Your data matches 47 different statistics following compositions of up to 3 maps.
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Mp00183: Skew partitions inner shapeInteger partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[1],[]]
=> []
=> 0
[[2],[]]
=> []
=> 0
[[1,1],[]]
=> []
=> 0
[[2,1],[1]]
=> [1]
=> 1
[[3],[]]
=> []
=> 0
[[2,1],[]]
=> []
=> 0
[[3,1],[1]]
=> [1]
=> 1
[[2,2],[1]]
=> [1]
=> 1
[[3,2],[2]]
=> [2]
=> 1
[[1,1,1],[]]
=> []
=> 0
[[2,2,1],[1,1]]
=> [1,1]
=> 2
[[2,1,1],[1]]
=> [1]
=> 1
[[3,2,1],[2,1]]
=> [2,1]
=> 2
[[4],[]]
=> []
=> 0
[[3,1],[]]
=> []
=> 0
[[4,1],[1]]
=> [1]
=> 1
[[2,2],[]]
=> []
=> 0
[[3,2],[1]]
=> [1]
=> 1
[[4,2],[2]]
=> [2]
=> 1
[[2,1,1],[]]
=> []
=> 0
[[3,2,1],[1,1]]
=> [1,1]
=> 2
[[3,1,1],[1]]
=> [1]
=> 1
[[4,2,1],[2,1]]
=> [2,1]
=> 2
[[3,3],[2]]
=> [2]
=> 1
[[4,3],[3]]
=> [3]
=> 1
[[2,2,1],[1]]
=> [1]
=> 1
[[3,3,1],[2,1]]
=> [2,1]
=> 2
[[3,2,1],[2]]
=> [2]
=> 1
[[4,3,1],[3,1]]
=> [3,1]
=> 2
[[2,2,2],[1,1]]
=> [1,1]
=> 2
[[3,3,2],[2,2]]
=> [2,2]
=> 2
[[3,2,2],[2,1]]
=> [2,1]
=> 2
[[4,3,2],[3,2]]
=> [3,2]
=> 2
[[1,1,1,1],[]]
=> []
=> 0
[[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 3
[[2,2,1,1],[1,1]]
=> [1,1]
=> 2
[[3,3,2,1],[2,2,1]]
=> [2,2,1]
=> 3
[[2,1,1,1],[1]]
=> [1]
=> 1
[[3,2,2,1],[2,1,1]]
=> [2,1,1]
=> 3
[[3,2,1,1],[2,1]]
=> [2,1]
=> 2
[[4,3,2,1],[3,2,1]]
=> [3,2,1]
=> 3
[[5],[]]
=> []
=> 0
[[4,1],[]]
=> []
=> 0
[[5,1],[1]]
=> [1]
=> 1
[[3,2],[]]
=> []
=> 0
[[4,2],[1]]
=> [1]
=> 1
[[5,2],[2]]
=> [2]
=> 1
[[3,1,1],[]]
=> []
=> 0
[[4,2,1],[1,1]]
=> [1,1]
=> 2
[[4,1,1],[1]]
=> [1]
=> 1
Description
The length of the partition.
Mp00183: Skew partitions inner shapeInteger partitions
Mp00044: Integer partitions conjugateInteger partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[1],[]]
=> []
=> []
=> 0
[[2],[]]
=> []
=> []
=> 0
[[1,1],[]]
=> []
=> []
=> 0
[[2,1],[1]]
=> [1]
=> [1]
=> 1
[[3],[]]
=> []
=> []
=> 0
[[2,1],[]]
=> []
=> []
=> 0
[[3,1],[1]]
=> [1]
=> [1]
=> 1
[[2,2],[1]]
=> [1]
=> [1]
=> 1
[[3,2],[2]]
=> [2]
=> [1,1]
=> 1
[[1,1,1],[]]
=> []
=> []
=> 0
[[2,2,1],[1,1]]
=> [1,1]
=> [2]
=> 2
[[2,1,1],[1]]
=> [1]
=> [1]
=> 1
[[3,2,1],[2,1]]
=> [2,1]
=> [2,1]
=> 2
[[4],[]]
=> []
=> []
=> 0
[[3,1],[]]
=> []
=> []
=> 0
[[4,1],[1]]
=> [1]
=> [1]
=> 1
[[2,2],[]]
=> []
=> []
=> 0
[[3,2],[1]]
=> [1]
=> [1]
=> 1
[[4,2],[2]]
=> [2]
=> [1,1]
=> 1
[[2,1,1],[]]
=> []
=> []
=> 0
[[3,2,1],[1,1]]
=> [1,1]
=> [2]
=> 2
[[3,1,1],[1]]
=> [1]
=> [1]
=> 1
[[4,2,1],[2,1]]
=> [2,1]
=> [2,1]
=> 2
[[3,3],[2]]
=> [2]
=> [1,1]
=> 1
[[4,3],[3]]
=> [3]
=> [1,1,1]
=> 1
[[2,2,1],[1]]
=> [1]
=> [1]
=> 1
[[3,3,1],[2,1]]
=> [2,1]
=> [2,1]
=> 2
[[3,2,1],[2]]
=> [2]
=> [1,1]
=> 1
[[4,3,1],[3,1]]
=> [3,1]
=> [2,1,1]
=> 2
[[2,2,2],[1,1]]
=> [1,1]
=> [2]
=> 2
[[3,3,2],[2,2]]
=> [2,2]
=> [2,2]
=> 2
[[3,2,2],[2,1]]
=> [2,1]
=> [2,1]
=> 2
[[4,3,2],[3,2]]
=> [3,2]
=> [2,2,1]
=> 2
[[1,1,1,1],[]]
=> []
=> []
=> 0
[[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [3]
=> 3
[[2,2,1,1],[1,1]]
=> [1,1]
=> [2]
=> 2
[[3,3,2,1],[2,2,1]]
=> [2,2,1]
=> [3,2]
=> 3
[[2,1,1,1],[1]]
=> [1]
=> [1]
=> 1
[[3,2,2,1],[2,1,1]]
=> [2,1,1]
=> [3,1]
=> 3
[[3,2,1,1],[2,1]]
=> [2,1]
=> [2,1]
=> 2
[[4,3,2,1],[3,2,1]]
=> [3,2,1]
=> [3,2,1]
=> 3
[[5],[]]
=> []
=> []
=> 0
[[4,1],[]]
=> []
=> []
=> 0
[[5,1],[1]]
=> [1]
=> [1]
=> 1
[[3,2],[]]
=> []
=> []
=> 0
[[4,2],[1]]
=> [1]
=> [1]
=> 1
[[5,2],[2]]
=> [2]
=> [1,1]
=> 1
[[3,1,1],[]]
=> []
=> []
=> 0
[[4,2,1],[1,1]]
=> [1,1]
=> [2]
=> 2
[[4,1,1],[1]]
=> [1]
=> [1]
=> 1
[[7,6,5,3,2,1],[6,5,3,2,1]]
=> [6,5,3,2,1]
=> [5,4,3,2,2,1]
=> ? = 5
[[7,6,5,4,2],[6,5,4,2]]
=> [6,5,4,2]
=> [4,4,3,3,2,1]
=> ? = 4
[[6,6,5,4,2,1],[5,5,4,2,1]]
=> [5,5,4,2,1]
=> [5,4,3,3,2]
=> ? = 5
[[6,6,5,4,3],[5,5,4,3]]
=> [5,5,4,3]
=> [4,4,4,3,2]
=> ? = 4
[[6,5,5,4,3,1],[5,4,4,3,1]]
=> [5,4,4,3,1]
=> [5,4,4,3,1]
=> ? = 5
Description
The largest part of an integer partition.
Mp00183: Skew partitions inner shapeInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00142: Dyck paths promotionDyck paths
St000676: Dyck paths ⟶ ℤResult quality: 96% values known / values provided: 96%distinct values known / distinct values provided: 100%
Values
[[1],[]]
=> []
=> []
=> []
=> 0
[[2],[]]
=> []
=> []
=> []
=> 0
[[1,1],[]]
=> []
=> []
=> []
=> 0
[[2,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[3],[]]
=> []
=> []
=> []
=> 0
[[2,1],[]]
=> []
=> []
=> []
=> 0
[[3,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[2,2],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[3,2],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[[1,1,1],[]]
=> []
=> []
=> []
=> 0
[[2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[2,1,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[[4],[]]
=> []
=> []
=> []
=> 0
[[3,1],[]]
=> []
=> []
=> []
=> 0
[[4,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[2,2],[]]
=> []
=> []
=> []
=> 0
[[3,2],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[4,2],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[[2,1,1],[]]
=> []
=> []
=> []
=> 0
[[3,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[3,1,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[4,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[[3,3],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[[4,3],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[[2,2,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[3,3,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[[3,2,1],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[[4,3,1],[3,1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[[2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[3,3,2],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> 2
[[3,2,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[[4,3,2],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[1,1,1,1],[]]
=> []
=> []
=> []
=> 0
[[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3
[[2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[3,3,2,1],[2,2,1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[[2,1,1,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[3,2,2,1],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[[3,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[[4,3,2,1],[3,2,1]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[[5],[]]
=> []
=> []
=> []
=> 0
[[4,1],[]]
=> []
=> []
=> []
=> 0
[[5,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[3,2],[]]
=> []
=> []
=> []
=> 0
[[4,2],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[5,2],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[[3,1,1],[]]
=> []
=> []
=> []
=> 0
[[4,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[4,1,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[6,5,3,2,1],[5,3,2,1]]
=> [5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 4
[[6,5,4,3,2,1],[5,4,3,2,1]]
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 5
[[7,5,3,2,1],[5,3,2,1]]
=> [5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 4
[[7,5,4,3,2,1],[5,4,3,2,1]]
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 5
[[7,6,2,1],[6,2,1]]
=> [6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 3
[[7,6,3,1],[6,3,1]]
=> [6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 3
[[7,6,3,2],[6,3,2]]
=> [6,3,2]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> ? = 3
[[6,6,3,2,1],[5,3,2,1]]
=> [5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 4
[[6,5,3,2,1],[5,2,2,1]]
=> [5,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 4
[[6,5,2,2,1],[5,2,1,1]]
=> [5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 4
[[7,6,3,2,1],[6,3,2,1]]
=> [6,3,2,1]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ?
=> ? = 4
[[7,6,4,1],[6,4,1]]
=> [6,4,1]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 3
[[7,6,4,2],[6,4,2]]
=> [6,4,2]
=> [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> ? = 3
[[6,5,4,2,1],[5,3,2,1]]
=> [5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 4
[[6,5,3,2,1],[5,3,1,1]]
=> [5,3,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 4
[[7,6,4,2,1],[6,4,2,1]]
=> [6,4,2,1]
=> [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> ?
=> ? = 4
[[7,6,4,3],[6,4,3]]
=> [6,4,3]
=> [1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> ? = 3
[[6,5,4,3,1],[5,3,3,1]]
=> [5,3,3,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 4
[[6,5,3,3,1],[5,3,2,1]]
=> [5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 4
[[7,6,4,3,1],[6,4,3,1]]
=> [6,4,3,1]
=> [1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> ?
=> ? = 4
[[6,5,4,3,2],[5,3,3,2]]
=> [5,3,3,2]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 4
[[6,5,3,3,2],[5,3,2,2]]
=> [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> ? = 4
[[6,5,3,2,2],[5,3,2,1]]
=> [5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 4
[[7,6,4,3,2],[6,4,3,2]]
=> [6,4,3,2]
=> [1,0,1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> ?
=> ? = 4
[[6,6,4,3,2,1],[5,4,3,2,1]]
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 5
[[5,4,3,3,2,1],[4,2,2,2,1]]
=> [4,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> ? = 5
[[5,4,3,2,2,1],[4,2,2,1,1]]
=> [4,2,2,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 5
[[6,5,4,3,2,1],[5,3,3,2,1]]
=> [5,3,3,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 5
[[5,4,2,2,2,1],[4,2,1,1,1]]
=> [4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 5
[[6,5,3,3,2,1],[5,3,2,2,1]]
=> [5,3,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> ? = 5
[[6,5,3,2,2,1],[5,3,2,1,1]]
=> [5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> ? = 5
[[6,5,3,2,1,1],[5,3,2,1]]
=> [5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 4
[[7,6,4,3,2,1],[6,4,3,2,1]]
=> [6,4,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 5
[[7,6,5,2,1],[6,5,2,1]]
=> [6,5,2,1]
=> [1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,1,0,0]
=> ?
=> ? = 4
[[7,6,5,3,1],[6,5,3,1]]
=> [6,5,3,1]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,1,0,0]
=> ?
=> ? = 4
[[7,6,5,3,2],[6,5,3,2]]
=> [6,5,3,2]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,0]
=> ?
=> ? = 4
[[6,6,5,3,2,1],[5,5,3,2,1]]
=> [5,5,3,2,1]
=> [1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ?
=> ? = 5
[[6,5,5,3,2,1],[5,4,3,2,1]]
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 5
[[6,5,4,3,2,1],[5,4,2,2,1]]
=> [5,4,2,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> ? = 5
[[6,5,4,2,2,1],[5,4,2,1,1]]
=> [5,4,2,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> ? = 5
[[7,6,5,3,2,1],[6,5,3,2,1]]
=> [6,5,3,2,1]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ?
=> ? = 5
[[7,6,5,4,1],[6,5,4,1]]
=> [6,5,4,1]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,1,0,0]
=> ?
=> ? = 4
[[7,6,5,4,2],[6,5,4,2]]
=> [6,5,4,2]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0]
=> ?
=> ? = 4
[[6,6,5,4,2,1],[5,5,4,2,1]]
=> [5,5,4,2,1]
=> [1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0]
=> ?
=> ? = 5
[[6,5,5,4,2,1],[5,4,4,2,1]]
=> [5,4,4,2,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,1,0,0]
=> ?
=> ? = 5
[[6,5,4,4,2,1],[5,4,3,2,1]]
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 5
[[6,5,4,3,2,1],[5,4,3,1,1]]
=> [5,4,3,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 5
[[6,5,5,4,3,1],[5,4,4,3,1]]
=> [5,4,4,3,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0]
=> ?
=> ? = 5
[[6,5,4,4,3,1],[5,4,3,3,1]]
=> [5,4,3,3,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,0]
=> ?
=> ? = 5
[[6,5,4,3,3,1],[5,4,3,2,1]]
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 5
Description
The number of odd rises of a Dyck path. This is the number of ones at an odd position, with the initial position equal to 1. The number of Dyck paths of semilength $n$ with $k$ up steps in odd positions and $k$ returns to the main diagonal are counted by the binomial coefficient $\binom{n-1}{k-1}$ [3,4].
Mp00183: Skew partitions inner shapeInteger partitions
Mp00322: Integer partitions Loehr-WarringtonInteger partitions
St000378: Integer partitions ⟶ ℤResult quality: 96% values known / values provided: 96%distinct values known / distinct values provided: 100%
Values
[[1],[]]
=> []
=> []
=> 0
[[2],[]]
=> []
=> []
=> 0
[[1,1],[]]
=> []
=> []
=> 0
[[2,1],[1]]
=> [1]
=> [1]
=> 1
[[3],[]]
=> []
=> []
=> 0
[[2,1],[]]
=> []
=> []
=> 0
[[3,1],[1]]
=> [1]
=> [1]
=> 1
[[2,2],[1]]
=> [1]
=> [1]
=> 1
[[3,2],[2]]
=> [2]
=> [1,1]
=> 1
[[1,1,1],[]]
=> []
=> []
=> 0
[[2,2,1],[1,1]]
=> [1,1]
=> [2]
=> 2
[[2,1,1],[1]]
=> [1]
=> [1]
=> 1
[[3,2,1],[2,1]]
=> [2,1]
=> [3]
=> 2
[[4],[]]
=> []
=> []
=> 0
[[3,1],[]]
=> []
=> []
=> 0
[[4,1],[1]]
=> [1]
=> [1]
=> 1
[[2,2],[]]
=> []
=> []
=> 0
[[3,2],[1]]
=> [1]
=> [1]
=> 1
[[4,2],[2]]
=> [2]
=> [1,1]
=> 1
[[2,1,1],[]]
=> []
=> []
=> 0
[[3,2,1],[1,1]]
=> [1,1]
=> [2]
=> 2
[[3,1,1],[1]]
=> [1]
=> [1]
=> 1
[[4,2,1],[2,1]]
=> [2,1]
=> [3]
=> 2
[[3,3],[2]]
=> [2]
=> [1,1]
=> 1
[[4,3],[3]]
=> [3]
=> [1,1,1]
=> 1
[[2,2,1],[1]]
=> [1]
=> [1]
=> 1
[[3,3,1],[2,1]]
=> [2,1]
=> [3]
=> 2
[[3,2,1],[2]]
=> [2]
=> [1,1]
=> 1
[[4,3,1],[3,1]]
=> [3,1]
=> [2,1,1]
=> 2
[[2,2,2],[1,1]]
=> [1,1]
=> [2]
=> 2
[[3,3,2],[2,2]]
=> [2,2]
=> [4]
=> 2
[[3,2,2],[2,1]]
=> [2,1]
=> [3]
=> 2
[[4,3,2],[3,2]]
=> [3,2]
=> [5]
=> 2
[[1,1,1,1],[]]
=> []
=> []
=> 0
[[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [2,1]
=> 3
[[2,2,1,1],[1,1]]
=> [1,1]
=> [2]
=> 2
[[3,3,2,1],[2,2,1]]
=> [2,2,1]
=> [2,2,1]
=> 3
[[2,1,1,1],[1]]
=> [1]
=> [1]
=> 1
[[3,2,2,1],[2,1,1]]
=> [2,1,1]
=> [2,2]
=> 3
[[3,2,1,1],[2,1]]
=> [2,1]
=> [3]
=> 2
[[4,3,2,1],[3,2,1]]
=> [3,2,1]
=> [5,1]
=> 3
[[5],[]]
=> []
=> []
=> 0
[[4,1],[]]
=> []
=> []
=> 0
[[5,1],[1]]
=> [1]
=> [1]
=> 1
[[3,2],[]]
=> []
=> []
=> 0
[[4,2],[1]]
=> [1]
=> [1]
=> 1
[[5,2],[2]]
=> [2]
=> [1,1]
=> 1
[[3,1,1],[]]
=> []
=> []
=> 0
[[4,2,1],[1,1]]
=> [1,1]
=> [2]
=> 2
[[4,1,1],[1]]
=> [1]
=> [1]
=> 1
[[6,5,4,3,1],[5,4,3,1]]
=> [5,4,3,1]
=> [9,2,1,1]
=> ? = 4
[[5,4,4,3,2,1],[4,3,3,2,1]]
=> [4,3,3,2,1]
=> [7,2,2,2]
=> ? = 5
[[7,5,4,3,1],[5,4,3,1]]
=> [5,4,3,1]
=> [9,2,1,1]
=> ? = 4
[[6,4,4,3,2,1],[4,3,3,2,1]]
=> [4,3,3,2,1]
=> [7,2,2,2]
=> ? = 5
[[7,6,4,2,1],[6,4,2,1]]
=> [6,4,2,1]
=> [7,2,2,1,1]
=> ? = 4
[[7,6,4,3],[6,4,3]]
=> [6,4,3]
=> [9,1,1,1,1]
=> ? = 3
[[6,6,4,3,1],[5,4,3,1]]
=> [5,4,3,1]
=> [9,2,1,1]
=> ? = 4
[[7,6,4,3,1],[6,4,3,1]]
=> [6,4,3,1]
=> [9,2,1,1,1]
=> ? = 4
[[7,6,4,3,2],[6,4,3,2]]
=> [6,4,3,2]
=> [9,2,2,1,1]
=> ? = 4
[[5,5,4,3,2,1],[4,3,3,2,1]]
=> [4,3,3,2,1]
=> [7,2,2,2]
=> ? = 5
[[7,6,4,3,2,1],[6,4,3,2,1]]
=> [6,4,3,2,1]
=> [9,5,1,1]
=> ? = 5
[[7,6,5,2],[6,5,2]]
=> [6,5,2]
=> [11,1,1]
=> ? = 3
[[7,6,5,2,1],[6,5,2,1]]
=> [6,5,2,1]
=> [11,3]
=> ? = 4
[[6,6,5,3],[5,5,3]]
=> [5,5,3]
=> [10,1,1,1]
=> ? = 3
[[7,6,5,3],[6,5,3]]
=> [6,5,3]
=> [11,1,1,1]
=> ? = 3
[[6,6,5,3,1],[5,5,3,1]]
=> [5,5,3,1]
=> [10,2,1,1]
=> ? = 4
[[6,5,5,3,1],[5,4,3,1]]
=> [5,4,3,1]
=> [9,2,1,1]
=> ? = 4
[[7,6,5,3,1],[6,5,3,1]]
=> [6,5,3,1]
=> [11,2,1,1]
=> ? = 4
[[6,6,5,3,2],[5,5,3,2]]
=> [5,5,3,2]
=> [10,5]
=> ? = 4
[[6,5,4,3,2],[5,4,2,2]]
=> [5,4,2,2]
=> [9,4]
=> ? = 4
[[7,6,5,3,2],[6,5,3,2]]
=> [6,5,3,2]
=> [11,5]
=> ? = 4
[[6,6,5,3,2,1],[5,5,3,2,1]]
=> [5,5,3,2,1]
=> [10,5,1]
=> ? = 5
[[6,5,4,3,2,1],[5,4,2,2,1]]
=> [5,4,2,2,1]
=> [9,2,2,1]
=> ? = 5
[[7,6,5,3,2,1],[6,5,3,2,1]]
=> [6,5,3,2,1]
=> [11,5,1]
=> ? = 5
[[7,6,5,4],[6,5,4]]
=> [6,5,4]
=> [11,1,1,1,1]
=> ? = 3
[[6,6,5,4,1],[5,5,4,1]]
=> [5,5,4,1]
=> [3,2,2,2,2,2,2]
=> ? = 4
[[6,5,5,4,1],[5,4,4,1]]
=> [5,4,4,1]
=> [3,2,2,2,2,2,1]
=> ? = 4
[[6,5,4,4,1],[5,4,3,1]]
=> [5,4,3,1]
=> [9,2,1,1]
=> ? = 4
[[7,6,5,4,1],[6,5,4,1]]
=> [6,5,4,1]
=> [11,2,1,1,1]
=> ? = 4
[[6,5,5,4,2],[5,4,4,2]]
=> [5,4,4,2]
=> [3,3,2,2,2,2,1]
=> ? = 4
[[6,5,4,3,2],[5,4,3,1]]
=> [5,4,3,1]
=> [9,2,1,1]
=> ? = 4
[[7,6,5,4,2],[6,5,4,2]]
=> [6,5,4,2]
=> [11,2,2,1,1]
=> ? = 4
[[5,5,5,4,2,1],[4,4,4,2,1]]
=> [4,4,4,2,1]
=> [5,2,2,2,2,2]
=> ? = 5
[[5,5,4,3,2,1],[4,4,3,1,1]]
=> [4,4,3,1,1]
=> [4,2,2,2,2,1]
=> ? = 5
[[6,6,5,4,2,1],[5,5,4,2,1]]
=> [5,5,4,2,1]
=> [5,2,2,2,2,2,2]
=> ? = 5
[[5,4,4,4,2,1],[4,3,3,2,1]]
=> [4,3,3,2,1]
=> [7,2,2,2]
=> ? = 5
[[6,5,5,4,2,1],[5,4,4,2,1]]
=> [5,4,4,2,1]
=> [5,2,2,2,2,2,1]
=> ? = 5
[[6,5,4,3,2,1],[5,4,3,1,1]]
=> [5,4,3,1,1]
=> [9,4,1]
=> ? = 5
[[6,5,4,3,1,1],[5,4,3,1]]
=> [5,4,3,1]
=> [9,2,1,1]
=> ? = 4
[[5,5,5,4,3],[4,4,4,3]]
=> [4,4,4,3]
=> [8,1,1,1,1,1,1,1]
=> ? = 4
[[5,5,4,4,3],[4,4,3,3]]
=> [4,4,3,3]
=> [8,1,1,1,1,1,1]
=> ? = 4
[[6,6,5,4,3],[5,5,4,3]]
=> [5,5,4,3]
=> [3,3,3,2,2,2,2]
=> ? = 4
[[5,4,4,4,3],[4,3,3,3]]
=> [4,3,3,3]
=> [7,1,1,1,1,1,1]
=> ? = 4
[[6,5,5,4,3],[5,4,4,3]]
=> [5,4,4,3]
=> [3,3,3,2,2,2,1]
=> ? = 4
[[4,4,4,4,3,1],[3,3,3,3,1]]
=> [3,3,3,3,1]
=> [7,2,1,1,1,1]
=> ? = 5
[[5,5,5,4,3,1],[4,4,4,3,1]]
=> [4,4,4,3,1]
=> [8,2,1,1,1,1,1,1]
=> ? = 5
[[5,5,4,4,3,1],[4,4,3,3,1]]
=> [4,4,3,3,1]
=> [8,2,1,1,1,1,1]
=> ? = 5
[[5,4,4,4,3,1],[4,3,3,3,1]]
=> [4,3,3,3,1]
=> [7,2,1,1,1,1,1]
=> ? = 5
[[5,4,4,3,3,1],[4,3,3,2,1]]
=> [4,3,3,2,1]
=> [7,2,2,2]
=> ? = 5
[[6,5,5,4,3,1],[5,4,4,3,1]]
=> [5,4,4,3,1]
=> [4,3,3,2,2,2,1]
=> ? = 5
Description
The diagonal inversion number of an integer partition. The dinv of a partition is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \in \{0,1\}$. See also exercise 3.19 of [2]. This statistic is equidistributed with the length of the partition, see [3].
Mp00183: Skew partitions inner shapeInteger partitions
Mp00095: Integer partitions to binary wordBinary words
Mp00224: Binary words runsortBinary words
St000288: Binary words ⟶ ℤResult quality: 87% values known / values provided: 87%distinct values known / distinct values provided: 89%
Values
[[1],[]]
=> []
=> => => ? = 0
[[2],[]]
=> []
=> => => ? = 0
[[1,1],[]]
=> []
=> => => ? = 0
[[2,1],[1]]
=> [1]
=> 10 => 01 => 1
[[3],[]]
=> []
=> => => ? = 0
[[2,1],[]]
=> []
=> => => ? = 0
[[3,1],[1]]
=> [1]
=> 10 => 01 => 1
[[2,2],[1]]
=> [1]
=> 10 => 01 => 1
[[3,2],[2]]
=> [2]
=> 100 => 001 => 1
[[1,1,1],[]]
=> []
=> => => ? = 0
[[2,2,1],[1,1]]
=> [1,1]
=> 110 => 011 => 2
[[2,1,1],[1]]
=> [1]
=> 10 => 01 => 1
[[3,2,1],[2,1]]
=> [2,1]
=> 1010 => 0011 => 2
[[4],[]]
=> []
=> => => ? = 0
[[3,1],[]]
=> []
=> => => ? = 0
[[4,1],[1]]
=> [1]
=> 10 => 01 => 1
[[2,2],[]]
=> []
=> => => ? = 0
[[3,2],[1]]
=> [1]
=> 10 => 01 => 1
[[4,2],[2]]
=> [2]
=> 100 => 001 => 1
[[2,1,1],[]]
=> []
=> => => ? = 0
[[3,2,1],[1,1]]
=> [1,1]
=> 110 => 011 => 2
[[3,1,1],[1]]
=> [1]
=> 10 => 01 => 1
[[4,2,1],[2,1]]
=> [2,1]
=> 1010 => 0011 => 2
[[3,3],[2]]
=> [2]
=> 100 => 001 => 1
[[4,3],[3]]
=> [3]
=> 1000 => 0001 => 1
[[2,2,1],[1]]
=> [1]
=> 10 => 01 => 1
[[3,3,1],[2,1]]
=> [2,1]
=> 1010 => 0011 => 2
[[3,2,1],[2]]
=> [2]
=> 100 => 001 => 1
[[4,3,1],[3,1]]
=> [3,1]
=> 10010 => 00011 => 2
[[2,2,2],[1,1]]
=> [1,1]
=> 110 => 011 => 2
[[3,3,2],[2,2]]
=> [2,2]
=> 1100 => 0011 => 2
[[3,2,2],[2,1]]
=> [2,1]
=> 1010 => 0011 => 2
[[4,3,2],[3,2]]
=> [3,2]
=> 10100 => 00011 => 2
[[1,1,1,1],[]]
=> []
=> => => ? = 0
[[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 1110 => 0111 => 3
[[2,2,1,1],[1,1]]
=> [1,1]
=> 110 => 011 => 2
[[3,3,2,1],[2,2,1]]
=> [2,2,1]
=> 11010 => 00111 => 3
[[2,1,1,1],[1]]
=> [1]
=> 10 => 01 => 1
[[3,2,2,1],[2,1,1]]
=> [2,1,1]
=> 10110 => 00111 => 3
[[3,2,1,1],[2,1]]
=> [2,1]
=> 1010 => 0011 => 2
[[4,3,2,1],[3,2,1]]
=> [3,2,1]
=> 101010 => 001011 => 3
[[5],[]]
=> []
=> => => ? = 0
[[4,1],[]]
=> []
=> => => ? = 0
[[5,1],[1]]
=> [1]
=> 10 => 01 => 1
[[3,2],[]]
=> []
=> => => ? = 0
[[4,2],[1]]
=> [1]
=> 10 => 01 => 1
[[5,2],[2]]
=> [2]
=> 100 => 001 => 1
[[3,1,1],[]]
=> []
=> => => ? = 0
[[4,2,1],[1,1]]
=> [1,1]
=> 110 => 011 => 2
[[4,1,1],[1]]
=> [1]
=> 10 => 01 => 1
[[5,2,1],[2,1]]
=> [2,1]
=> 1010 => 0011 => 2
[[3,3],[1]]
=> [1]
=> 10 => 01 => 1
[[4,3],[2]]
=> [2]
=> 100 => 001 => 1
[[5,3],[3]]
=> [3]
=> 1000 => 0001 => 1
[[2,2,1],[]]
=> []
=> => => ? = 0
[[3,3,1],[1,1]]
=> [1,1]
=> 110 => 011 => 2
[[3,2,1],[1]]
=> [1]
=> 10 => 01 => 1
[[4,3,1],[2,1]]
=> [2,1]
=> 1010 => 0011 => 2
[[4,2,1],[2]]
=> [2]
=> 100 => 001 => 1
[[5,3,1],[3,1]]
=> [3,1]
=> 10010 => 00011 => 2
[[3,2,2],[1,1]]
=> [1,1]
=> 110 => 011 => 2
[[4,3,2],[2,2]]
=> [2,2]
=> 1100 => 0011 => 2
[[4,2,2],[2,1]]
=> [2,1]
=> 1010 => 0011 => 2
[[5,3,2],[3,2]]
=> [3,2]
=> 10100 => 00011 => 2
[[2,1,1,1],[]]
=> []
=> => => ? = 0
[[3,2,2,1],[1,1,1]]
=> [1,1,1]
=> 1110 => 0111 => 3
[[3,2,1,1],[1,1]]
=> [1,1]
=> 110 => 011 => 2
[[1,1,1,1,1],[]]
=> []
=> => => ? = 0
[[6],[]]
=> []
=> => => ? = 0
[[5,1],[]]
=> []
=> => => ? = 0
[[4,2],[]]
=> []
=> => => ? = 0
[[4,1,1],[]]
=> []
=> => => ? = 0
[[3,3],[]]
=> []
=> => => ? = 0
[[3,2,1],[]]
=> []
=> => => ? = 0
[[3,1,1,1],[]]
=> []
=> => => ? = 0
[[2,2,2],[]]
=> []
=> => => ? = 0
[[2,2,1,1],[]]
=> []
=> => => ? = 0
[[2,1,1,1,1],[]]
=> []
=> => => ? = 0
[[1,1,1,1,1,1],[]]
=> []
=> => => ? = 0
[[7],[]]
=> []
=> => => ? = 0
[[6,1],[]]
=> []
=> => => ? = 0
[[5,2],[]]
=> []
=> => => ? = 0
[[5,1,1],[]]
=> []
=> => => ? = 0
[[4,3],[]]
=> []
=> => => ? = 0
[[4,2,1],[]]
=> []
=> => => ? = 0
[[4,1,1,1],[]]
=> []
=> => => ? = 0
[[3,3,1],[]]
=> []
=> => => ? = 0
[[3,2,2],[]]
=> []
=> => => ? = 0
[[3,2,1,1],[]]
=> []
=> => => ? = 0
[[3,1,1,1,1],[]]
=> []
=> => => ? = 0
[[2,2,2,1],[]]
=> []
=> => => ? = 0
[[2,2,1,1,1],[]]
=> []
=> => => ? = 0
[[2,1,1,1,1,1],[]]
=> []
=> => => ? = 0
[[6,5,3,2,1],[5,2,2,1]]
=> [5,2,2,1]
=> 100011010 => 000011011 => ? = 4
[[7,6,3,2,1],[6,3,2,1]]
=> [6,3,2,1]
=> 1000101010 => ? => ? = 4
[[7,6,4,2,1],[6,4,2,1]]
=> [6,4,2,1]
=> 1001001010 => ? => ? = 4
[[7,6,4,3,1],[6,4,3,1]]
=> [6,4,3,1]
=> 1001010010 => ? => ? = 4
[[6,5,4,3,2],[5,3,3,2]]
=> [5,3,3,2]
=> 100110100 => 000011011 => ? = 4
[[7,6,4,3,2],[6,4,3,2]]
=> [6,4,3,2]
=> 1001010100 => ? => ? = 4
[[5,4,3,3,2,1],[4,2,2,2,1]]
=> [4,2,2,2,1]
=> 100111010 => 000111011 => ? = 5
Description
The number of ones in a binary word. This is also known as the Hamming weight of the word.
Matching statistic: St000734
Mp00183: Skew partitions inner shapeInteger partitions
Mp00044: Integer partitions conjugateInteger partitions
Mp00042: Integer partitions initial tableauStandard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 85% values known / values provided: 85%distinct values known / distinct values provided: 89%
Values
[[1],[]]
=> []
=> []
=> []
=> ? = 0
[[2],[]]
=> []
=> []
=> []
=> ? = 0
[[1,1],[]]
=> []
=> []
=> []
=> ? = 0
[[2,1],[1]]
=> [1]
=> [1]
=> [[1]]
=> 1
[[3],[]]
=> []
=> []
=> []
=> ? = 0
[[2,1],[]]
=> []
=> []
=> []
=> ? = 0
[[3,1],[1]]
=> [1]
=> [1]
=> [[1]]
=> 1
[[2,2],[1]]
=> [1]
=> [1]
=> [[1]]
=> 1
[[3,2],[2]]
=> [2]
=> [1,1]
=> [[1],[2]]
=> 1
[[1,1,1],[]]
=> []
=> []
=> []
=> ? = 0
[[2,2,1],[1,1]]
=> [1,1]
=> [2]
=> [[1,2]]
=> 2
[[2,1,1],[1]]
=> [1]
=> [1]
=> [[1]]
=> 1
[[3,2,1],[2,1]]
=> [2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
[[4],[]]
=> []
=> []
=> []
=> ? = 0
[[3,1],[]]
=> []
=> []
=> []
=> ? = 0
[[4,1],[1]]
=> [1]
=> [1]
=> [[1]]
=> 1
[[2,2],[]]
=> []
=> []
=> []
=> ? = 0
[[3,2],[1]]
=> [1]
=> [1]
=> [[1]]
=> 1
[[4,2],[2]]
=> [2]
=> [1,1]
=> [[1],[2]]
=> 1
[[2,1,1],[]]
=> []
=> []
=> []
=> ? = 0
[[3,2,1],[1,1]]
=> [1,1]
=> [2]
=> [[1,2]]
=> 2
[[3,1,1],[1]]
=> [1]
=> [1]
=> [[1]]
=> 1
[[4,2,1],[2,1]]
=> [2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
[[3,3],[2]]
=> [2]
=> [1,1]
=> [[1],[2]]
=> 1
[[4,3],[3]]
=> [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 1
[[2,2,1],[1]]
=> [1]
=> [1]
=> [[1]]
=> 1
[[3,3,1],[2,1]]
=> [2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
[[3,2,1],[2]]
=> [2]
=> [1,1]
=> [[1],[2]]
=> 1
[[4,3,1],[3,1]]
=> [3,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[[2,2,2],[1,1]]
=> [1,1]
=> [2]
=> [[1,2]]
=> 2
[[3,3,2],[2,2]]
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 2
[[3,2,2],[2,1]]
=> [2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
[[4,3,2],[3,2]]
=> [3,2]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
[[1,1,1,1],[]]
=> []
=> []
=> []
=> ? = 0
[[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [3]
=> [[1,2,3]]
=> 3
[[2,2,1,1],[1,1]]
=> [1,1]
=> [2]
=> [[1,2]]
=> 2
[[3,3,2,1],[2,2,1]]
=> [2,2,1]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 3
[[2,1,1,1],[1]]
=> [1]
=> [1]
=> [[1]]
=> 1
[[3,2,2,1],[2,1,1]]
=> [2,1,1]
=> [3,1]
=> [[1,2,3],[4]]
=> 3
[[3,2,1,1],[2,1]]
=> [2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
[[4,3,2,1],[3,2,1]]
=> [3,2,1]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 3
[[5],[]]
=> []
=> []
=> []
=> ? = 0
[[4,1],[]]
=> []
=> []
=> []
=> ? = 0
[[5,1],[1]]
=> [1]
=> [1]
=> [[1]]
=> 1
[[3,2],[]]
=> []
=> []
=> []
=> ? = 0
[[4,2],[1]]
=> [1]
=> [1]
=> [[1]]
=> 1
[[5,2],[2]]
=> [2]
=> [1,1]
=> [[1],[2]]
=> 1
[[3,1,1],[]]
=> []
=> []
=> []
=> ? = 0
[[4,2,1],[1,1]]
=> [1,1]
=> [2]
=> [[1,2]]
=> 2
[[4,1,1],[1]]
=> [1]
=> [1]
=> [[1]]
=> 1
[[5,2,1],[2,1]]
=> [2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
[[3,3],[1]]
=> [1]
=> [1]
=> [[1]]
=> 1
[[4,3],[2]]
=> [2]
=> [1,1]
=> [[1],[2]]
=> 1
[[5,3],[3]]
=> [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 1
[[2,2,1],[]]
=> []
=> []
=> []
=> ? = 0
[[3,3,1],[1,1]]
=> [1,1]
=> [2]
=> [[1,2]]
=> 2
[[3,2,1],[1]]
=> [1]
=> [1]
=> [[1]]
=> 1
[[4,3,1],[2,1]]
=> [2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
[[4,2,1],[2]]
=> [2]
=> [1,1]
=> [[1],[2]]
=> 1
[[5,3,1],[3,1]]
=> [3,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[[3,2,2],[1,1]]
=> [1,1]
=> [2]
=> [[1,2]]
=> 2
[[4,3,2],[2,2]]
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 2
[[4,2,2],[2,1]]
=> [2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
[[5,3,2],[3,2]]
=> [3,2]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
[[2,1,1,1],[]]
=> []
=> []
=> []
=> ? = 0
[[3,2,2,1],[1,1,1]]
=> [1,1,1]
=> [3]
=> [[1,2,3]]
=> 3
[[3,2,1,1],[1,1]]
=> [1,1]
=> [2]
=> [[1,2]]
=> 2
[[1,1,1,1,1],[]]
=> []
=> []
=> []
=> ? = 0
[[6],[]]
=> []
=> []
=> []
=> ? = 0
[[5,1],[]]
=> []
=> []
=> []
=> ? = 0
[[4,2],[]]
=> []
=> []
=> []
=> ? = 0
[[4,1,1],[]]
=> []
=> []
=> []
=> ? = 0
[[3,3],[]]
=> []
=> []
=> []
=> ? = 0
[[3,2,1],[]]
=> []
=> []
=> []
=> ? = 0
[[3,1,1,1],[]]
=> []
=> []
=> []
=> ? = 0
[[2,2,2],[]]
=> []
=> []
=> []
=> ? = 0
[[2,2,1,1],[]]
=> []
=> []
=> []
=> ? = 0
[[2,1,1,1,1],[]]
=> []
=> []
=> []
=> ? = 0
[[1,1,1,1,1,1],[]]
=> []
=> []
=> []
=> ? = 0
[[7],[]]
=> []
=> []
=> []
=> ? = 0
[[6,1],[]]
=> []
=> []
=> []
=> ? = 0
[[5,2],[]]
=> []
=> []
=> []
=> ? = 0
[[5,1,1],[]]
=> []
=> []
=> []
=> ? = 0
[[4,3],[]]
=> []
=> []
=> []
=> ? = 0
[[4,2,1],[]]
=> []
=> []
=> []
=> ? = 0
[[4,1,1,1],[]]
=> []
=> []
=> []
=> ? = 0
[[3,3,1],[]]
=> []
=> []
=> []
=> ? = 0
[[3,2,2],[]]
=> []
=> []
=> []
=> ? = 0
[[3,2,1,1],[]]
=> []
=> []
=> []
=> ? = 0
[[3,1,1,1,1],[]]
=> []
=> []
=> []
=> ? = 0
[[2,2,2,1],[]]
=> []
=> []
=> []
=> ? = 0
[[2,2,1,1,1],[]]
=> []
=> []
=> []
=> ? = 0
[[2,1,1,1,1,1],[]]
=> []
=> []
=> []
=> ? = 0
[[7,6,3,2],[6,3,2]]
=> [6,3,2]
=> [3,3,2,1,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10],[11]]
=> ? = 3
[[7,6,3,2,1],[6,3,2,1]]
=> [6,3,2,1]
=> [4,3,2,1,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11],[12]]
=> ? = 4
[[7,6,4,1],[6,4,1]]
=> [6,4,1]
=> [3,2,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10],[11]]
=> ? = 3
[[7,6,4,2,1],[6,4,2,1]]
=> [6,4,2,1]
=> [4,3,2,2,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11],[12],[13]]
=> ? = 4
[[7,6,4,3],[6,4,3]]
=> [6,4,3]
=> [3,3,3,2,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11],[12],[13]]
=> ? = 3
[[7,6,4,3,1],[6,4,3,1]]
=> [6,4,3,1]
=> [4,3,3,2,1,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12],[13],[14]]
=> ? = 4
[[7,6,4,3,2],[6,4,3,2]]
=> [6,4,3,2]
=> [4,4,3,2,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13],[14],[15]]
=> ? = 4
Description
The last entry in the first row of a standard tableau.
Mp00183: Skew partitions inner shapeInteger partitions
Mp00045: Integer partitions reading tableauStandard tableaux
Mp00084: Standard tableaux conjugateStandard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 85% values known / values provided: 85%distinct values known / distinct values provided: 100%
Values
[[1],[]]
=> []
=> []
=> []
=> 0
[[2],[]]
=> []
=> []
=> []
=> 0
[[1,1],[]]
=> []
=> []
=> []
=> 0
[[2,1],[1]]
=> [1]
=> [[1]]
=> [[1]]
=> 1
[[3],[]]
=> []
=> []
=> []
=> 0
[[2,1],[]]
=> []
=> []
=> []
=> 0
[[3,1],[1]]
=> [1]
=> [[1]]
=> [[1]]
=> 1
[[2,2],[1]]
=> [1]
=> [[1]]
=> [[1]]
=> 1
[[3,2],[2]]
=> [2]
=> [[1,2]]
=> [[1],[2]]
=> 1
[[1,1,1],[]]
=> []
=> []
=> []
=> 0
[[2,2,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 2
[[2,1,1],[1]]
=> [1]
=> [[1]]
=> [[1]]
=> 1
[[3,2,1],[2,1]]
=> [2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 2
[[4],[]]
=> []
=> []
=> []
=> 0
[[3,1],[]]
=> []
=> []
=> []
=> 0
[[4,1],[1]]
=> [1]
=> [[1]]
=> [[1]]
=> 1
[[2,2],[]]
=> []
=> []
=> []
=> 0
[[3,2],[1]]
=> [1]
=> [[1]]
=> [[1]]
=> 1
[[4,2],[2]]
=> [2]
=> [[1,2]]
=> [[1],[2]]
=> 1
[[2,1,1],[]]
=> []
=> []
=> []
=> 0
[[3,2,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 2
[[3,1,1],[1]]
=> [1]
=> [[1]]
=> [[1]]
=> 1
[[4,2,1],[2,1]]
=> [2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 2
[[3,3],[2]]
=> [2]
=> [[1,2]]
=> [[1],[2]]
=> 1
[[4,3],[3]]
=> [3]
=> [[1,2,3]]
=> [[1],[2],[3]]
=> 1
[[2,2,1],[1]]
=> [1]
=> [[1]]
=> [[1]]
=> 1
[[3,3,1],[2,1]]
=> [2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 2
[[3,2,1],[2]]
=> [2]
=> [[1,2]]
=> [[1],[2]]
=> 1
[[4,3,1],[3,1]]
=> [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 2
[[2,2,2],[1,1]]
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 2
[[3,3,2],[2,2]]
=> [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2
[[3,2,2],[2,1]]
=> [2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 2
[[4,3,2],[3,2]]
=> [3,2]
=> [[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 2
[[1,1,1,1],[]]
=> []
=> []
=> []
=> 0
[[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 3
[[2,2,1,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 2
[[3,3,2,1],[2,2,1]]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 3
[[2,1,1,1],[1]]
=> [1]
=> [[1]]
=> [[1]]
=> 1
[[3,2,2,1],[2,1,1]]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 3
[[3,2,1,1],[2,1]]
=> [2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 2
[[4,3,2,1],[3,2,1]]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [[1,2,4],[3,5],[6]]
=> 3
[[5],[]]
=> []
=> []
=> []
=> 0
[[4,1],[]]
=> []
=> []
=> []
=> 0
[[5,1],[1]]
=> [1]
=> [[1]]
=> [[1]]
=> 1
[[3,2],[]]
=> []
=> []
=> []
=> 0
[[4,2],[1]]
=> [1]
=> [[1]]
=> [[1]]
=> 1
[[5,2],[2]]
=> [2]
=> [[1,2]]
=> [[1],[2]]
=> 1
[[3,1,1],[]]
=> []
=> []
=> []
=> 0
[[4,2,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 2
[[4,1,1],[1]]
=> [1]
=> [[1]]
=> [[1]]
=> 1
[[6,5,3,2,1],[5,3,2,1]]
=> [5,3,2,1]
=> [[1,3,6,10,11],[2,5,9],[4,8],[7]]
=> [[1,2,4,7],[3,5,8],[6,9],[10],[11]]
=> ? = 4
[[6,5,4,2],[5,4,2]]
=> [5,4,2]
=> [[1,2,5,6,11],[3,4,9,10],[7,8]]
=> [[1,3,7],[2,4,8],[5,9],[6,10],[11]]
=> ? = 3
[[5,5,4,2,1],[4,4,2,1]]
=> [4,4,2,1]
=> [[1,3,6,7],[2,5,10,11],[4,9],[8]]
=> [[1,2,4,8],[3,5,9],[6,10],[7,11]]
=> ? = 4
[[6,5,4,2,1],[5,4,2,1]]
=> [5,4,2,1]
=> [[1,3,6,7,12],[2,5,10,11],[4,9],[8]]
=> [[1,2,4,8],[3,5,9],[6,10],[7,11],[12]]
=> ? = 4
[[5,5,4,3],[4,4,3]]
=> [4,4,3]
=> [[1,2,3,7],[4,5,6,11],[8,9,10]]
=> [[1,4,8],[2,5,9],[3,6,10],[7,11]]
=> ? = 3
[[6,5,4,3],[5,4,3]]
=> [5,4,3]
=> [[1,2,3,7,12],[4,5,6,11],[8,9,10]]
=> [[1,4,8],[2,5,9],[3,6,10],[7,11],[12]]
=> ? = 3
[[5,5,4,3,1],[4,4,3,1]]
=> [4,4,3,1]
=> [[1,3,4,8],[2,6,7,12],[5,10,11],[9]]
=> [[1,2,5,9],[3,6,10],[4,7,11],[8,12]]
=> ? = 4
[[5,4,4,3,1],[4,3,3,1]]
=> [4,3,3,1]
=> [[1,3,4,11],[2,6,7],[5,9,10],[8]]
=> [[1,2,5,8],[3,6,9],[4,7,10],[11]]
=> ? = 4
[[6,5,4,3,1],[5,4,3,1]]
=> [5,4,3,1]
=> [[1,3,4,8,13],[2,6,7,12],[5,10,11],[9]]
=> [[1,2,5,9],[3,6,10],[4,7,11],[8,12],[13]]
=> ? = 4
[[4,4,4,3,2],[3,3,3,2]]
=> [3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10]]
=> [[1,3,6,9],[2,4,7,10],[5,8,11]]
=> ? = 4
[[5,5,4,3,2],[4,4,3,2]]
=> [4,4,3,2]
=> [[1,2,5,9],[3,4,8,13],[6,7,12],[10,11]]
=> [[1,3,6,10],[2,4,7,11],[5,8,12],[9,13]]
=> ? = 4
[[5,4,4,3,2],[4,3,3,2]]
=> [4,3,3,2]
=> [[1,2,5,12],[3,4,8],[6,7,11],[9,10]]
=> [[1,3,6,9],[2,4,7,10],[5,8,11],[12]]
=> ? = 4
[[5,4,3,3,2],[4,3,2,2]]
=> [4,3,2,2]
=> [[1,2,7,11],[3,4,10],[5,6],[8,9]]
=> [[1,3,5,8],[2,4,6,9],[7,10],[11]]
=> ? = 4
[[6,5,4,3,2],[5,4,3,2]]
=> [5,4,3,2]
=> [[1,2,5,9,14],[3,4,8,13],[6,7,12],[10,11]]
=> [[1,3,6,10],[2,4,7,11],[5,8,12],[9,13],[14]]
=> ? = 4
[[4,4,4,3,2,1],[3,3,3,2,1]]
=> [3,3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8,12],[7,11],[10]]
=> [[1,2,4,7,10],[3,5,8,11],[6,9,12]]
=> ? = 5
[[4,4,3,3,2,1],[3,3,2,2,1]]
=> [3,3,2,2,1]
=> [[1,3,8],[2,5,11],[4,7],[6,10],[9]]
=> [[1,2,4,6,9],[3,5,7,10],[8,11]]
=> ? = 5
[[5,5,4,3,2,1],[4,4,3,2,1]]
=> [4,4,3,2,1]
=> [[1,3,6,10],[2,5,9,14],[4,8,13],[7,12],[11]]
=> [[1,2,4,7,11],[3,5,8,12],[6,9,13],[10,14]]
=> ? = 5
[[5,4,4,3,2,1],[4,3,3,2,1]]
=> [4,3,3,2,1]
=> [[1,3,6,13],[2,5,9],[4,8,12],[7,11],[10]]
=> [[1,2,4,7,10],[3,5,8,11],[6,9,12],[13]]
=> ? = 5
[[5,4,3,3,2,1],[4,3,2,2,1]]
=> [4,3,2,2,1]
=> [[1,3,8,12],[2,5,11],[4,7],[6,10],[9]]
=> [[1,2,4,6,9],[3,5,7,10],[8,11],[12]]
=> ? = 5
[[5,4,3,2,2,1],[4,3,2,1,1]]
=> [4,3,2,1,1]
=> [[1,4,7,11],[2,6,10],[3,9],[5],[8]]
=> [[1,2,3,5,8],[4,6,9],[7,10],[11]]
=> ? = 5
[[6,5,4,3,2,1],[5,4,3,2,1]]
=> [5,4,3,2,1]
=> [[1,3,6,10,15],[2,5,9,14],[4,8,13],[7,12],[11]]
=> [[1,2,4,7,11],[3,5,8,12],[6,9,13],[10,14],[15]]
=> ? = 5
[[7,5,3,2,1],[5,3,2,1]]
=> [5,3,2,1]
=> [[1,3,6,10,11],[2,5,9],[4,8],[7]]
=> [[1,2,4,7],[3,5,8],[6,9],[10],[11]]
=> ? = 4
[[7,5,4,2],[5,4,2]]
=> [5,4,2]
=> [[1,2,5,6,11],[3,4,9,10],[7,8]]
=> [[1,3,7],[2,4,8],[5,9],[6,10],[11]]
=> ? = 3
[[6,5,4,2,1],[4,4,2,1]]
=> [4,4,2,1]
=> [[1,3,6,7],[2,5,10,11],[4,9],[8]]
=> [[1,2,4,8],[3,5,9],[6,10],[7,11]]
=> ? = 4
[[7,5,4,2,1],[5,4,2,1]]
=> [5,4,2,1]
=> [[1,3,6,7,12],[2,5,10,11],[4,9],[8]]
=> [[1,2,4,8],[3,5,9],[6,10],[7,11],[12]]
=> ? = 4
[[6,5,4,3],[4,4,3]]
=> [4,4,3]
=> [[1,2,3,7],[4,5,6,11],[8,9,10]]
=> [[1,4,8],[2,5,9],[3,6,10],[7,11]]
=> ? = 3
[[7,5,4,3],[5,4,3]]
=> [5,4,3]
=> [[1,2,3,7,12],[4,5,6,11],[8,9,10]]
=> [[1,4,8],[2,5,9],[3,6,10],[7,11],[12]]
=> ? = 3
[[6,5,4,3,1],[4,4,3,1]]
=> [4,4,3,1]
=> [[1,3,4,8],[2,6,7,12],[5,10,11],[9]]
=> [[1,2,5,9],[3,6,10],[4,7,11],[8,12]]
=> ? = 4
[[6,4,4,3,1],[4,3,3,1]]
=> [4,3,3,1]
=> [[1,3,4,11],[2,6,7],[5,9,10],[8]]
=> [[1,2,5,8],[3,6,9],[4,7,10],[11]]
=> ? = 4
[[7,5,4,3,1],[5,4,3,1]]
=> [5,4,3,1]
=> [[1,3,4,8,13],[2,6,7,12],[5,10,11],[9]]
=> [[1,2,5,9],[3,6,10],[4,7,11],[8,12],[13]]
=> ? = 4
[[5,4,4,3,2],[3,3,3,2]]
=> [3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10]]
=> [[1,3,6,9],[2,4,7,10],[5,8,11]]
=> ? = 4
[[6,5,4,3,2],[4,4,3,2]]
=> [4,4,3,2]
=> [[1,2,5,9],[3,4,8,13],[6,7,12],[10,11]]
=> [[1,3,6,10],[2,4,7,11],[5,8,12],[9,13]]
=> ? = 4
[[6,4,4,3,2],[4,3,3,2]]
=> [4,3,3,2]
=> [[1,2,5,12],[3,4,8],[6,7,11],[9,10]]
=> [[1,3,6,9],[2,4,7,10],[5,8,11],[12]]
=> ? = 4
[[6,4,3,3,2],[4,3,2,2]]
=> [4,3,2,2]
=> [[1,2,7,11],[3,4,10],[5,6],[8,9]]
=> [[1,3,5,8],[2,4,6,9],[7,10],[11]]
=> ? = 4
[[7,5,4,3,2],[5,4,3,2]]
=> [5,4,3,2]
=> [[1,2,5,9,14],[3,4,8,13],[6,7,12],[10,11]]
=> [[1,3,6,10],[2,4,7,11],[5,8,12],[9,13],[14]]
=> ? = 4
[[5,4,4,3,2,1],[3,3,3,2,1]]
=> [3,3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8,12],[7,11],[10]]
=> [[1,2,4,7,10],[3,5,8,11],[6,9,12]]
=> ? = 5
[[5,4,3,3,2,1],[3,3,2,2,1]]
=> [3,3,2,2,1]
=> [[1,3,8],[2,5,11],[4,7],[6,10],[9]]
=> [[1,2,4,6,9],[3,5,7,10],[8,11]]
=> ? = 5
[[6,5,4,3,2,1],[4,4,3,2,1]]
=> [4,4,3,2,1]
=> [[1,3,6,10],[2,5,9,14],[4,8,13],[7,12],[11]]
=> [[1,2,4,7,11],[3,5,8,12],[6,9,13],[10,14]]
=> ? = 5
[[6,4,4,3,2,1],[4,3,3,2,1]]
=> [4,3,3,2,1]
=> [[1,3,6,13],[2,5,9],[4,8,12],[7,11],[10]]
=> [[1,2,4,7,10],[3,5,8,11],[6,9,12],[13]]
=> ? = 5
[[6,4,3,3,2,1],[4,3,2,2,1]]
=> [4,3,2,2,1]
=> [[1,3,8,12],[2,5,11],[4,7],[6,10],[9]]
=> [[1,2,4,6,9],[3,5,7,10],[8,11],[12]]
=> ? = 5
[[6,4,3,2,2,1],[4,3,2,1,1]]
=> [4,3,2,1,1]
=> [[1,4,7,11],[2,6,10],[3,9],[5],[8]]
=> [[1,2,3,5,8],[4,6,9],[7,10],[11]]
=> ? = 5
[[7,5,4,3,2,1],[5,4,3,2,1]]
=> [5,4,3,2,1]
=> [[1,3,6,10,15],[2,5,9,14],[4,8,13],[7,12],[11]]
=> [[1,2,4,7,11],[3,5,8,12],[6,9,13],[10,14],[15]]
=> ? = 5
[[7,6,3,2],[6,3,2]]
=> [6,3,2]
=> [[1,2,5,9,10,11],[3,4,8],[6,7]]
=> ?
=> ? = 3
[[6,6,3,2,1],[5,3,2,1]]
=> [5,3,2,1]
=> [[1,3,6,10,11],[2,5,9],[4,8],[7]]
=> [[1,2,4,7],[3,5,8],[6,9],[10],[11]]
=> ? = 4
[[7,6,3,2,1],[6,3,2,1]]
=> [6,3,2,1]
=> [[1,3,6,10,11,12],[2,5,9],[4,8],[7]]
=> ?
=> ? = 4
[[7,6,4,1],[6,4,1]]
=> [6,4,1]
=> [[1,3,4,5,10,11],[2,7,8,9],[6]]
=> ?
=> ? = 3
[[6,6,4,2],[5,4,2]]
=> [5,4,2]
=> [[1,2,5,6,11],[3,4,9,10],[7,8]]
=> [[1,3,7],[2,4,8],[5,9],[6,10],[11]]
=> ? = 3
[[7,6,4,2],[6,4,2]]
=> [6,4,2]
=> [[1,2,5,6,11,12],[3,4,9,10],[7,8]]
=> [[1,3,7],[2,4,8],[5,9],[6,10],[11],[12]]
=> ? = 3
[[6,6,4,2,1],[5,4,2,1]]
=> [5,4,2,1]
=> [[1,3,6,7,12],[2,5,10,11],[4,9],[8]]
=> [[1,2,4,8],[3,5,9],[6,10],[7,11],[12]]
=> ? = 4
[[6,5,4,2,1],[5,3,2,1]]
=> [5,3,2,1]
=> [[1,3,6,10,11],[2,5,9],[4,8],[7]]
=> [[1,2,4,7],[3,5,8],[6,9],[10],[11]]
=> ? = 4
Description
The number of ascents of a standard tableau. Entry $i$ of a standard Young tableau is an '''ascent''' if $i+1$ appears to the right or above $i$ in the tableau (with respect to the English notation for tableaux).
Matching statistic: St000007
Mp00183: Skew partitions inner shapeInteger partitions
Mp00042: Integer partitions initial tableauStandard tableaux
Mp00081: Standard tableaux reading word permutationPermutations
St000007: Permutations ⟶ ℤResult quality: 85% values known / values provided: 85%distinct values known / distinct values provided: 100%
Values
[[1],[]]
=> []
=> []
=> [] => 0
[[2],[]]
=> []
=> []
=> [] => 0
[[1,1],[]]
=> []
=> []
=> [] => 0
[[2,1],[1]]
=> [1]
=> [[1]]
=> [1] => 1
[[3],[]]
=> []
=> []
=> [] => 0
[[2,1],[]]
=> []
=> []
=> [] => 0
[[3,1],[1]]
=> [1]
=> [[1]]
=> [1] => 1
[[2,2],[1]]
=> [1]
=> [[1]]
=> [1] => 1
[[3,2],[2]]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[[1,1,1],[]]
=> []
=> []
=> [] => 0
[[2,2,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 2
[[2,1,1],[1]]
=> [1]
=> [[1]]
=> [1] => 1
[[3,2,1],[2,1]]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 2
[[4],[]]
=> []
=> []
=> [] => 0
[[3,1],[]]
=> []
=> []
=> [] => 0
[[4,1],[1]]
=> [1]
=> [[1]]
=> [1] => 1
[[2,2],[]]
=> []
=> []
=> [] => 0
[[3,2],[1]]
=> [1]
=> [[1]]
=> [1] => 1
[[4,2],[2]]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[[2,1,1],[]]
=> []
=> []
=> [] => 0
[[3,2,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 2
[[3,1,1],[1]]
=> [1]
=> [[1]]
=> [1] => 1
[[4,2,1],[2,1]]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 2
[[3,3],[2]]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[[4,3],[3]]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 1
[[2,2,1],[1]]
=> [1]
=> [[1]]
=> [1] => 1
[[3,3,1],[2,1]]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 2
[[3,2,1],[2]]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[[4,3,1],[3,1]]
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 2
[[2,2,2],[1,1]]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 2
[[3,3,2],[2,2]]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[[3,2,2],[2,1]]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 2
[[4,3,2],[3,2]]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 2
[[1,1,1,1],[]]
=> []
=> []
=> [] => 0
[[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 3
[[2,2,1,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 2
[[3,3,2,1],[2,2,1]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 3
[[2,1,1,1],[1]]
=> [1]
=> [[1]]
=> [1] => 1
[[3,2,2,1],[2,1,1]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 3
[[3,2,1,1],[2,1]]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 2
[[4,3,2,1],[3,2,1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 3
[[5],[]]
=> []
=> []
=> [] => 0
[[4,1],[]]
=> []
=> []
=> [] => 0
[[5,1],[1]]
=> [1]
=> [[1]]
=> [1] => 1
[[3,2],[]]
=> []
=> []
=> [] => 0
[[4,2],[1]]
=> [1]
=> [[1]]
=> [1] => 1
[[5,2],[2]]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[[3,1,1],[]]
=> []
=> []
=> [] => 0
[[4,2,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 2
[[4,1,1],[1]]
=> [1]
=> [[1]]
=> [1] => 1
[[6,5,3,2,1],[5,3,2,1]]
=> [5,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11]]
=> [11,9,10,6,7,8,1,2,3,4,5] => ? = 4
[[6,5,4,2],[5,4,2]]
=> [5,4,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11]]
=> [10,11,6,7,8,9,1,2,3,4,5] => ? = 3
[[5,5,4,2,1],[4,4,2,1]]
=> [4,4,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11]]
=> [11,9,10,5,6,7,8,1,2,3,4] => ? = 4
[[6,5,4,2,1],[5,4,2,1]]
=> [5,4,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12]]
=> [12,10,11,6,7,8,9,1,2,3,4,5] => ? = 4
[[5,5,4,3],[4,4,3]]
=> [4,4,3]
=> [[1,2,3,4],[5,6,7,8],[9,10,11]]
=> [9,10,11,5,6,7,8,1,2,3,4] => ? = 3
[[6,5,4,3],[5,4,3]]
=> [5,4,3]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12]]
=> [10,11,12,6,7,8,9,1,2,3,4,5] => ? = 3
[[5,5,4,3,1],[4,4,3,1]]
=> [4,4,3,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12]]
=> [12,9,10,11,5,6,7,8,1,2,3,4] => ? = 4
[[5,4,4,3,1],[4,3,3,1]]
=> [4,3,3,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11]]
=> [11,8,9,10,5,6,7,1,2,3,4] => ? = 4
[[6,5,4,3,1],[5,4,3,1]]
=> [5,4,3,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13]]
=> [13,10,11,12,6,7,8,9,1,2,3,4,5] => ? = 4
[[4,4,4,3,2],[3,3,3,2]]
=> [3,3,3,2]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11]]
=> [10,11,7,8,9,4,5,6,1,2,3] => ? = 4
[[5,5,4,3,2],[4,4,3,2]]
=> [4,4,3,2]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13]]
=> [12,13,9,10,11,5,6,7,8,1,2,3,4] => ? = 4
[[5,4,4,3,2],[4,3,3,2]]
=> [4,3,3,2]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12]]
=> [11,12,8,9,10,5,6,7,1,2,3,4] => ? = 4
[[5,4,3,3,2],[4,3,2,2]]
=> [4,3,2,2]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11]]
=> [10,11,8,9,5,6,7,1,2,3,4] => ? = 4
[[6,5,4,3,2],[5,4,3,2]]
=> [5,4,3,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14]]
=> [13,14,10,11,12,6,7,8,9,1,2,3,4,5] => ? = 4
[[4,4,4,3,2,1],[3,3,3,2,1]]
=> [3,3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11],[12]]
=> [12,10,11,7,8,9,4,5,6,1,2,3] => ? = 5
[[4,4,3,3,2,1],[3,3,2,2,1]]
=> [3,3,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
=> [11,9,10,7,8,4,5,6,1,2,3] => ? = 5
[[5,5,4,3,2,1],[4,4,3,2,1]]
=> [4,4,3,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13],[14]]
=> [14,12,13,9,10,11,5,6,7,8,1,2,3,4] => ? = 5
[[5,4,4,3,2,1],[4,3,3,2,1]]
=> [4,3,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12],[13]]
=> [13,11,12,8,9,10,5,6,7,1,2,3,4] => ? = 5
[[5,4,3,3,2,1],[4,3,2,2,1]]
=> [4,3,2,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11],[12]]
=> [12,10,11,8,9,5,6,7,1,2,3,4] => ? = 5
[[5,4,3,2,2,1],[4,3,2,1,1]]
=> [4,3,2,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11]]
=> [11,10,8,9,5,6,7,1,2,3,4] => ? = 5
[[6,5,4,3,2,1],[5,4,3,2,1]]
=> [5,4,3,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
=> [15,13,14,10,11,12,6,7,8,9,1,2,3,4,5] => ? = 5
[[7,5,3,2,1],[5,3,2,1]]
=> [5,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11]]
=> [11,9,10,6,7,8,1,2,3,4,5] => ? = 4
[[7,5,4,2],[5,4,2]]
=> [5,4,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11]]
=> [10,11,6,7,8,9,1,2,3,4,5] => ? = 3
[[6,5,4,2,1],[4,4,2,1]]
=> [4,4,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11]]
=> [11,9,10,5,6,7,8,1,2,3,4] => ? = 4
[[7,5,4,2,1],[5,4,2,1]]
=> [5,4,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12]]
=> [12,10,11,6,7,8,9,1,2,3,4,5] => ? = 4
[[6,5,4,3],[4,4,3]]
=> [4,4,3]
=> [[1,2,3,4],[5,6,7,8],[9,10,11]]
=> [9,10,11,5,6,7,8,1,2,3,4] => ? = 3
[[7,5,4,3],[5,4,3]]
=> [5,4,3]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12]]
=> [10,11,12,6,7,8,9,1,2,3,4,5] => ? = 3
[[6,5,4,3,1],[4,4,3,1]]
=> [4,4,3,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12]]
=> [12,9,10,11,5,6,7,8,1,2,3,4] => ? = 4
[[6,4,4,3,1],[4,3,3,1]]
=> [4,3,3,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11]]
=> [11,8,9,10,5,6,7,1,2,3,4] => ? = 4
[[7,5,4,3,1],[5,4,3,1]]
=> [5,4,3,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13]]
=> [13,10,11,12,6,7,8,9,1,2,3,4,5] => ? = 4
[[5,4,4,3,2],[3,3,3,2]]
=> [3,3,3,2]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11]]
=> [10,11,7,8,9,4,5,6,1,2,3] => ? = 4
[[6,5,4,3,2],[4,4,3,2]]
=> [4,4,3,2]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13]]
=> [12,13,9,10,11,5,6,7,8,1,2,3,4] => ? = 4
[[6,4,4,3,2],[4,3,3,2]]
=> [4,3,3,2]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12]]
=> [11,12,8,9,10,5,6,7,1,2,3,4] => ? = 4
[[6,4,3,3,2],[4,3,2,2]]
=> [4,3,2,2]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11]]
=> [10,11,8,9,5,6,7,1,2,3,4] => ? = 4
[[7,5,4,3,2],[5,4,3,2]]
=> [5,4,3,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14]]
=> [13,14,10,11,12,6,7,8,9,1,2,3,4,5] => ? = 4
[[5,4,4,3,2,1],[3,3,3,2,1]]
=> [3,3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11],[12]]
=> [12,10,11,7,8,9,4,5,6,1,2,3] => ? = 5
[[5,4,3,3,2,1],[3,3,2,2,1]]
=> [3,3,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
=> [11,9,10,7,8,4,5,6,1,2,3] => ? = 5
[[6,5,4,3,2,1],[4,4,3,2,1]]
=> [4,4,3,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13],[14]]
=> [14,12,13,9,10,11,5,6,7,8,1,2,3,4] => ? = 5
[[6,4,4,3,2,1],[4,3,3,2,1]]
=> [4,3,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12],[13]]
=> [13,11,12,8,9,10,5,6,7,1,2,3,4] => ? = 5
[[6,4,3,3,2,1],[4,3,2,2,1]]
=> [4,3,2,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11],[12]]
=> [12,10,11,8,9,5,6,7,1,2,3,4] => ? = 5
[[6,4,3,2,2,1],[4,3,2,1,1]]
=> [4,3,2,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11]]
=> [11,10,8,9,5,6,7,1,2,3,4] => ? = 5
[[7,5,4,3,2,1],[5,4,3,2,1]]
=> [5,4,3,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
=> [15,13,14,10,11,12,6,7,8,9,1,2,3,4,5] => ? = 5
[[7,6,3,2],[6,3,2]]
=> [6,3,2]
=> [[1,2,3,4,5,6],[7,8,9],[10,11]]
=> ? => ? = 3
[[6,6,3,2,1],[5,3,2,1]]
=> [5,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11]]
=> [11,9,10,6,7,8,1,2,3,4,5] => ? = 4
[[7,6,3,2,1],[6,3,2,1]]
=> [6,3,2,1]
=> [[1,2,3,4,5,6],[7,8,9],[10,11],[12]]
=> ? => ? = 4
[[7,6,4,1],[6,4,1]]
=> [6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ? => ? = 3
[[6,6,4,2],[5,4,2]]
=> [5,4,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11]]
=> [10,11,6,7,8,9,1,2,3,4,5] => ? = 3
[[7,6,4,2],[6,4,2]]
=> [6,4,2]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12]]
=> [11,12,7,8,9,10,1,2,3,4,5,6] => ? = 3
[[6,6,4,2,1],[5,4,2,1]]
=> [5,4,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12]]
=> [12,10,11,6,7,8,9,1,2,3,4,5] => ? = 4
[[6,5,4,2,1],[5,3,2,1]]
=> [5,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11]]
=> [11,9,10,6,7,8,1,2,3,4,5] => ? = 4
Description
The number of saliances of the permutation. A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
Mp00183: Skew partitions inner shapeInteger partitions
Mp00042: Integer partitions initial tableauStandard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 84% values known / values provided: 84%distinct values known / distinct values provided: 89%
Values
[[1],[]]
=> []
=> []
=> ? = 0
[[2],[]]
=> []
=> []
=> ? = 0
[[1,1],[]]
=> []
=> []
=> ? = 0
[[2,1],[1]]
=> [1]
=> [[1]]
=> 1
[[3],[]]
=> []
=> []
=> ? = 0
[[2,1],[]]
=> []
=> []
=> ? = 0
[[3,1],[1]]
=> [1]
=> [[1]]
=> 1
[[2,2],[1]]
=> [1]
=> [[1]]
=> 1
[[3,2],[2]]
=> [2]
=> [[1,2]]
=> 1
[[1,1,1],[]]
=> []
=> []
=> ? = 0
[[2,2,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 2
[[2,1,1],[1]]
=> [1]
=> [[1]]
=> 1
[[3,2,1],[2,1]]
=> [2,1]
=> [[1,2],[3]]
=> 2
[[4],[]]
=> []
=> []
=> ? = 0
[[3,1],[]]
=> []
=> []
=> ? = 0
[[4,1],[1]]
=> [1]
=> [[1]]
=> 1
[[2,2],[]]
=> []
=> []
=> ? = 0
[[3,2],[1]]
=> [1]
=> [[1]]
=> 1
[[4,2],[2]]
=> [2]
=> [[1,2]]
=> 1
[[2,1,1],[]]
=> []
=> []
=> ? = 0
[[3,2,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 2
[[3,1,1],[1]]
=> [1]
=> [[1]]
=> 1
[[4,2,1],[2,1]]
=> [2,1]
=> [[1,2],[3]]
=> 2
[[3,3],[2]]
=> [2]
=> [[1,2]]
=> 1
[[4,3],[3]]
=> [3]
=> [[1,2,3]]
=> 1
[[2,2,1],[1]]
=> [1]
=> [[1]]
=> 1
[[3,3,1],[2,1]]
=> [2,1]
=> [[1,2],[3]]
=> 2
[[3,2,1],[2]]
=> [2]
=> [[1,2]]
=> 1
[[4,3,1],[3,1]]
=> [3,1]
=> [[1,2,3],[4]]
=> 2
[[2,2,2],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 2
[[3,3,2],[2,2]]
=> [2,2]
=> [[1,2],[3,4]]
=> 2
[[3,2,2],[2,1]]
=> [2,1]
=> [[1,2],[3]]
=> 2
[[4,3,2],[3,2]]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 2
[[1,1,1,1],[]]
=> []
=> []
=> ? = 0
[[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3
[[2,2,1,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 2
[[3,3,2,1],[2,2,1]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 3
[[2,1,1,1],[1]]
=> [1]
=> [[1]]
=> 1
[[3,2,2,1],[2,1,1]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 3
[[3,2,1,1],[2,1]]
=> [2,1]
=> [[1,2],[3]]
=> 2
[[4,3,2,1],[3,2,1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 3
[[5],[]]
=> []
=> []
=> ? = 0
[[4,1],[]]
=> []
=> []
=> ? = 0
[[5,1],[1]]
=> [1]
=> [[1]]
=> 1
[[3,2],[]]
=> []
=> []
=> ? = 0
[[4,2],[1]]
=> [1]
=> [[1]]
=> 1
[[5,2],[2]]
=> [2]
=> [[1,2]]
=> 1
[[3,1,1],[]]
=> []
=> []
=> ? = 0
[[4,2,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 2
[[4,1,1],[1]]
=> [1]
=> [[1]]
=> 1
[[5,2,1],[2,1]]
=> [2,1]
=> [[1,2],[3]]
=> 2
[[3,3],[1]]
=> [1]
=> [[1]]
=> 1
[[4,3],[2]]
=> [2]
=> [[1,2]]
=> 1
[[5,3],[3]]
=> [3]
=> [[1,2,3]]
=> 1
[[2,2,1],[]]
=> []
=> []
=> ? = 0
[[3,3,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 2
[[3,2,1],[1]]
=> [1]
=> [[1]]
=> 1
[[4,3,1],[2,1]]
=> [2,1]
=> [[1,2],[3]]
=> 2
[[4,2,1],[2]]
=> [2]
=> [[1,2]]
=> 1
[[5,3,1],[3,1]]
=> [3,1]
=> [[1,2,3],[4]]
=> 2
[[3,2,2],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 2
[[4,3,2],[2,2]]
=> [2,2]
=> [[1,2],[3,4]]
=> 2
[[4,2,2],[2,1]]
=> [2,1]
=> [[1,2],[3]]
=> 2
[[5,3,2],[3,2]]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 2
[[2,1,1,1],[]]
=> []
=> []
=> ? = 0
[[3,2,2,1],[1,1,1]]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3
[[3,2,1,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 2
[[1,1,1,1,1],[]]
=> []
=> []
=> ? = 0
[[6],[]]
=> []
=> []
=> ? = 0
[[5,1],[]]
=> []
=> []
=> ? = 0
[[4,2],[]]
=> []
=> []
=> ? = 0
[[4,1,1],[]]
=> []
=> []
=> ? = 0
[[3,3],[]]
=> []
=> []
=> ? = 0
[[3,2,1],[]]
=> []
=> []
=> ? = 0
[[3,1,1,1],[]]
=> []
=> []
=> ? = 0
[[2,2,2],[]]
=> []
=> []
=> ? = 0
[[2,2,1,1],[]]
=> []
=> []
=> ? = 0
[[2,1,1,1,1],[]]
=> []
=> []
=> ? = 0
[[1,1,1,1,1,1],[]]
=> []
=> []
=> ? = 0
[[7],[]]
=> []
=> []
=> ? = 0
[[6,1],[]]
=> []
=> []
=> ? = 0
[[5,2],[]]
=> []
=> []
=> ? = 0
[[5,1,1],[]]
=> []
=> []
=> ? = 0
[[4,3],[]]
=> []
=> []
=> ? = 0
[[4,2,1],[]]
=> []
=> []
=> ? = 0
[[4,1,1,1],[]]
=> []
=> []
=> ? = 0
[[3,3,1],[]]
=> []
=> []
=> ? = 0
[[3,2,2],[]]
=> []
=> []
=> ? = 0
[[3,2,1,1],[]]
=> []
=> []
=> ? = 0
[[3,1,1,1,1],[]]
=> []
=> []
=> ? = 0
[[2,2,2,1],[]]
=> []
=> []
=> ? = 0
[[2,2,1,1,1],[]]
=> []
=> []
=> ? = 0
[[2,1,1,1,1,1],[]]
=> []
=> []
=> ? = 0
[[7,6,3,2],[6,3,2]]
=> [6,3,2]
=> [[1,2,3,4,5,6],[7,8,9],[10,11]]
=> ? = 3
[[7,6,3,2,1],[6,3,2,1]]
=> [6,3,2,1]
=> [[1,2,3,4,5,6],[7,8,9],[10,11],[12]]
=> ? = 4
[[7,6,4,1],[6,4,1]]
=> [6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ? = 3
[[7,6,4,2,1],[6,4,2,1]]
=> [6,4,2,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12],[13]]
=> ? = 4
[[7,6,4,3],[6,4,3]]
=> [6,4,3]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12,13]]
=> ? = 3
[[7,6,4,3,1],[6,4,3,1]]
=> [6,4,3,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12,13],[14]]
=> ? = 4
[[7,6,4,3,2],[6,4,3,2]]
=> [6,4,3,2]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12,13],[14,15]]
=> ? = 4
Description
The row containing the largest entry of a standard tableau.
Mp00183: Skew partitions inner shapeInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
St001227: Dyck paths ⟶ ℤResult quality: 56% values known / values provided: 82%distinct values known / distinct values provided: 56%
Values
[[1],[]]
=> []
=> []
=> ? = 0
[[2],[]]
=> []
=> []
=> ? = 0
[[1,1],[]]
=> []
=> []
=> ? = 0
[[2,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[[3],[]]
=> []
=> []
=> ? = 0
[[2,1],[]]
=> []
=> []
=> ? = 0
[[3,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[[2,2],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[[3,2],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[1,1,1],[]]
=> []
=> []
=> ? = 0
[[2,2,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[2,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[[3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[[4],[]]
=> []
=> []
=> ? = 0
[[3,1],[]]
=> []
=> []
=> ? = 0
[[4,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[[2,2],[]]
=> []
=> []
=> ? = 0
[[3,2],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[[4,2],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[2,1,1],[]]
=> []
=> []
=> ? = 0
[[3,2,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[3,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[[4,2,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[[3,3],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[4,3],[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[[2,2,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[[3,3,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[[3,2,1],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[4,3,1],[3,1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
[[2,2,2],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[3,3,2],[2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2
[[3,2,2],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[[4,3,2],[3,2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 2
[[1,1,1,1],[]]
=> []
=> []
=> ? = 0
[[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3
[[2,2,1,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[3,3,2,1],[2,2,1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[[2,1,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[[3,2,2,1],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3
[[3,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[[4,3,2,1],[3,2,1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 3
[[5],[]]
=> []
=> []
=> ? = 0
[[4,1],[]]
=> []
=> []
=> ? = 0
[[5,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[[3,2],[]]
=> []
=> []
=> ? = 0
[[4,2],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[[5,2],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[3,1,1],[]]
=> []
=> []
=> ? = 0
[[4,2,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[4,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[[5,2,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[[3,3],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[[4,3],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[5,3],[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[[2,2,1],[]]
=> []
=> []
=> ? = 0
[[3,3,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[3,2,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[[4,3,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[[4,2,1],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[5,3,1],[3,1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
[[3,2,2],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[4,3,2],[2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2
[[4,2,2],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[[5,3,2],[3,2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 2
[[2,1,1,1],[]]
=> []
=> []
=> ? = 0
[[3,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3
[[3,2,1,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[1,1,1,1,1],[]]
=> []
=> []
=> ? = 0
[[6],[]]
=> []
=> []
=> ? = 0
[[5,1],[]]
=> []
=> []
=> ? = 0
[[4,2],[]]
=> []
=> []
=> ? = 0
[[4,1,1],[]]
=> []
=> []
=> ? = 0
[[3,3],[]]
=> []
=> []
=> ? = 0
[[3,2,1],[]]
=> []
=> []
=> ? = 0
[[3,1,1,1],[]]
=> []
=> []
=> ? = 0
[[2,2,2],[]]
=> []
=> []
=> ? = 0
[[2,2,1,1],[]]
=> []
=> []
=> ? = 0
[[2,1,1,1,1],[]]
=> []
=> []
=> ? = 0
[[1,1,1,1,1,1],[]]
=> []
=> []
=> ? = 0
[[7],[]]
=> []
=> []
=> ? = 0
[[6,1],[]]
=> []
=> []
=> ? = 0
[[5,2],[]]
=> []
=> []
=> ? = 0
[[5,1,1],[]]
=> []
=> []
=> ? = 0
[[4,3],[]]
=> []
=> []
=> ? = 0
[[4,2,1],[]]
=> []
=> []
=> ? = 0
[[4,1,1,1],[]]
=> []
=> []
=> ? = 0
[[3,3,1],[]]
=> []
=> []
=> ? = 0
[[3,2,2],[]]
=> []
=> []
=> ? = 0
[[3,2,1,1],[]]
=> []
=> []
=> ? = 0
[[3,1,1,1,1],[]]
=> []
=> []
=> ? = 0
[[2,2,2,1],[]]
=> []
=> []
=> ? = 0
[[2,2,1,1,1],[]]
=> []
=> []
=> ? = 0
[[2,1,1,1,1,1],[]]
=> []
=> []
=> ? = 0
[[7,6],[6]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[[7,6,1],[6,1]]
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 2
[[7,6,2],[6,2]]
=> [6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> ? = 2
[[7,6,2,1],[6,2,1]]
=> [6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> ? = 3
[[7,6,3],[6,3]]
=> [6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> ? = 2
[[7,6,3,1],[6,3,1]]
=> [6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> ? = 3
[[7,6,3,2],[6,3,2]]
=> [6,3,2]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> ? = 3
Description
The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra.
The following 37 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000653The last descent of a permutation. St001480The number of simple summands of the module J^2/J^3. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St000053The number of valleys of the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St001777The number of weak descents in an integer composition. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000015The number of peaks of a Dyck path. St000155The number of exceedances (also excedences) of a permutation. St000331The number of upper interactions of a Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000809The reduced reflection length of the permutation. St001462The number of factors of a standard tableaux under concatenation. St000216The absolute length of a permutation. St000006The dinv of a Dyck path. St000105The number of blocks in the set partition. St001152The number of pairs with even minimum in a perfect matching. St000925The number of topologically connected components of a set partition. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000325The width of the tree associated to a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000354The number of recoils of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000831The number of indices that are either descents or recoils. St001061The number of indices that are both descents and recoils of a permutation. St000225Difference between largest and smallest parts in a partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001435The number of missing boxes in the first row.