- FindStat

Identifier

Mp00098: Alternating sign matrices —link pattern⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St001087: Permutations ⟶ ℤ

Values

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

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$F_{1} = 1$

$F_{2} = q + q^{2}$

$F_{3} = 2\ q^{2} + 4\ q^{3} + q^{4}$

Description

The number of occurrences of the vincular pattern |12-3 in a permutation.
This is the number of occurrences of the pattern

$123$ , where the first matched entry is the first entry of the permutation and the other two matched entries are consecutive.
In other words, this is the number of ascents whose bottom value is strictly larger than the first entry of the permutation.

Map

runsort

Description

The permutation obtained by sorting the increasing runs lexicographically.

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.