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Your data matches 15 different statistics following compositions of up to 3 maps.
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Matching statistic: St001481
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001481: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck 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.
Matching statistic: St001487
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001487: Skew partitions ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 50%
Mp00179: Integer partitions —to skew partition⟶ Skew 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.
Matching statistic: St001490
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001490: Skew partitions ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 50%
Mp00179: Integer partitions —to skew partition⟶ Skew 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.
Matching statistic: St001435
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001435: Skew partitions ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 50%
Mp00179: Integer partitions —to skew partition⟶ Skew 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.
Matching statistic: St001438
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001438: Skew partitions ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 50%
Mp00179: Integer partitions —to skew partition⟶ Skew 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 partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000181: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 50%
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
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.
Matching statistic: St001208
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001208: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 50%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
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 partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St001890: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 50%
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
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 partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001811: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 50%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
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 partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 50%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck 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.
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