Your data matches 15 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001481
Mp00311: Plane partitions to partitionInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
St001481: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[1],[1]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1
[[1],[1],[1]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[2],[1]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1
[[1,1],[1]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[2],[2]]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[1,1],[1,1]]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[3],[1]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1
[[2,1],[1]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1
[[1,1,1],[1]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[[2],[2],[1]]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1
[[3],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[3],[2]]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[2,1],[2]]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[2,1],[1,1]]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[1,1,1],[1,1]]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[4],[1]]
=> [4,1]
=> [1]
=> [1,0,1,0]
=> 1
[[3,1],[1]]
=> [4,1]
=> [1]
=> [1,0,1,0]
=> 1
[[2,2],[1]]
=> [4,1]
=> [1]
=> [1,0,1,0]
=> 1
[[2,1,1],[1]]
=> [4,1]
=> [1]
=> [1,0,1,0]
=> 1
[[1,1,1,1],[1]]
=> [4,1]
=> [1]
=> [1,0,1,0]
=> 1
[[1],[1],[1],[1],[1],[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],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[[2],[2],[2]]
=> [2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[[3],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1
[[3],[3]]
=> [3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1
[[2,1],[2,1]]
=> [3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1
[[1,1,1],[1,1,1]]
=> [3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[[4],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[4],[2]]
=> [4,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[3,1],[2]]
=> [4,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
Description
The minimal height of a peak of a Dyck path.
Mp00311: Plane partitions to partitionInteger partitions
Mp00179: Integer partitions to skew partitionSkew partitions
St001487: Skew partitions ⟶ ℤResult quality: 3% values known / values provided: 3%distinct values known / distinct values provided: 50%
Values
[[1],[1]]
=> [1,1]
=> [[1,1],[]]
=> 1
[[1],[1],[1]]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1
[[2],[1]]
=> [2,1]
=> [[2,1],[]]
=> 1
[[1,1],[1]]
=> [2,1]
=> [[2,1],[]]
=> 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 1
[[2],[1],[1]]
=> [2,1,1]
=> [[2,1,1],[]]
=> 1
[[2],[2]]
=> [2,2]
=> [[2,2],[]]
=> 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [[2,1,1],[]]
=> 1
[[1,1],[1,1]]
=> [2,2]
=> [[2,2],[]]
=> 1
[[3],[1]]
=> [3,1]
=> [[3,1],[]]
=> 1
[[2,1],[1]]
=> [3,1]
=> [[3,1],[]]
=> 1
[[1,1,1],[1]]
=> [3,1]
=> [[3,1],[]]
=> 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> 1
[[2],[2],[1]]
=> [2,2,1]
=> [[2,2,1],[]]
=> 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [[2,2,1],[]]
=> 1
[[3],[1],[1]]
=> [3,1,1]
=> [[3,1,1],[]]
=> 1
[[3],[2]]
=> [3,2]
=> [[3,2],[]]
=> 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [[3,1,1],[]]
=> 1
[[2,1],[2]]
=> [3,2]
=> [[3,2],[]]
=> 1
[[2,1],[1,1]]
=> [3,2]
=> [[3,2],[]]
=> 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [[3,1,1],[]]
=> 1
[[1,1,1],[1,1]]
=> [3,2]
=> [[3,2],[]]
=> 1
[[4],[1]]
=> [4,1]
=> [[4,1],[]]
=> 1
[[3,1],[1]]
=> [4,1]
=> [[4,1],[]]
=> 1
[[2,2],[1]]
=> [4,1]
=> [[4,1],[]]
=> 1
[[2,1,1],[1]]
=> [4,1]
=> [[4,1],[]]
=> 1
[[1,1,1,1],[1]]
=> [4,1]
=> [[4,1],[]]
=> 1
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ? = 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ? = 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ? = 1
[[2],[2],[2]]
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 2
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ? = 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ? = 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 2
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ? = 1
[[3],[2],[1]]
=> [3,2,1]
=> [[3,2,1],[]]
=> ? = 1
[[3],[3]]
=> [3,3]
=> [[3,3],[]]
=> ? = 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ? = 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [[3,2,1],[]]
=> ? = 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [[3,2,1],[]]
=> ? = 1
[[2,1],[2,1]]
=> [3,3]
=> [[3,3],[]]
=> ? = 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ? = 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [[3,2,1],[]]
=> ? = 1
[[1,1,1],[1,1,1]]
=> [3,3]
=> [[3,3],[]]
=> ? = 1
[[4],[1],[1]]
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 1
[[4],[2]]
=> [4,2]
=> [[4,2],[]]
=> ? = 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 1
[[3,1],[2]]
=> [4,2]
=> [[4,2],[]]
=> ? = 1
[[3,1],[1,1]]
=> [4,2]
=> [[4,2],[]]
=> ? = 1
[[2,2],[1],[1]]
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 1
[[2,2],[2]]
=> [4,2]
=> [[4,2],[]]
=> ? = 1
[[2,2],[1,1]]
=> [4,2]
=> [[4,2],[]]
=> ? = 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 1
[[2,1,1],[2]]
=> [4,2]
=> [[4,2],[]]
=> ? = 1
[[2,1,1],[1,1]]
=> [4,2]
=> [[4,2],[]]
=> ? = 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 1
[[1,1,1,1],[1,1]]
=> [4,2]
=> [[4,2],[]]
=> ? = 1
[[5],[1]]
=> [5,1]
=> [[5,1],[]]
=> ? = 1
[[4,1],[1]]
=> [5,1]
=> [[5,1],[]]
=> ? = 1
[[3,2],[1]]
=> [5,1]
=> [[5,1],[]]
=> ? = 1
[[3,1,1],[1]]
=> [5,1]
=> [[5,1],[]]
=> ? = 1
[[2,2,1],[1]]
=> [5,1]
=> [[5,1],[]]
=> ? = 1
[[2,1,1,1],[1]]
=> [5,1]
=> [[5,1],[]]
=> ? = 1
[[1,1,1,1,1],[1]]
=> [5,1]
=> [[5,1],[]]
=> ? = 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [[2,1,1,1,1,1],[]]
=> ? = 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> ? = 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [[2,2,2,1],[]]
=> ? = 1
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [[2,1,1,1,1,1],[]]
=> ? = 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> ? = 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [[2,2,2,1],[]]
=> ? = 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[3,1,1,1,1],[]]
=> ? = 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ? = 1
[[3],[2],[2]]
=> [3,2,2]
=> [[3,2,2],[]]
=> ? = 2
[[3],[3],[1]]
=> [3,3,1]
=> [[3,3,1],[]]
=> ? = 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[3,1,1,1,1],[]]
=> ? = 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ? = 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [[3,2,2],[]]
=> ? = 2
Description
The number of inner corners of a skew partition.
Mp00311: Plane partitions to partitionInteger partitions
Mp00179: Integer partitions to skew partitionSkew partitions
St001490: Skew partitions ⟶ ℤResult quality: 3% values known / values provided: 3%distinct values known / distinct values provided: 50%
Values
[[1],[1]]
=> [1,1]
=> [[1,1],[]]
=> 1
[[1],[1],[1]]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1
[[2],[1]]
=> [2,1]
=> [[2,1],[]]
=> 1
[[1,1],[1]]
=> [2,1]
=> [[2,1],[]]
=> 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 1
[[2],[1],[1]]
=> [2,1,1]
=> [[2,1,1],[]]
=> 1
[[2],[2]]
=> [2,2]
=> [[2,2],[]]
=> 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [[2,1,1],[]]
=> 1
[[1,1],[1,1]]
=> [2,2]
=> [[2,2],[]]
=> 1
[[3],[1]]
=> [3,1]
=> [[3,1],[]]
=> 1
[[2,1],[1]]
=> [3,1]
=> [[3,1],[]]
=> 1
[[1,1,1],[1]]
=> [3,1]
=> [[3,1],[]]
=> 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> 1
[[2],[2],[1]]
=> [2,2,1]
=> [[2,2,1],[]]
=> 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [[2,2,1],[]]
=> 1
[[3],[1],[1]]
=> [3,1,1]
=> [[3,1,1],[]]
=> 1
[[3],[2]]
=> [3,2]
=> [[3,2],[]]
=> 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [[3,1,1],[]]
=> 1
[[2,1],[2]]
=> [3,2]
=> [[3,2],[]]
=> 1
[[2,1],[1,1]]
=> [3,2]
=> [[3,2],[]]
=> 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [[3,1,1],[]]
=> 1
[[1,1,1],[1,1]]
=> [3,2]
=> [[3,2],[]]
=> 1
[[4],[1]]
=> [4,1]
=> [[4,1],[]]
=> 1
[[3,1],[1]]
=> [4,1]
=> [[4,1],[]]
=> 1
[[2,2],[1]]
=> [4,1]
=> [[4,1],[]]
=> 1
[[2,1,1],[1]]
=> [4,1]
=> [[4,1],[]]
=> 1
[[1,1,1,1],[1]]
=> [4,1]
=> [[4,1],[]]
=> 1
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ? = 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ? = 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ? = 1
[[2],[2],[2]]
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 2
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ? = 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ? = 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 2
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ? = 1
[[3],[2],[1]]
=> [3,2,1]
=> [[3,2,1],[]]
=> ? = 1
[[3],[3]]
=> [3,3]
=> [[3,3],[]]
=> ? = 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ? = 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [[3,2,1],[]]
=> ? = 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [[3,2,1],[]]
=> ? = 1
[[2,1],[2,1]]
=> [3,3]
=> [[3,3],[]]
=> ? = 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ? = 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [[3,2,1],[]]
=> ? = 1
[[1,1,1],[1,1,1]]
=> [3,3]
=> [[3,3],[]]
=> ? = 1
[[4],[1],[1]]
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 1
[[4],[2]]
=> [4,2]
=> [[4,2],[]]
=> ? = 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 1
[[3,1],[2]]
=> [4,2]
=> [[4,2],[]]
=> ? = 1
[[3,1],[1,1]]
=> [4,2]
=> [[4,2],[]]
=> ? = 1
[[2,2],[1],[1]]
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 1
[[2,2],[2]]
=> [4,2]
=> [[4,2],[]]
=> ? = 1
[[2,2],[1,1]]
=> [4,2]
=> [[4,2],[]]
=> ? = 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 1
[[2,1,1],[2]]
=> [4,2]
=> [[4,2],[]]
=> ? = 1
[[2,1,1],[1,1]]
=> [4,2]
=> [[4,2],[]]
=> ? = 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 1
[[1,1,1,1],[1,1]]
=> [4,2]
=> [[4,2],[]]
=> ? = 1
[[5],[1]]
=> [5,1]
=> [[5,1],[]]
=> ? = 1
[[4,1],[1]]
=> [5,1]
=> [[5,1],[]]
=> ? = 1
[[3,2],[1]]
=> [5,1]
=> [[5,1],[]]
=> ? = 1
[[3,1,1],[1]]
=> [5,1]
=> [[5,1],[]]
=> ? = 1
[[2,2,1],[1]]
=> [5,1]
=> [[5,1],[]]
=> ? = 1
[[2,1,1,1],[1]]
=> [5,1]
=> [[5,1],[]]
=> ? = 1
[[1,1,1,1,1],[1]]
=> [5,1]
=> [[5,1],[]]
=> ? = 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [[2,1,1,1,1,1],[]]
=> ? = 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> ? = 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [[2,2,2,1],[]]
=> ? = 1
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [[2,1,1,1,1,1],[]]
=> ? = 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> ? = 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [[2,2,2,1],[]]
=> ? = 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[3,1,1,1,1],[]]
=> ? = 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ? = 1
[[3],[2],[2]]
=> [3,2,2]
=> [[3,2,2],[]]
=> ? = 2
[[3],[3],[1]]
=> [3,3,1]
=> [[3,3,1],[]]
=> ? = 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[3,1,1,1,1],[]]
=> ? = 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ? = 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [[3,2,2],[]]
=> ? = 2
Description
The number of connected components of a skew partition.
Mp00311: Plane partitions to partitionInteger partitions
Mp00179: Integer partitions to skew partitionSkew partitions
St001435: Skew partitions ⟶ ℤResult quality: 3% values known / values provided: 3%distinct values known / distinct values provided: 50%
Values
[[1],[1]]
=> [1,1]
=> [[1,1],[]]
=> 0 = 1 - 1
[[1],[1],[1]]
=> [1,1,1]
=> [[1,1,1],[]]
=> 0 = 1 - 1
[[2],[1]]
=> [2,1]
=> [[2,1],[]]
=> 0 = 1 - 1
[[1,1],[1]]
=> [2,1]
=> [[2,1],[]]
=> 0 = 1 - 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 0 = 1 - 1
[[2],[1],[1]]
=> [2,1,1]
=> [[2,1,1],[]]
=> 0 = 1 - 1
[[2],[2]]
=> [2,2]
=> [[2,2],[]]
=> 0 = 1 - 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [[2,1,1],[]]
=> 0 = 1 - 1
[[1,1],[1,1]]
=> [2,2]
=> [[2,2],[]]
=> 0 = 1 - 1
[[3],[1]]
=> [3,1]
=> [[3,1],[]]
=> 0 = 1 - 1
[[2,1],[1]]
=> [3,1]
=> [[3,1],[]]
=> 0 = 1 - 1
[[1,1,1],[1]]
=> [3,1]
=> [[3,1],[]]
=> 0 = 1 - 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 0 = 1 - 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> 0 = 1 - 1
[[2],[2],[1]]
=> [2,2,1]
=> [[2,2,1],[]]
=> 0 = 1 - 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> 0 = 1 - 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [[2,2,1],[]]
=> 0 = 1 - 1
[[3],[1],[1]]
=> [3,1,1]
=> [[3,1,1],[]]
=> 0 = 1 - 1
[[3],[2]]
=> [3,2]
=> [[3,2],[]]
=> 0 = 1 - 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [[3,1,1],[]]
=> 0 = 1 - 1
[[2,1],[2]]
=> [3,2]
=> [[3,2],[]]
=> 0 = 1 - 1
[[2,1],[1,1]]
=> [3,2]
=> [[3,2],[]]
=> 0 = 1 - 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [[3,1,1],[]]
=> 0 = 1 - 1
[[1,1,1],[1,1]]
=> [3,2]
=> [[3,2],[]]
=> 0 = 1 - 1
[[4],[1]]
=> [4,1]
=> [[4,1],[]]
=> 0 = 1 - 1
[[3,1],[1]]
=> [4,1]
=> [[4,1],[]]
=> 0 = 1 - 1
[[2,2],[1]]
=> [4,1]
=> [[4,1],[]]
=> 0 = 1 - 1
[[2,1,1],[1]]
=> [4,1]
=> [[4,1],[]]
=> 0 = 1 - 1
[[1,1,1,1],[1]]
=> [4,1]
=> [[4,1],[]]
=> 0 = 1 - 1
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ? = 1 - 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ? = 1 - 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ? = 1 - 1
[[2],[2],[2]]
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 2 - 1
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ? = 1 - 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ? = 1 - 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 2 - 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ? = 1 - 1
[[3],[2],[1]]
=> [3,2,1]
=> [[3,2,1],[]]
=> ? = 1 - 1
[[3],[3]]
=> [3,3]
=> [[3,3],[]]
=> ? = 1 - 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ? = 1 - 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [[3,2,1],[]]
=> ? = 1 - 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [[3,2,1],[]]
=> ? = 1 - 1
[[2,1],[2,1]]
=> [3,3]
=> [[3,3],[]]
=> ? = 1 - 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ? = 1 - 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [[3,2,1],[]]
=> ? = 1 - 1
[[1,1,1],[1,1,1]]
=> [3,3]
=> [[3,3],[]]
=> ? = 1 - 1
[[4],[1],[1]]
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 1 - 1
[[4],[2]]
=> [4,2]
=> [[4,2],[]]
=> ? = 1 - 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 1 - 1
[[3,1],[2]]
=> [4,2]
=> [[4,2],[]]
=> ? = 1 - 1
[[3,1],[1,1]]
=> [4,2]
=> [[4,2],[]]
=> ? = 1 - 1
[[2,2],[1],[1]]
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 1 - 1
[[2,2],[2]]
=> [4,2]
=> [[4,2],[]]
=> ? = 1 - 1
[[2,2],[1,1]]
=> [4,2]
=> [[4,2],[]]
=> ? = 1 - 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 1 - 1
[[2,1,1],[2]]
=> [4,2]
=> [[4,2],[]]
=> ? = 1 - 1
[[2,1,1],[1,1]]
=> [4,2]
=> [[4,2],[]]
=> ? = 1 - 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 1 - 1
[[1,1,1,1],[1,1]]
=> [4,2]
=> [[4,2],[]]
=> ? = 1 - 1
[[5],[1]]
=> [5,1]
=> [[5,1],[]]
=> ? = 1 - 1
[[4,1],[1]]
=> [5,1]
=> [[5,1],[]]
=> ? = 1 - 1
[[3,2],[1]]
=> [5,1]
=> [[5,1],[]]
=> ? = 1 - 1
[[3,1,1],[1]]
=> [5,1]
=> [[5,1],[]]
=> ? = 1 - 1
[[2,2,1],[1]]
=> [5,1]
=> [[5,1],[]]
=> ? = 1 - 1
[[2,1,1,1],[1]]
=> [5,1]
=> [[5,1],[]]
=> ? = 1 - 1
[[1,1,1,1,1],[1]]
=> [5,1]
=> [[5,1],[]]
=> ? = 1 - 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [[2,1,1,1,1,1],[]]
=> ? = 1 - 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> ? = 1 - 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [[2,2,2,1],[]]
=> ? = 1 - 1
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [[2,1,1,1,1,1],[]]
=> ? = 1 - 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> ? = 1 - 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [[2,2,2,1],[]]
=> ? = 1 - 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[3,1,1,1,1],[]]
=> ? = 1 - 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ? = 1 - 1
[[3],[2],[2]]
=> [3,2,2]
=> [[3,2,2],[]]
=> ? = 2 - 1
[[3],[3],[1]]
=> [3,3,1]
=> [[3,3,1],[]]
=> ? = 1 - 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[3,1,1,1,1],[]]
=> ? = 1 - 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ? = 1 - 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [[3,2,2],[]]
=> ? = 2 - 1
Description
The number of missing boxes in the first row.
Mp00311: Plane partitions to partitionInteger partitions
Mp00179: Integer partitions to skew partitionSkew partitions
St001438: Skew partitions ⟶ ℤResult quality: 3% values known / values provided: 3%distinct values known / distinct values provided: 50%
Values
[[1],[1]]
=> [1,1]
=> [[1,1],[]]
=> 0 = 1 - 1
[[1],[1],[1]]
=> [1,1,1]
=> [[1,1,1],[]]
=> 0 = 1 - 1
[[2],[1]]
=> [2,1]
=> [[2,1],[]]
=> 0 = 1 - 1
[[1,1],[1]]
=> [2,1]
=> [[2,1],[]]
=> 0 = 1 - 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 0 = 1 - 1
[[2],[1],[1]]
=> [2,1,1]
=> [[2,1,1],[]]
=> 0 = 1 - 1
[[2],[2]]
=> [2,2]
=> [[2,2],[]]
=> 0 = 1 - 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [[2,1,1],[]]
=> 0 = 1 - 1
[[1,1],[1,1]]
=> [2,2]
=> [[2,2],[]]
=> 0 = 1 - 1
[[3],[1]]
=> [3,1]
=> [[3,1],[]]
=> 0 = 1 - 1
[[2,1],[1]]
=> [3,1]
=> [[3,1],[]]
=> 0 = 1 - 1
[[1,1,1],[1]]
=> [3,1]
=> [[3,1],[]]
=> 0 = 1 - 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 0 = 1 - 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> 0 = 1 - 1
[[2],[2],[1]]
=> [2,2,1]
=> [[2,2,1],[]]
=> 0 = 1 - 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> 0 = 1 - 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [[2,2,1],[]]
=> 0 = 1 - 1
[[3],[1],[1]]
=> [3,1,1]
=> [[3,1,1],[]]
=> 0 = 1 - 1
[[3],[2]]
=> [3,2]
=> [[3,2],[]]
=> 0 = 1 - 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [[3,1,1],[]]
=> 0 = 1 - 1
[[2,1],[2]]
=> [3,2]
=> [[3,2],[]]
=> 0 = 1 - 1
[[2,1],[1,1]]
=> [3,2]
=> [[3,2],[]]
=> 0 = 1 - 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [[3,1,1],[]]
=> 0 = 1 - 1
[[1,1,1],[1,1]]
=> [3,2]
=> [[3,2],[]]
=> 0 = 1 - 1
[[4],[1]]
=> [4,1]
=> [[4,1],[]]
=> 0 = 1 - 1
[[3,1],[1]]
=> [4,1]
=> [[4,1],[]]
=> 0 = 1 - 1
[[2,2],[1]]
=> [4,1]
=> [[4,1],[]]
=> 0 = 1 - 1
[[2,1,1],[1]]
=> [4,1]
=> [[4,1],[]]
=> 0 = 1 - 1
[[1,1,1,1],[1]]
=> [4,1]
=> [[4,1],[]]
=> 0 = 1 - 1
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ? = 1 - 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ? = 1 - 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ? = 1 - 1
[[2],[2],[2]]
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 2 - 1
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ? = 1 - 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ? = 1 - 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 2 - 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ? = 1 - 1
[[3],[2],[1]]
=> [3,2,1]
=> [[3,2,1],[]]
=> ? = 1 - 1
[[3],[3]]
=> [3,3]
=> [[3,3],[]]
=> ? = 1 - 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ? = 1 - 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [[3,2,1],[]]
=> ? = 1 - 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [[3,2,1],[]]
=> ? = 1 - 1
[[2,1],[2,1]]
=> [3,3]
=> [[3,3],[]]
=> ? = 1 - 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ? = 1 - 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [[3,2,1],[]]
=> ? = 1 - 1
[[1,1,1],[1,1,1]]
=> [3,3]
=> [[3,3],[]]
=> ? = 1 - 1
[[4],[1],[1]]
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 1 - 1
[[4],[2]]
=> [4,2]
=> [[4,2],[]]
=> ? = 1 - 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 1 - 1
[[3,1],[2]]
=> [4,2]
=> [[4,2],[]]
=> ? = 1 - 1
[[3,1],[1,1]]
=> [4,2]
=> [[4,2],[]]
=> ? = 1 - 1
[[2,2],[1],[1]]
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 1 - 1
[[2,2],[2]]
=> [4,2]
=> [[4,2],[]]
=> ? = 1 - 1
[[2,2],[1,1]]
=> [4,2]
=> [[4,2],[]]
=> ? = 1 - 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 1 - 1
[[2,1,1],[2]]
=> [4,2]
=> [[4,2],[]]
=> ? = 1 - 1
[[2,1,1],[1,1]]
=> [4,2]
=> [[4,2],[]]
=> ? = 1 - 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 1 - 1
[[1,1,1,1],[1,1]]
=> [4,2]
=> [[4,2],[]]
=> ? = 1 - 1
[[5],[1]]
=> [5,1]
=> [[5,1],[]]
=> ? = 1 - 1
[[4,1],[1]]
=> [5,1]
=> [[5,1],[]]
=> ? = 1 - 1
[[3,2],[1]]
=> [5,1]
=> [[5,1],[]]
=> ? = 1 - 1
[[3,1,1],[1]]
=> [5,1]
=> [[5,1],[]]
=> ? = 1 - 1
[[2,2,1],[1]]
=> [5,1]
=> [[5,1],[]]
=> ? = 1 - 1
[[2,1,1,1],[1]]
=> [5,1]
=> [[5,1],[]]
=> ? = 1 - 1
[[1,1,1,1,1],[1]]
=> [5,1]
=> [[5,1],[]]
=> ? = 1 - 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [[2,1,1,1,1,1],[]]
=> ? = 1 - 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> ? = 1 - 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [[2,2,2,1],[]]
=> ? = 1 - 1
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [[2,1,1,1,1,1],[]]
=> ? = 1 - 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> ? = 1 - 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [[2,2,2,1],[]]
=> ? = 1 - 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[3,1,1,1,1],[]]
=> ? = 1 - 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ? = 1 - 1
[[3],[2],[2]]
=> [3,2,2]
=> [[3,2,2],[]]
=> ? = 2 - 1
[[3],[3],[1]]
=> [3,3,1]
=> [[3,3,1],[]]
=> ? = 1 - 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[3,1,1,1,1],[]]
=> ? = 1 - 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ? = 1 - 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [[3,2,2],[]]
=> ? = 2 - 1
Description
The number of missing boxes of a skew partition.
Matching statistic: St000181
Mp00311: Plane partitions to partitionInteger partitions
Mp00179: Integer partitions to skew partitionSkew partitions
Mp00185: Skew partitions cell posetPosets
St000181: Posets ⟶ ℤResult quality: 3% values known / values provided: 3%distinct values known / distinct values provided: 50%
Values
[[1],[1]]
=> [1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> 1
[[1],[1],[1]]
=> [1,1,1]
=> [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 1
[[2],[1]]
=> [2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> 1
[[1,1],[1]]
=> [2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[2],[1],[1]]
=> [2,1,1]
=> [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 1
[[2],[2]]
=> [2,2]
=> [[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 1
[[1,1],[1,1]]
=> [2,2]
=> [[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[3],[1]]
=> [3,1]
=> [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 1
[[2,1],[1]]
=> [3,1]
=> [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 1
[[1,1,1],[1]]
=> [3,1]
=> [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 1
[[2],[2],[1]]
=> [2,2,1]
=> [[2,2,1],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [[2,2,1],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 1
[[3],[1],[1]]
=> [3,1,1]
=> [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 1
[[3],[2]]
=> [3,2]
=> [[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 1
[[2,1],[2]]
=> [3,2]
=> [[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 1
[[2,1],[1,1]]
=> [3,2]
=> [[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 1
[[1,1,1],[1,1]]
=> [3,2]
=> [[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 1
[[4],[1]]
=> [4,1]
=> [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 1
[[3,1],[1]]
=> [4,1]
=> [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 1
[[2,2],[1]]
=> [4,1]
=> [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 1
[[2,1,1],[1]]
=> [4,1]
=> [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 1
[[1,1,1,1],[1]]
=> [4,1]
=> [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 1
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ? = 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> ? = 1
[[2],[2],[2]]
=> [2,2,2]
=> [[2,2,2],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ? = 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> ? = 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [[2,2,2],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ? = 1
[[3],[2],[1]]
=> [3,2,1]
=> [[3,2,1],[]]
=> ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> ? = 1
[[3],[3]]
=> [3,3]
=> [[3,3],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ? = 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [[3,2,1],[]]
=> ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> ? = 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [[3,2,1],[]]
=> ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> ? = 1
[[2,1],[2,1]]
=> [3,3]
=> [[3,3],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ? = 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [[3,2,1],[]]
=> ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> ? = 1
[[1,1,1],[1,1,1]]
=> [3,3]
=> [[3,3],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
[[4],[1],[1]]
=> [4,1,1]
=> [[4,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ? = 1
[[4],[2]]
=> [4,2]
=> [[4,2],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> ? = 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [[4,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ? = 1
[[3,1],[2]]
=> [4,2]
=> [[4,2],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> ? = 1
[[3,1],[1,1]]
=> [4,2]
=> [[4,2],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> ? = 1
[[2,2],[1],[1]]
=> [4,1,1]
=> [[4,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ? = 1
[[2,2],[2]]
=> [4,2]
=> [[4,2],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> ? = 1
[[2,2],[1,1]]
=> [4,2]
=> [[4,2],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> ? = 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [[4,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ? = 1
[[2,1,1],[2]]
=> [4,2]
=> [[4,2],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> ? = 1
[[2,1,1],[1,1]]
=> [4,2]
=> [[4,2],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> ? = 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [[4,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ? = 1
[[1,1,1,1],[1,1]]
=> [4,2]
=> [[4,2],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> ? = 1
[[5],[1]]
=> [5,1]
=> [[5,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ? = 1
[[4,1],[1]]
=> [5,1]
=> [[5,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ? = 1
[[3,2],[1]]
=> [5,1]
=> [[5,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ? = 1
[[3,1,1],[1]]
=> [5,1]
=> [[5,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ? = 1
[[2,2,1],[1]]
=> [5,1]
=> [[5,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ? = 1
[[2,1,1,1],[1]]
=> [5,1]
=> [[5,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ? = 1
[[1,1,1,1,1],[1]]
=> [5,1]
=> [[5,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ? = 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [[2,1,1,1,1,1],[]]
=> ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
=> ? = 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
=> ? = 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [[2,2,2,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> ? = 1
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [[2,1,1,1,1,1],[]]
=> ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
=> ? = 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
=> ? = 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [[2,2,2,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> ? = 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[3,1,1,1,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
=> ? = 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1
[[3],[2],[2]]
=> [3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 2
[[3],[3],[1]]
=> [3,3,1]
=> [[3,3,1],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[3,1,1,1,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
=> ? = 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 2
Description
The number of connected components of the Hasse diagram for the poset.
Mp00311: Plane partitions to partitionInteger partitions
Mp00042: Integer partitions initial tableauStandard tableaux
Mp00081: Standard tableaux reading word permutationPermutations
St001208: Permutations ⟶ ℤResult quality: 3% values known / values provided: 3%distinct values known / distinct values provided: 50%
Values
[[1],[1]]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 1
[[1],[1],[1]]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[[2],[1]]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[[1,1],[1]]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 1
[[2],[1],[1]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[[2],[2]]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[[1,1],[1,1]]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 1
[[3],[1]]
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[[2,1],[1]]
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[[1,1,1],[1]]
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[[2],[2],[1]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 1
[[3],[1],[1]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[[3],[2]]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[[2,1],[2]]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 1
[[2,1],[1,1]]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[[1,1,1],[1,1]]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 1
[[4],[1]]
=> [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1
[[3,1],[1]]
=> [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1
[[2,2],[1]]
=> [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1
[[2,1,1],[1]]
=> [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1
[[1,1,1,1],[1]]
=> [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => ? = 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => ? = 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => ? = 1
[[2],[2],[2]]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => ? = 2
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => ? = 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => ? = 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => ? = 2
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => ? = 1
[[3],[2],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => ? = 1
[[3],[3]]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => ? = 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => ? = 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => ? = 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => ? = 1
[[2,1],[2,1]]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => ? = 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => ? = 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => ? = 1
[[1,1,1],[1,1,1]]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => ? = 1
[[4],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => ? = 1
[[4],[2]]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => ? = 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => ? = 1
[[3,1],[2]]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => ? = 1
[[3,1],[1,1]]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => ? = 1
[[2,2],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => ? = 1
[[2,2],[2]]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => ? = 1
[[2,2],[1,1]]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => ? = 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => ? = 1
[[2,1,1],[2]]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => ? = 1
[[2,1,1],[1,1]]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => ? = 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => ? = 1
[[1,1,1,1],[1,1]]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => ? = 1
[[5],[1]]
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ? = 1
[[4,1],[1]]
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ? = 1
[[3,2],[1]]
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ? = 1
[[3,1,1],[1]]
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ? = 1
[[2,2,1],[1]]
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ? = 1
[[2,1,1,1],[1]]
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ? = 1
[[1,1,1,1,1],[1]]
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ? = 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? = 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ? = 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => ? = 1
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? = 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ? = 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => ? = 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 1
[[3],[2],[2]]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => ? = 2
[[3],[3],[1]]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => ? = 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => ? = 2
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: St001890
Mp00311: Plane partitions to partitionInteger partitions
Mp00179: Integer partitions to skew partitionSkew partitions
Mp00185: Skew partitions cell posetPosets
St001890: Posets ⟶ ℤResult quality: 3% values known / values provided: 3%distinct values known / distinct values provided: 50%
Values
[[1],[1]]
=> [1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> 1
[[1],[1],[1]]
=> [1,1,1]
=> [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 1
[[2],[1]]
=> [2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> 1
[[1,1],[1]]
=> [2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[2],[1],[1]]
=> [2,1,1]
=> [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 1
[[2],[2]]
=> [2,2]
=> [[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 1
[[1,1],[1,1]]
=> [2,2]
=> [[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[3],[1]]
=> [3,1]
=> [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 1
[[2,1],[1]]
=> [3,1]
=> [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 1
[[1,1,1],[1]]
=> [3,1]
=> [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 1
[[2],[2],[1]]
=> [2,2,1]
=> [[2,2,1],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [[2,2,1],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 1
[[3],[1],[1]]
=> [3,1,1]
=> [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 1
[[3],[2]]
=> [3,2]
=> [[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 1
[[2,1],[2]]
=> [3,2]
=> [[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 1
[[2,1],[1,1]]
=> [3,2]
=> [[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 1
[[1,1,1],[1,1]]
=> [3,2]
=> [[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 1
[[4],[1]]
=> [4,1]
=> [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 1
[[3,1],[1]]
=> [4,1]
=> [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 1
[[2,2],[1]]
=> [4,1]
=> [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 1
[[2,1,1],[1]]
=> [4,1]
=> [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 1
[[1,1,1,1],[1]]
=> [4,1]
=> [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 1
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ? = 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> ? = 1
[[2],[2],[2]]
=> [2,2,2]
=> [[2,2,2],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ? = 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> ? = 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [[2,2,2],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ? = 1
[[3],[2],[1]]
=> [3,2,1]
=> [[3,2,1],[]]
=> ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> ? = 1
[[3],[3]]
=> [3,3]
=> [[3,3],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ? = 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [[3,2,1],[]]
=> ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> ? = 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [[3,2,1],[]]
=> ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> ? = 1
[[2,1],[2,1]]
=> [3,3]
=> [[3,3],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ? = 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [[3,2,1],[]]
=> ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> ? = 1
[[1,1,1],[1,1,1]]
=> [3,3]
=> [[3,3],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
[[4],[1],[1]]
=> [4,1,1]
=> [[4,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ? = 1
[[4],[2]]
=> [4,2]
=> [[4,2],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> ? = 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [[4,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ? = 1
[[3,1],[2]]
=> [4,2]
=> [[4,2],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> ? = 1
[[3,1],[1,1]]
=> [4,2]
=> [[4,2],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> ? = 1
[[2,2],[1],[1]]
=> [4,1,1]
=> [[4,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ? = 1
[[2,2],[2]]
=> [4,2]
=> [[4,2],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> ? = 1
[[2,2],[1,1]]
=> [4,2]
=> [[4,2],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> ? = 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [[4,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ? = 1
[[2,1,1],[2]]
=> [4,2]
=> [[4,2],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> ? = 1
[[2,1,1],[1,1]]
=> [4,2]
=> [[4,2],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> ? = 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [[4,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ? = 1
[[1,1,1,1],[1,1]]
=> [4,2]
=> [[4,2],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> ? = 1
[[5],[1]]
=> [5,1]
=> [[5,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ? = 1
[[4,1],[1]]
=> [5,1]
=> [[5,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ? = 1
[[3,2],[1]]
=> [5,1]
=> [[5,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ? = 1
[[3,1,1],[1]]
=> [5,1]
=> [[5,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ? = 1
[[2,2,1],[1]]
=> [5,1]
=> [[5,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ? = 1
[[2,1,1,1],[1]]
=> [5,1]
=> [[5,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ? = 1
[[1,1,1,1,1],[1]]
=> [5,1]
=> [[5,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ? = 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [[2,1,1,1,1,1],[]]
=> ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
=> ? = 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
=> ? = 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [[2,2,2,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> ? = 1
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [[2,1,1,1,1,1],[]]
=> ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
=> ? = 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
=> ? = 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [[2,2,2,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> ? = 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[3,1,1,1,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
=> ? = 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1
[[3],[2],[2]]
=> [3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 2
[[3],[3],[1]]
=> [3,3,1]
=> [[3,3,1],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[3,1,1,1,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
=> ? = 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 2
Description
The maximum magnitude of the Möbius function of a poset. The '''Möbius function''' of a poset is the multiplicative inverse of the zeta function in the incidence algebra. The Möbius value $\mu(x, y)$ is equal to the signed sum of chains from $x$ to $y$, where odd-length chains are counted with a minus sign, so this statistic is bounded above by the total number of chains in the poset.
Matching statistic: St001811
Mp00311: Plane partitions to partitionInteger partitions
Mp00042: Integer partitions initial tableauStandard tableaux
Mp00081: Standard tableaux reading word permutationPermutations
St001811: Permutations ⟶ ℤResult quality: 3% values known / values provided: 3%distinct values known / distinct values provided: 50%
Values
[[1],[1]]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0 = 1 - 1
[[1],[1],[1]]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0 = 1 - 1
[[2],[1]]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0 = 1 - 1
[[1,1],[1]]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0 = 1 - 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0 = 1 - 1
[[2],[1],[1]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0 = 1 - 1
[[2],[2]]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 0 = 1 - 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0 = 1 - 1
[[1,1],[1,1]]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 0 = 1 - 1
[[3],[1]]
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0 = 1 - 1
[[2,1],[1]]
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0 = 1 - 1
[[1,1,1],[1]]
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0 = 1 - 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 0 = 1 - 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 0 = 1 - 1
[[2],[2],[1]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 0 = 1 - 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 0 = 1 - 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 0 = 1 - 1
[[3],[1],[1]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 0 = 1 - 1
[[3],[2]]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 0 = 1 - 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 0 = 1 - 1
[[2,1],[2]]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 0 = 1 - 1
[[2,1],[1,1]]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 0 = 1 - 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 0 = 1 - 1
[[1,1,1],[1,1]]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 0 = 1 - 1
[[4],[1]]
=> [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 0 = 1 - 1
[[3,1],[1]]
=> [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 0 = 1 - 1
[[2,2],[1]]
=> [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 0 = 1 - 1
[[2,1,1],[1]]
=> [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 0 = 1 - 1
[[1,1,1,1],[1]]
=> [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 0 = 1 - 1
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => ? = 1 - 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => ? = 1 - 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => ? = 1 - 1
[[2],[2],[2]]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => ? = 2 - 1
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => ? = 1 - 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => ? = 1 - 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => ? = 2 - 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => ? = 1 - 1
[[3],[2],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => ? = 1 - 1
[[3],[3]]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => ? = 1 - 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => ? = 1 - 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => ? = 1 - 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => ? = 1 - 1
[[2,1],[2,1]]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => ? = 1 - 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => ? = 1 - 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => ? = 1 - 1
[[1,1,1],[1,1,1]]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => ? = 1 - 1
[[4],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => ? = 1 - 1
[[4],[2]]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => ? = 1 - 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => ? = 1 - 1
[[3,1],[2]]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => ? = 1 - 1
[[3,1],[1,1]]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => ? = 1 - 1
[[2,2],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => ? = 1 - 1
[[2,2],[2]]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => ? = 1 - 1
[[2,2],[1,1]]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => ? = 1 - 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => ? = 1 - 1
[[2,1,1],[2]]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => ? = 1 - 1
[[2,1,1],[1,1]]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => ? = 1 - 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => ? = 1 - 1
[[1,1,1,1],[1,1]]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => ? = 1 - 1
[[5],[1]]
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ? = 1 - 1
[[4,1],[1]]
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ? = 1 - 1
[[3,2],[1]]
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ? = 1 - 1
[[3,1,1],[1]]
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ? = 1 - 1
[[2,2,1],[1]]
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ? = 1 - 1
[[2,1,1,1],[1]]
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ? = 1 - 1
[[1,1,1,1,1],[1]]
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ? = 1 - 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? = 1 - 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ? = 1 - 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => ? = 1 - 1
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? = 1 - 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ? = 1 - 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => ? = 1 - 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 1 - 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 1 - 1
[[3],[2],[2]]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => ? = 2 - 1
[[3],[3],[1]]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => ? = 1 - 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 1 - 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 1 - 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => ? = 2 - 1
Description
The Castelnuovo-Mumford regularity of a permutation. The ''Castelnuovo-Mumford regularity'' of a permutation $\sigma$ is the ''Castelnuovo-Mumford regularity'' of the ''matrix Schubert variety'' $X_\sigma$. Equivalently, it is the difference between the degrees of the ''Grothendieck polynomial'' and the ''Schubert polynomial'' for $\sigma$. It can be computed by subtracting the ''Coxeter length'' [[St000018]] from the ''Rajchgot index'' [[St001759]].
Matching statistic: St001195
Mp00311: Plane partitions to partitionInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00199: Dyck paths prime Dyck pathDyck paths
St001195: Dyck paths ⟶ ℤResult quality: 3% values known / values provided: 3%distinct values known / distinct values provided: 50%
Values
[[1],[1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 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
[[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
[[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
[[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]
=> ? = 2
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 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]
=> ? = 2
[[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
[[1,1,1],[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
[[1,1,1],[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
[[1,1,1],[1,1,1]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 1
[[4],[1],[1]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1
[[4],[2]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1
[[3,1],[2]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 1
[[3,1],[1,1]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 1
[[2,2],[1],[1]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1
[[2,2],[2]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 1
[[2,2],[1,1]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1
[[2,1,1],[2]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 1
[[2,1,1],[1,1]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1
[[1,1,1,1],[1,1]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 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
[[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
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 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
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 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
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 1
Description
The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$.
The following 5 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. 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. 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.