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Your data matches 34 different statistics following compositions of up to 3 maps.
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Matching statistic: St000459
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Mp00182: Skew partitions —outer shape⟶ Integer partitions
St000459: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000459: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
=> [1]
=> 1 = 0 + 1
[[2],[]]
=> [2]
=> 2 = 1 + 1
[[1,1],[]]
=> [1,1]
=> 2 = 1 + 1
[[3],[]]
=> [3]
=> 3 = 2 + 1
[[2,1],[]]
=> [2,1]
=> 3 = 2 + 1
[[2,2],[1]]
=> [2,2]
=> 3 = 2 + 1
[[1,1,1],[]]
=> [1,1,1]
=> 3 = 2 + 1
[[4],[]]
=> [4]
=> 4 = 3 + 1
[[3,1],[]]
=> [3,1]
=> 4 = 3 + 1
[[2,2],[]]
=> [2,2]
=> 3 = 2 + 1
[[3,2],[1]]
=> [3,2]
=> 4 = 3 + 1
[[2,1,1],[]]
=> [2,1,1]
=> 4 = 3 + 1
[[3,3],[2]]
=> [3,3]
=> 4 = 3 + 1
[[2,2,1],[1]]
=> [2,2,1]
=> 4 = 3 + 1
[[2,2,2],[1,1]]
=> [2,2,2]
=> 4 = 3 + 1
[[1,1,1,1],[]]
=> [1,1,1,1]
=> 4 = 3 + 1
[[5],[]]
=> [5]
=> 5 = 4 + 1
[[4,1],[]]
=> [4,1]
=> 5 = 4 + 1
[[3,2],[]]
=> [3,2]
=> 4 = 3 + 1
[[4,2],[1]]
=> [4,2]
=> 5 = 4 + 1
[[3,1,1],[]]
=> [3,1,1]
=> 5 = 4 + 1
[[3,3],[1]]
=> [3,3]
=> 4 = 3 + 1
[[4,3],[2]]
=> [4,3]
=> 5 = 4 + 1
[[2,2,1],[]]
=> [2,2,1]
=> 4 = 3 + 1
[[3,2,1],[1]]
=> [3,2,1]
=> 5 = 4 + 1
[[3,2,2],[1,1]]
=> [3,2,2]
=> 5 = 4 + 1
[[2,1,1,1],[]]
=> [2,1,1,1]
=> 5 = 4 + 1
[[4,4],[3]]
=> [4,4]
=> 5 = 4 + 1
[[3,3,1],[2]]
=> [3,3,1]
=> 5 = 4 + 1
[[2,2,2],[1]]
=> [2,2,2]
=> 4 = 3 + 1
[[3,3,2],[2,1]]
=> [3,3,2]
=> 5 = 4 + 1
[[2,2,1,1],[1]]
=> [2,2,1,1]
=> 5 = 4 + 1
[[3,3,3],[2,2]]
=> [3,3,3]
=> 5 = 4 + 1
[[2,2,2,1],[1,1]]
=> [2,2,2,1]
=> 5 = 4 + 1
[[2,2,2,2],[1,1,1]]
=> [2,2,2,2]
=> 5 = 4 + 1
[[1,1,1,1,1],[]]
=> [1,1,1,1,1]
=> 5 = 4 + 1
[[6],[]]
=> [6]
=> 6 = 5 + 1
[[5,1],[]]
=> [5,1]
=> 6 = 5 + 1
[[4,2],[]]
=> [4,2]
=> 5 = 4 + 1
[[5,2],[1]]
=> [5,2]
=> 6 = 5 + 1
[[4,1,1],[]]
=> [4,1,1]
=> 6 = 5 + 1
[[3,3],[]]
=> [3,3]
=> 4 = 3 + 1
[[4,3],[1]]
=> [4,3]
=> 5 = 4 + 1
[[5,3],[2]]
=> [5,3]
=> 6 = 5 + 1
[[3,2,1],[]]
=> [3,2,1]
=> 5 = 4 + 1
[[4,2,1],[1]]
=> [4,2,1]
=> 6 = 5 + 1
[[4,2,2],[1,1]]
=> [4,2,2]
=> 6 = 5 + 1
[[3,1,1,1],[]]
=> [3,1,1,1]
=> 6 = 5 + 1
[[4,4],[2]]
=> [4,4]
=> 5 = 4 + 1
[[5,4],[3]]
=> [5,4]
=> 6 = 5 + 1
Description
The hook length of the base cell of a partition.
This is also known as the perimeter of a partition. In particular, the perimeter of the empty partition is zero.
Matching statistic: St000259
Values
[[1],[]]
=> ([],1)
=> ([],1)
=> 0
[[2],[]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1],[]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[3],[]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[[3,3],[1]]
=> ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[[2,2,1],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[[3,2,1],[1]]
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[[3,2,2],[1,1]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[[3,3,1],[2]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[[2,2,2],[1]]
=> ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[[2,2,1,1],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[[3,3,3],[2,2]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[[2,2,2,1],[1,1]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[[2,2,2,2],[1,1,1]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 5
[[5,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 5
[[4,2],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 4
[[5,2],[1]]
=> ([(0,5),(1,4),(1,5),(3,2),(4,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 5
[[4,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 5
[[3,3],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> 3
[[4,3],[1]]
=> ([(0,4),(1,2),(1,4),(2,3),(2,5),(4,5)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[[5,3],[2]]
=> ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 5
[[3,2,1],[]]
=> ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[[4,2,1],[1]]
=> ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 5
[[4,2,2],[1,1]]
=> ([(0,5),(1,3),(1,4),(3,5),(4,2)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 5
[[3,1,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 5
[[4,4],[2]]
=> ([(0,3),(1,2),(1,4),(2,5),(3,4),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 4
[[5,4],[3]]
=> ([(0,4),(1,3),(1,5),(2,5),(4,2)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 5
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St000395
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000395: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000395: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
=> [1]
=> [1,0]
=> [1,0]
=> 1 = 0 + 1
[[2],[]]
=> [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2 = 1 + 1
[[1,1],[]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2 = 1 + 1
[[3],[]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[[2,1],[]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3 = 2 + 1
[[2,2],[1]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 3 = 2 + 1
[[1,1,1],[]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 3 = 2 + 1
[[4],[]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[[3,1],[]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[[2,2],[]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 3 = 2 + 1
[[3,2],[1]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 4 = 3 + 1
[[2,1,1],[]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[[3,3],[2]]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 4 = 3 + 1
[[2,2,1],[1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[[2,2,2],[1,1]]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[[1,1,1,1],[]]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[[5],[]]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[[4,1],[]]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[[3,2],[]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 4 = 3 + 1
[[4,2],[1]]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 5 = 4 + 1
[[3,1,1],[]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[[3,3],[1]]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 4 = 3 + 1
[[4,3],[2]]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 5 = 4 + 1
[[2,2,1],[]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[[3,2,1],[1]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 5 = 4 + 1
[[3,2,2],[1,1]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[[2,1,1,1],[]]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 5 = 4 + 1
[[4,4],[3]]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 5 = 4 + 1
[[3,3,1],[2]]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 5 = 4 + 1
[[2,2,2],[1]]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[[3,3,2],[2,1]]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[[2,2,1,1],[1]]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 5 = 4 + 1
[[3,3,3],[2,2]]
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 4 + 1
[[2,2,2,1],[1,1]]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 5 = 4 + 1
[[2,2,2,2],[1,1,1]]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[[1,1,1,1,1],[]]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 5 = 4 + 1
[[6],[]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[[5,1],[]]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 6 = 5 + 1
[[4,2],[]]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 5 = 4 + 1
[[5,2],[1]]
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 6 = 5 + 1
[[4,1,1],[]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 6 = 5 + 1
[[3,3],[]]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 4 = 3 + 1
[[4,3],[1]]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 5 = 4 + 1
[[5,3],[2]]
=> [5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 6 = 5 + 1
[[3,2,1],[]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 5 = 4 + 1
[[4,2,1],[1]]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 6 = 5 + 1
[[4,2,2],[1,1]]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 6 = 5 + 1
[[3,1,1,1],[]]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 6 = 5 + 1
[[4,4],[2]]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 5 = 4 + 1
[[5,4],[3]]
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 6 = 5 + 1
Description
The sum of the heights of the peaks of a Dyck path.
Matching statistic: St001020
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001020: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001020: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
=> [1]
=> [1,0]
=> [1,0]
=> 1 = 0 + 1
[[2],[]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2 = 1 + 1
[[1,1],[]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
[[3],[]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3 = 2 + 1
[[2,1],[]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3 = 2 + 1
[[2,2],[1]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[[1,1,1],[]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[[4],[]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[[3,1],[]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 4 = 3 + 1
[[2,2],[]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[[3,2],[1]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[[2,1,1],[]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[[3,3],[2]]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[[2,2,1],[1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[[2,2,2],[1,1]]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[[1,1,1,1],[]]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[[5],[]]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 4 + 1
[[4,1],[]]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[[3,2],[]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[[4,2],[1]]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 5 = 4 + 1
[[3,1,1],[]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 5 = 4 + 1
[[3,3],[1]]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[[4,3],[2]]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[[2,2,1],[]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[[3,2,1],[1]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 5 = 4 + 1
[[3,2,2],[1,1]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[[2,1,1,1],[]]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 5 = 4 + 1
[[4,4],[3]]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 5 = 4 + 1
[[3,3,1],[2]]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 5 = 4 + 1
[[2,2,2],[1]]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[[3,3,2],[2,1]]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 5 = 4 + 1
[[2,2,1,1],[1]]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 5 = 4 + 1
[[3,3,3],[2,2]]
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[[2,2,2,1],[1,1]]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 5 = 4 + 1
[[2,2,2,2],[1,1,1]]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[[1,1,1,1,1],[]]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[[6],[]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6 = 5 + 1
[[5,1],[]]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[[4,2],[]]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 5 = 4 + 1
[[5,2],[1]]
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 6 = 5 + 1
[[4,1,1],[]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 6 = 5 + 1
[[3,3],[]]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[[4,3],[1]]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[[5,3],[2]]
=> [5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> 6 = 5 + 1
[[3,2,1],[]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 5 = 4 + 1
[[4,2,1],[1]]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> 6 = 5 + 1
[[4,2,2],[1,1]]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 6 = 5 + 1
[[3,1,1,1],[]]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 6 = 5 + 1
[[4,4],[2]]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 5 = 4 + 1
[[5,4],[3]]
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> 6 = 5 + 1
Description
Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001348
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
St001348: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
St001348: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
=> [1]
=> [1,0]
=> [1,0]
=> 1 = 0 + 1
[[2],[]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2 = 1 + 1
[[1,1],[]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
[[3],[]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 3 = 2 + 1
[[2,1],[]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,1,0,0,0]
=> 3 = 2 + 1
[[2,2],[1]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[[1,1,1],[]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[[4],[]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 4 = 3 + 1
[[3,1],[]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 4 = 3 + 1
[[2,2],[]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[[3,2],[1]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[[2,1,1],[]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[[3,3],[2]]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[[2,2,1],[1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[[2,2,2],[1,1]]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[[1,1,1,1],[]]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[[5],[]]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 5 = 4 + 1
[[4,1],[]]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 5 = 4 + 1
[[3,2],[]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[[4,2],[1]]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[[3,1,1],[]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 5 = 4 + 1
[[3,3],[1]]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[[4,3],[2]]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 5 = 4 + 1
[[2,2,1],[]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[[3,2,1],[1]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 5 = 4 + 1
[[3,2,2],[1,1]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 4 + 1
[[2,1,1,1],[]]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 5 = 4 + 1
[[4,4],[3]]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 5 = 4 + 1
[[3,3,1],[2]]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 5 = 4 + 1
[[2,2,2],[1]]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[[3,3,2],[2,1]]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 5 = 4 + 1
[[2,2,1,1],[1]]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[[3,3,3],[2,2]]
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[[2,2,2,1],[1,1]]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 5 = 4 + 1
[[2,2,2,2],[1,1,1]]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 5 = 4 + 1
[[1,1,1,1,1],[]]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[[6],[]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 6 = 5 + 1
[[5,1],[]]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 6 = 5 + 1
[[4,2],[]]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[[5,2],[1]]
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 6 = 5 + 1
[[4,1,1],[]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 6 = 5 + 1
[[3,3],[]]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[[4,3],[1]]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 5 = 4 + 1
[[5,3],[2]]
=> [5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 6 = 5 + 1
[[3,2,1],[]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 5 = 4 + 1
[[4,2,1],[1]]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 6 = 5 + 1
[[4,2,2],[1,1]]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[[3,1,1,1],[]]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 6 = 5 + 1
[[4,4],[2]]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 5 = 4 + 1
[[5,4],[3]]
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 6 = 5 + 1
Description
The bounce of the parallelogram polyomino associated with the Dyck path.
A bijection due to Delest and Viennot [1] associates a Dyck path with a parallelogram polyomino. The bounce statistic is defined in [2].
Matching statistic: St000543
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St000543: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St000543: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
=> [1]
=> 10 => 01 => 2 = 0 + 2
[[2],[]]
=> [2]
=> 100 => 001 => 3 = 1 + 2
[[1,1],[]]
=> [1,1]
=> 110 => 011 => 3 = 1 + 2
[[3],[]]
=> [3]
=> 1000 => 0001 => 4 = 2 + 2
[[2,1],[]]
=> [2,1]
=> 1010 => 0011 => 4 = 2 + 2
[[2,2],[1]]
=> [2,2]
=> 1100 => 0011 => 4 = 2 + 2
[[1,1,1],[]]
=> [1,1,1]
=> 1110 => 0111 => 4 = 2 + 2
[[4],[]]
=> [4]
=> 10000 => 00001 => 5 = 3 + 2
[[3,1],[]]
=> [3,1]
=> 10010 => 00011 => 5 = 3 + 2
[[2,2],[]]
=> [2,2]
=> 1100 => 0011 => 4 = 2 + 2
[[3,2],[1]]
=> [3,2]
=> 10100 => 00011 => 5 = 3 + 2
[[2,1,1],[]]
=> [2,1,1]
=> 10110 => 00111 => 5 = 3 + 2
[[3,3],[2]]
=> [3,3]
=> 11000 => 00011 => 5 = 3 + 2
[[2,2,1],[1]]
=> [2,2,1]
=> 11010 => 00111 => 5 = 3 + 2
[[2,2,2],[1,1]]
=> [2,2,2]
=> 11100 => 00111 => 5 = 3 + 2
[[1,1,1,1],[]]
=> [1,1,1,1]
=> 11110 => 01111 => 5 = 3 + 2
[[5],[]]
=> [5]
=> 100000 => 000001 => 6 = 4 + 2
[[4,1],[]]
=> [4,1]
=> 100010 => 000011 => 6 = 4 + 2
[[3,2],[]]
=> [3,2]
=> 10100 => 00011 => 5 = 3 + 2
[[4,2],[1]]
=> [4,2]
=> 100100 => 000011 => 6 = 4 + 2
[[3,1,1],[]]
=> [3,1,1]
=> 100110 => 000111 => 6 = 4 + 2
[[3,3],[1]]
=> [3,3]
=> 11000 => 00011 => 5 = 3 + 2
[[4,3],[2]]
=> [4,3]
=> 101000 => 000011 => 6 = 4 + 2
[[2,2,1],[]]
=> [2,2,1]
=> 11010 => 00111 => 5 = 3 + 2
[[3,2,1],[1]]
=> [3,2,1]
=> 101010 => 001011 => 6 = 4 + 2
[[3,2,2],[1,1]]
=> [3,2,2]
=> 101100 => 000111 => 6 = 4 + 2
[[2,1,1,1],[]]
=> [2,1,1,1]
=> 101110 => 001111 => 6 = 4 + 2
[[4,4],[3]]
=> [4,4]
=> 110000 => 000011 => 6 = 4 + 2
[[3,3,1],[2]]
=> [3,3,1]
=> 110010 => 000111 => 6 = 4 + 2
[[2,2,2],[1]]
=> [2,2,2]
=> 11100 => 00111 => 5 = 3 + 2
[[3,3,2],[2,1]]
=> [3,3,2]
=> 110100 => 000111 => 6 = 4 + 2
[[2,2,1,1],[1]]
=> [2,2,1,1]
=> 110110 => 001111 => 6 = 4 + 2
[[3,3,3],[2,2]]
=> [3,3,3]
=> 111000 => 000111 => 6 = 4 + 2
[[2,2,2,1],[1,1]]
=> [2,2,2,1]
=> 111010 => 001111 => 6 = 4 + 2
[[2,2,2,2],[1,1,1]]
=> [2,2,2,2]
=> 111100 => 001111 => 6 = 4 + 2
[[1,1,1,1,1],[]]
=> [1,1,1,1,1]
=> 111110 => 011111 => 6 = 4 + 2
[[6],[]]
=> [6]
=> 1000000 => 0000001 => 7 = 5 + 2
[[5,1],[]]
=> [5,1]
=> 1000010 => 0000011 => 7 = 5 + 2
[[4,2],[]]
=> [4,2]
=> 100100 => 000011 => 6 = 4 + 2
[[5,2],[1]]
=> [5,2]
=> 1000100 => 0000011 => 7 = 5 + 2
[[4,1,1],[]]
=> [4,1,1]
=> 1000110 => 0000111 => 7 = 5 + 2
[[3,3],[]]
=> [3,3]
=> 11000 => 00011 => 5 = 3 + 2
[[4,3],[1]]
=> [4,3]
=> 101000 => 000011 => 6 = 4 + 2
[[5,3],[2]]
=> [5,3]
=> 1001000 => 0000011 => 7 = 5 + 2
[[3,2,1],[]]
=> [3,2,1]
=> 101010 => 001011 => 6 = 4 + 2
[[4,2,1],[1]]
=> [4,2,1]
=> 1001010 => 0001011 => 7 = 5 + 2
[[4,2,2],[1,1]]
=> [4,2,2]
=> 1001100 => 0000111 => 7 = 5 + 2
[[3,1,1,1],[]]
=> [3,1,1,1]
=> 1001110 => 0001111 => 7 = 5 + 2
[[4,4],[2]]
=> [4,4]
=> 110000 => 000011 => 6 = 4 + 2
[[5,4],[3]]
=> [5,4]
=> 1010000 => 0000011 => 7 = 5 + 2
Description
The size of the conjugacy class of a binary word.
Two words $u$ and $v$ are conjugate, if $u=w_1 w_2$ and $v=w_2 w_1$, see Section 1.3 of [1].
Matching statistic: St000626
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St000626: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St000626: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
=> [1]
=> 10 => 01 => 2 = 0 + 2
[[2],[]]
=> [2]
=> 100 => 001 => 3 = 1 + 2
[[1,1],[]]
=> [1,1]
=> 110 => 011 => 3 = 1 + 2
[[3],[]]
=> [3]
=> 1000 => 0001 => 4 = 2 + 2
[[2,1],[]]
=> [2,1]
=> 1010 => 0011 => 4 = 2 + 2
[[2,2],[1]]
=> [2,2]
=> 1100 => 0011 => 4 = 2 + 2
[[1,1,1],[]]
=> [1,1,1]
=> 1110 => 0111 => 4 = 2 + 2
[[4],[]]
=> [4]
=> 10000 => 00001 => 5 = 3 + 2
[[3,1],[]]
=> [3,1]
=> 10010 => 00011 => 5 = 3 + 2
[[2,2],[]]
=> [2,2]
=> 1100 => 0011 => 4 = 2 + 2
[[3,2],[1]]
=> [3,2]
=> 10100 => 00011 => 5 = 3 + 2
[[2,1,1],[]]
=> [2,1,1]
=> 10110 => 00111 => 5 = 3 + 2
[[3,3],[2]]
=> [3,3]
=> 11000 => 00011 => 5 = 3 + 2
[[2,2,1],[1]]
=> [2,2,1]
=> 11010 => 00111 => 5 = 3 + 2
[[2,2,2],[1,1]]
=> [2,2,2]
=> 11100 => 00111 => 5 = 3 + 2
[[1,1,1,1],[]]
=> [1,1,1,1]
=> 11110 => 01111 => 5 = 3 + 2
[[5],[]]
=> [5]
=> 100000 => 000001 => 6 = 4 + 2
[[4,1],[]]
=> [4,1]
=> 100010 => 000011 => 6 = 4 + 2
[[3,2],[]]
=> [3,2]
=> 10100 => 00011 => 5 = 3 + 2
[[4,2],[1]]
=> [4,2]
=> 100100 => 000011 => 6 = 4 + 2
[[3,1,1],[]]
=> [3,1,1]
=> 100110 => 000111 => 6 = 4 + 2
[[3,3],[1]]
=> [3,3]
=> 11000 => 00011 => 5 = 3 + 2
[[4,3],[2]]
=> [4,3]
=> 101000 => 000011 => 6 = 4 + 2
[[2,2,1],[]]
=> [2,2,1]
=> 11010 => 00111 => 5 = 3 + 2
[[3,2,1],[1]]
=> [3,2,1]
=> 101010 => 001011 => 6 = 4 + 2
[[3,2,2],[1,1]]
=> [3,2,2]
=> 101100 => 000111 => 6 = 4 + 2
[[2,1,1,1],[]]
=> [2,1,1,1]
=> 101110 => 001111 => 6 = 4 + 2
[[4,4],[3]]
=> [4,4]
=> 110000 => 000011 => 6 = 4 + 2
[[3,3,1],[2]]
=> [3,3,1]
=> 110010 => 000111 => 6 = 4 + 2
[[2,2,2],[1]]
=> [2,2,2]
=> 11100 => 00111 => 5 = 3 + 2
[[3,3,2],[2,1]]
=> [3,3,2]
=> 110100 => 000111 => 6 = 4 + 2
[[2,2,1,1],[1]]
=> [2,2,1,1]
=> 110110 => 001111 => 6 = 4 + 2
[[3,3,3],[2,2]]
=> [3,3,3]
=> 111000 => 000111 => 6 = 4 + 2
[[2,2,2,1],[1,1]]
=> [2,2,2,1]
=> 111010 => 001111 => 6 = 4 + 2
[[2,2,2,2],[1,1,1]]
=> [2,2,2,2]
=> 111100 => 001111 => 6 = 4 + 2
[[1,1,1,1,1],[]]
=> [1,1,1,1,1]
=> 111110 => 011111 => 6 = 4 + 2
[[6],[]]
=> [6]
=> 1000000 => 0000001 => 7 = 5 + 2
[[5,1],[]]
=> [5,1]
=> 1000010 => 0000011 => 7 = 5 + 2
[[4,2],[]]
=> [4,2]
=> 100100 => 000011 => 6 = 4 + 2
[[5,2],[1]]
=> [5,2]
=> 1000100 => 0000011 => 7 = 5 + 2
[[4,1,1],[]]
=> [4,1,1]
=> 1000110 => 0000111 => 7 = 5 + 2
[[3,3],[]]
=> [3,3]
=> 11000 => 00011 => 5 = 3 + 2
[[4,3],[1]]
=> [4,3]
=> 101000 => 000011 => 6 = 4 + 2
[[5,3],[2]]
=> [5,3]
=> 1001000 => 0000011 => 7 = 5 + 2
[[3,2,1],[]]
=> [3,2,1]
=> 101010 => 001011 => 6 = 4 + 2
[[4,2,1],[1]]
=> [4,2,1]
=> 1001010 => 0001011 => 7 = 5 + 2
[[4,2,2],[1,1]]
=> [4,2,2]
=> 1001100 => 0000111 => 7 = 5 + 2
[[3,1,1,1],[]]
=> [3,1,1,1]
=> 1001110 => 0001111 => 7 = 5 + 2
[[4,4],[2]]
=> [4,4]
=> 110000 => 000011 => 6 = 4 + 2
[[5,4],[3]]
=> [5,4]
=> 1010000 => 0000011 => 7 = 5 + 2
Description
The minimal period of a binary word.
This is the smallest natural number $p$ such that $w_i=w_{i+p}$ for all $i\in\{1,\dots,|w|-p\}$.
Matching statistic: St000998
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000998: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000998: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
=> [1]
=> [1,0]
=> [1,0]
=> 2 = 0 + 2
[[2],[]]
=> [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 3 = 1 + 2
[[1,1],[]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 3 = 1 + 2
[[3],[]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 4 = 2 + 2
[[2,1],[]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 4 = 2 + 2
[[2,2],[1]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 4 = 2 + 2
[[1,1,1],[]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 4 = 2 + 2
[[4],[]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 5 = 3 + 2
[[3,1],[]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 5 = 3 + 2
[[2,2],[]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 4 = 2 + 2
[[3,2],[1]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 5 = 3 + 2
[[2,1,1],[]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 5 = 3 + 2
[[3,3],[2]]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 5 = 3 + 2
[[2,2,1],[1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 5 = 3 + 2
[[2,2,2],[1,1]]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 5 = 3 + 2
[[1,1,1,1],[]]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 5 = 3 + 2
[[5],[]]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 6 = 4 + 2
[[4,1],[]]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 6 = 4 + 2
[[3,2],[]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 5 = 3 + 2
[[4,2],[1]]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 6 = 4 + 2
[[3,1,1],[]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 6 = 4 + 2
[[3,3],[1]]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 5 = 3 + 2
[[4,3],[2]]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 6 = 4 + 2
[[2,2,1],[]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 5 = 3 + 2
[[3,2,1],[1]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 6 = 4 + 2
[[3,2,2],[1,1]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 6 = 4 + 2
[[2,1,1,1],[]]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 6 = 4 + 2
[[4,4],[3]]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 6 = 4 + 2
[[3,3,1],[2]]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 6 = 4 + 2
[[2,2,2],[1]]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 5 = 3 + 2
[[3,3,2],[2,1]]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 6 = 4 + 2
[[2,2,1,1],[1]]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 6 = 4 + 2
[[3,3,3],[2,2]]
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 6 = 4 + 2
[[2,2,2,1],[1,1]]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 6 = 4 + 2
[[2,2,2,2],[1,1,1]]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 6 = 4 + 2
[[1,1,1,1,1],[]]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 6 = 4 + 2
[[6],[]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 7 = 5 + 2
[[5,1],[]]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 7 = 5 + 2
[[4,2],[]]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 6 = 4 + 2
[[5,2],[1]]
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 7 = 5 + 2
[[4,1,1],[]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 7 = 5 + 2
[[3,3],[]]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 5 = 3 + 2
[[4,3],[1]]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 6 = 4 + 2
[[5,3],[2]]
=> [5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 7 = 5 + 2
[[3,2,1],[]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 6 = 4 + 2
[[4,2,1],[1]]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 7 = 5 + 2
[[4,2,2],[1,1]]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 7 = 5 + 2
[[3,1,1,1],[]]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 7 = 5 + 2
[[4,4],[2]]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 6 = 4 + 2
[[5,4],[3]]
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 7 = 5 + 2
Description
Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000806
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St000806: Integer compositions ⟶ ℤResult quality: 86% ●values known / values provided: 99%●distinct values known / distinct values provided: 86%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St000806: Integer compositions ⟶ ℤResult quality: 86% ●values known / values provided: 99%●distinct values known / distinct values provided: 86%
Values
[[1],[]]
=> [1]
=> [1,0]
=> [1] => ? = 0 + 2
[[2],[]]
=> [2]
=> [1,0,1,0]
=> [1,1] => 3 = 1 + 2
[[1,1],[]]
=> [1,1]
=> [1,1,0,0]
=> [2] => 3 = 1 + 2
[[3],[]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1] => 4 = 2 + 2
[[2,1],[]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,2] => 4 = 2 + 2
[[2,2],[1]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3] => 4 = 2 + 2
[[1,1,1],[]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [3] => 4 = 2 + 2
[[4],[]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 5 = 3 + 2
[[3,1],[]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 5 = 3 + 2
[[2,2],[]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3] => 4 = 2 + 2
[[3,2],[1]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 5 = 3 + 2
[[2,1,1],[]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => 5 = 3 + 2
[[3,3],[2]]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [4] => 5 = 3 + 2
[[2,2,1],[1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [4] => 5 = 3 + 2
[[2,2,2],[1,1]]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4] => 5 = 3 + 2
[[1,1,1,1],[]]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [4] => 5 = 3 + 2
[[5],[]]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 6 = 4 + 2
[[4,1],[]]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 6 = 4 + 2
[[3,2],[]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 5 = 3 + 2
[[4,2],[1]]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 6 = 4 + 2
[[3,1,1],[]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => 6 = 4 + 2
[[3,3],[1]]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [4] => 5 = 3 + 2
[[4,3],[2]]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4] => 6 = 4 + 2
[[2,2,1],[]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [4] => 5 = 3 + 2
[[3,2,1],[1]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4] => 6 = 4 + 2
[[3,2,2],[1,1]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 6 = 4 + 2
[[2,1,1,1],[]]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,4] => 6 = 4 + 2
[[4,4],[3]]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5] => 6 = 4 + 2
[[3,3,1],[2]]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [5] => 6 = 4 + 2
[[2,2,2],[1]]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4] => 5 = 3 + 2
[[3,3,2],[2,1]]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5] => 6 = 4 + 2
[[2,2,1,1],[1]]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [5] => 6 = 4 + 2
[[3,3,3],[2,2]]
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 6 = 4 + 2
[[2,2,2,1],[1,1]]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [5] => 6 = 4 + 2
[[2,2,2,2],[1,1,1]]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 6 = 4 + 2
[[1,1,1,1,1],[]]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5] => 6 = 4 + 2
[[6],[]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1] => 7 = 5 + 2
[[5,1],[]]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => 7 = 5 + 2
[[4,2],[]]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 6 = 4 + 2
[[5,2],[1]]
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => 7 = 5 + 2
[[4,1,1],[]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,3] => 7 = 5 + 2
[[3,3],[]]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [4] => 5 = 3 + 2
[[4,3],[1]]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4] => 6 = 4 + 2
[[5,3],[2]]
=> [5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,4] => 7 = 5 + 2
[[3,2,1],[]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4] => 6 = 4 + 2
[[4,2,1],[1]]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,4] => 7 = 5 + 2
[[4,2,2],[1,1]]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,4] => 7 = 5 + 2
[[3,1,1,1],[]]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,4] => 7 = 5 + 2
[[4,4],[2]]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5] => 6 = 4 + 2
[[5,4],[3]]
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,5] => 7 = 5 + 2
[[3,3,1],[1]]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [5] => 6 = 4 + 2
Description
The semiperimeter of the associated bargraph.
Interpret the composition as the sequence of heights of the bars of a bargraph. This statistic is the semiperimeter of the polygon determined by the axis and the bargraph. Put differently, it is the sum of the number of up steps and the number of horizontal steps when regarding the bargraph as a path with up, horizontal and down steps.
Matching statistic: St001004
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001004: Permutations ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001004: Permutations ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
=> [1]
=> [1,0]
=> [1] => 1 = 0 + 1
[[2],[]]
=> [2]
=> [1,0,1,0]
=> [1,2] => 2 = 1 + 1
[[1,1],[]]
=> [1,1]
=> [1,1,0,0]
=> [2,1] => 2 = 1 + 1
[[3],[]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 3 = 2 + 1
[[2,1],[]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 3 = 2 + 1
[[2,2],[1]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,1,2] => 3 = 2 + 1
[[1,1,1],[]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 3 = 2 + 1
[[4],[]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 4 = 3 + 1
[[3,1],[]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 4 = 3 + 1
[[2,2],[]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,1,2] => 3 = 2 + 1
[[3,2],[1]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 4 = 3 + 1
[[2,1,1],[]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 4 = 3 + 1
[[3,3],[2]]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 4 = 3 + 1
[[2,2,1],[1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 4 = 3 + 1
[[2,2,2],[1,1]]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 4 = 3 + 1
[[1,1,1,1],[]]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 4 = 3 + 1
[[5],[]]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 5 = 4 + 1
[[4,1],[]]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 5 = 4 + 1
[[3,2],[]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 4 = 3 + 1
[[4,2],[1]]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 5 = 4 + 1
[[3,1,1],[]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 5 = 4 + 1
[[3,3],[1]]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 4 = 3 + 1
[[4,3],[2]]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => 5 = 4 + 1
[[2,2,1],[]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 4 = 3 + 1
[[3,2,1],[1]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => 5 = 4 + 1
[[3,2,2],[1,1]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 5 = 4 + 1
[[2,1,1,1],[]]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 5 = 4 + 1
[[4,4],[3]]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => 5 = 4 + 1
[[3,3,1],[2]]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => 5 = 4 + 1
[[2,2,2],[1]]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 4 = 3 + 1
[[3,3,2],[2,1]]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4] => 5 = 4 + 1
[[2,2,1,1],[1]]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => 5 = 4 + 1
[[3,3,3],[2,2]]
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => 5 = 4 + 1
[[2,2,2,1],[1,1]]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => 5 = 4 + 1
[[2,2,2,2],[1,1,1]]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => 5 = 4 + 1
[[1,1,1,1,1],[]]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 5 = 4 + 1
[[6],[]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => 6 = 5 + 1
[[5,1],[]]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 6 = 5 + 1
[[4,2],[]]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 5 = 4 + 1
[[5,2],[1]]
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,4,5] => 6 = 5 + 1
[[4,1,1],[]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => 6 = 5 + 1
[[3,3],[]]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 4 = 3 + 1
[[4,3],[1]]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => 5 = 4 + 1
[[5,3],[2]]
=> [5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,5,6,3,4] => 6 = 5 + 1
[[3,2,1],[]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => 5 = 4 + 1
[[4,2,1],[1]]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,3,6,4] => 6 = 5 + 1
[[4,2,2],[1,1]]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,3,4,5] => 6 = 5 + 1
[[3,1,1,1],[]]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => 6 = 5 + 1
[[4,4],[2]]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => 5 = 4 + 1
[[5,4],[3]]
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,4,5,6,2,3] => 6 = 5 + 1
[[3,2,2,1,1],[1,1]]
=> [3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,5,2,3,6,7,4] => ? = 6 + 1
[[3,2,2,2,1],[1,1,1]]
=> [3,2,2,2,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,5,6,2,3,7,4] => ? = 6 + 1
[[5,5,1],[4]]
=> [5,5,1]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [3,4,5,6,1,7,2] => ? = 6 + 1
[[5,5,2],[4,1]]
=> [5,5,2]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [3,4,5,7,1,2,6] => ? = 6 + 1
[[4,4,1,1],[3]]
=> [4,4,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [3,4,5,1,6,7,2] => ? = 6 + 1
[[5,5,3],[4,2]]
=> [5,5,3]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [3,4,7,1,2,5,6] => ? = 6 + 1
[[4,4,2,1],[3,1]]
=> [4,4,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [3,4,6,1,2,7,5] => ? = 6 + 1
[[4,4,2,2],[3,1,1]]
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [3,4,6,7,1,2,5] => ? = 6 + 1
[[3,3,1,1,1],[2]]
=> [3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [3,4,1,5,6,7,2] => ? = 6 + 1
[[5,5,4],[4,3]]
=> [5,5,4]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [3,6,7,1,2,4,5] => ? = 6 + 1
[[4,4,3,1],[3,2]]
=> [4,4,3,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [3,6,1,2,4,7,5] => ? = 6 + 1
[[4,4,3,2],[3,2,1]]
=> [4,4,3,2]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [3,6,1,7,2,4,5] => ? = 6 + 1
[[3,3,2,1,1],[2,1]]
=> [3,3,2,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [3,5,1,2,6,7,4] => ? = 6 + 1
[[4,4,3,3],[3,2,2]]
=> [4,4,3,3]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [3,7,1,2,4,5,6] => ? = 6 + 1
[[3,3,2,2,1],[2,1,1]]
=> [3,3,2,2,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [3,5,6,1,2,7,4] => ? = 6 + 1
[[3,3,2,2,2],[2,1,1,1]]
=> [3,3,2,2,2]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [3,5,6,7,1,2,4] => ? = 6 + 1
[[2,2,1,1,1,1],[1]]
=> [2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,1,4,5,6,7,2] => ? = 6 + 1
[[4,4,4,2],[3,3,1]]
=> [4,4,4,2]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [5,6,1,7,2,3,4] => ? = 6 + 1
[[4,4,4,3],[3,3,2]]
=> [4,4,4,3]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [5,7,1,2,3,4,6] => ? = 6 + 1
[[3,3,3,2,2],[2,2,1,1]]
=> [3,3,3,2,2]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [5,1,6,7,2,3,4] => ? = 6 + 1
[[2,2,2,1,1,1],[1,1]]
=> [2,2,2,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [4,1,2,5,6,7,3] => ? = 6 + 1
[[2,2,2,2,1,1],[1,1,1]]
=> [2,2,2,2,1,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [4,5,1,2,6,7,3] => ? = 6 + 1
[[2,2,2,2,2,1],[1,1,1,1]]
=> [2,2,2,2,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [4,5,6,1,2,7,3] => ? = 6 + 1
[[2,2,2,2,2,2],[1,1,1,1,1]]
=> [2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [4,5,6,7,1,2,3] => ? = 6 + 1
Description
The number of indices that are either left-to-right maxima or right-to-left minima.
The (bivariate) generating function for this statistic is (essentially) given in [1], the mid points of a $321$ pattern in the permutation are those elements which are neither left-to-right maxima nor a right-to-left minima, see [[St000371]] and [[St000372]].
The following 24 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001267The length of the Lyndon factorization of the binary word. St000288The number of ones in a binary word. St000336The leg major index of a standard tableau. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000144The pyramid weight of the Dyck path. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St000026The position of the first return of a Dyck path. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St000019The cardinality of the support of a permutation. St000924The number of topologically connected components of a perfect matching. St001480The number of simple summands of the module J^2/J^3. St001958The degree of the polynomial interpolating the values of a permutation. St000501The size of the first part in the decomposition of a permutation. St000673The number of non-fixed points of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St001468The smallest fixpoint of a permutation. St000863The length of the first row of the shifted shape of a permutation. St000744The length of the path to the largest entry in a standard Young tableau. St000044The number of vertices of the unicellular map given by a perfect matching.
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