,4,1,2] => 000000
[(1,6),(2,8),(3,5),(4,7)] => [6,8,5,7,3,1,4,2] => [6,5,7,3,1,4,2] => 000000
[(1,5),(2,8),(3,6),(4,7)] => [5,8,6,7,1,3,4,2] => [5,6,7,1,3,4,2] => 000000
[(1,4),(2,8),(3,6),(5,7)] => [4,8,6,1,7,3,5,2] => [4,6,1,7,3,5,2] => 000000
[(1,3),(2,8),(4,6),(5,7)] => [3,8,1,6,7,4,5,2] => [3,1,6,7,4,5,2] => 000000
[(1,2),(3,8),(4,6),(5,7)] => [2,1,8,6,7,4,5,3] => [2,1,6,7,4,5,3] => 010000
[(1,2),(3,7),(4,6),(5,8)] => [2,1,7,6,8,4,3,5] => [2,1,7,6,4,3,5] => 010000
[(1,3),(2,7),(4,6),(5,8)] => [3,7,1,6,8,4,2,5] => [3,7,1,6,4,2,5] => 000000
[(1,4),(2,7),(3,6),(5,8)] => [4,7,6,1,8,3,2,5] => [4,7,6,1,3,2,5] => 000000
[(1,5),(2,7),(3,6),(4,8)] => [5,7,6,8,1,3,2,4] => [5,7,6,1,3,2,4] => 000000
[(1,6),(2,7),(3,5),(4,8)] => [6,7,5,8,3,1,2,4] => [6,7,5,3,1,2,4] => 000000
[(1,7),(2,6),(3,5),(4,8)] => [7,6,5,8,3,2,1,4] => [7,6,5,3,2,1,4] => 000000
[(1,8),(2,6),(3,5),(4,7)] => [8,6,5,7,3,2,4,1] => [6,5,7,3,2,4,1] => 000000
[(1,8),(2,5),(3,6),(4,7)] => [8,5,6,7,2,3,4,1] => [5,6,7,2,3,4,1] => 000000
[(1,7),(2,5),(3,6),(4,8)] => [7,5,6,8,2,3,1,4] => [7,5,6,2,3,1,4] => 000000
[(1,6),(2,5),(3,7),(4,8)] => [6,5,7,8,2,1,3,4] => [6,5,7,2,1,3,4] => 000000
[(1,5),(2,6),(3,7),(4,8)] => [5,6,7,8,1,2,3,4] => [5,6,7,1,2,3,4] => 000000
[(1,4),(2,6),(3,7),(5,8)] => [4,6,7,1,8,2,3,5] => [4,6,7,1,2,3,5] => 000000
[(1,3),(2,6),(4,7),(5,8)] => [3,6,1,7,8,2,4,5] => [3,6,1,7,2,4,5] => 000000
[(1,2),(3,6),(4,7),(5,8)] => [2,1,6,7,8,3,4,5] => [2,1,6,7,3,4,5] => 010000
[(1,2),(3,5),(4,7),(6,8)] => [2,1,5,7,3,8,4,6] => [2,1,5,7,3,4,6] => 010000
[(1,3),(2,5),(4,7),(6,8)] => [3,5,1,7,2,8,4,6] => [3,5,1,7,2,4,6] => 000000
[(1,4),(2,5),(3,7),(6,8)] => [4,5,7,1,2,8,3,6] => [4,5,7,1,2,3,6] => 000000
[(1,5),(2,4),(3,7),(6,8)] => [5,4,7,2,1,8,3,6] => [5,4,7,2,1,3,6] => 000000
[(1,6),(2,4),(3,7),(5,8)] => [6,4,7,2,8,1,3,5] => [6,4,7,2,1,3,5] => 000000
[(1,7),(2,4),(3,6),(5,8)] => [7,4,6,2,8,3,1,5] => [7,4,6,2,3,1,5] => 000000
[(1,8),(2,4),(3,6),(5,7)] => [8,4,6,2,7,3,5,1] => [4,6,2,7,3,5,1] => 000000
[(1,8),(2,3),(4,6),(5,7)] => [8,3,2,6,7,4,5,1] => [3,2,6,7,4,5,1] => 000000
[(1,7),(2,3),(4,6),(5,8)] => [7,3,2,6,8,4,1,5] => [7,3,2,6,4,1,5] => 000000
[(1,6),(2,3),(4,7),(5,8)] => [6,3,2,7,8,1,4,5] => [6,3,2,7,1,4,5] => 000000
[(1,5),(2,3),(4,7),(6,8)] => [5,3,2,7,1,8,4,6] => [5,3,2,7,1,4,6] => 000000
[(1,4),(2,3),(5,7),(6,8)] => [4,3,2,1,7,8,5,6] => [4,3,2,1,7,5,6] => 000100
[(1,3),(2,4),(5,7),(6,8)] => [3,4,1,2,7,8,5,6] => [3,4,1,2,7,5,6] => 000100
[(1,2),(3,4),(5,7),(6,8)] => [2,1,4,3,7,8,5,6] => [2,1,4,3,7,5,6] => 010100
[(1,2),(3,4),(5,8),(6,7)] => [2,1,4,3,8,7,6,5] => [2,1,4,3,7,6,5] => 010100
[(1,3),(2,4),(5,8),(6,7)] => [3,4,1,2,8,7,6,5] => [3,4,1,2,7,6,5] => 000100
[(1,4),(2,3),(5,8),(6,7)] => [4,3,2,1,8,7,6,5] => [4,3,2,1,7,6,5] => 000100
[(1,5),(2,3),(4,8),(6,7)] => [5,3,2,8,1,7,6,4] => [5,3,2,1,7,6,4] => 000000
[(1,6),(2,3),(4,8),(5,7)] => [6,3,2,8,7,1,5,4] => [6,3,2,7,1,5,4] => 000000
[(1,7),(2,3),(4,8),(5,6)] => [7,3,2,8,6,5,1,4] => [7,3,2,6,5,1,4] => 000000
[(1,8),(2,3),(4,7),(5,6)] => [8,3,2,7,6,5,4,1] => [3,2,7,6,5,4,1] => 000000
[(1,8),(2,4),(3,7),(5,6)] => [8,4,7,2,6,5,3,1] => [4,7,2,6,5,3,1] => 000000
[(1,7),(2,4),(3,8),(5,6)] => [7,4,8,2,6,5,1,3] => [7,4,2,6,5,1,3] => 000000
[(1,6),(2,4),(3,8),(5,7)] => [6,4,8,2,7,1,5,3] => [6,4,2,7,1,5,3] => 000000
[(1,5),(2,4),(3,8),(6,7)] => [5,4,8,2,1,7,6,3] => [5,4,2,1,7,6,3] => 000000
[(1,4),(2,5),(3,8),(6,7)] => [4,5,8,1,2,7,6,3] => [4,5,1,2,7,6,3] => 000000
>>> Load all 146 entries. <<<
[(1,3),(2,5),(4,8),(6,7)] => [3,5,1,8,2,7,6,4] => [3,5,1,2,7,6,4] => 000000
[(1,2),(3,5),(4,8),(6,7)] => [2,1,5,8,3,7,6,4] => [2,1,5,3,7,6,4] => 010000
[(1,2),(3,6),(4,8),(5,7)] => [2,1,6,8,7,3,5,4] => [2,1,6,7,3,5,4] => 010000
[(1,3),(2,6),(4,8),(5,7)] => [3,6,1,8,7,2,5,4] => [3,6,1,7,2,5,4] => 000000
[(1,4),(2,6),(3,8),(5,7)] => [4,6,8,1,7,2,5,3] => [4,6,1,7,2,5,3] => 000000
[(1,5),(2,6),(3,8),(4,7)] => [5,6,8,7,1,2,4,3] => [5,6,7,1,2,4,3] => 000000
[(1,6),(2,5),(3,8),(4,7)] => [6,5,8,7,2,1,4,3] => [6,5,7,2,1,4,3] => 000000
[(1,7),(2,5),(3,8),(4,6)] => [7,5,8,6,2,4,1,3] => [7,5,6,2,4,1,3] => 000000
[(1,8),(2,5),(3,7),(4,6)] => [8,5,7,6,2,4,3,1] => [5,7,6,2,4,3,1] => 000000
[(1,8),(2,6),(3,7),(4,5)] => [8,6,7,5,4,2,3,1] => [6,7,5,4,2,3,1] => 000000
[(1,7),(2,6),(3,8),(4,5)] => [7,6,8,5,4,2,1,3] => [7,6,5,4,2,1,3] => 000000
[(1,6),(2,7),(3,8),(4,5)] => [6,7,8,5,4,1,2,3] => [6,7,5,4,1,2,3] => 000000
[(1,5),(2,7),(3,8),(4,6)] => [5,7,8,6,1,4,2,3] => [5,7,6,1,4,2,3] => 000000
[(1,4),(2,7),(3,8),(5,6)] => [4,7,8,1,6,5,2,3] => [4,7,1,6,5,2,3] => 000000
[(1,3),(2,7),(4,8),(5,6)] => [3,7,1,8,6,5,2,4] => [3,7,1,6,5,2,4] => 000000
[(1,2),(3,7),(4,8),(5,6)] => [2,1,7,8,6,5,3,4] => [2,1,7,6,5,3,4] => 010000
[(1,2),(3,8),(4,7),(5,6)] => [2,1,8,7,6,5,4,3] => [2,1,7,6,5,4,3] => 010000
[(1,3),(2,8),(4,7),(5,6)] => [3,8,1,7,6,5,4,2] => [3,1,7,6,5,4,2] => 000000
[(1,4),(2,8),(3,7),(5,6)] => [4,8,7,1,6,5,3,2] => [4,7,1,6,5,3,2] => 000000
[(1,5),(2,8),(3,7),(4,6)] => [5,8,7,6,1,4,3,2] => [5,7,6,1,4,3,2] => 000000
[(1,6),(2,8),(3,7),(4,5)] => [6,8,7,5,4,1,3,2] => [6,7,5,4,1,3,2] => 000000
[(1,7),(2,8),(3,6),(4,5)] => [7,8,6,5,4,3,1,2] => [7,6,5,4,3,1,2] => 000000
[(1,8),(2,7),(3,6),(4,5)] => [8,7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => 000000
[(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [2,1,4,3,6,5,8,7,9] => 01010101
[(1,4),(2,3),(5,6),(7,8),(9,10)] => [4,3,2,1,6,5,8,7,10,9] => [4,3,2,1,6,5,8,7,9] => 00010101
[(1,10),(2,3),(4,5),(6,7),(8,9)] => [10,3,2,5,4,7,6,9,8,1] => [3,2,5,4,7,6,9,8,1] => 00000000
[(1,6),(2,5),(3,4),(7,8),(9,10)] => [6,5,4,3,2,1,8,7,10,9] => [6,5,4,3,2,1,8,7,9] => 00000101
[(1,8),(2,9),(3,5),(4,6),(7,10)] => [8,9,5,6,3,4,10,1,2,7] => [8,9,5,6,3,4,1,2,7] => 00000000
[(1,4),(2,3),(5,8),(6,7),(9,10)] => [4,3,2,1,8,7,6,5,10,9] => [4,3,2,1,8,7,6,5,9] => 00010001
[(1,2),(3,8),(4,7),(5,6),(9,10)] => [2,1,8,7,6,5,4,3,10,9] => [2,1,8,7,6,5,4,3,9] => 01000001
[(1,8),(2,7),(3,6),(4,5),(9,10)] => [8,7,6,5,4,3,2,1,10,9] => [8,7,6,5,4,3,2,1,9] => 00000001
[(1,9),(2,7),(3,6),(4,5),(8,10)] => [9,7,6,5,4,3,2,10,1,8] => [9,7,6,5,4,3,2,1,8] => 00000000
[(1,10),(2,7),(3,6),(4,5),(8,9)] => [10,7,6,5,4,3,2,9,8,1] => [7,6,5,4,3,2,9,8,1] => 00000000
[(1,6),(2,7),(3,8),(4,9),(5,10)] => [6,7,8,9,10,1,2,3,4,5] => [6,7,8,9,1,2,3,4,5] => 00000000
[(1,2),(3,4),(5,6),(7,10),(8,9)] => [2,1,4,3,6,5,10,9,8,7] => [2,1,4,3,6,5,9,8,7] => 01010100
[(1,6),(2,5),(3,4),(7,10),(8,9)] => [6,5,4,3,2,1,10,9,8,7] => [6,5,4,3,2,1,9,8,7] => 00000100
[(1,10),(2,5),(3,4),(6,9),(7,8)] => [10,5,4,3,2,9,8,7,6,1] => [5,4,3,2,9,8,7,6,1] => 00000000
[(1,2),(3,4),(5,10),(6,9),(7,8)] => [2,1,4,3,10,9,8,7,6,5] => [2,1,4,3,9,8,7,6,5] => 01010000
[(1,4),(2,3),(5,10),(6,9),(7,8)] => [4,3,2,1,10,9,8,7,6,5] => [4,3,2,1,9,8,7,6,5] => 00010000
[(1,10),(2,3),(4,9),(5,8),(6,7)] => [10,3,2,9,8,7,6,5,4,1] => [3,2,9,8,7,6,5,4,1] => 00000000
[(1,2),(3,10),(4,9),(5,8),(6,7)] => [2,1,10,9,8,7,6,5,4,3] => [2,1,9,8,7,6,5,4,3] => 01000000
[(1,10),(2,9),(3,8),(4,7),(5,6)] => [10,9,8,7,6,5,4,3,2,1] => [9,8,7,6,5,4,3,2,1] => 00000000
[(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)] => [12,11,10,9,8,7,6,5,4,3,2,1] => [11,10,9,8,7,6,5,4,3,2,1] => 0000000000
[(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)] => [2,1,12,11,10,9,8,7,6,5,4,3] => [2,1,11,10,9,8,7,6,5,4,3] => 0100000000
[(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)] => [10,9,8,7,6,5,4,3,2,1,12,11] => [10,9,8,7,6,5,4,3,2,1,11] => 0000000001
Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.
Map
restriction
Description
The permutation obtained by removing the largest letter.
This map is undefined for the empty permutation.
Map
connectivity set
Description
The connectivity set of a permutation as a binary word.
According to [2], also known as the global ascent set.
The connectivity set is
$$C(\pi)=\{i\in [n-1] | \forall 1 \leq j \leq i < k \leq n : \pi(j) < \pi(k)\}.$$
For $n > 1$ it can also be described as the set of occurrences of the mesh pattern
$$([1,2], \{(0,2),(1,0),(1,1),(2,0),(2,1) \})$$
or equivalently
$$([1,2], \{(0,1),(0,2),(1,1),(1,2),(2,0) \}),$$
see [3].
The permutation is connected, when the connectivity set is empty.