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Your data matches 83 different statistics following compositions of up to 3 maps.
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Matching statistic: St001451
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St001451: Plane partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> 1
[[1],[1]]
=> 1
[[2]]
=> 1
[[1,1]]
=> 1
[[1],[1],[1]]
=> 1
[[2],[1]]
=> 1
[[1,1],[1]]
=> 1
[[3]]
=> 1
[[2,1]]
=> 1
[[1,1,1]]
=> 1
[[1],[1],[1],[1]]
=> 1
[[2],[1],[1]]
=> 1
[[2],[2]]
=> 1
[[1,1],[1],[1]]
=> 1
[[1,1],[1,1]]
=> 1
[[3],[1]]
=> 1
[[2,1],[1]]
=> 1
[[1,1,1],[1]]
=> 1
[[4]]
=> 1
[[3,1]]
=> 1
[[2,2]]
=> 1
[[2,1,1]]
=> 1
[[1,1,1,1]]
=> 1
[[1],[1],[1],[1],[1]]
=> 1
[[2],[1],[1],[1]]
=> 1
[[2],[2],[1]]
=> 1
[[1,1],[1],[1],[1]]
=> 1
[[1,1],[1,1],[1]]
=> 1
[[3],[1],[1]]
=> 1
[[3],[2]]
=> 1
[[2,1],[1],[1]]
=> 1
[[2,1],[2]]
=> 1
[[2,1],[1,1]]
=> 1
[[1,1,1],[1],[1]]
=> 1
[[1,1,1],[1,1]]
=> 1
[[4],[1]]
=> 1
[[3,1],[1]]
=> 1
[[2,2],[1]]
=> 1
[[2,1,1],[1]]
=> 1
[[1,1,1,1],[1]]
=> 1
[[5]]
=> 1
[[4,1]]
=> 1
[[3,2]]
=> 1
[[3,1,1]]
=> 1
[[2,2,1]]
=> 1
[[2,1,1,1]]
=> 1
[[1,1,1,1,1]]
=> 1
[[1],[1],[1],[1],[1],[1]]
=> 1
[[2],[1],[1],[1],[1]]
=> 1
[[2],[2],[1],[1]]
=> 1
Description
The side length of the largest cube contained in a plane partition.
A plane partition analogue of the Durfee square of an integer partition ([[St000183]]).
Matching statistic: St000091
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000091: Integer compositions ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 50%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000091: Integer compositions ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 50%
Values
[[1]]
=> [1]
=> [[1]]
=> [1] => 0 = 1 - 1
[[1],[1]]
=> [1,1]
=> [[1],[2]]
=> [1,1] => 0 = 1 - 1
[[2]]
=> [2]
=> [[1,2]]
=> [2] => 0 = 1 - 1
[[1,1]]
=> [2]
=> [[1,2]]
=> [2] => 0 = 1 - 1
[[1],[1],[1]]
=> [1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => 0 = 1 - 1
[[2],[1]]
=> [2,1]
=> [[1,2],[3]]
=> [2,1] => 0 = 1 - 1
[[1,1],[1]]
=> [2,1]
=> [[1,2],[3]]
=> [2,1] => 0 = 1 - 1
[[3]]
=> [3]
=> [[1,2,3]]
=> [3] => 0 = 1 - 1
[[2,1]]
=> [3]
=> [[1,2,3]]
=> [3] => 0 = 1 - 1
[[1,1,1]]
=> [3]
=> [[1,2,3]]
=> [3] => 0 = 1 - 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => 0 = 1 - 1
[[2],[1],[1]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => 0 = 1 - 1
[[2],[2]]
=> [2,2]
=> [[1,2],[3,4]]
=> [2,2] => 0 = 1 - 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => 0 = 1 - 1
[[1,1],[1,1]]
=> [2,2]
=> [[1,2],[3,4]]
=> [2,2] => 0 = 1 - 1
[[3],[1]]
=> [3,1]
=> [[1,2,3],[4]]
=> [3,1] => 0 = 1 - 1
[[2,1],[1]]
=> [3,1]
=> [[1,2,3],[4]]
=> [3,1] => 0 = 1 - 1
[[1,1,1],[1]]
=> [3,1]
=> [[1,2,3],[4]]
=> [3,1] => 0 = 1 - 1
[[4]]
=> [4]
=> [[1,2,3,4]]
=> [4] => 0 = 1 - 1
[[3,1]]
=> [4]
=> [[1,2,3,4]]
=> [4] => 0 = 1 - 1
[[2,2]]
=> [4]
=> [[1,2,3,4]]
=> [4] => 0 = 1 - 1
[[2,1,1]]
=> [4]
=> [[1,2,3,4]]
=> [4] => 0 = 1 - 1
[[1,1,1,1]]
=> [4]
=> [[1,2,3,4]]
=> [4] => 0 = 1 - 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => 0 = 1 - 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => 0 = 1 - 1
[[2],[2],[1]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => 0 = 1 - 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => 0 = 1 - 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => 0 = 1 - 1
[[3],[1],[1]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => 0 = 1 - 1
[[3],[2]]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [3,2] => 0 = 1 - 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => 0 = 1 - 1
[[2,1],[2]]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [3,2] => 0 = 1 - 1
[[2,1],[1,1]]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [3,2] => 0 = 1 - 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => 0 = 1 - 1
[[1,1,1],[1,1]]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [3,2] => 0 = 1 - 1
[[4],[1]]
=> [4,1]
=> [[1,2,3,4],[5]]
=> [4,1] => 0 = 1 - 1
[[3,1],[1]]
=> [4,1]
=> [[1,2,3,4],[5]]
=> [4,1] => 0 = 1 - 1
[[2,2],[1]]
=> [4,1]
=> [[1,2,3,4],[5]]
=> [4,1] => 0 = 1 - 1
[[2,1,1],[1]]
=> [4,1]
=> [[1,2,3,4],[5]]
=> [4,1] => 0 = 1 - 1
[[1,1,1,1],[1]]
=> [4,1]
=> [[1,2,3,4],[5]]
=> [4,1] => 0 = 1 - 1
[[5]]
=> [5]
=> [[1,2,3,4,5]]
=> [5] => 0 = 1 - 1
[[4,1]]
=> [5]
=> [[1,2,3,4,5]]
=> [5] => 0 = 1 - 1
[[3,2]]
=> [5]
=> [[1,2,3,4,5]]
=> [5] => 0 = 1 - 1
[[3,1,1]]
=> [5]
=> [[1,2,3,4,5]]
=> [5] => 0 = 1 - 1
[[2,2,1]]
=> [5]
=> [[1,2,3,4,5]]
=> [5] => 0 = 1 - 1
[[2,1,1,1]]
=> [5]
=> [[1,2,3,4,5]]
=> [5] => 0 = 1 - 1
[[1,1,1,1,1]]
=> [5]
=> [[1,2,3,4,5]]
=> [5] => 0 = 1 - 1
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => 0 = 1 - 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1] => 0 = 1 - 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [2,2,1,1] => 0 = 1 - 1
[[1],[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [1,1,1,1,1,1,1,1] => ? = 1 - 1
[[2],[1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [2,1,1,1,1,1,1] => ? = 1 - 1
[[2],[2],[1],[1],[1],[1]]
=> [2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [2,2,1,1,1,1] => ? = 1 - 1
[[2],[2],[2],[1],[1]]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [2,2,2,1,1] => ? = 1 - 1
[[2],[2],[2],[2]]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [2,2,2,2] => ? = 1 - 1
[[1,1],[1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [2,1,1,1,1,1,1] => ? = 1 - 1
[[1,1],[1,1],[1],[1],[1],[1]]
=> [2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [2,2,1,1,1,1] => ? = 1 - 1
[[1,1],[1,1],[1,1],[1],[1]]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [2,2,2,1,1] => ? = 1 - 1
[[1,1],[1,1],[1,1],[1,1]]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [2,2,2,2] => ? = 1 - 1
[[3],[1],[1],[1],[1],[1]]
=> [3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> [3,1,1,1,1,1] => ? = 1 - 1
[[3],[2],[1],[1],[1]]
=> [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [3,2,1,1,1] => ? = 1 - 1
[[3],[2],[2],[1]]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [3,2,2,1] => ? = 1 - 1
[[3],[3],[1],[1]]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [3,3,1,1] => ? = 1 - 1
[[3],[3],[2]]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [3,3,2] => ? = 1 - 1
[[2,1],[1],[1],[1],[1],[1]]
=> [3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> [3,1,1,1,1,1] => ? = 1 - 1
[[2,1],[2],[1],[1],[1]]
=> [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [3,2,1,1,1] => ? = 1 - 1
[[2,1],[2],[2],[1]]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [3,2,2,1] => ? = 1 - 1
[[2,1],[1,1],[1],[1],[1]]
=> [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [3,2,1,1,1] => ? = 1 - 1
[[2,1],[1,1],[1,1],[1]]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [3,2,2,1] => ? = 1 - 1
[[2,1],[2,1],[1],[1]]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [3,3,1,1] => ? = 1 - 1
[[2,1],[2,1],[2]]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [3,3,2] => ? = 1 - 1
[[2,1],[2,1],[1,1]]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [3,3,2] => ? = 1 - 1
[[1,1,1],[1],[1],[1],[1],[1]]
=> [3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> [3,1,1,1,1,1] => ? = 1 - 1
[[1,1,1],[1,1],[1],[1],[1]]
=> [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [3,2,1,1,1] => ? = 1 - 1
[[1,1,1],[1,1],[1,1],[1]]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [3,2,2,1] => ? = 1 - 1
[[1,1,1],[1,1,1],[1],[1]]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [3,3,1,1] => ? = 1 - 1
[[1,1,1],[1,1,1],[1,1]]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [3,3,2] => ? = 1 - 1
[[4],[1],[1],[1],[1]]
=> [4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [4,1,1,1,1] => ? = 1 - 1
[[4],[2],[1],[1]]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [4,2,1,1] => ? = 1 - 1
[[4],[2],[2]]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [4,2,2] => ? = 1 - 1
[[4],[3],[1]]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [4,3,1] => ? = 1 - 1
[[4],[4]]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [4,4] => ? = 1 - 1
[[3,1],[1],[1],[1],[1]]
=> [4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [4,1,1,1,1] => ? = 1 - 1
[[3,1],[2],[1],[1]]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [4,2,1,1] => ? = 1 - 1
[[3,1],[2],[2]]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [4,2,2] => ? = 1 - 1
[[3,1],[1,1],[1],[1]]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [4,2,1,1] => ? = 1 - 1
[[3,1],[1,1],[1,1]]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [4,2,2] => ? = 1 - 1
[[3,1],[3],[1]]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [4,3,1] => ? = 1 - 1
[[3,1],[2,1],[1]]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [4,3,1] => ? = 1 - 1
[[3,1],[3,1]]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [4,4] => ? = 1 - 1
[[2,2],[1],[1],[1],[1]]
=> [4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [4,1,1,1,1] => ? = 1 - 1
[[2,2],[2],[1],[1]]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [4,2,1,1] => ? = 1 - 1
[[2,2],[2],[2]]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [4,2,2] => ? = 1 - 1
[[2,2],[1,1],[1],[1]]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [4,2,1,1] => ? = 1 - 1
[[2,2],[1,1],[1,1]]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [4,2,2] => ? = 1 - 1
[[2,2],[2,1],[1]]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [4,3,1] => ? = 1 - 1
[[2,2],[2,2]]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [4,4] => ? = 2 - 1
[[2,1,1],[1],[1],[1],[1]]
=> [4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [4,1,1,1,1] => ? = 1 - 1
[[2,1,1],[2],[1],[1]]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [4,2,1,1] => ? = 1 - 1
[[2,1,1],[2],[2]]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [4,2,2] => ? = 1 - 1
Description
The descent variation of a composition.
Defined in [1].
Matching statistic: St000788
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000788: Perfect matchings ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 50%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000788: Perfect matchings ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 50%
Values
[[1]]
=> [1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 1
[[1],[1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[[1,1]]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[[1],[1],[1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 1
[[2],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[[1,1],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> 1
[[2,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> 1
[[1,1,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> 1
[[2],[2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> 1
[[1,1],[1,1]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> 1
[[3],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> 1
[[2,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> 1
[[1,1,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> 1
[[4]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> 1
[[3,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> 1
[[2,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> 1
[[2,1,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> 1
[[1,1,1,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 1
[[2],[2],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> 1
[[3],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> 1
[[3],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> 1
[[2,1],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> 1
[[2,1],[1,1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> 1
[[1,1,1],[1,1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> 1
[[4],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> 1
[[3,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> 1
[[2,2],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> 1
[[2,1,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> 1
[[1,1,1,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> 1
[[5]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
=> ? = 1
[[4,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
=> ? = 1
[[3,2]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
=> ? = 1
[[3,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
=> ? = 1
[[2,2,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
=> ? = 1
[[2,1,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
=> ? = 1
[[1,1,1,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
=> ? = 1
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
=> ? = 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> 1
[[2],[2],[2]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> 1
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
=> ? = 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> 1
[[3],[2],[1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 1
[[3],[3]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 1
[[2,1],[2,1]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> 1
[[5],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7),(11,12)]
=> ? = 1
[[4,1],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7),(11,12)]
=> ? = 1
[[3,2],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7),(11,12)]
=> ? = 1
[[3,1,1],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7),(11,12)]
=> ? = 1
[[2,2,1],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7),(11,12)]
=> ? = 1
[[2,1,1,1],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7),(11,12)]
=> ? = 1
[[1,1,1,1,1],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7),(11,12)]
=> ? = 1
[[6]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14)]
=> ? = 1
[[5,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14)]
=> ? = 1
[[4,2]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14)]
=> ? = 1
[[4,1,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14)]
=> ? = 1
[[3,3]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14)]
=> ? = 1
[[3,2,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14)]
=> ? = 1
[[3,1,1,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14)]
=> ? = 1
[[2,2,2]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14)]
=> ? = 1
[[2,2,1,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14)]
=> ? = 1
[[2,1,1,1,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14)]
=> ? = 1
[[1,1,1,1,1,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14)]
=> ? = 1
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [(1,2),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> ? = 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,8),(9,10)]
=> ? = 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,6),(7,10),(8,9)]
=> ? = 1
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,8),(9,10)]
=> ? = 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,6),(7,10),(8,9)]
=> ? = 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [(1,2),(3,12),(4,11),(5,8),(6,7),(9,10)]
=> ? = 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [(1,2),(3,12),(4,11),(5,8),(6,7),(9,10)]
=> ? = 1
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [(1,2),(3,12),(4,11),(5,8),(6,7),(9,10)]
=> ? = 1
[[5],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7),(11,12)]
=> ? = 1
[[5],[2]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8),(11,12)]
=> ? = 1
[[4,1],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7),(11,12)]
=> ? = 1
[[4,1],[2]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8),(11,12)]
=> ? = 1
[[4,1],[1,1]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8),(11,12)]
=> ? = 1
[[3,2],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7),(11,12)]
=> ? = 1
[[3,2],[2]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8),(11,12)]
=> ? = 1
[[3,2],[1,1]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8),(11,12)]
=> ? = 1
[[3,1,1],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7),(11,12)]
=> ? = 1
[[3,1,1],[2]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8),(11,12)]
=> ? = 1
[[3,1,1],[1,1]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8),(11,12)]
=> ? = 1
[[2,2,1],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7),(11,12)]
=> ? = 1
[[2,2,1],[2]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8),(11,12)]
=> ? = 1
Description
The number of nesting-similar perfect matchings of a perfect matching.
Consider the infinite tree $T$ defined in [1] as follows. $T$ has the perfect matchings on $\{1,\dots,2n\}$ on level $n$, with children obtained by inserting an arc with opener $1$. For example, the matching $[(1,2)]$ has the three children $[(1,2),(3,4)]$, $[(1,3),(2,4)]$ and $[(1,4),(2,3)]$.
Two perfect matchings $M$ and $N$ on $\{1,\dots,2n\}$ are nesting-similar, if the distribution of the number of nestings agrees on all levels of the subtrees of $T$ rooted at $M$ and $N$.
[thm 1.2, 1] shows that to find out whether $M$ and $N$ are nesting-similar, it is enough to check that $M$ and $N$ have the same number of nestings, and that the distribution of nestings agrees for their direct children.
[thm 3.5, 1], see also [2], gives the number of equivalence classes of nesting-similar matchings with $n$ arcs as $$2\cdot 4^{n-1} - \frac{3n-1}{2n+2}\binom{2n}{n}.$$ [prop 3.6, 1] has further interpretations of this number.
Matching statistic: St000787
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000787: Perfect matchings ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 50%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000787: Perfect matchings ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 50%
Values
[[1]]
=> [1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 0 = 1 - 1
[[1],[1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 0 = 1 - 1
[[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 0 = 1 - 1
[[1,1]]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 0 = 1 - 1
[[1],[1],[1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 0 = 1 - 1
[[2],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 0 = 1 - 1
[[1,1],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 0 = 1 - 1
[[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> 0 = 1 - 1
[[2,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> 0 = 1 - 1
[[1,1,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> 0 = 1 - 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 0 = 1 - 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> 0 = 1 - 1
[[2],[2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> 0 = 1 - 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> 0 = 1 - 1
[[1,1],[1,1]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> 0 = 1 - 1
[[3],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> 0 = 1 - 1
[[2,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> 0 = 1 - 1
[[1,1,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> 0 = 1 - 1
[[4]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> 0 = 1 - 1
[[3,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> 0 = 1 - 1
[[2,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> 0 = 1 - 1
[[2,1,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> 0 = 1 - 1
[[1,1,1,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> 0 = 1 - 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 1 - 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 0 = 1 - 1
[[2],[2],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> 0 = 1 - 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 0 = 1 - 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> 0 = 1 - 1
[[3],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> 0 = 1 - 1
[[3],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> 0 = 1 - 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> 0 = 1 - 1
[[2,1],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> 0 = 1 - 1
[[2,1],[1,1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> 0 = 1 - 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> 0 = 1 - 1
[[1,1,1],[1,1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> 0 = 1 - 1
[[4],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> 0 = 1 - 1
[[3,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> 0 = 1 - 1
[[2,2],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> 0 = 1 - 1
[[2,1,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> 0 = 1 - 1
[[1,1,1,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> 0 = 1 - 1
[[5]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
=> ? = 1 - 1
[[4,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
=> ? = 1 - 1
[[3,2]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
=> ? = 1 - 1
[[3,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
=> ? = 1 - 1
[[2,2,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
=> ? = 1 - 1
[[2,1,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
=> ? = 1 - 1
[[1,1,1,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
=> ? = 1 - 1
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 1 - 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
=> ? = 1 - 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> 0 = 1 - 1
[[2],[2],[2]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> 0 = 1 - 1
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
=> ? = 1 - 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> 0 = 1 - 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> 0 = 1 - 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> 0 = 1 - 1
[[3],[2],[1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 0 = 1 - 1
[[3],[3]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> 0 = 1 - 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> 0 = 1 - 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 0 = 1 - 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 0 = 1 - 1
[[2,1],[2,1]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> 0 = 1 - 1
[[5],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7),(11,12)]
=> ? = 1 - 1
[[4,1],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7),(11,12)]
=> ? = 1 - 1
[[3,2],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7),(11,12)]
=> ? = 1 - 1
[[3,1,1],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7),(11,12)]
=> ? = 1 - 1
[[2,2,1],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7),(11,12)]
=> ? = 1 - 1
[[2,1,1,1],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7),(11,12)]
=> ? = 1 - 1
[[1,1,1,1,1],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7),(11,12)]
=> ? = 1 - 1
[[6]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14)]
=> ? = 1 - 1
[[5,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14)]
=> ? = 1 - 1
[[4,2]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14)]
=> ? = 1 - 1
[[4,1,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14)]
=> ? = 1 - 1
[[3,3]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14)]
=> ? = 1 - 1
[[3,2,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14)]
=> ? = 1 - 1
[[3,1,1,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14)]
=> ? = 1 - 1
[[2,2,2]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14)]
=> ? = 1 - 1
[[2,2,1,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14)]
=> ? = 1 - 1
[[2,1,1,1,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14)]
=> ? = 1 - 1
[[1,1,1,1,1,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14)]
=> ? = 1 - 1
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [(1,2),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> ? = 1 - 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,8),(9,10)]
=> ? = 1 - 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,6),(7,10),(8,9)]
=> ? = 1 - 1
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,8),(9,10)]
=> ? = 1 - 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,6),(7,10),(8,9)]
=> ? = 1 - 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [(1,2),(3,12),(4,11),(5,8),(6,7),(9,10)]
=> ? = 1 - 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [(1,2),(3,12),(4,11),(5,8),(6,7),(9,10)]
=> ? = 1 - 1
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [(1,2),(3,12),(4,11),(5,8),(6,7),(9,10)]
=> ? = 1 - 1
[[5],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7),(11,12)]
=> ? = 1 - 1
[[5],[2]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8),(11,12)]
=> ? = 1 - 1
[[4,1],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7),(11,12)]
=> ? = 1 - 1
[[4,1],[2]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8),(11,12)]
=> ? = 1 - 1
[[4,1],[1,1]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8),(11,12)]
=> ? = 1 - 1
[[3,2],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7),(11,12)]
=> ? = 1 - 1
[[3,2],[2]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8),(11,12)]
=> ? = 1 - 1
[[3,2],[1,1]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8),(11,12)]
=> ? = 1 - 1
[[3,1,1],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7),(11,12)]
=> ? = 1 - 1
[[3,1,1],[2]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8),(11,12)]
=> ? = 1 - 1
[[3,1,1],[1,1]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8),(11,12)]
=> ? = 1 - 1
[[2,2,1],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7),(11,12)]
=> ? = 1 - 1
[[2,2,1],[2]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8),(11,12)]
=> ? = 1 - 1
Description
The number of flips required to make a perfect matching noncrossing.
A crossing in a perfect matching is a pair of arcs $\{a,b\}$ and $\{c,d\}$ such that $a < c < b < d$. Replacing any such pair by either $\{a,c\}$ and $\{b,d\}$ or by $\{a,d\}$, $\{b,c\}$ produces a perfect matching with fewer crossings.
This statistic is the minimal number of such flips required to turn a given matching into a noncrossing matching.
Matching statistic: St000056
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000056: Permutations ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 50%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000056: Permutations ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 50%
Values
[[1]]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[[1],[1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 1
[[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 1
[[1,1]]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 1
[[1],[1],[1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1
[[2],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1
[[1,1],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1
[[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 1
[[2,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 1
[[1,1,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1
[[2],[2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1
[[1,1],[1,1]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 1
[[3],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1
[[2,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1
[[1,1,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1
[[4]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 1
[[3,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 1
[[2,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 1
[[2,1,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 1
[[1,1,1,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 1
[[2],[2],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1
[[3],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[[3],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[[2,1],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1
[[2,1],[1,1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[[1,1,1],[1,1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1
[[4],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 1
[[3,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 1
[[2,2],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 1
[[2,1,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 1
[[1,1,1,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 1
[[5]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? = 1
[[4,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? = 1
[[3,2]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? = 1
[[3,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? = 1
[[2,2,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? = 1
[[2,1,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? = 1
[[1,1,1,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? = 1
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => ? = 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? = 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => 1
[[2],[2],[2]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 1
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? = 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => 1
[[3],[2],[1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
[[3],[3]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
[[2,1],[2,1]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 1
[[5],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? = 1
[[4,1],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? = 1
[[3,2],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? = 1
[[3,1,1],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? = 1
[[2,2,1],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? = 1
[[2,1,1,1],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? = 1
[[1,1,1,1,1],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? = 1
[[6]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 1
[[5,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 1
[[4,2]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 1
[[4,1,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 1
[[3,3]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 1
[[3,2,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 1
[[3,1,1,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 1
[[2,2,2]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 1
[[2,2,1,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 1
[[2,1,1,1,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 1
[[1,1,1,1,1,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 1
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,2] => ? = 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [8,1,4,5,6,7,2,3] => ? = 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? = 1
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [8,1,4,5,6,7,2,3] => ? = 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? = 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => ? = 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => ? = 1
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => ? = 1
[[5],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => ? = 1
[[5],[2]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => ? = 1
[[4,1],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => ? = 1
[[4,1],[2]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => ? = 1
[[4,1],[1,1]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => ? = 1
[[3,2],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => ? = 1
[[3,2],[2]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => ? = 1
[[3,2],[1,1]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => ? = 1
[[3,1,1],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => ? = 1
[[3,1,1],[2]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => ? = 1
[[3,1,1],[1,1]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => ? = 1
[[2,2,1],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => ? = 1
[[2,2,1],[2]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => ? = 1
Description
The decomposition (or block) number of a permutation.
For $\pi \in \mathcal{S}_n$, this is given by
$$\#\big\{ 1 \leq k \leq n : \{\pi_1,\ldots,\pi_k\} = \{1,\ldots,k\} \big\}.$$
This is also known as the number of connected components [1] or the number of blocks [2] of the permutation, considering it as a direct sum.
This is one plus [[St000234]].
Matching statistic: St000486
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000486: Permutations ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 50%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000486: Permutations ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 50%
Values
[[1]]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[[1],[1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 1
[[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 1
[[1,1]]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 1
[[1],[1],[1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1
[[2],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1
[[1,1],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1
[[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 1
[[2,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 1
[[1,1,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1
[[2],[2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1
[[1,1],[1,1]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 1
[[3],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1
[[2,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1
[[1,1,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1
[[4]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 1
[[3,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 1
[[2,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 1
[[2,1,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 1
[[1,1,1,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 1
[[2],[2],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1
[[3],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[[3],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[[2,1],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1
[[2,1],[1,1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[[1,1,1],[1,1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1
[[4],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 1
[[3,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 1
[[2,2],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 1
[[2,1,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 1
[[1,1,1,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 1
[[5]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? = 1
[[4,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? = 1
[[3,2]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? = 1
[[3,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? = 1
[[2,2,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? = 1
[[2,1,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? = 1
[[1,1,1,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? = 1
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => ? = 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? = 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => 1
[[2],[2],[2]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 1
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? = 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => 1
[[3],[2],[1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
[[3],[3]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
[[2,1],[2,1]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 1
[[5],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? = 1
[[4,1],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? = 1
[[3,2],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? = 1
[[3,1,1],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? = 1
[[2,2,1],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? = 1
[[2,1,1,1],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? = 1
[[1,1,1,1,1],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? = 1
[[6]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 1
[[5,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 1
[[4,2]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 1
[[4,1,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 1
[[3,3]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 1
[[3,2,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 1
[[3,1,1,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 1
[[2,2,2]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 1
[[2,2,1,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 1
[[2,1,1,1,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 1
[[1,1,1,1,1,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 1
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,2] => ? = 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [8,1,4,5,6,7,2,3] => ? = 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? = 1
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [8,1,4,5,6,7,2,3] => ? = 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? = 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => ? = 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => ? = 1
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => ? = 1
[[5],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => ? = 1
[[5],[2]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => ? = 1
[[4,1],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => ? = 1
[[4,1],[2]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => ? = 1
[[4,1],[1,1]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => ? = 1
[[3,2],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => ? = 1
[[3,2],[2]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => ? = 1
[[3,2],[1,1]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => ? = 1
[[3,1,1],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => ? = 1
[[3,1,1],[2]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => ? = 1
[[3,1,1],[1,1]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => ? = 1
[[2,2,1],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => ? = 1
[[2,2,1],[2]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => ? = 1
Description
The number of cycles of length at least 3 of a permutation.
Matching statistic: St000694
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000694: Permutations ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 50%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000694: Permutations ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 50%
Values
[[1]]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[[1],[1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 1
[[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 1
[[1,1]]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 1
[[1],[1],[1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1
[[2],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1
[[1,1],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1
[[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 1
[[2,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 1
[[1,1,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1
[[2],[2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1
[[1,1],[1,1]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 1
[[3],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1
[[2,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1
[[1,1,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1
[[4]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 1
[[3,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 1
[[2,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 1
[[2,1,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 1
[[1,1,1,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 1
[[2],[2],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1
[[3],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[[3],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[[2,1],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1
[[2,1],[1,1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[[1,1,1],[1,1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1
[[4],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 1
[[3,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 1
[[2,2],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 1
[[2,1,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 1
[[1,1,1,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 1
[[5]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? = 1
[[4,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? = 1
[[3,2]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? = 1
[[3,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? = 1
[[2,2,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? = 1
[[2,1,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? = 1
[[1,1,1,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? = 1
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => ? = 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? = 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => 1
[[2],[2],[2]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 1
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? = 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => 1
[[3],[2],[1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
[[3],[3]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
[[2,1],[2,1]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 1
[[5],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? = 1
[[4,1],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? = 1
[[3,2],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? = 1
[[3,1,1],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? = 1
[[2,2,1],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? = 1
[[2,1,1,1],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? = 1
[[1,1,1,1,1],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? = 1
[[6]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 1
[[5,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 1
[[4,2]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 1
[[4,1,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 1
[[3,3]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 1
[[3,2,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 1
[[3,1,1,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 1
[[2,2,2]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 1
[[2,2,1,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 1
[[2,1,1,1,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 1
[[1,1,1,1,1,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 1
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,2] => ? = 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [8,1,4,5,6,7,2,3] => ? = 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? = 1
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [8,1,4,5,6,7,2,3] => ? = 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? = 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => ? = 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => ? = 1
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => ? = 1
[[5],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => ? = 1
[[5],[2]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => ? = 1
[[4,1],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => ? = 1
[[4,1],[2]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => ? = 1
[[4,1],[1,1]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => ? = 1
[[3,2],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => ? = 1
[[3,2],[2]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => ? = 1
[[3,2],[1,1]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => ? = 1
[[3,1,1],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => ? = 1
[[3,1,1],[2]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => ? = 1
[[3,1,1],[1,1]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => ? = 1
[[2,2,1],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => ? = 1
[[2,2,1],[2]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => ? = 1
Description
The number of affine bounded permutations that project to a given permutation.
As affine bounded permutations are in bijection with usual permutations where fix-points come in two colors, this statistic is $2^k$ where $k$ is the number of fixed points [[St000022]].
Matching statistic: St001174
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001174: Permutations ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 50%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001174: Permutations ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 50%
Values
[[1]]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[[1],[1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 1
[[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 1
[[1,1]]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 1
[[1],[1],[1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1
[[2],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1
[[1,1],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1
[[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 1
[[2,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 1
[[1,1,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1
[[2],[2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1
[[1,1],[1,1]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 1
[[3],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1
[[2,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1
[[1,1,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1
[[4]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 1
[[3,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 1
[[2,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 1
[[2,1,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 1
[[1,1,1,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 1
[[2],[2],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1
[[3],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[[3],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[[2,1],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1
[[2,1],[1,1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[[1,1,1],[1,1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1
[[4],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 1
[[3,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 1
[[2,2],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 1
[[2,1,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 1
[[1,1,1,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 1
[[5]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? = 1
[[4,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? = 1
[[3,2]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? = 1
[[3,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? = 1
[[2,2,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? = 1
[[2,1,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? = 1
[[1,1,1,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? = 1
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => ? = 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? = 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => 1
[[2],[2],[2]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 1
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? = 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => 1
[[3],[2],[1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
[[3],[3]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
[[2,1],[2,1]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 1
[[5],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? = 1
[[4,1],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? = 1
[[3,2],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? = 1
[[3,1,1],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? = 1
[[2,2,1],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? = 1
[[2,1,1,1],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? = 1
[[1,1,1,1,1],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? = 1
[[6]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 1
[[5,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 1
[[4,2]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 1
[[4,1,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 1
[[3,3]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 1
[[3,2,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 1
[[3,1,1,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 1
[[2,2,2]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 1
[[2,2,1,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 1
[[2,1,1,1,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 1
[[1,1,1,1,1,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 1
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,2] => ? = 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [8,1,4,5,6,7,2,3] => ? = 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? = 1
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [8,1,4,5,6,7,2,3] => ? = 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? = 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => ? = 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => ? = 1
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => ? = 1
[[5],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => ? = 1
[[5],[2]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => ? = 1
[[4,1],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => ? = 1
[[4,1],[2]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => ? = 1
[[4,1],[1,1]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => ? = 1
[[3,2],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => ? = 1
[[3,2],[2]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => ? = 1
[[3,2],[1,1]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => ? = 1
[[3,1,1],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => ? = 1
[[3,1,1],[2]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => ? = 1
[[3,1,1],[1,1]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => ? = 1
[[2,2,1],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => ? = 1
[[2,2,1],[2]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => ? = 1
Description
The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
Matching statistic: St001208
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001208: Permutations ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 50%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001208: Permutations ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 50%
Values
[[1]]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[[1],[1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 1
[[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 1
[[1,1]]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 1
[[1],[1],[1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[[2],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 1
[[1,1],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 1
[[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[[2,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[[1,1,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 1
[[2],[2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 1
[[1,1],[1,1]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 1
[[3],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 1
[[2,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 1
[[1,1,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 1
[[4]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1
[[3,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1
[[2,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1
[[2,1,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1
[[1,1,1,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ? = 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 1
[[2],[2],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 1
[[3],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 1
[[3],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 1
[[2,1],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 1
[[2,1],[1,1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 1
[[1,1,1],[1,1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 1
[[4],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 1
[[3,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 1
[[2,2],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 1
[[2,1,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 1
[[1,1,1,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 1
[[5]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => ? = 1
[[4,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => ? = 1
[[3,2]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => ? = 1
[[3,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => ? = 1
[[2,2,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => ? = 1
[[2,1,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => ? = 1
[[1,1,1,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => ? = 1
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => ? = 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 1
[[2],[2],[2]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 1
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => ? = 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 1
[[3],[2],[1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 1
[[3],[3]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 1
[[2,1],[2,1]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 1
[[5],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => ? = 1
[[4,1],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => ? = 1
[[3,2],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => ? = 1
[[3,1,1],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => ? = 1
[[2,2,1],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => ? = 1
[[2,1,1,1],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => ? = 1
[[1,1,1,1,1],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => ? = 1
[[6]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 1
[[5,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 1
[[4,2]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 1
[[4,1,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 1
[[3,3]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 1
[[3,2,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 1
[[3,1,1,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 1
[[2,2,2]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 1
[[2,2,1,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 1
[[2,1,1,1,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 1
[[1,1,1,1,1,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 1
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => ? = 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => ? = 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => ? = 1
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => ? = 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => ? = 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => ? = 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => ? = 1
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => ? = 1
[[5],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => ? = 1
[[5],[2]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => ? = 1
[[4,1],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => ? = 1
[[4,1],[2]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => ? = 1
[[4,1],[1,1]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => ? = 1
[[3,2],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => ? = 1
[[3,2],[2]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => ? = 1
[[3,2],[1,1]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => ? = 1
[[3,1,1],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => ? = 1
[[3,1,1],[2]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => ? = 1
[[3,1,1],[1,1]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => ? = 1
[[2,2,1],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => ? = 1
[[2,2,1],[2]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => ? = 1
Description
The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$.
Matching statistic: St001256
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001256: Dyck paths ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 50%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001256: Dyck paths ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 50%
Values
[[1]]
=> [1]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[[1],[1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[[1,1]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[[1],[1],[1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[[2],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[[1,1],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[[2,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[[1,1,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[[2],[2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[[1,1],[1,1]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[[3],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[[2,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[[1,1,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[[4]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
[[3,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
[[2,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
[[2,1,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
[[1,1,1,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 1
[[2],[2],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[[3],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[[3],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[[2,1],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[[2,1],[1,1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[[1,1,1],[1,1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[[4],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> 1
[[3,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> 1
[[2,2],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> 1
[[2,1,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> 1
[[1,1,1,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> 1
[[5]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 1
[[4,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 1
[[3,2]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 1
[[3,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 1
[[2,2,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 1
[[2,1,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 1
[[1,1,1,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 1
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 1
[[2],[2],[2]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 1
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 1
[[3],[2],[1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[[3],[3]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[[2,1],[2,1]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 1
[[5],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 1
[[4,1],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 1
[[3,2],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 1
[[3,1,1],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 1
[[2,2,1],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 1
[[2,1,1,1],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 1
[[1,1,1,1,1],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 1
[[6]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1
[[5,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1
[[4,2]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1
[[4,1,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1
[[3,3]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1
[[3,2,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1
[[3,1,1,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1
[[2,2,2]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1
[[2,2,1,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1
[[2,1,1,1,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1
[[1,1,1,1,1,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 1
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 1
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 1
[[5],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> ? = 1
[[5],[2]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 1
[[4,1],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> ? = 1
[[4,1],[2]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 1
[[4,1],[1,1]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 1
[[3,2],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> ? = 1
[[3,2],[2]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 1
[[3,2],[1,1]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 1
[[3,1,1],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> ? = 1
[[3,1,1],[2]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 1
[[3,1,1],[1,1]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 1
[[2,2,1],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> ? = 1
[[2,2,1],[2]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 1
Description
Number of simple reflexive modules that are 2-stable reflexive.
See Definition 3.1. in the reference for the definition of 2-stable reflexive.
The following 73 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001461The number of topologically connected components of the chord diagram of a permutation. St001590The crossing number of a perfect matching. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000221The number of strong fixed points of a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000623The number of occurrences of the pattern 52341 in a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001381The fertility of a permutation. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001549The number of restricted non-inversions between exceedances. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001850The number of Hecke atoms of a permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000255The number of reduced Kogan faces with the permutation as type. St000078The number of alternating sign matrices whose left key is the permutation. St001735The number of permutations with the same set of runs. St000355The number of occurrences of the pattern 21-3. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000663The number of right floats of a permutation. St000664The number of right ropes of a permutation. St000666The number of right tethers of a permutation. St001550The number of inversions between exceedances where the greater exceedance is linked. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001715The number of non-records in a permutation. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St001847The number of occurrences of the pattern 1432 in a permutation. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St000516The number of stretching pairs of a permutation. St000646The number of big ascents of a permutation. St000650The number of 3-rises of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000732The number of double deficiencies of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000908The length of the shortest maximal antichain in a poset. St000317The cycle descent number of a permutation. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001301The first Betti number of the order complex associated with the poset. St001705The number of occurrences of the pattern 2413 in a permutation. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000570The Edelman-Greene number of a permutation. St000914The sum of the values of the Möbius function of a poset. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau.
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