- FindStat

Identifier

Mp00098: Alternating sign matrices —link pattern⟶ Perfect matchings
St000746: Perfect matchings ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

[[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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[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

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

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

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

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

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

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

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

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

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

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

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

[[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

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

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

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

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

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

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

[[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)] => 3

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

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

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

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

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

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

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

[[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)] => 5

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

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

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

[[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)] => 5

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

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

[[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)] => 5

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

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

[[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)] => 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)] => 5

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

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

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

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

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

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

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

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

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

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

[[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)] => 5

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

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

[[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)] => 5

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = q$

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

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

$F_{4} = 7\ q + 14\ q^{2} + 14\ q^{3} + 7\ q^{4}$

$F_{5} = 42\ q + 105\ q^{2} + 135\ q^{3} + 105\ q^{4} + 42\ q^{5}$

Description

The number of pairs with odd minimum in a perfect matching.

Map

link pattern

Description

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