Your data matches 22 different statistics following compositions of up to 3 maps.
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Matching statistic: St001449
St001449: Plane partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[1]]
=> 2
[[1],[1]]
=> 2
[[2]]
=> 1
[[1,1]]
=> 2
[[1],[1],[1]]
=> 2
[[2],[1]]
=> 3
[[1,1],[1]]
=> 2
[[3]]
=> 1
[[2,1]]
=> 3
[[1,1,1]]
=> 2
[[1],[1],[1],[1]]
=> 2
[[2],[1],[1]]
=> 3
[[2],[2]]
=> 1
[[1,1],[1],[1]]
=> 2
[[1,1],[1,1]]
=> 2
[[3],[1]]
=> 2
[[2,1],[1]]
=> 3
[[1,1,1],[1]]
=> 2
[[4]]
=> 1
[[3,1]]
=> 2
[[2,2]]
=> 1
[[2,1,1]]
=> 3
[[1,1,1,1]]
=> 2
[[1],[1],[1],[1],[1]]
=> 2
[[2],[1],[1],[1]]
=> 3
[[2],[2],[1]]
=> 3
[[1,1],[1],[1],[1]]
=> 2
[[1,1],[1,1],[1]]
=> 2
[[3],[1],[1]]
=> 2
[[3],[2]]
=> 1
[[2,1],[1],[1]]
=> 3
[[2,1],[2]]
=> 3
[[2,1],[1,1]]
=> 3
[[1,1,1],[1],[1]]
=> 2
[[1,1,1],[1,1]]
=> 2
[[4],[1]]
=> 2
[[3,1],[1]]
=> 2
[[2,2],[1]]
=> 3
[[2,1,1],[1]]
=> 3
[[1,1,1,1],[1]]
=> 2
[[5]]
=> 1
[[4,1]]
=> 2
[[3,2]]
=> 1
[[3,1,1]]
=> 2
[[2,2,1]]
=> 3
[[2,1,1,1]]
=> 3
[[1,1,1,1,1]]
=> 2
[[1],[1],[1],[1],[1],[1]]
=> 2
[[2],[1],[1],[1],[1]]
=> 3
[[2],[2],[1],[1]]
=> 3
Description
The smallest missing nonzero part in the plane partition.
Mp00311: Plane partitions to partitionInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
St000678: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1]
=> [1,0,1,0]
=> 2
[[1],[1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 2
[[1,1]]
=> [2]
=> [1,1,0,0,1,0]
=> 2
[[1],[1],[1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[[2],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 3
[[1,1],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 3
[[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 2
[[2,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 2
[[1,1,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 2
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[[2],[2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[[1,1],[1,1]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[[3],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 3
[[2,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 3
[[1,1,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 3
[[4]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[[3,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[[2,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[[2,1,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[[1,1,1,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[[2],[2],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[[3],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
[[3],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 3
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
[[2,1],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 3
[[2,1],[1,1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 3
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
[[1,1,1],[1,1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 3
[[4],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3
[[3,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3
[[2,2],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3
[[2,1,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3
[[1,1,1,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3
[[5]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2
[[4,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2
[[3,2]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2
[[3,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2
[[2,2,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2
[[2,1,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2
[[1,1,1,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2
[[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],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 2
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
Description
The number of up steps after the last double rise of a Dyck path.
Matching statistic: St000759
Mp00311: Plane partitions to partitionInteger partitions
Mp00044: Integer partitions conjugateInteger partitions
St000759: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1]
=> [1]
=> 2
[[1],[1]]
=> [1,1]
=> [2]
=> 1
[[2]]
=> [2]
=> [1,1]
=> 2
[[1,1]]
=> [2]
=> [1,1]
=> 2
[[1],[1],[1]]
=> [1,1,1]
=> [3]
=> 1
[[2],[1]]
=> [2,1]
=> [2,1]
=> 3
[[1,1],[1]]
=> [2,1]
=> [2,1]
=> 3
[[3]]
=> [3]
=> [1,1,1]
=> 2
[[2,1]]
=> [3]
=> [1,1,1]
=> 2
[[1,1,1]]
=> [3]
=> [1,1,1]
=> 2
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [4]
=> 1
[[2],[1],[1]]
=> [2,1,1]
=> [3,1]
=> 2
[[2],[2]]
=> [2,2]
=> [2,2]
=> 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [3,1]
=> 2
[[1,1],[1,1]]
=> [2,2]
=> [2,2]
=> 1
[[3],[1]]
=> [3,1]
=> [2,1,1]
=> 3
[[2,1],[1]]
=> [3,1]
=> [2,1,1]
=> 3
[[1,1,1],[1]]
=> [3,1]
=> [2,1,1]
=> 3
[[4]]
=> [4]
=> [1,1,1,1]
=> 2
[[3,1]]
=> [4]
=> [1,1,1,1]
=> 2
[[2,2]]
=> [4]
=> [1,1,1,1]
=> 2
[[2,1,1]]
=> [4]
=> [1,1,1,1]
=> 2
[[1,1,1,1]]
=> [4]
=> [1,1,1,1]
=> 2
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [5]
=> 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [4,1]
=> 2
[[2],[2],[1]]
=> [2,2,1]
=> [3,2]
=> 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [4,1]
=> 2
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [3,2]
=> 1
[[3],[1],[1]]
=> [3,1,1]
=> [3,1,1]
=> 2
[[3],[2]]
=> [3,2]
=> [2,2,1]
=> 3
[[2,1],[1],[1]]
=> [3,1,1]
=> [3,1,1]
=> 2
[[2,1],[2]]
=> [3,2]
=> [2,2,1]
=> 3
[[2,1],[1,1]]
=> [3,2]
=> [2,2,1]
=> 3
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [3,1,1]
=> 2
[[1,1,1],[1,1]]
=> [3,2]
=> [2,2,1]
=> 3
[[4],[1]]
=> [4,1]
=> [2,1,1,1]
=> 3
[[3,1],[1]]
=> [4,1]
=> [2,1,1,1]
=> 3
[[2,2],[1]]
=> [4,1]
=> [2,1,1,1]
=> 3
[[2,1,1],[1]]
=> [4,1]
=> [2,1,1,1]
=> 3
[[1,1,1,1],[1]]
=> [4,1]
=> [2,1,1,1]
=> 3
[[5]]
=> [5]
=> [1,1,1,1,1]
=> 2
[[4,1]]
=> [5]
=> [1,1,1,1,1]
=> 2
[[3,2]]
=> [5]
=> [1,1,1,1,1]
=> 2
[[3,1,1]]
=> [5]
=> [1,1,1,1,1]
=> 2
[[2,2,1]]
=> [5]
=> [1,1,1,1,1]
=> 2
[[2,1,1,1]]
=> [5]
=> [1,1,1,1,1]
=> 2
[[1,1,1,1,1]]
=> [5]
=> [1,1,1,1,1]
=> 2
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [6]
=> 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [5,1]
=> 2
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [4,2]
=> 1
Description
The smallest missing part in an integer partition. In [3], this is referred to as the mex, the minimal excluded part of the partition. For compositions, this is studied in [sec.3.2., 1].
Matching statistic: St000011
Mp00311: Plane partitions to partitionInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00132: Dyck paths switch returns and last double riseDyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2
[[1],[1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[[1,1]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[[1],[1],[1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[[2],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3
[[1,1],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3
[[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[[2,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[[1,1,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[[2],[2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[[1,1],[1,1]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[[3],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[[2,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[[1,1,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[[4]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[[3,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[[2,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[[2,1,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[[1,1,1,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[[2],[2],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[[3],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[[3],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[[2,1],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[[2,1],[1,1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[[1,1,1],[1,1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[[4],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3
[[3,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3
[[2,2],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3
[[2,1,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3
[[1,1,1,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3
[[5]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2
[[4,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2
[[3,2]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2
[[3,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2
[[2,2,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2
[[2,1,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2
[[1,1,1,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2
[[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]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 2
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
Description
The number of touch points (or returns) of a Dyck path. This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St001050
Mp00311: Plane partitions to partitionInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00138: Dyck paths to noncrossing partitionSet partitions
St001050: Set partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1]
=> [1,0,1,0]
=> {{1},{2}}
=> 2
[[1],[1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
[[2]]
=> [2]
=> [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 2
[[1,1]]
=> [2]
=> [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 2
[[1],[1],[1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[[2],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 3
[[1,1],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 3
[[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 2
[[2,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 2
[[1,1,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 2
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 2
[[2],[2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 2
[[1,1],[1,1]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 1
[[3],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 3
[[2,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 3
[[1,1,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 3
[[4]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 2
[[3,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 2
[[2,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 2
[[2,1,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 2
[[1,1,1,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 2
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2,3,4,5,6}}
=> 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> 2
[[2],[2],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> 2
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1
[[3],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 2
[[3],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 3
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 2
[[2,1],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 3
[[2,1],[1,1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 3
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 2
[[1,1,1],[1,1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 3
[[4],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> {{1,2,4},{3},{5}}
=> 3
[[3,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> {{1,2,4},{3},{5}}
=> 3
[[2,2],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> {{1,2,4},{3},{5}}
=> 3
[[2,1,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> {{1,2,4},{3},{5}}
=> 3
[[1,1,1,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> {{1,2,4},{3},{5}}
=> 3
[[5]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> {{1,2,3,4,5},{6}}
=> 2
[[4,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> {{1,2,3,4,5},{6}}
=> 2
[[3,2]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> {{1,2,3,4,5},{6}}
=> 2
[[3,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> {{1,2,3,4,5},{6}}
=> 2
[[2,2,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> {{1,2,3,4,5},{6}}
=> 2
[[2,1,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> {{1,2,3,4,5},{6}}
=> 2
[[1,1,1,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> {{1,2,3,4,5},{6}}
=> 2
[[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},{2,3,4,5,6,7}}
=> 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> {{1},{2,3,4,6},{5}}
=> 2
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> 1
Description
The number of terminal closers of a set partition. A closer of a set partition is a number that is maximal in its block. In particular, a singleton is a closer. This statistic counts the number of terminal closers. In other words, this is the number of closers such that all larger elements are also closers.
Matching statistic: St000990
Mp00311: Plane partitions to partitionInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00025: Dyck paths to 132-avoiding permutationPermutations
St000990: Permutations ⟶ ℤResult quality: 87% values known / values provided: 87%distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1]
=> [1,0,1,0]
=> [2,1] => 2
[[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] => 2
[[1,1]]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[[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] => 3
[[1,1],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 3
[[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 2
[[2,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 2
[[1,1,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 2
[[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] => 2
[[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] => 2
[[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] => 3
[[2,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 3
[[1,1,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 3
[[4]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 2
[[3,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 2
[[2,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 2
[[2,1,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 2
[[1,1,1,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 2
[[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] => 2
[[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] => 2
[[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] => 2
[[3],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 3
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 2
[[2,1],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 3
[[2,1],[1,1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 3
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 2
[[1,1,1],[1,1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 3
[[4],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 3
[[3,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 3
[[2,2],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 3
[[2,1,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 3
[[1,1,1,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 3
[[5]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 2
[[4,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 2
[[3,2]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 2
[[3,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 2
[[2,2,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 2
[[2,1,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 2
[[1,1,1,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 2
[[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] => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2}
[[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] => 2
[[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
[[6]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2}
[[5,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2}
[[4,2]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2}
[[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] => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2}
[[3,3]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2}
[[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] => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2}
[[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] => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2}
[[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] => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2}
[[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] => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2}
[[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] => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2}
[[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] => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2}
Description
The first ascent of a permutation. For a permutation $\pi$, this is the smallest index such that $\pi(i) < \pi(i+1)$. For the first descent, see [[St000654]].
Matching statistic: St001184
Mp00311: Plane partitions to partitionInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00120: Dyck paths Lalanne-Kreweras involutionDyck paths
St001184: Dyck paths ⟶ ℤResult quality: 87% values known / values provided: 87%distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[[1],[1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[[1,1]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[[1],[1],[1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[[2],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[[1,1],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[[2,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[[1,1,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[[2],[2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[[1,1],[1,1]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[[3],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 3
[[2,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 3
[[1,1,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 3
[[4]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[[3,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[[2,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[[2,1,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[[1,1,1,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[[2],[2],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[[3],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[3],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[2,1],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[[2,1],[1,1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[1,1,1],[1,1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[[4],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3
[[3,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3
[[2,2],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3
[[2,1,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3
[[1,1,1,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3
[[5]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 2
[[4,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 2
[[3,2]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 2
[[3,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 2
[[2,2,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 2
[[2,1,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 2
[[1,1,1,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 2
[[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]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2}
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 2
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
[[2],[2],[2]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[[6]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2}
[[5,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2}
[[4,2]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2}
[[4,1,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2}
[[3,3]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2}
[[3,2,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2}
[[3,1,1,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2}
[[2,2,2]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2}
[[2,2,1,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2}
[[2,1,1,1,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2}
[[1,1,1,1,1,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2}
Description
Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra.
Matching statistic: St000989
Mp00311: Plane partitions to partitionInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00201: Dyck paths RingelPermutations
St000989: Permutations ⟶ ℤResult quality: 69% values known / values provided: 69%distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 1 = 2 - 1
[[1],[1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 0 = 1 - 1
[[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 1 = 2 - 1
[[1,1]]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 1 = 2 - 1
[[1],[1],[1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 0 = 1 - 1
[[2],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 2 = 3 - 1
[[1,1],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 2 = 3 - 1
[[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 1 = 2 - 1
[[2,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 1 = 2 - 1
[[1,1,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 1 = 2 - 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] => 0 = 1 - 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 2 - 1
[[2],[2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 0 = 1 - 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 2 - 1
[[1,1],[1,1]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 0 = 1 - 1
[[3],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 2 = 3 - 1
[[2,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 2 = 3 - 1
[[1,1,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 2 = 3 - 1
[[4]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 1 = 2 - 1
[[3,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 1 = 2 - 1
[[2,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 1 = 2 - 1
[[2,1,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 1 = 2 - 1
[[1,1,1,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 1 = 2 - 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] => ? ∊ {1,2,2,2,2,2,2,2} - 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 - 1
[[2],[2],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0 = 1 - 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 = 2 - 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0 = 1 - 1
[[3],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 2 - 1
[[3],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 2 = 3 - 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 2 - 1
[[2,1],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 2 = 3 - 1
[[2,1],[1,1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 2 = 3 - 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 2 - 1
[[1,1,1],[1,1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 2 = 3 - 1
[[4],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 2 = 3 - 1
[[3,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 2 = 3 - 1
[[2,2],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 2 = 3 - 1
[[2,1,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 2 = 3 - 1
[[1,1,1,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 2 = 3 - 1
[[5]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? ∊ {1,2,2,2,2,2,2,2} - 1
[[4,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? ∊ {1,2,2,2,2,2,2,2} - 1
[[3,2]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? ∊ {1,2,2,2,2,2,2,2} - 1
[[3,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? ∊ {1,2,2,2,2,2,2,2} - 1
[[2,2,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? ∊ {1,2,2,2,2,2,2,2} - 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] => ? ∊ {1,2,2,2,2,2,2,2} - 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] => ? ∊ {1,2,2,2,2,2,2,2} - 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] => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} - 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] => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} - 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] => 0 = 1 - 1
[[2],[2],[2]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 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]
=> [7,1,4,5,6,2,3] => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} - 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] => 0 = 1 - 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] => 0 = 1 - 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 = 2 - 1
[[3],[2],[1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 3 = 4 - 1
[[3],[3]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 0 = 1 - 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],[2],[1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 3 = 4 - 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 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] => 0 = 1 - 1
[[5],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} - 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] => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} - 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] => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} - 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] => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} - 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] => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} - 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] => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} - 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] => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} - 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] => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} - 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] => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} - 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] => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} - 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] => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} - 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] => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} - 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] => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} - 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] => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} - 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] => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} - 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] => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} - 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] => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} - 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] => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} - 1
Description
The number of final rises of a permutation. For a permutation $\pi$ of length $n$, this is the maximal $k$ such that $$\pi(n-k) \leq \pi(n-k+1) \leq \cdots \leq \pi(n-1) \leq \pi(n).$$ Equivalently, this is $n-1$ minus the position of the last descent [[St000653]].
Matching statistic: St001640
Mp00311: Plane partitions to partitionInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00201: Dyck paths RingelPermutations
St001640: Permutations ⟶ ℤResult quality: 69% values known / values provided: 69%distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 1 = 2 - 1
[[1],[1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 0 = 1 - 1
[[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 1 = 2 - 1
[[1,1]]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 1 = 2 - 1
[[1],[1],[1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 0 = 1 - 1
[[2],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 2 = 3 - 1
[[1,1],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 2 = 3 - 1
[[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 1 = 2 - 1
[[2,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 1 = 2 - 1
[[1,1,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 1 = 2 - 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] => 0 = 1 - 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 2 - 1
[[2],[2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 0 = 1 - 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 2 - 1
[[1,1],[1,1]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 0 = 1 - 1
[[3],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 2 = 3 - 1
[[2,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 2 = 3 - 1
[[1,1,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 2 = 3 - 1
[[4]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 1 = 2 - 1
[[3,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 1 = 2 - 1
[[2,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 1 = 2 - 1
[[2,1,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 1 = 2 - 1
[[1,1,1,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 1 = 2 - 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] => ? ∊ {1,1,1,2,2,2,2,2} - 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 - 1
[[2],[2],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1 = 2 - 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 = 2 - 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1 = 2 - 1
[[3],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 2 - 1
[[3],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 2 = 3 - 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 2 - 1
[[2,1],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 2 = 3 - 1
[[2,1],[1,1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 2 = 3 - 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 2 - 1
[[1,1,1],[1,1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 2 = 3 - 1
[[4],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 2 = 3 - 1
[[3,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 2 = 3 - 1
[[2,2],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 2 = 3 - 1
[[2,1,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 2 = 3 - 1
[[1,1,1,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 2 = 3 - 1
[[5]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? ∊ {1,1,1,2,2,2,2,2} - 1
[[4,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? ∊ {1,1,1,2,2,2,2,2} - 1
[[3,2]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? ∊ {1,1,1,2,2,2,2,2} - 1
[[3,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? ∊ {1,1,1,2,2,2,2,2} - 1
[[2,2,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? ∊ {1,1,1,2,2,2,2,2} - 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] => ? ∊ {1,1,1,2,2,2,2,2} - 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] => ? ∊ {1,1,1,2,2,2,2,2} - 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] => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} - 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] => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} - 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] => 0 = 1 - 1
[[2],[2],[2]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 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]
=> [7,1,4,5,6,2,3] => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} - 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] => 0 = 1 - 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] => 0 = 1 - 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 = 2 - 1
[[3],[2],[1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 3 = 4 - 1
[[3],[3]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 0 = 1 - 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],[2],[1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 3 = 4 - 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 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] => 0 = 1 - 1
[[5],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} - 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] => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} - 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] => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} - 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] => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} - 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] => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} - 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] => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} - 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] => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} - 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] => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} - 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] => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} - 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] => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} - 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] => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} - 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] => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} - 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] => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} - 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] => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} - 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] => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} - 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] => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} - 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] => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} - 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] => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} - 1
Description
The number of ascent tops in the permutation such that all smaller elements appear before.
Mp00311: Plane partitions to partitionInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00142: Dyck paths promotionDyck paths
St001200: Dyck paths ⟶ ℤResult quality: 50% values known / values provided: 58%distinct values known / distinct values provided: 50%
Values
[[1]]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> ? = 2
[[1],[1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> ? ∊ {1,2}
[[1,1]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> ? ∊ {1,2}
[[1],[1],[1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[2],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[[1,1],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,3,3}
[[2,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,3,3}
[[1,1,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,3,3}
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[[2],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[[2],[2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[[1,1],[1,1]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[[3],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[[2,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[[1,1,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[[4]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,2,3}
[[3,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,2,3}
[[2,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,2,3}
[[2,1,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,2,3}
[[1,1,1,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,2,3}
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,3,3,3,3,3}
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3
[[2],[2],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[[3],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[[3],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[[2,1],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[[2,1],[1,1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[[1,1,1],[1,1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[[4],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
[[3,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
[[2,2],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
[[2,1,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
[[1,1,1,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
[[5]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,3,3,3,3,3}
[[4,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,3,3,3,3,3}
[[3,2]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,3,3,3,3,3}
[[3,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,3,3,3,3,3}
[[2,2,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,3,3,3,3,3}
[[2,1,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,3,3,3,3,3}
[[1,1,1,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,3,3,3,3,3}
[[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]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4}
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4}
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3
[[2],[2],[2]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4}
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[[3],[2],[1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[[3],[3]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[[2,1],[2],[1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[[2,1],[2,1]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[[1,1,1],[1,1,1]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[[4],[1],[1]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[[4],[2]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[[3,1],[1],[1]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[[3,1],[2]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[[3,1],[1,1]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[[2,2],[1],[1]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[[2,2],[2]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[[2,2],[1,1]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[[5],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4}
[[4,1],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4}
[[3,2],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4}
[[3,1,1],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4}
[[2,2,1],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4}
[[2,1,1,1],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4}
[[1,1,1,1,1],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4}
[[6]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4}
[[5,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4}
[[4,2]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4}
[[4,1,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4}
[[3,3]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4}
[[3,2,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4}
[[3,1,1,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4}
[[2,2,2]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4}
[[2,2,1,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4}
[[2,1,1,1,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4}
[[1,1,1,1,1,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4}
Description
The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
The following 12 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000668The least common multiple of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition.