Identifier
-
Mp00058:
Perfect matchings
—to permutation⟶
Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000891: Permutations ⟶ ℤ
Values
[(1,2)] => [2,1] => [1,2] => [1,2] => 2
[(1,2),(3,4)] => [2,1,4,3] => [3,4,1,2] => [4,3,2,1] => 2
[(1,3),(2,4)] => [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 2
[(1,4),(2,3)] => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 2
[(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [5,6,3,4,1,2] => [6,5,4,3,2,1] => 2
[(1,3),(2,4),(5,6)] => [3,4,1,2,6,5] => [5,6,2,1,4,3] => [6,5,4,3,2,1] => 2
[(1,4),(2,3),(5,6)] => [4,3,2,1,6,5] => [5,6,1,2,3,4] => [6,5,3,4,2,1] => 3
[(1,5),(2,3),(4,6)] => [5,3,2,6,1,4] => [4,1,6,2,3,5] => [5,2,6,4,1,3] => 3
[(1,6),(2,3),(4,5)] => [6,3,2,5,4,1] => [1,4,5,2,3,6] => [1,5,4,3,2,6] => 3
[(1,6),(2,4),(3,5)] => [6,4,5,2,3,1] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => 2
[(1,5),(2,4),(3,6)] => [5,4,6,2,1,3] => [3,1,2,6,4,5] => [3,2,1,6,5,4] => 2
[(1,4),(2,5),(3,6)] => [4,5,6,1,2,3] => [3,2,1,6,5,4] => [3,2,1,6,5,4] => 2
[(1,3),(2,5),(4,6)] => [3,5,1,6,2,4] => [4,2,6,1,5,3] => [6,4,5,2,3,1] => 3
[(1,2),(3,5),(4,6)] => [2,1,5,6,3,4] => [4,3,6,5,1,2] => [6,5,4,3,2,1] => 2
[(1,2),(3,6),(4,5)] => [2,1,6,5,4,3] => [3,4,5,6,1,2] => [6,5,3,4,2,1] => 3
[(1,3),(2,6),(4,5)] => [3,6,1,5,4,2] => [2,4,5,1,6,3] => [4,6,3,1,5,2] => 3
[(1,4),(2,6),(3,5)] => [4,6,5,1,3,2] => [2,3,1,5,6,4] => [3,2,1,6,5,4] => 2
[(1,5),(2,6),(3,4)] => [5,6,4,3,1,2] => [2,1,3,4,6,5] => [2,1,3,4,6,5] => 2
[(1,6),(2,5),(3,4)] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 2
[(1,2),(3,4),(5,6),(7,8)] => [2,1,4,3,6,5,8,7] => [7,8,5,6,3,4,1,2] => [8,7,6,5,4,3,2,1] => 2
[(1,3),(2,4),(5,6),(7,8)] => [3,4,1,2,6,5,8,7] => [7,8,5,6,2,1,4,3] => [8,7,6,5,4,3,2,1] => 2
[(1,4),(2,3),(5,6),(7,8)] => [4,3,2,1,6,5,8,7] => [7,8,5,6,1,2,3,4] => [8,7,6,5,4,3,2,1] => 2
[(1,3),(2,5),(4,6),(7,8)] => [3,5,1,6,2,4,8,7] => [7,8,4,2,6,1,5,3] => [8,7,6,5,4,3,2,1] => 2
[(1,2),(3,5),(4,6),(7,8)] => [2,1,5,6,3,4,8,7] => [7,8,4,3,6,5,1,2] => [8,7,6,5,4,3,2,1] => 2
[(1,2),(3,6),(4,5),(7,8)] => [2,1,6,5,4,3,8,7] => [7,8,3,4,5,6,1,2] => [8,7,6,5,4,3,2,1] => 2
[(1,3),(2,6),(4,5),(7,8)] => [3,6,1,5,4,2,8,7] => [7,8,2,4,5,1,6,3] => [8,7,6,5,4,3,2,1] => 2
[(1,3),(2,7),(4,5),(6,8)] => [3,7,1,5,4,8,2,6] => [6,2,8,4,5,1,7,3] => [8,6,7,5,4,2,3,1] => 3
[(1,2),(3,7),(4,5),(6,8)] => [2,1,7,5,4,8,3,6] => [6,3,8,4,5,7,1,2] => [8,7,6,5,4,3,2,1] => 2
[(1,7),(2,8),(3,4),(5,6)] => [7,8,4,3,6,5,1,2] => [2,1,5,6,3,4,8,7] => [2,1,6,5,4,3,8,7] => 3
[(1,8),(2,7),(3,5),(4,6)] => [8,7,5,6,3,4,2,1] => [1,2,4,3,6,5,7,8] => [1,2,4,3,6,5,7,8] => 3
[(1,7),(2,8),(3,5),(4,6)] => [7,8,5,6,3,4,1,2] => [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => 2
[(1,5),(2,7),(3,6),(4,8)] => [5,7,6,8,1,3,2,4] => [4,2,3,1,8,6,7,5] => [4,3,2,1,8,7,6,5] => 2
[(1,7),(2,6),(3,5),(4,8)] => [7,6,5,8,3,2,1,4] => [4,1,2,3,8,5,6,7] => [4,2,3,1,8,6,7,5] => 3
[(1,8),(2,6),(3,5),(4,7)] => [8,6,5,7,3,2,4,1] => [1,4,2,3,7,5,6,8] => [1,4,3,2,7,6,5,8] => 3
[(1,8),(2,5),(3,6),(4,7)] => [8,5,6,7,2,3,4,1] => [1,4,3,2,7,6,5,8] => [1,4,3,2,7,6,5,8] => 3
[(1,6),(2,5),(3,7),(4,8)] => [6,5,7,8,2,1,3,4] => [4,3,1,2,8,7,5,6] => [4,3,2,1,8,7,6,5] => 2
[(1,5),(2,6),(3,7),(4,8)] => [5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5] => [4,3,2,1,8,7,6,5] => 2
[(1,2),(3,5),(4,7),(6,8)] => [2,1,5,7,3,8,4,6] => [6,4,8,3,7,5,1,2] => [8,7,6,5,4,3,2,1] => 2
[(1,3),(2,5),(4,7),(6,8)] => [3,5,1,7,2,8,4,6] => [6,4,8,2,7,1,5,3] => [8,7,6,5,4,3,2,1] => 2
[(1,6),(2,4),(3,7),(5,8)] => [6,4,7,2,8,1,3,5] => [5,3,1,8,2,7,4,6] => [7,5,3,8,2,6,1,4] => 3
[(1,4),(2,3),(5,7),(6,8)] => [4,3,2,1,7,8,5,6] => [6,5,8,7,1,2,3,4] => [8,7,6,5,4,3,2,1] => 2
[(1,3),(2,4),(5,7),(6,8)] => [3,4,1,2,7,8,5,6] => [6,5,8,7,2,1,4,3] => [8,7,6,5,4,3,2,1] => 2
[(1,2),(3,4),(5,7),(6,8)] => [2,1,4,3,7,8,5,6] => [6,5,8,7,3,4,1,2] => [8,7,6,5,4,3,2,1] => 2
[(1,2),(3,4),(5,8),(6,7)] => [2,1,4,3,8,7,6,5] => [5,6,7,8,3,4,1,2] => [8,7,6,5,4,3,2,1] => 2
[(1,3),(2,4),(5,8),(6,7)] => [3,4,1,2,8,7,6,5] => [5,6,7,8,2,1,4,3] => [8,7,6,5,4,3,2,1] => 2
[(1,4),(2,3),(5,8),(6,7)] => [4,3,2,1,8,7,6,5] => [5,6,7,8,1,2,3,4] => [8,7,6,5,4,3,2,1] => 2
[(1,5),(2,6),(3,8),(4,7)] => [5,6,8,7,1,2,4,3] => [3,4,2,1,7,8,6,5] => [4,3,2,1,8,7,6,5] => 2
[(1,6),(2,5),(3,8),(4,7)] => [6,5,8,7,2,1,4,3] => [3,4,1,2,7,8,5,6] => [4,3,2,1,8,7,6,5] => 2
[(1,8),(2,5),(3,7),(4,6)] => [8,5,7,6,2,4,3,1] => [1,3,4,2,6,7,5,8] => [1,4,3,2,7,6,5,8] => 3
[(1,8),(2,6),(3,7),(4,5)] => [8,6,7,5,4,2,3,1] => [1,3,2,4,5,7,6,8] => [1,3,2,4,5,7,6,8] => 3
[(1,7),(2,6),(3,8),(4,5)] => [7,6,8,5,4,2,1,3] => [3,1,2,4,5,8,6,7] => [3,2,1,4,5,8,7,6] => 3
[(1,6),(2,7),(3,8),(4,5)] => [6,7,8,5,4,1,2,3] => [3,2,1,4,5,8,7,6] => [3,2,1,4,5,8,7,6] => 3
[(1,5),(2,8),(3,7),(4,6)] => [5,8,7,6,1,4,3,2] => [2,3,4,1,6,7,8,5] => [4,2,3,1,8,6,7,5] => 3
[(1,6),(2,8),(3,7),(4,5)] => [6,8,7,5,4,1,3,2] => [2,3,1,4,5,7,8,6] => [3,2,1,4,5,8,7,6] => 3
[(1,7),(2,8),(3,6),(4,5)] => [7,8,6,5,4,3,1,2] => [2,1,3,4,5,6,8,7] => [2,1,3,4,5,6,8,7] => 3
[(1,8),(2,7),(3,6),(4,5)] => [8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => 2
[(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [9,10,7,8,5,6,3,4,1,2] => [10,9,8,7,6,5,4,3,2,1] => 2
[(1,4),(2,3),(5,6),(7,8),(9,10)] => [4,3,2,1,6,5,8,7,10,9] => [9,10,7,8,5,6,1,2,3,4] => [10,9,8,7,6,5,4,3,2,1] => 2
[(1,2),(3,6),(4,5),(7,8),(9,10)] => [2,1,6,5,4,3,8,7,10,9] => [9,10,7,8,3,4,5,6,1,2] => [10,9,8,7,6,5,4,3,2,1] => 2
[(1,2),(3,8),(4,5),(6,7),(9,10)] => [2,1,8,5,4,7,6,3,10,9] => [9,10,3,6,7,4,5,8,1,2] => [10,9,8,7,6,5,4,3,2,1] => 2
[(1,2),(3,4),(5,8),(6,7),(9,10)] => [2,1,4,3,8,7,6,5,10,9] => [9,10,5,6,7,8,3,4,1,2] => [10,9,8,7,6,5,4,3,2,1] => 2
[(1,4),(2,3),(5,8),(6,7),(9,10)] => [4,3,2,1,8,7,6,5,10,9] => [9,10,5,6,7,8,1,2,3,4] => [10,9,8,7,6,5,4,3,2,1] => 2
[(1,2),(3,4),(5,6),(7,10),(8,9)] => [2,1,4,3,6,5,10,9,8,7] => [7,8,9,10,5,6,3,4,1,2] => [10,9,8,7,6,5,4,3,2,1] => 2
[(1,4),(2,3),(5,6),(7,10),(8,9)] => [4,3,2,1,6,5,10,9,8,7] => [7,8,9,10,5,6,1,2,3,4] => [10,9,8,7,6,5,4,3,2,1] => 2
[(1,2),(3,6),(4,5),(7,10),(8,9)] => [2,1,6,5,4,3,10,9,8,7] => [7,8,9,10,3,4,5,6,1,2] => [10,9,8,7,6,5,4,3,2,1] => 2
[(1,10),(2,9),(3,8),(4,7),(5,6)] => [10,9,8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,10] => 2
[(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)] => [2,1,4,3,8,7,6,5,10,9,12,11] => [11,12,9,10,5,6,7,8,3,4,1,2] => [12,11,10,9,8,7,6,5,4,3,2,1] => 2
[(1,4),(2,3),(5,8),(6,7),(9,10),(11,12)] => [4,3,2,1,8,7,6,5,10,9,12,11] => [11,12,9,10,5,6,7,8,1,2,3,4] => [12,11,10,9,8,7,6,5,4,3,2,1] => 2
[(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)] => [2,1,4,3,8,7,6,5,12,11,10,9] => [9,10,11,12,5,6,7,8,3,4,1,2] => [12,11,10,9,8,7,6,5,4,3,2,1] => 2
[(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)] => [4,3,2,1,8,7,6,5,12,11,10,9] => [9,10,11,12,5,6,7,8,1,2,3,4] => [12,11,10,9,8,7,6,5,4,3,2,1] => 2
[(1,2),(3,10),(4,5),(6,7),(8,9),(11,12)] => [2,1,10,5,4,7,6,9,8,3,12,11] => [11,12,3,8,9,6,7,4,5,10,1,2] => [12,11,10,9,8,7,6,5,4,3,2,1] => 2
[(1,2),(3,4),(5,10),(6,7),(8,9),(11,12)] => [2,1,4,3,10,7,6,9,8,5,12,11] => [11,12,5,8,9,6,7,10,3,4,1,2] => [12,11,10,9,8,7,6,5,4,3,2,1] => 2
[(1,4),(2,3),(5,10),(6,7),(8,9),(11,12)] => [4,3,2,1,10,7,6,9,8,5,12,11] => [11,12,5,8,9,6,7,10,1,2,3,4] => [12,11,10,9,8,7,6,5,4,3,2,1] => 2
[(1,2),(3,8),(4,5),(6,7),(9,10),(11,12)] => [2,1,8,5,4,7,6,3,10,9,12,11] => [11,12,9,10,3,6,7,4,5,8,1,2] => [12,11,10,9,8,7,6,5,4,3,2,1] => 2
[(1,2),(3,8),(4,5),(6,7),(9,12),(10,11)] => [2,1,8,5,4,7,6,3,12,11,10,9] => [9,10,11,12,3,6,7,4,5,8,1,2] => [12,11,10,9,8,7,6,5,4,3,2,1] => 2
[(1,2),(3,10),(4,9),(5,6),(7,8),(11,12)] => [2,1,10,9,6,5,8,7,4,3,12,11] => [11,12,3,4,7,8,5,6,9,10,1,2] => [12,11,10,9,8,7,6,5,4,3,2,1] => 2
[(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => [2,1,4,3,6,5,8,7,10,9,12,11] => [11,12,9,10,7,8,5,6,3,4,1,2] => [12,11,10,9,8,7,6,5,4,3,2,1] => 2
[(1,4),(2,3),(5,6),(7,8),(9,10),(11,12)] => [4,3,2,1,6,5,8,7,10,9,12,11] => [11,12,9,10,7,8,5,6,1,2,3,4] => [12,11,10,9,8,7,6,5,4,3,2,1] => 2
[(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)] => [2,1,4,3,6,5,8,7,12,11,10,9] => [9,10,11,12,7,8,5,6,3,4,1,2] => [12,11,10,9,8,7,6,5,4,3,2,1] => 2
[(1,4),(2,3),(5,6),(7,8),(9,12),(10,11)] => [4,3,2,1,6,5,8,7,12,11,10,9] => [9,10,11,12,7,8,5,6,1,2,3,4] => [12,11,10,9,8,7,6,5,4,3,2,1] => 2
[(1,2),(3,10),(4,5),(6,9),(7,8),(11,12)] => [2,1,10,5,4,9,8,7,6,3,12,11] => [11,12,3,6,7,8,9,4,5,10,1,2] => [12,11,10,9,8,7,6,5,4,3,2,1] => 2
[(1,2),(3,4),(5,10),(6,9),(7,8),(11,12)] => [2,1,4,3,10,9,8,7,6,5,12,11] => [11,12,5,6,7,8,9,10,3,4,1,2] => [12,11,10,9,8,7,6,5,4,3,2,1] => 2
[(1,4),(2,3),(5,10),(6,9),(7,8),(11,12)] => [4,3,2,1,10,9,8,7,6,5,12,11] => [11,12,5,6,7,8,9,10,1,2,3,4] => [12,11,10,9,8,7,6,5,4,3,2,1] => 2
[(1,2),(3,6),(4,5),(7,8),(9,10),(11,12)] => [2,1,6,5,4,3,8,7,10,9,12,11] => [11,12,9,10,7,8,3,4,5,6,1,2] => [12,11,10,9,8,7,6,5,4,3,2,1] => 2
[(1,6),(2,3),(4,5),(7,8),(9,10),(11,12)] => [6,3,2,5,4,1,8,7,10,9,12,11] => [11,12,9,10,7,8,1,4,5,2,3,6] => [12,11,10,9,8,7,6,5,4,3,2,1] => 2
[(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)] => [2,1,6,5,4,3,8,7,12,11,10,9] => [9,10,11,12,7,8,3,4,5,6,1,2] => [12,11,10,9,8,7,6,5,4,3,2,1] => 2
[(1,6),(2,3),(4,5),(7,8),(9,12),(10,11)] => [6,3,2,5,4,1,8,7,12,11,10,9] => [9,10,11,12,7,8,1,4,5,2,3,6] => [12,11,10,9,8,7,6,5,4,3,2,1] => 2
[(1,6),(2,5),(3,4),(7,8),(9,10),(11,12)] => [6,5,4,3,2,1,8,7,10,9,12,11] => [11,12,9,10,7,8,1,2,3,4,5,6] => [12,11,10,9,8,7,6,5,4,3,2,1] => 2
[(1,6),(2,5),(3,4),(7,8),(9,12),(10,11)] => [6,5,4,3,2,1,8,7,12,11,10,9] => [9,10,11,12,7,8,1,2,3,4,5,6] => [12,11,10,9,8,7,6,5,4,3,2,1] => 2
[(1,2),(3,10),(4,7),(5,6),(8,9),(11,12)] => [2,1,10,7,6,5,4,9,8,3,12,11] => [11,12,3,8,9,4,5,6,7,10,1,2] => [12,11,10,9,8,7,6,5,4,3,2,1] => 2
[(1,2),(3,4),(5,6),(7,10),(8,9),(11,12)] => [2,1,4,3,6,5,10,9,8,7,12,11] => [11,12,7,8,9,10,5,6,3,4,1,2] => [12,11,10,9,8,7,6,5,4,3,2,1] => 2
[(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)] => [4,3,2,1,6,5,10,9,8,7,12,11] => [11,12,7,8,9,10,5,6,1,2,3,4] => [12,11,10,9,8,7,6,5,4,3,2,1] => 2
[(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)] => [2,1,4,3,6,5,12,9,8,11,10,7] => [7,10,11,8,9,12,5,6,3,4,1,2] => [12,11,10,9,8,7,6,5,4,3,2,1] => 2
[(1,4),(2,3),(5,6),(7,12),(8,9),(10,11)] => [4,3,2,1,6,5,12,9,8,11,10,7] => [7,10,11,8,9,12,5,6,1,2,3,4] => [12,11,10,9,8,7,6,5,4,3,2,1] => 2
[(1,2),(3,8),(4,7),(5,6),(9,10),(11,12)] => [2,1,8,7,6,5,4,3,10,9,12,11] => [11,12,9,10,3,4,5,6,7,8,1,2] => [12,11,10,9,8,7,6,5,4,3,2,1] => 2
[(1,2),(3,8),(4,7),(5,6),(9,12),(10,11)] => [2,1,8,7,6,5,4,3,12,11,10,9] => [9,10,11,12,3,4,5,6,7,8,1,2] => [12,11,10,9,8,7,6,5,4,3,2,1] => 2
[(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)] => [2,1,4,3,6,5,12,11,10,9,8,7] => [7,8,9,10,11,12,5,6,3,4,1,2] => [12,11,10,9,8,7,6,5,4,3,2,1] => 2
[(1,4),(2,3),(5,6),(7,12),(8,11),(9,10)] => [4,3,2,1,6,5,12,11,10,9,8,7] => [7,8,9,10,11,12,5,6,1,2,3,4] => [12,11,10,9,8,7,6,5,4,3,2,1] => 2
[(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)] => [2,1,6,5,4,3,10,9,8,7,12,11] => [11,12,7,8,9,10,3,4,5,6,1,2] => [12,11,10,9,8,7,6,5,4,3,2,1] => 2
[(1,6),(2,3),(4,5),(7,10),(8,9),(11,12)] => [6,3,2,5,4,1,10,9,8,7,12,11] => [11,12,7,8,9,10,1,4,5,2,3,6] => [12,11,10,9,8,7,6,5,4,3,2,1] => 2
[(1,2),(3,6),(4,5),(7,12),(8,9),(10,11)] => [2,1,6,5,4,3,12,9,8,11,10,7] => [7,10,11,8,9,12,3,4,5,6,1,2] => [12,11,10,9,8,7,6,5,4,3,2,1] => 2
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Description
The number of distinct diagonal sums of a permutation matrix.
For example, the sums of the diagonals of the matrix $$\left(\begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{array}\right)$$
are $(1,0,1,0,2,0)$, so the statistic is $3$.
For example, the sums of the diagonals of the matrix $$\left(\begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{array}\right)$$
are $(1,0,1,0,2,0)$, so the statistic is $3$.
Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.
Map
reverse
Description
Sends a permutation to its reverse.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.
Map
Demazure product with inverse
Description
This map sends a permutation $\pi$ to $\pi^{-1} \star \pi$ where $\star$ denotes the Demazure product on permutations.
This map is a surjection onto the set of involutions, i.e., the set of permutations $\pi$ for which $\pi = \pi^{-1}$.
This map is a surjection onto the set of involutions, i.e., the set of permutations $\pi$ for which $\pi = \pi^{-1}$.
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