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Your data matches 27 different statistics following compositions of up to 3 maps.
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Matching statistic: St000938
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000938: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000938: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1},{2},{3}}
=> [1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,2,3,4}}
=> [4]
=> [2,2]
=> [2]
=> 0
{{1,2,3},{4}}
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,2,4},{3}}
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,3,4},{2}}
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1},{2,3,4}}
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
{{1,2,3,4,5}}
=> [5]
=> [2,2,1]
=> [2,1]
=> 1
{{1,2,3,4},{5}}
=> [4,1]
=> [3,2]
=> [2]
=> 0
{{1,2,3,5},{4}}
=> [4,1]
=> [3,2]
=> [2]
=> 0
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,2,4,5},{3}}
=> [4,1]
=> [3,2]
=> [2]
=> 0
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
{{1,3,4,5},{2}}
=> [4,1]
=> [3,2]
=> [2]
=> 0
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
{{1},{2,3,4,5}}
=> [4,1]
=> [3,2]
=> [2]
=> 0
{{1},{2,3,4},{5}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1},{2,3,5},{4}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
{{1,4,5},{2},{3}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
{{1},{2,4,5},{3}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
{{1},{2},{3,4,5}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
{{1,2,3,4,5,6}}
=> [6]
=> [2,2,2]
=> [2,2]
=> 2
{{1,2,3,4,5},{6}}
=> [5,1]
=> [2,2,1,1]
=> [2,1,1]
=> 1
{{1,2,3,4,6},{5}}
=> [5,1]
=> [2,2,1,1]
=> [2,1,1]
=> 1
{{1,2,3,4},{5,6}}
=> [4,2]
=> [4,2]
=> [2]
=> 0
{{1,2,3,4},{5},{6}}
=> [4,1,1]
=> [4,1,1]
=> [1,1]
=> 0
{{1,2,3,5,6},{4}}
=> [5,1]
=> [2,2,1,1]
=> [2,1,1]
=> 1
{{1,2,3,5},{4,6}}
=> [4,2]
=> [4,2]
=> [2]
=> 0
{{1,2,3,5},{4},{6}}
=> [4,1,1]
=> [4,1,1]
=> [1,1]
=> 0
{{1,2,3,6},{4,5}}
=> [4,2]
=> [4,2]
=> [2]
=> 0
{{1,2,3},{4,5,6}}
=> [3,3]
=> [3,2,1]
=> [2,1]
=> 1
{{1,2,3},{4,5},{6}}
=> [3,2,1]
=> [3,3]
=> [3]
=> 0
{{1,2,3,6},{4},{5}}
=> [4,1,1]
=> [4,1,1]
=> [1,1]
=> 0
{{1,2,3},{4,6},{5}}
=> [3,2,1]
=> [3,3]
=> [3]
=> 0
{{1,2,3},{4},{5,6}}
=> [3,2,1]
=> [3,3]
=> [3]
=> 0
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
{{1,2,4,5,6},{3}}
=> [5,1]
=> [2,2,1,1]
=> [2,1,1]
=> 1
Description
The number of zeros of the symmetric group character corresponding to the partition.
For example, the character values of the irreducible representation $S^{(2,2)}$ are $2$ on the conjugacy classes $(4)$ and $(2,2)$, $0$ on the conjugacy classes $(3,1)$ and $(1,1,1,1)$, and $-1$ on the conjugacy class $(2,1,1)$. Therefore, the statistic on the partition $(2,2)$ is $2$.
Matching statistic: St001570
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001570: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 8%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001570: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 8%
Values
{{1},{2},{3}}
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
{{1,2,3,4}}
=> [4] => ([],4)
=> ([],1)
=> ? = 0
{{1,2,3},{4}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 0
{{1,2,4},{3}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 0
{{1,3,4},{2}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 0
{{1},{2,3,4}}
=> [1,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 0
{{1},{2},{3},{4}}
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
{{1,2,3,4,5}}
=> [5] => ([],5)
=> ([],1)
=> ? = 1
{{1,2,3,4},{5}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 0
{{1,2,3,5},{4}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 0
{{1,2,3},{4},{5}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
{{1,2,4,5},{3}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 0
{{1,2,4},{3},{5}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
{{1,2,5},{3},{4}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
{{1,3,4,5},{2}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 0
{{1,3,4},{2},{5}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
{{1,3,5},{2},{4}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
{{1},{2,3,4,5}}
=> [1,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 0
{{1},{2,3,4},{5}}
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
{{1},{2,3,5},{4}}
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
{{1},{2,3},{4},{5}}
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
{{1,4,5},{2},{3}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
{{1,4},{2},{3},{5}}
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
{{1},{2,4,5},{3}}
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
{{1},{2,4},{3},{5}}
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
{{1},{2},{3,4,5}}
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
{{1},{2},{3,4},{5}}
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
{{1,5},{2},{3},{4}}
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
{{1},{2,5},{3},{4}}
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
{{1},{2},{3,5},{4}}
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
{{1},{2},{3},{4,5}}
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
{{1,2,3,4,5,6}}
=> [6] => ([],6)
=> ([],1)
=> ? = 2
{{1,2,3,4,5},{6}}
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1
{{1,2,3,4,6},{5}}
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1
{{1,2,3,4},{5,6}}
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 0
{{1,2,3,4},{5},{6}}
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
{{1,2,3,5,6},{4}}
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1
{{1,2,3,5},{4,6}}
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 0
{{1,2,3,5},{4},{6}}
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
{{1,2,3,6},{4,5}}
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 0
{{1,2,3},{4,5,6}}
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1
{{1,2,3},{4,5},{6}}
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
{{1,2,3,6},{4},{5}}
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
{{1,2,3},{4,6},{5}}
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
{{1,2,3},{4},{5,6}}
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
{{1,2,4,5,6},{3}}
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1
{{1,2,4,5},{3,6}}
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 0
{{1,2,4,5},{3},{6}}
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
{{1,2,4,6},{3,5}}
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 0
{{1,2,4},{3,5,6}}
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1
{{1,2,4},{3,5},{6}}
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
{{1,2,4,6},{3},{5}}
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
{{1,2,4},{3,6},{5}}
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
{{1,2,4},{3},{5,6}}
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
{{1,2,5,6},{3,4}}
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 0
{{1,2,5},{3,4,6}}
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1
{{1,2,5},{3,4},{6}}
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
{{1,2,6},{3,4,5}}
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1
{{1,2},{3,4,5,6}}
=> [2,4] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 0
{{1,2},{3,4,5},{6}}
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
{{1,2,6},{3,4},{5}}
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
{{1,2},{3,4,6},{5}}
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
{{1,2,5,6},{3},{4}}
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
{{1,2,5},{3,6},{4}}
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
{{1,2,5},{3},{4,6}}
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
{{1,2,6},{3,5},{4}}
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
{{1,2},{3,5,6},{4}}
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
{{1,2,6},{3},{4,5}}
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
{{1,2},{3},{4,5,6}}
=> [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
{{1,3,4,5,6},{2}}
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1
{{1,3,4,5},{2,6}}
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 0
{{1,3,4,6},{2,5}}
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 0
{{1,3,4},{2,5,6}}
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1
{{1,3,5,6},{2,4}}
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 0
{{1,3,5},{2,4,6}}
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1
{{1,3,6},{2,4,5}}
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1
{{1,3},{2,4,5,6}}
=> [2,4] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 0
{{1,4,5,6},{2,3}}
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 0
{{1,4,5},{2,3,6}}
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1
{{1,4,6},{2,3,5}}
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1
{{1,4},{2,3,5,6}}
=> [2,4] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 0
{{1,5,6},{2,3,4}}
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1
{{1,5},{2,3,4,6}}
=> [2,4] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 0
{{1,6},{2,3,4,5}}
=> [2,4] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 0
{{1},{2,3,4,5,6}}
=> [1,5] => ([(4,5)],6)
=> ([(0,1)],2)
=> ? = 1
{{1,2,3,4,5,6,7}}
=> [7] => ([],7)
=> ?
=> ? = 1
{{1,2,3,4,5,6},{7}}
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
{{1,2,3,4,5,7},{6}}
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
{{1,2,3,4,5},{6,7}}
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 1
{{1,2,3,4,5},{6},{7}}
=> [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 2
{{1,2,3,4,6,7},{5}}
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
{{1,2,3,4,6},{5,7}}
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 1
Description
The minimal number of edges to add to make a graph Hamiltonian.
A graph is Hamiltonian if it contains a cycle as a subgraph, which contains all vertices.
Matching statistic: St000264
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 8%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 8%
Values
{{1},{2},{3}}
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,2,3,4}}
=> [4] => ([],4)
=> ([],1)
=> ? = 0 + 3
{{1,2,3},{4}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 0 + 3
{{1,2,4},{3}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 0 + 3
{{1,3,4},{2}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 0 + 3
{{1},{2,3,4}}
=> [1,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 0 + 3
{{1},{2},{3},{4}}
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
{{1,2,3,4,5}}
=> [5] => ([],5)
=> ([],1)
=> ? = 1 + 3
{{1,2,3,4},{5}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 0 + 3
{{1,2,3,5},{4}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 0 + 3
{{1,2,3},{4},{5}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,2,4,5},{3}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 0 + 3
{{1,2,4},{3},{5}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,2,5},{3},{4}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
{{1,3,4,5},{2}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 0 + 3
{{1,3,4},{2},{5}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,3,5},{2},{4}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
{{1},{2,3,4,5}}
=> [1,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 0 + 3
{{1},{2,3,4},{5}}
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1},{2,3,5},{4}}
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1},{2,3},{4},{5}}
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
{{1,4,5},{2},{3}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,4},{2},{3},{5}}
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
{{1},{2,4,5},{3}}
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1},{2,4},{3},{5}}
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
{{1},{2},{3,4,5}}
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1},{2},{3,4},{5}}
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
{{1,5},{2},{3},{4}}
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
{{1},{2,5},{3},{4}}
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
{{1},{2},{3,5},{4}}
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
{{1},{2},{3},{4,5}}
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
{{1,2,3,4,5,6}}
=> [6] => ([],6)
=> ([],1)
=> ? = 2 + 3
{{1,2,3,4,5},{6}}
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1 + 3
{{1,2,3,4,6},{5}}
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1 + 3
{{1,2,3,4},{5,6}}
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 0 + 3
{{1,2,3,4},{5},{6}}
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,2,3,5,6},{4}}
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1 + 3
{{1,2,3,5},{4,6}}
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 0 + 3
{{1,2,3,5},{4},{6}}
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,2,3,6},{4,5}}
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 0 + 3
{{1,2,3},{4,5,6}}
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1 + 3
{{1,2,3},{4,5},{6}}
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,2,3,6},{4},{5}}
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,2,3},{4,6},{5}}
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,2,3},{4},{5,6}}
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
{{1,2,4,5,6},{3}}
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1 + 3
{{1,2,4,5},{3,6}}
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 0 + 3
{{1,2,4,5},{3},{6}}
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,2,4,6},{3,5}}
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 0 + 3
{{1,2,4},{3,5,6}}
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1 + 3
{{1,2,4},{3,5},{6}}
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,2,4,6},{3},{5}}
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,2,4},{3,6},{5}}
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,2,4},{3},{5,6}}
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
{{1,2,5,6},{3,4}}
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 0 + 3
{{1,2,5},{3,4,6}}
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1 + 3
{{1,2,5},{3,4},{6}}
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,2,6},{3,4,5}}
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1 + 3
{{1,2},{3,4,5,6}}
=> [2,4] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 0 + 3
{{1,2},{3,4,5},{6}}
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,2,6},{3,4},{5}}
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,2},{3,4,6},{5}}
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,2,5,6},{3},{4}}
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,2,5},{3,6},{4}}
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,2,5},{3},{4,6}}
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
{{1,2,6},{3,5},{4}}
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,2},{3,5,6},{4}}
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,2,6},{3},{4,5}}
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,2},{3},{4,5,6}}
=> [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
{{1,3,4,5,6},{2}}
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1 + 3
{{1,3,4,5},{2,6}}
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 0 + 3
{{1,3,4,6},{2,5}}
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 0 + 3
{{1,3,4},{2,5,6}}
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1 + 3
{{1,3,5,6},{2,4}}
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 0 + 3
{{1,3,5},{2,4,6}}
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1 + 3
{{1,3,6},{2,4,5}}
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1 + 3
{{1,3},{2,4,5,6}}
=> [2,4] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 0 + 3
{{1,4,5,6},{2,3}}
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 0 + 3
{{1,4,5},{2,3,6}}
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1 + 3
{{1,4,6},{2,3,5}}
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1 + 3
{{1,4},{2,3,5,6}}
=> [2,4] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 0 + 3
{{1,5,6},{2,3,4}}
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1 + 3
{{1,5},{2,3,4,6}}
=> [2,4] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 0 + 3
{{1,6},{2,3,4,5}}
=> [2,4] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 0 + 3
{{1},{2,3,4,5,6}}
=> [1,5] => ([(4,5)],6)
=> ([(0,1)],2)
=> ? = 1 + 3
{{1,2,3,4,5,6,7}}
=> [7] => ([],7)
=> ?
=> ? = 1 + 3
{{1,2,3,4,5,6},{7}}
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2 + 3
{{1,2,3,4,5,7},{6}}
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2 + 3
{{1,2,3,4,5},{6,7}}
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 1 + 3
{{1,2,3,4,5},{6},{7}}
=> [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 2 + 3
{{1,2,3,4,6,7},{5}}
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2 + 3
{{1,2,3,4,6},{5,7}}
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 1 + 3
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
Matching statistic: St001001
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001001: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 8%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001001: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 8%
Values
{{1},{2},{3}}
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0
{{1,2,3,4}}
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 0
{{1,2,3},{4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
{{1,2,4},{3}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
{{1,3,4},{2}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
{{1},{2,3,4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0
{{1,2,3,4,5}}
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 1
{{1,2,3,4},{5}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
{{1,2,3,5},{4}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
{{1,2,4,5},{3}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0
{{1,3,4,5},{2}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0
{{1},{2,3,4,5}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
{{1},{2,3,4},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
{{1},{2,3,5},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0
{{1,4,5},{2},{3}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0
{{1},{2,4,5},{3}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0
{{1},{2},{3,4,5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
{{1,2,3,4,5,6}}
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 2
{{1,2,3,4,5},{6}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 1
{{1,2,3,4,6},{5}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 1
{{1,2,3,4},{5,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 0
{{1,2,3,4},{5},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0
{{1,2,3,5,6},{4}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 1
{{1,2,3,5},{4,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 0
{{1,2,3,5},{4},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0
{{1,2,3,6},{4,5}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 0
{{1,2,3},{4,5,6}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 1
{{1,2,3},{4,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
{{1,2,3,6},{4},{5}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0
{{1,2,3},{4,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
{{1,2,3},{4},{5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 0
{{1,2,4,5,6},{3}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 1
{{1,2,4,5},{3,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 0
{{1,2,4,5},{3},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0
{{1,2,4,6},{3,5}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 0
{{1,2,4},{3,5,6}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 1
{{1,2,4},{3,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
{{1,2,4,6},{3},{5}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0
{{1,2,4},{3,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
{{1,2,4},{3},{5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 0
{{1,2,5,6},{3,4}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 0
{{1,2,5},{3,4,6}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 1
{{1,2,5},{3,4},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
{{1,2,6},{3,4,5}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 1
{{1,2},{3,4,5,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 0
{{1,2},{3,4,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
{{1,2,6},{3,4},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
{{1,2},{3,4,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
{{1,2,5,6},{3},{4}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0
{{1,2,5},{3,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
{{1,2,5},{3},{4,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 0
{{1,2,6},{3,5},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
{{1,2},{3,5,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
{{1,2,6},{3},{4,5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
{{1,2},{3},{4,5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 0
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 0
{{1,3,4,5,6},{2}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 1
{{1,3,4,5},{2,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 0
{{1,3,4,5},{2},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0
{{1,3,4,6},{2,5}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 0
{{1,3,4},{2,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
{{1,3,4},{2,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
{{1,3,4},{2},{5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
{{1,3,5},{2,4},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
{{1,3},{2,4,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
{{1,3,6},{2,4},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
{{1,3},{2,4,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
{{1,3,5},{2,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
{{1,3,5},{2},{4,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
{{1,3,6},{2,5},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
{{1,3},{2,5,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
{{1,3,6},{2},{4,5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
{{1,3},{2},{4,5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
{{1,4,5},{2,3},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
{{1,4},{2,3,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
{{1,4,6},{2,3},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
{{1,4},{2,3,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
{{1,5},{2,3,4},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
{{1,6},{2,3,4},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
Description
The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001371
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001371: Binary words ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 8%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001371: Binary words ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 8%
Values
{{1},{2},{3}}
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
{{1,2,3,4}}
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 0
{{1,2,3},{4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 0
{{1,2,4},{3}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 0
{{1,3,4},{2}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 0
{{1},{2,3,4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 0
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 0
{{1,2,3,4,5}}
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 111110000010 => ? = 1
{{1,2,3,4},{5}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
{{1,2,3,5},{4}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
{{1,2,4,5},{3}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
{{1,3,4,5},{2}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
{{1},{2,3,4,5}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
{{1},{2,3,4},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
{{1},{2,3,5},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
{{1,4,5},{2},{3}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
{{1},{2,4,5},{3}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
{{1},{2},{3,4,5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 101111100000 => ? = 0
{{1,2,3,4,5,6}}
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 11111100000010 => ? = 2
{{1,2,3,4,5},{6}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 111101000010 => ? = 1
{{1,2,3,4,6},{5}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 111101000010 => ? = 1
{{1,2,3,4},{5,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? = 0
{{1,2,3,4},{5},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 0
{{1,2,3,5,6},{4}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 111101000010 => ? = 1
{{1,2,3,5},{4,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? = 0
{{1,2,3,5},{4},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 0
{{1,2,3,6},{4,5}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? = 0
{{1,2,3},{4,5,6}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => ? = 1
{{1,2,3},{4,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,2,3,6},{4},{5}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 0
{{1,2,3},{4,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,2,3},{4},{5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 0
{{1,2,4,5,6},{3}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 111101000010 => ? = 1
{{1,2,4,5},{3,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? = 0
{{1,2,4,5},{3},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 0
{{1,2,4,6},{3,5}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? = 0
{{1,2,4},{3,5,6}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => ? = 1
{{1,2,4},{3,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,2,4,6},{3},{5}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 0
{{1,2,4},{3,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,2,4},{3},{5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 0
{{1,2,5,6},{3,4}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? = 0
{{1,2,5},{3,4,6}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => ? = 1
{{1,2,5},{3,4},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,2,6},{3,4,5}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => ? = 1
{{1,2},{3,4,5,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? = 0
{{1,2},{3,4,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,2,6},{3,4},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,2},{3,4,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,2,5,6},{3},{4}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 0
{{1,2,5},{3,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,2,5},{3},{4,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 0
{{1,2,6},{3,5},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,2},{3,5,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,2,6},{3},{4,5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,2},{3},{4,5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 0
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 101111010000 => ? = 0
{{1,3,4,5,6},{2}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 111101000010 => ? = 1
{{1,3,4,5},{2,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? = 0
{{1,3,4,5},{2},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 0
{{1,3,4,6},{2,5}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? = 0
{{1,3,4},{2,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,3,4},{2,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,3,4},{2},{5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,3,5},{2,4},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,3},{2,4,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,3,6},{2,4},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,3},{2,4,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,3,5},{2,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,3,5},{2},{4,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,3,6},{2,5},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,3},{2,5,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,3,6},{2},{4,5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,3},{2},{4,5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,4,5},{2,3},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,4},{2,3,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,4,6},{2,3},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,4},{2,3,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,5},{2,3,4},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,6},{2,3,4},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
Description
The length of the longest Yamanouchi prefix of a binary word.
This is the largest index $i$ such that in each of the prefixes $w_1$, $w_1w_2$, $w_1w_2\dots w_i$ the number of zeros is greater than or equal to the number of ones.
Matching statistic: St001730
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001730: Binary words ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 8%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001730: Binary words ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 8%
Values
{{1},{2},{3}}
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
{{1,2,3,4}}
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 0
{{1,2,3},{4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 0
{{1,2,4},{3}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 0
{{1,3,4},{2}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 0
{{1},{2,3,4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 0
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 0
{{1,2,3,4,5}}
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 111110000010 => ? = 1
{{1,2,3,4},{5}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
{{1,2,3,5},{4}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
{{1,2,4,5},{3}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
{{1,3,4,5},{2}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
{{1},{2,3,4,5}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
{{1},{2,3,4},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
{{1},{2,3,5},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
{{1,4,5},{2},{3}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
{{1},{2,4,5},{3}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
{{1},{2},{3,4,5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 101111100000 => ? = 0
{{1,2,3,4,5,6}}
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 11111100000010 => ? = 2
{{1,2,3,4,5},{6}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 111101000010 => ? = 1
{{1,2,3,4,6},{5}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 111101000010 => ? = 1
{{1,2,3,4},{5,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? = 0
{{1,2,3,4},{5},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 0
{{1,2,3,5,6},{4}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 111101000010 => ? = 1
{{1,2,3,5},{4,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? = 0
{{1,2,3,5},{4},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 0
{{1,2,3,6},{4,5}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? = 0
{{1,2,3},{4,5,6}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => ? = 1
{{1,2,3},{4,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,2,3,6},{4},{5}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 0
{{1,2,3},{4,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,2,3},{4},{5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 0
{{1,2,4,5,6},{3}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 111101000010 => ? = 1
{{1,2,4,5},{3,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? = 0
{{1,2,4,5},{3},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 0
{{1,2,4,6},{3,5}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? = 0
{{1,2,4},{3,5,6}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => ? = 1
{{1,2,4},{3,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,2,4,6},{3},{5}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 0
{{1,2,4},{3,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,2,4},{3},{5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 0
{{1,2,5,6},{3,4}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? = 0
{{1,2,5},{3,4,6}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => ? = 1
{{1,2,5},{3,4},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,2,6},{3,4,5}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => ? = 1
{{1,2},{3,4,5,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? = 0
{{1,2},{3,4,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,2,6},{3,4},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,2},{3,4,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,2,5,6},{3},{4}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 0
{{1,2,5},{3,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,2,5},{3},{4,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 0
{{1,2,6},{3,5},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,2},{3,5,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,2,6},{3},{4,5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,2},{3},{4,5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 0
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 101111010000 => ? = 0
{{1,3,4,5,6},{2}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 111101000010 => ? = 1
{{1,3,4,5},{2,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? = 0
{{1,3,4,5},{2},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 0
{{1,3,4,6},{2,5}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? = 0
{{1,3,4},{2,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,3,4},{2,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,3,4},{2},{5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,3,5},{2,4},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,3},{2,4,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,3,6},{2,4},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,3},{2,4,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,3,5},{2,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,3,5},{2},{4,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,3,6},{2,5},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,3},{2,5,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,3,6},{2},{4,5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,3},{2},{4,5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,4,5},{2,3},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,4},{2,3,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,4,6},{2,3},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,4},{2,3,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,5},{2,3,4},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,6},{2,3,4},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
Description
The number of times the path corresponding to a binary word crosses the base line.
Interpret each $0$ as a step $(1,-1)$ and $1$ as a step $(1,1)$. Then this statistic counts the number of times the path crosses the $x$-axis.
Matching statistic: St001803
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001803: Standard tableaux ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 8%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001803: Standard tableaux ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 8%
Values
{{1},{2},{3}}
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 0
{{1,2,3,4}}
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 0
{{1,2,3},{4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 0
{{1,2,4},{3}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 0
{{1,3,4},{2}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 0
{{1},{2,3,4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 0
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 0
{{1,2,3,4,5}}
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,11],[6,7,8,9,10,12]]
=> ? = 1
{{1,2,3,4},{5}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
{{1,2,3,5},{4}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
{{1,2,4,5},{3}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0
{{1,3,4,5},{2}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0
{{1},{2,3,4,5}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
{{1},{2,3,4},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
{{1},{2,3,5},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0
{{1,4,5},{2},{3}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0
{{1},{2,4,5},{3}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0
{{1},{2},{3,4,5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,4,5,6,7],[2,8,9,10,11,12]]
=> ? = 0
{{1,2,3,4,5,6}}
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,6,13],[7,8,9,10,11,12,14]]
=> ? = 2
{{1,2,3,4,5},{6}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[1,2,3,4,6,11],[5,7,8,9,10,12]]
=> ? = 1
{{1,2,3,4,6},{5}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[1,2,3,4,6,11],[5,7,8,9,10,12]]
=> ? = 1
{{1,2,3,4},{5,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? = 0
{{1,2,3,4},{5},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 0
{{1,2,3,5,6},{4}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[1,2,3,4,6,11],[5,7,8,9,10,12]]
=> ? = 1
{{1,2,3,5},{4,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? = 0
{{1,2,3,5},{4},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 0
{{1,2,3,6},{4,5}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? = 0
{{1,2,3},{4,5,6}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> ? = 1
{{1,2,3},{4,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0
{{1,2,3,6},{4},{5}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 0
{{1,2,3},{4,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0
{{1,2,3},{4},{5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 0
{{1,2,4,5,6},{3}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[1,2,3,4,6,11],[5,7,8,9,10,12]]
=> ? = 1
{{1,2,4,5},{3,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? = 0
{{1,2,4,5},{3},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 0
{{1,2,4,6},{3,5}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? = 0
{{1,2,4},{3,5,6}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> ? = 1
{{1,2,4},{3,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0
{{1,2,4,6},{3},{5}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 0
{{1,2,4},{3,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0
{{1,2,4},{3},{5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 0
{{1,2,5,6},{3,4}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? = 0
{{1,2,5},{3,4,6}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> ? = 1
{{1,2,5},{3,4},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0
{{1,2,6},{3,4,5}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> ? = 1
{{1,2},{3,4,5,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? = 0
{{1,2},{3,4,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0
{{1,2,6},{3,4},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0
{{1,2},{3,4,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0
{{1,2,5,6},{3},{4}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 0
{{1,2,5},{3,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0
{{1,2,5},{3},{4,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 0
{{1,2,6},{3,5},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0
{{1,2},{3,5,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0
{{1,2,6},{3},{4,5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0
{{1,2},{3},{4,5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 0
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[1,3,4,5,6,8],[2,7,9,10,11,12]]
=> ? = 0
{{1,3,4,5,6},{2}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[1,2,3,4,6,11],[5,7,8,9,10,12]]
=> ? = 1
{{1,3,4,5},{2,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? = 0
{{1,3,4,5},{2},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 0
{{1,3,4,6},{2,5}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? = 0
{{1,3,4},{2,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0
{{1,3,4},{2,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0
{{1,3,4},{2},{5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0
{{1,3,5},{2,4},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0
{{1,3},{2,4,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0
{{1,3,6},{2,4},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0
{{1,3},{2,4,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0
{{1,3,5},{2,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0
{{1,3,5},{2},{4,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0
{{1,3,6},{2,5},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0
{{1,3},{2,5,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0
{{1,3,6},{2},{4,5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0
{{1,3},{2},{4,5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0
{{1,4,5},{2,3},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0
{{1,4},{2,3,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0
{{1,4,6},{2,3},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0
{{1,4},{2,3,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0
{{1,5},{2,3,4},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0
{{1,6},{2,3,4},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0
Description
The maximal overlap of the cylindrical tableau associated with a tableau.
A cylindrical tableau associated with a standard Young tableau $T$ is the skew row-strict tableau obtained by gluing two copies of $T$ such that the inner shape is a rectangle.
The overlap, recorded in this statistic, equals $\max_C\big(2\ell(T) - \ell(C)\big)$, where $\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux.
In particular, the statistic equals $0$, if and only if the last entry of the first row is larger than or equal to the first entry of the last row. Moreover, the statistic attains its maximal value, the number of rows of the tableau minus 1, if and only if the tableau consists of a single column.
Matching statistic: St001195
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 8%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 8%
Values
{{1},{2},{3}}
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
{{1,2,3,4}}
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 0 + 1
{{1,2,3},{4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1 = 0 + 1
{{1,2,4},{3}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1 = 0 + 1
{{1,3,4},{2}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1 = 0 + 1
{{1},{2,3,4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1 = 0 + 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
{{1,2,3,4,5}}
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 1 + 1
{{1,2,3,4},{5}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
{{1,2,3,5},{4}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
{{1,2,4,5},{3}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 1
{{1,3,4,5},{2}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 1
{{1},{2,3,4,5}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
{{1},{2,3,4},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
{{1},{2,3,5},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 1
{{1,4,5},{2},{3}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 1
{{1},{2,4,5},{3}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 1
{{1},{2},{3,4,5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 1
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 1
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 1
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 1
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 1
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
{{1,2,3,4,5,6}}
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 2 + 1
{{1,2,3,4,5},{6}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 1 + 1
{{1,2,3,4,6},{5}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 1 + 1
{{1,2,3,4},{5,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 0 + 1
{{1,2,3,4},{5},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0 + 1
{{1,2,3,5,6},{4}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 1 + 1
{{1,2,3,5},{4,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 0 + 1
{{1,2,3,5},{4},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0 + 1
{{1,2,3,6},{4,5}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 0 + 1
{{1,2,3},{4,5,6}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 1 + 1
{{1,2,3},{4,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,2,3,6},{4},{5}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0 + 1
{{1,2,3},{4,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,2,3},{4},{5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 0 + 1
{{1,2,4,5,6},{3}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 1 + 1
{{1,2,4,5},{3,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 0 + 1
{{1,2,4,5},{3},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0 + 1
{{1,2,4,6},{3,5}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 0 + 1
{{1,2,4},{3,5,6}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 1 + 1
{{1,2,4},{3,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,2,4,6},{3},{5}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0 + 1
{{1,2,4},{3,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,2,4},{3},{5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 0 + 1
{{1,2,5,6},{3,4}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 0 + 1
{{1,2,5},{3,4,6}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 1 + 1
{{1,2,5},{3,4},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,2,6},{3,4,5}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 1 + 1
{{1,2},{3,4,5,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 0 + 1
{{1,2},{3,4,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,2,6},{3,4},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,2},{3,4,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,2,5,6},{3},{4}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0 + 1
{{1,2,5},{3,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,2,5},{3},{4,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 0 + 1
{{1,2,6},{3,5},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,2},{3,5,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,2,6},{3},{4,5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,2},{3},{4,5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 0 + 1
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 0 + 1
{{1,3,4,5,6},{2}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 1 + 1
{{1,3,4,5},{2,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 0 + 1
{{1,3,4,5},{2},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0 + 1
{{1,3,4,6},{2,5}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 0 + 1
{{1,3,4},{2,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,3,4},{2,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,3,4},{2},{5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,3,5},{2,4},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,3},{2,4,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,3,6},{2,4},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,3},{2,4,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,3,5},{2,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,3,5},{2},{4,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,3,6},{2,5},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,3},{2,5,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,3,6},{2},{4,5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,3},{2},{4,5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,4,5},{2,3},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,4},{2,3,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,4,6},{2,3},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,4},{2,3,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,5},{2,3,4},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,6},{2,3,4},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
Description
The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$.
Matching statistic: St001208
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001208: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 8%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001208: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 8%
Values
{{1},{2},{3}}
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1 = 0 + 1
{{1,2,3,4}}
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 0 + 1
{{1,2,3},{4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1 = 0 + 1
{{1,2,4},{3}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1 = 0 + 1
{{1,3,4},{2}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1 = 0 + 1
{{1},{2,3,4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1 = 0 + 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ? = 0 + 1
{{1,2,3,4,5}}
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? = 1 + 1
{{1,2,3,4},{5}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
{{1,2,3,5},{4}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
{{1,2,4,5},{3}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0 + 1
{{1,3,4,5},{2}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0 + 1
{{1},{2,3,4,5}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
{{1},{2,3,4},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
{{1},{2,3,5},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0 + 1
{{1,4,5},{2},{3}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0 + 1
{{1},{2,4,5},{3}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0 + 1
{{1},{2},{3,4,5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0 + 1
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0 + 1
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0 + 1
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0 + 1
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0 + 1
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 0 + 1
{{1,2,3,4,5,6}}
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 2 + 1
{{1,2,3,4,5},{6}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? = 1 + 1
{{1,2,3,4,6},{5}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? = 1 + 1
{{1,2,3,4},{5,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ? = 0 + 1
{{1,2,3,4},{5},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => ? = 0 + 1
{{1,2,3,5,6},{4}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? = 1 + 1
{{1,2,3,5},{4,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ? = 0 + 1
{{1,2,3,5},{4},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => ? = 0 + 1
{{1,2,3,6},{4,5}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ? = 0 + 1
{{1,2,3},{4,5,6}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => ? = 1 + 1
{{1,2,3},{4,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1 = 0 + 1
{{1,2,3,6},{4},{5}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => ? = 0 + 1
{{1,2,3},{4,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1 = 0 + 1
{{1,2,3},{4},{5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1 = 0 + 1
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? = 0 + 1
{{1,2,4,5,6},{3}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? = 1 + 1
{{1,2,4,5},{3,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ? = 0 + 1
{{1,2,4,5},{3},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => ? = 0 + 1
{{1,2,4,6},{3,5}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ? = 0 + 1
{{1,2,4},{3,5,6}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => ? = 1 + 1
{{1,2,4},{3,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1 = 0 + 1
{{1,2,4,6},{3},{5}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => ? = 0 + 1
{{1,2,4},{3,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1 = 0 + 1
{{1,2,4},{3},{5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1 = 0 + 1
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? = 0 + 1
{{1,2,5,6},{3,4}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ? = 0 + 1
{{1,2,5},{3,4,6}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => ? = 1 + 1
{{1,2,5},{3,4},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1 = 0 + 1
{{1,2,6},{3,4,5}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => ? = 1 + 1
{{1,2},{3,4,5,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ? = 0 + 1
{{1,2},{3,4,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1 = 0 + 1
{{1,2,6},{3,4},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1 = 0 + 1
{{1,2},{3,4,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1 = 0 + 1
{{1,2,5,6},{3},{4}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => ? = 0 + 1
{{1,2,5},{3,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1 = 0 + 1
{{1,2,5},{3},{4,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1 = 0 + 1
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? = 0 + 1
{{1,2,6},{3,5},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1 = 0 + 1
{{1,2},{3,5,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1 = 0 + 1
{{1,2,6},{3},{4,5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1 = 0 + 1
{{1,2},{3},{4,5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1 = 0 + 1
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? = 0 + 1
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? = 0 + 1
{{1,3,4,5,6},{2}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? = 1 + 1
{{1,3,4,5},{2,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ? = 0 + 1
{{1,3,4,5},{2},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => ? = 0 + 1
{{1,3,4,6},{2,5}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ? = 0 + 1
{{1,3,4},{2,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1 = 0 + 1
{{1,3,4},{2,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1 = 0 + 1
{{1,3,4},{2},{5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1 = 0 + 1
{{1,3,5},{2,4},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1 = 0 + 1
{{1,3},{2,4,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1 = 0 + 1
{{1,3,6},{2,4},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1 = 0 + 1
{{1,3},{2,4,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1 = 0 + 1
{{1,3,5},{2,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1 = 0 + 1
{{1,3,5},{2},{4,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1 = 0 + 1
{{1,3,6},{2,5},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1 = 0 + 1
{{1,3},{2,5,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1 = 0 + 1
{{1,3,6},{2},{4,5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1 = 0 + 1
{{1,3},{2},{4,5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1 = 0 + 1
{{1,4,5},{2,3},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1 = 0 + 1
{{1,4},{2,3,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1 = 0 + 1
{{1,4,6},{2,3},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1 = 0 + 1
{{1,4},{2,3,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1 = 0 + 1
{{1,5},{2,3,4},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1 = 0 + 1
{{1,6},{2,3,4},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1 = 0 + 1
Description
The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$.
Matching statistic: St001804
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001804: Standard tableaux ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 8%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001804: Standard tableaux ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 8%
Values
{{1},{2},{3}}
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 2 = 0 + 2
{{1,2,3,4}}
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 0 + 2
{{1,2,3},{4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 2 = 0 + 2
{{1,2,4},{3}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 2 = 0 + 2
{{1,3,4},{2}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 2 = 0 + 2
{{1},{2,3,4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 2 = 0 + 2
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 0 + 2
{{1,2,3,4,5}}
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,11],[6,7,8,9,10,12]]
=> ? = 1 + 2
{{1,2,3,4},{5}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
{{1,2,3,5},{4}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 0 + 2
{{1,2,4,5},{3}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 0 + 2
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 0 + 2
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0 + 2
{{1,3,4,5},{2}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 0 + 2
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 0 + 2
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0 + 2
{{1},{2,3,4,5}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
{{1},{2,3,4},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 0 + 2
{{1},{2,3,5},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 0 + 2
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0 + 2
{{1,4,5},{2},{3}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 0 + 2
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0 + 2
{{1},{2,4,5},{3}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 0 + 2
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0 + 2
{{1},{2},{3,4,5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 0 + 2
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0 + 2
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0 + 2
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0 + 2
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0 + 2
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0 + 2
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,4,5,6,7],[2,8,9,10,11,12]]
=> ? = 0 + 2
{{1,2,3,4,5,6}}
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,6,13],[7,8,9,10,11,12,14]]
=> ? = 2 + 2
{{1,2,3,4,5},{6}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[1,2,3,4,6,11],[5,7,8,9,10,12]]
=> ? = 1 + 2
{{1,2,3,4,6},{5}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[1,2,3,4,6,11],[5,7,8,9,10,12]]
=> ? = 1 + 2
{{1,2,3,4},{5,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? = 0 + 2
{{1,2,3,4},{5},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 0 + 2
{{1,2,3,5,6},{4}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[1,2,3,4,6,11],[5,7,8,9,10,12]]
=> ? = 1 + 2
{{1,2,3,5},{4,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? = 0 + 2
{{1,2,3,5},{4},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 0 + 2
{{1,2,3,6},{4,5}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? = 0 + 2
{{1,2,3},{4,5,6}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> ? = 1 + 2
{{1,2,3},{4,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 0 + 2
{{1,2,3,6},{4},{5}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 0 + 2
{{1,2,3},{4,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 0 + 2
{{1,2,3},{4},{5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 0 + 2
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 0 + 2
{{1,2,4,5,6},{3}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[1,2,3,4,6,11],[5,7,8,9,10,12]]
=> ? = 1 + 2
{{1,2,4,5},{3,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? = 0 + 2
{{1,2,4,5},{3},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 0 + 2
{{1,2,4,6},{3,5}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? = 0 + 2
{{1,2,4},{3,5,6}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> ? = 1 + 2
{{1,2,4},{3,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 0 + 2
{{1,2,4,6},{3},{5}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 0 + 2
{{1,2,4},{3,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 0 + 2
{{1,2,4},{3},{5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 0 + 2
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 0 + 2
{{1,2,5,6},{3,4}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? = 0 + 2
{{1,2,5},{3,4,6}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> ? = 1 + 2
{{1,2,5},{3,4},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 0 + 2
{{1,2,6},{3,4,5}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> ? = 1 + 2
{{1,2},{3,4,5,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? = 0 + 2
{{1,2},{3,4,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 0 + 2
{{1,2,6},{3,4},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 0 + 2
{{1,2},{3,4,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 0 + 2
{{1,2,5,6},{3},{4}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 0 + 2
{{1,2,5},{3,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 0 + 2
{{1,2,5},{3},{4,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 0 + 2
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 0 + 2
{{1,2,6},{3,5},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 0 + 2
{{1,2},{3,5,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 0 + 2
{{1,2,6},{3},{4,5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 0 + 2
{{1,2},{3},{4,5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 0 + 2
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 0 + 2
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[1,3,4,5,6,8],[2,7,9,10,11,12]]
=> ? = 0 + 2
{{1,3,4,5,6},{2}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[1,2,3,4,6,11],[5,7,8,9,10,12]]
=> ? = 1 + 2
{{1,3,4,5},{2,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? = 0 + 2
{{1,3,4,5},{2},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 0 + 2
{{1,3,4,6},{2,5}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? = 0 + 2
{{1,3,4},{2,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 0 + 2
{{1,3,4},{2,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 0 + 2
{{1,3,4},{2},{5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 0 + 2
{{1,3,5},{2,4},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 0 + 2
{{1,3},{2,4,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 0 + 2
{{1,3,6},{2,4},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 0 + 2
{{1,3},{2,4,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 0 + 2
{{1,3,5},{2,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 0 + 2
{{1,3,5},{2},{4,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 0 + 2
{{1,3,6},{2,5},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 0 + 2
{{1,3},{2,5,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 0 + 2
{{1,3,6},{2},{4,5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 0 + 2
{{1,3},{2},{4,5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 0 + 2
{{1,4,5},{2,3},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 0 + 2
{{1,4},{2,3,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 0 + 2
{{1,4,6},{2,3},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 0 + 2
{{1,4},{2,3,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 0 + 2
{{1,5},{2,3,4},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 0 + 2
{{1,6},{2,3,4},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 0 + 2
Description
The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau.
A cylindrical tableau associated with a standard Young tableau $T$ is the skew row-strict tableau obtained by gluing two copies of $T$ such that the inner shape is a rectangle.
This statistic equals $\max_C\big(\ell(C) - \ell(T)\big)$, where $\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux.
The following 17 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000744The length of the path to the largest entry in a standard Young tableau. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St000044The number of vertices of the unicellular map given by a perfect matching. St000017The number of inversions of a standard tableau. St001721The degree of a binary word. St000016The number of attacking pairs of a standard tableau. St001549The number of restricted non-inversions between exceedances. St000486The number of cycles of length at least 3 of a permutation. St001845The number of join irreducibles minus the rank of a lattice. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001875The number of simple modules with projective dimension at most 1. St000068The number of minimal elements in a poset.
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