- FindStat

Identifier

Mp00098: Alternating sign matrices —link pattern⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000028: Permutations ⟶ ℤ

Values

[[1]] => [(1,2)] => [2,1] => [2,1] => 1

[[1,0],[0,1]] => [(1,4),(2,3)] => [4,3,2,1] => [4,3,2,1] => 1

[[0,1],[1,0]] => [(1,2),(3,4)] => [2,1,4,3] => [2,4,1,3] => 2

[[1,0,0],[0,1,0],[0,0,1]] => [(1,6),(2,5),(3,4)] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 1

[[0,1,0],[1,0,0],[0,0,1]] => [(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [2,4,6,1,3,5] => 3

[[1,0,0],[0,0,1],[0,1,0]] => [(1,6),(2,3),(4,5)] => [6,3,2,5,4,1] => [3,6,2,5,4,1] => 2

[[0,1,0],[1,-1,1],[0,1,0]] => [(1,2),(3,6),(4,5)] => [2,1,6,5,4,3] => [6,5,2,4,1,3] => 2

[[0,0,1],[1,0,0],[0,1,0]] => [(1,6),(2,3),(4,5)] => [6,3,2,5,4,1] => [3,6,2,5,4,1] => 2

[[0,1,0],[0,0,1],[1,0,0]] => [(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [2,4,6,1,3,5] => 3

[[0,0,1],[0,1,0],[1,0,0]] => [(1,4),(2,3),(5,6)] => [4,3,2,1,6,5] => [4,3,2,6,1,5] => 4

[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]] => [(1,8),(2,7),(3,6),(4,5)] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1

[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]] => [(1,2),(3,4),(5,6),(7,8)] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => 4

[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]] => [(1,2),(3,4),(5,6),(7,8)] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => 4

[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]] => [(1,2),(3,4),(5,6),(7,8)] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => 4

[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]] => [(1,2),(3,4),(5,6),(7,8)] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => 4

[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]] => [(1,2),(3,4),(5,6),(7,8)] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => 4

[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]] => [(1,2),(3,4),(5,6),(7,8)] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => 4

[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]] => [(1,2),(3,4),(5,6),(7,8)] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => 4

[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]] => [(1,10),(2,9),(3,8),(4,7),(5,6)] => [10,9,8,7,6,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 1

[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]] => [(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [2,4,6,8,10,1,3,5,7,9] => 5

[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]] => [(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [2,4,6,8,10,1,3,5,7,9] => 5

[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]] => [(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [2,4,6,8,10,1,3,5,7,9] => 5

[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]] => [(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [2,4,6,8,10,1,3,5,7,9] => 5

[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]] => [(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [2,4,6,8,10,1,3,5,7,9] => 5

[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]] => [(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [2,4,6,8,10,1,3,5,7,9] => 5

[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]] => [(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [2,4,6,8,10,1,3,5,7,9] => 5

[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [2,4,6,8,10,1,3,5,7,9] => 5

[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,0,0,1],[0,0,1,0,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [2,4,6,8,10,1,3,5,7,9] => 5

[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [2,4,6,8,10,1,3,5,7,9] => 5

[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [2,4,6,8,10,1,3,5,7,9] => 5

[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [2,4,6,8,10,1,3,5,7,9] => 5

[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [2,4,6,8,10,1,3,5,7,9] => 5

[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,-1,0,1],[0,0,1,0,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [2,4,6,8,10,1,3,5,7,9] => 5

[[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [2,4,6,8,10,1,3,5,7,9] => 5

[[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [2,4,6,8,10,1,3,5,7,9] => 5

[[0,0,1,0,0],[0,1,-1,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [2,4,6,8,10,1,3,5,7,9] => 5

[[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [2,4,6,8,10,1,3,5,7,9] => 5

[[0,1,0,0,0],[0,0,0,0,1],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [2,4,6,8,10,1,3,5,7,9] => 5

[[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [2,4,6,8,10,1,3,5,7,9] => 5

[[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [2,4,6,8,10,1,3,5,7,9] => 5

[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,0,0,0,1],[0,1,0,0,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [2,4,6,8,10,1,3,5,7,9] => 5

[[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [2,4,6,8,10,1,3,5,7,9] => 5

[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,0,0,0,1],[0,1,0,0,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [2,4,6,8,10,1,3,5,7,9] => 5

[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,0,0,0,1],[0,1,0,0,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [2,4,6,8,10,1,3,5,7,9] => 5

[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [2,4,6,8,10,1,3,5,7,9] => 5

[[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,-1,0,0,1],[0,1,0,0,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [2,4,6,8,10,1,3,5,7,9] => 5

[[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [2,4,6,8,10,1,3,5,7,9] => 5

[[0,1,0,0,0],[1,-1,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [2,4,6,8,10,1,3,5,7,9] => 5

[[0,1,0,0,0],[0,0,0,0,1],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,0,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [2,4,6,8,10,1,3,5,7,9] => 5

[[0,0,1,0,0],[0,1,-1,0,1],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,0,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [2,4,6,8,10,1,3,5,7,9] => 5

[[0,0,0,0,1],[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,0,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [2,4,6,8,10,1,3,5,7,9] => 5

[[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [2,4,6,8,10,1,3,5,7,9] => 5

[[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [2,4,6,8,10,1,3,5,7,9] => 5

[[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,0,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [2,4,6,8,10,1,3,5,7,9] => 5

[[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [2,4,6,8,10,1,3,5,7,9] => 5

[[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [2,4,6,8,10,1,3,5,7,9] => 5

[[0,1,0,0,0],[0,0,0,0,1],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [2,4,6,8,10,1,3,5,7,9] => 5

[[0,0,1,0,0],[0,0,0,0,1],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,0,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [2,4,6,8,10,1,3,5,7,9] => 5

[[0,0,0,0,1],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,0,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [2,4,6,8,10,1,3,5,7,9] => 5

[[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [2,4,6,8,10,1,3,5,7,9] => 5

[[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [2,4,6,8,10,1,3,5,7,9] => 5

[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]] => [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)] => [12,11,10,9,8,7,6,5,4,3,2,1] => [12,11,10,9,8,7,6,5,4,3,2,1] => 1

[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,4,6,8,10,12,1,3,5,7,9,11] => 6

[[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,4,6,8,10,12,1,3,5,7,9,11] => 6

[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,4,6,8,10,12,1,3,5,7,9,11] => 6

[[0,1,0,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,4,6,8,10,12,1,3,5,7,9,11] => 6

[[0,0,1,0,0,0],[0,1,-1,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,4,6,8,10,12,1,3,5,7,9,11] => 6

[[0,1,0,0,0,0],[0,0,0,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,4,6,8,10,12,1,3,5,7,9,11] => 6

[[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,4,6,8,10,12,1,3,5,7,9,11] => 6

[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,4,6,8,10,12,1,3,5,7,9,11] => 6

[[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,0,0,0,1,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,4,6,8,10,12,1,3,5,7,9,11] => 6

[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,4,6,8,10,12,1,3,5,7,9,11] => 6

[[0,1,0,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,4,6,8,10,12,1,3,5,7,9,11] => 6

[[0,0,1,0,0,0],[0,1,-1,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,4,6,8,10,12,1,3,5,7,9,11] => 6

[[0,1,0,0,0,0],[0,0,0,1,0,0],[1,-1,1,0,0,0],[0,1,-1,0,1,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,4,6,8,10,12,1,3,5,7,9,11] => 6

[[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,-1,0,1,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,4,6,8,10,12,1,3,5,7,9,11] => 6

[[0,1,0,0,0,0],[1,-1,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,4,6,8,10,12,1,3,5,7,9,11] => 6

[[0,1,0,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,4,6,8,10,12,1,3,5,7,9,11] => 6

[[0,0,1,0,0,0],[0,1,-1,0,1,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,4,6,8,10,12,1,3,5,7,9,11] => 6

[[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,4,6,8,10,12,1,3,5,7,9,11] => 6

[[0,1,0,0,0,0],[0,0,0,0,1,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,4,6,8,10,12,1,3,5,7,9,11] => 6

[[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,4,6,8,10,12,1,3,5,7,9,11] => 6

[[0,0,0,0,1,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,4,6,8,10,12,1,3,5,7,9,11] => 6

[[0,1,0,0,0,0],[0,0,1,0,0,0],[1,-1,0,1,0,0],[0,0,0,0,1,0],[0,1,0,0,-1,1],[0,0,0,0,1,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,4,6,8,10,12,1,3,5,7,9,11] => 6

[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,-1,1],[0,0,0,0,1,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,4,6,8,10,12,1,3,5,7,9,11] => 6

[[0,1,0,0,0,0],[0,0,0,1,0,0],[1,-1,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,-1,1],[0,0,0,0,1,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,4,6,8,10,12,1,3,5,7,9,11] => 6

[[0,0,1,0,0,0],[0,1,-1,1,0,0],[1,-1,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,-1,1],[0,0,0,0,1,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,4,6,8,10,12,1,3,5,7,9,11] => 6

[[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,-1,1],[0,0,0,0,1,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,4,6,8,10,12,1,3,5,7,9,11] => 6

[[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[1,-1,0,0,1,0],[0,1,0,0,-1,1],[0,0,0,0,1,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,4,6,8,10,12,1,3,5,7,9,11] => 6

[[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[1,-1,0,0,1,0],[0,1,0,0,-1,1],[0,0,0,0,1,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,4,6,8,10,12,1,3,5,7,9,11] => 6

[[0,1,0,0,0,0],[1,-1,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,-1,1],[0,0,0,0,1,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,4,6,8,10,12,1,3,5,7,9,11] => 6

[[0,1,0,0,0,0],[0,0,0,0,1,0],[1,-1,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,-1,1],[0,0,0,0,1,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,4,6,8,10,12,1,3,5,7,9,11] => 6

[[0,0,1,0,0,0],[0,1,-1,0,1,0],[1,-1,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,-1,1],[0,0,0,0,1,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,4,6,8,10,12,1,3,5,7,9,11] => 6

[[0,0,0,0,1,0],[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,-1,1],[0,0,0,0,1,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,4,6,8,10,12,1,3,5,7,9,11] => 6

[[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,-1,1],[0,0,0,0,1,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,4,6,8,10,12,1,3,5,7,9,11] => 6

[[0,0,0,0,1,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,-1,1],[0,0,0,0,1,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,4,6,8,10,12,1,3,5,7,9,11] => 6

[[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,-1,1],[0,0,0,0,1,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,4,6,8,10,12,1,3,5,7,9,11] => 6

[[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,-1,1],[0,0,0,0,1,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,4,6,8,10,12,1,3,5,7,9,11] => 6

[[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,-1,1],[0,0,0,0,1,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,4,6,8,10,12,1,3,5,7,9,11] => 6

[[0,1,0,0,0,0],[0,0,0,0,1,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,-1,1],[0,0,0,0,1,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,4,6,8,10,12,1,3,5,7,9,11] => 6

[[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,-1,1,0,0],[0,0,1,0,0,0],[0,1,0,0,-1,1],[0,0,0,0,1,0]] => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,4,6,8,10,12,1,3,5,7,9,11] => 6

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Description

The number of stack-sorts needed to sort a permutation.
A permutation is (West) $t$-stack sortable if it is sortable using $t$ stacks in series.
Let $W_t(n,k)$ be the number of permutations of size $n$
with $k$ descents which are $t$-stack sortable. Then the polynomials $W_{n,t}(x) = \sum_{k=0}^n W_t(n,k)x^k$
are symmetric and unimodal.
We have $W_{n,1}(x) = A_n(x)$, the Eulerian polynomials. One can show that $W_{n,1}(x)$ and $W_{n,2}(x)$ are real-rooted.
Precisely the permutations that avoid the pattern $231$ have statistic at most $1$, see [3]. These are counted by $\frac{1}{n+1}\binom{2n}{n}$ (OEIS:A000108). Precisely the permutations that avoid the pattern $2341$ and the barred pattern $3\bar 5241$ have statistic at most $2$, see [4]. These are counted by $\frac{2(3n)!}{(n+1)!(2n+1)!}$ (OEIS:A000139).

Map

inverse Foata bijection

Description

The inverse of Foata's bijection.
See Mp00067Foata bijection.

Map

link pattern

Description

Sends an alternating sign matrix to the link pattern of the corresponding fully packed loop configuration.

Map

to permutation

Description

Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.