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Matching statistic: St000771
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1 => [1] => ([],1)
=> 1
[2]
=> 0 => [1] => ([],1)
=> 1
[3]
=> 1 => [1] => ([],1)
=> 1
[2,1]
=> 01 => [1,1] => ([(0,1)],2)
=> 1
[4]
=> 0 => [1] => ([],1)
=> 1
[5]
=> 1 => [1] => ([],1)
=> 1
[4,1]
=> 01 => [1,1] => ([(0,1)],2)
=> 1
[3,2]
=> 10 => [1,1] => ([(0,1)],2)
=> 1
[2,2,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 1
[6]
=> 0 => [1] => ([],1)
=> 1
[3,2,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[7]
=> 1 => [1] => ([],1)
=> 1
[6,1]
=> 01 => [1,1] => ([(0,1)],2)
=> 1
[5,2]
=> 10 => [1,1] => ([(0,1)],2)
=> 1
[4,3]
=> 01 => [1,1] => ([(0,1)],2)
=> 1
[4,2,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 1
[2,2,2,1]
=> 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[8]
=> 0 => [1] => ([],1)
=> 1
[5,2,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[3,3,2]
=> 110 => [2,1] => ([(0,2),(1,2)],3)
=> 1
[3,2,2,1]
=> 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[9]
=> 1 => [1] => ([],1)
=> 1
[8,1]
=> 01 => [1,1] => ([(0,1)],2)
=> 1
[7,2]
=> 10 => [1,1] => ([(0,1)],2)
=> 1
[6,3]
=> 01 => [1,1] => ([(0,1)],2)
=> 1
[6,2,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 1
[5,4]
=> 10 => [1,1] => ([(0,1)],2)
=> 1
[4,4,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 1
[4,3,2]
=> 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[4,2,2,1]
=> 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[3,3,2,1]
=> 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,2,2,2,1]
=> 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[10]
=> 0 => [1] => ([],1)
=> 1
[7,2,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[5,4,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[5,3,2]
=> 110 => [2,1] => ([(0,2),(1,2)],3)
=> 1
[5,2,2,1]
=> 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,3,2,1]
=> 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,2,2,2,1]
=> 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[11]
=> 1 => [1] => ([],1)
=> 1
[10,1]
=> 01 => [1,1] => ([(0,1)],2)
=> 1
[9,2]
=> 10 => [1,1] => ([(0,1)],2)
=> 1
[8,3]
=> 01 => [1,1] => ([(0,1)],2)
=> 1
[8,2,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 1
[7,4]
=> 10 => [1,1] => ([(0,1)],2)
=> 1
[6,5]
=> 01 => [1,1] => ([(0,1)],2)
=> 1
[6,4,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 1
[6,3,2]
=> 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[6,2,2,1]
=> 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[5,3,2,1]
=> 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
Description
The largest multiplicity of a distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
$$
\left(\begin{array}{rrrr}
4 & -1 & -2 & -1 \\
-1 & 4 & -1 & -2 \\
-2 & -1 & 4 & -1 \\
-1 & -2 & -1 & 4
\end{array}\right).
$$
Its eigenvalues are $0,4,4,6$, so the statistic is $2$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore statistic $1$.
Matching statistic: St000774
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000774: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000774: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1 => [1] => ([],1)
=> 1
[2]
=> 0 => [1] => ([],1)
=> 1
[3]
=> 1 => [1] => ([],1)
=> 1
[2,1]
=> 01 => [1,1] => ([(0,1)],2)
=> 1
[4]
=> 0 => [1] => ([],1)
=> 1
[5]
=> 1 => [1] => ([],1)
=> 1
[4,1]
=> 01 => [1,1] => ([(0,1)],2)
=> 1
[3,2]
=> 10 => [1,1] => ([(0,1)],2)
=> 1
[2,2,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 1
[6]
=> 0 => [1] => ([],1)
=> 1
[3,2,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[7]
=> 1 => [1] => ([],1)
=> 1
[6,1]
=> 01 => [1,1] => ([(0,1)],2)
=> 1
[5,2]
=> 10 => [1,1] => ([(0,1)],2)
=> 1
[4,3]
=> 01 => [1,1] => ([(0,1)],2)
=> 1
[4,2,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 1
[2,2,2,1]
=> 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[8]
=> 0 => [1] => ([],1)
=> 1
[5,2,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[3,3,2]
=> 110 => [2,1] => ([(0,2),(1,2)],3)
=> 1
[3,2,2,1]
=> 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[9]
=> 1 => [1] => ([],1)
=> 1
[8,1]
=> 01 => [1,1] => ([(0,1)],2)
=> 1
[7,2]
=> 10 => [1,1] => ([(0,1)],2)
=> 1
[6,3]
=> 01 => [1,1] => ([(0,1)],2)
=> 1
[6,2,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 1
[5,4]
=> 10 => [1,1] => ([(0,1)],2)
=> 1
[4,4,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 1
[4,3,2]
=> 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[4,2,2,1]
=> 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[3,3,2,1]
=> 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,2,2,2,1]
=> 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[10]
=> 0 => [1] => ([],1)
=> 1
[7,2,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[5,4,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[5,3,2]
=> 110 => [2,1] => ([(0,2),(1,2)],3)
=> 1
[5,2,2,1]
=> 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,3,2,1]
=> 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,2,2,2,1]
=> 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[11]
=> 1 => [1] => ([],1)
=> 1
[10,1]
=> 01 => [1,1] => ([(0,1)],2)
=> 1
[9,2]
=> 10 => [1,1] => ([(0,1)],2)
=> 1
[8,3]
=> 01 => [1,1] => ([(0,1)],2)
=> 1
[8,2,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 1
[7,4]
=> 10 => [1,1] => ([(0,1)],2)
=> 1
[6,5]
=> 01 => [1,1] => ([(0,1)],2)
=> 1
[6,4,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 1
[6,3,2]
=> 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[6,2,2,1]
=> 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[5,3,2,1]
=> 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
Description
The maximal multiplicity of a Laplacian eigenvalue in a graph.
Matching statistic: St000181
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000181: Posets ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 20%
Mp00262: Binary words —poset of factors⟶ Posets
St000181: Posets ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 20%
Values
[1]
=> 1 => ([(0,1)],2)
=> 1
[2]
=> 0 => ([(0,1)],2)
=> 1
[3]
=> 1 => ([(0,1)],2)
=> 1
[2,1]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[4]
=> 0 => ([(0,1)],2)
=> 1
[5]
=> 1 => ([(0,1)],2)
=> 1
[4,1]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[3,2]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,2,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
[6]
=> 0 => ([(0,1)],2)
=> 1
[3,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 2
[7]
=> 1 => ([(0,1)],2)
=> 1
[6,1]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[5,2]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[4,3]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[4,2,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
[2,2,2,1]
=> 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2
[8]
=> 0 => ([(0,1)],2)
=> 1
[5,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 2
[3,3,2]
=> 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
[3,2,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1
[9]
=> 1 => ([(0,1)],2)
=> 1
[8,1]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[7,2]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[6,3]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[6,2,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
[5,4]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[4,4,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
[4,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 2
[4,2,2,1]
=> 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2
[3,3,2,1]
=> 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 2
[2,2,2,2,1]
=> 00001 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 3
[10]
=> 0 => ([(0,1)],2)
=> 1
[7,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 2
[5,4,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 2
[5,3,2]
=> 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
[5,2,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1
[4,3,2,1]
=> 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 3
[3,2,2,2,1]
=> 10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ? = 2
[11]
=> 1 => ([(0,1)],2)
=> 1
[10,1]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[9,2]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[8,3]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[8,2,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
[7,4]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[6,5]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[6,4,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
[6,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 2
[6,2,2,1]
=> 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2
[5,3,2,1]
=> 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 2
[4,4,3]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
[4,4,2,1]
=> 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2
[4,2,2,2,1]
=> 00001 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 3
[3,3,3,2]
=> 1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2
[3,3,2,2,1]
=> 11001 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 1
[2,2,2,2,2,1]
=> 000001 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 4
[12]
=> 0 => ([(0,1)],2)
=> 1
[9,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 2
[7,4,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 2
[7,3,2]
=> 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
[7,2,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1
[6,3,2,1]
=> 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 3
[5,5,2]
=> 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
[5,4,3]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 2
[5,4,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1
[5,2,2,2,1]
=> 10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ? = 2
[4,3,3,2]
=> 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1
[4,3,2,2,1]
=> 01001 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ? = 2
[3,3,3,2,1]
=> 11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 2
[3,2,2,2,2,1]
=> 100001 => ([(0,4),(0,5),(1,3),(1,7),(1,8),(2,13),(2,14),(3,2),(3,11),(3,12),(4,9),(4,10),(5,1),(5,9),(5,10),(7,12),(8,11),(9,8),(10,7),(11,13),(12,14),(13,6),(14,6)],15)
=> ? = 3
[13]
=> 1 => ([(0,1)],2)
=> 1
[12,1]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[11,2]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[10,3]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[10,2,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
[9,4]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[8,5]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[8,4,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
[8,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 2
[8,2,2,1]
=> 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2
[7,6]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[7,3,2,1]
=> 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 2
[6,6,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
[6,5,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 2
[14]
=> 0 => ([(0,1)],2)
=> 1
[15]
=> 1 => ([(0,1)],2)
=> 1
[14,1]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[13,2]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[12,3]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[11,4]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[10,5]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[9,6]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[8,7]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[16]
=> 0 => ([(0,1)],2)
=> 1
[17]
=> 1 => ([(0,1)],2)
=> 1
[16,1]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[15,2]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[14,3]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[13,4]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[12,5]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
Description
The number of connected components of the Hasse diagram for the poset.
Matching statistic: St001890
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St001890: Posets ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 20%
Mp00262: Binary words —poset of factors⟶ Posets
St001890: Posets ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 20%
Values
[1]
=> 1 => ([(0,1)],2)
=> 1
[2]
=> 0 => ([(0,1)],2)
=> 1
[3]
=> 1 => ([(0,1)],2)
=> 1
[2,1]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[4]
=> 0 => ([(0,1)],2)
=> 1
[5]
=> 1 => ([(0,1)],2)
=> 1
[4,1]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[3,2]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,2,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
[6]
=> 0 => ([(0,1)],2)
=> 1
[3,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 2
[7]
=> 1 => ([(0,1)],2)
=> 1
[6,1]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[5,2]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[4,3]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[4,2,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
[2,2,2,1]
=> 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2
[8]
=> 0 => ([(0,1)],2)
=> 1
[5,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 2
[3,3,2]
=> 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
[3,2,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1
[9]
=> 1 => ([(0,1)],2)
=> 1
[8,1]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[7,2]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[6,3]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[6,2,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
[5,4]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[4,4,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
[4,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 2
[4,2,2,1]
=> 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2
[3,3,2,1]
=> 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 2
[2,2,2,2,1]
=> 00001 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 3
[10]
=> 0 => ([(0,1)],2)
=> 1
[7,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 2
[5,4,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 2
[5,3,2]
=> 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
[5,2,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1
[4,3,2,1]
=> 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 3
[3,2,2,2,1]
=> 10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ? = 2
[11]
=> 1 => ([(0,1)],2)
=> 1
[10,1]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[9,2]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[8,3]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[8,2,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
[7,4]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[6,5]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[6,4,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
[6,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 2
[6,2,2,1]
=> 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2
[5,3,2,1]
=> 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 2
[4,4,3]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
[4,4,2,1]
=> 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2
[4,2,2,2,1]
=> 00001 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 3
[3,3,3,2]
=> 1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2
[3,3,2,2,1]
=> 11001 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 1
[2,2,2,2,2,1]
=> 000001 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 4
[12]
=> 0 => ([(0,1)],2)
=> 1
[9,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 2
[7,4,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 2
[7,3,2]
=> 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
[7,2,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1
[6,3,2,1]
=> 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 3
[5,5,2]
=> 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
[5,4,3]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 2
[5,4,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1
[5,2,2,2,1]
=> 10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ? = 2
[4,3,3,2]
=> 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1
[4,3,2,2,1]
=> 01001 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ? = 2
[3,3,3,2,1]
=> 11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 2
[3,2,2,2,2,1]
=> 100001 => ([(0,4),(0,5),(1,3),(1,7),(1,8),(2,13),(2,14),(3,2),(3,11),(3,12),(4,9),(4,10),(5,1),(5,9),(5,10),(7,12),(8,11),(9,8),(10,7),(11,13),(12,14),(13,6),(14,6)],15)
=> ? = 3
[13]
=> 1 => ([(0,1)],2)
=> 1
[12,1]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[11,2]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[10,3]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[10,2,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
[9,4]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[8,5]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[8,4,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
[8,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 2
[8,2,2,1]
=> 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2
[7,6]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[7,3,2,1]
=> 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 2
[6,6,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
[6,5,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 2
[14]
=> 0 => ([(0,1)],2)
=> 1
[15]
=> 1 => ([(0,1)],2)
=> 1
[14,1]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[13,2]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[12,3]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[11,4]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[10,5]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[9,6]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[8,7]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[16]
=> 0 => ([(0,1)],2)
=> 1
[17]
=> 1 => ([(0,1)],2)
=> 1
[16,1]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[15,2]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[14,3]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[13,4]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[12,5]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
Description
The maximum magnitude of the Möbius function of a poset.
The '''Möbius function''' of a poset is the multiplicative inverse of the zeta function in the incidence algebra. The Möbius value $\mu(x, y)$ is equal to the signed sum of chains from $x$ to $y$, where odd-length chains are counted with a minus sign, so this statistic is bounded above by the total number of chains in the poset.
Matching statistic: St001964
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St001964: Posets ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 20%
Mp00262: Binary words —poset of factors⟶ Posets
St001964: Posets ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 20%
Values
[1]
=> 1 => ([(0,1)],2)
=> 0 = 1 - 1
[2]
=> 0 => ([(0,1)],2)
=> 0 = 1 - 1
[3]
=> 1 => ([(0,1)],2)
=> 0 = 1 - 1
[2,1]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[4]
=> 0 => ([(0,1)],2)
=> 0 = 1 - 1
[5]
=> 1 => ([(0,1)],2)
=> 0 = 1 - 1
[4,1]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[3,2]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[2,2,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1 - 1
[6]
=> 0 => ([(0,1)],2)
=> 0 = 1 - 1
[3,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 2 - 1
[7]
=> 1 => ([(0,1)],2)
=> 0 = 1 - 1
[6,1]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[5,2]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[4,3]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[4,2,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1 - 1
[2,2,2,1]
=> 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2 - 1
[8]
=> 0 => ([(0,1)],2)
=> 0 = 1 - 1
[5,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 2 - 1
[3,3,2]
=> 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1 - 1
[3,2,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1 - 1
[9]
=> 1 => ([(0,1)],2)
=> 0 = 1 - 1
[8,1]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[7,2]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[6,3]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[6,2,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1 - 1
[5,4]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[4,4,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1 - 1
[4,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 2 - 1
[4,2,2,1]
=> 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2 - 1
[3,3,2,1]
=> 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 2 - 1
[2,2,2,2,1]
=> 00001 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 3 - 1
[10]
=> 0 => ([(0,1)],2)
=> 0 = 1 - 1
[7,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 2 - 1
[5,4,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 2 - 1
[5,3,2]
=> 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1 - 1
[5,2,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1 - 1
[4,3,2,1]
=> 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 3 - 1
[3,2,2,2,1]
=> 10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ? = 2 - 1
[11]
=> 1 => ([(0,1)],2)
=> 0 = 1 - 1
[10,1]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[9,2]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[8,3]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[8,2,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1 - 1
[7,4]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[6,5]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[6,4,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1 - 1
[6,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 2 - 1
[6,2,2,1]
=> 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2 - 1
[5,3,2,1]
=> 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 2 - 1
[4,4,3]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1 - 1
[4,4,2,1]
=> 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2 - 1
[4,2,2,2,1]
=> 00001 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 3 - 1
[3,3,3,2]
=> 1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2 - 1
[3,3,2,2,1]
=> 11001 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 1 - 1
[2,2,2,2,2,1]
=> 000001 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 4 - 1
[12]
=> 0 => ([(0,1)],2)
=> 0 = 1 - 1
[9,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 2 - 1
[7,4,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 2 - 1
[7,3,2]
=> 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1 - 1
[7,2,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1 - 1
[6,3,2,1]
=> 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 3 - 1
[5,5,2]
=> 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1 - 1
[5,4,3]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 2 - 1
[5,4,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1 - 1
[5,2,2,2,1]
=> 10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ? = 2 - 1
[4,3,3,2]
=> 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1 - 1
[4,3,2,2,1]
=> 01001 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ? = 2 - 1
[3,3,3,2,1]
=> 11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 2 - 1
[3,2,2,2,2,1]
=> 100001 => ([(0,4),(0,5),(1,3),(1,7),(1,8),(2,13),(2,14),(3,2),(3,11),(3,12),(4,9),(4,10),(5,1),(5,9),(5,10),(7,12),(8,11),(9,8),(10,7),(11,13),(12,14),(13,6),(14,6)],15)
=> ? = 3 - 1
[13]
=> 1 => ([(0,1)],2)
=> 0 = 1 - 1
[12,1]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[11,2]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[10,3]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[10,2,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1 - 1
[9,4]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[8,5]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[8,4,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1 - 1
[8,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 2 - 1
[8,2,2,1]
=> 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2 - 1
[7,6]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[7,3,2,1]
=> 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 2 - 1
[6,6,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1 - 1
[6,5,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 2 - 1
[14]
=> 0 => ([(0,1)],2)
=> 0 = 1 - 1
[15]
=> 1 => ([(0,1)],2)
=> 0 = 1 - 1
[14,1]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[13,2]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[12,3]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[11,4]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[10,5]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[9,6]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[8,7]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[16]
=> 0 => ([(0,1)],2)
=> 0 = 1 - 1
[17]
=> 1 => ([(0,1)],2)
=> 0 = 1 - 1
[16,1]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[15,2]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[14,3]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[13,4]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[12,5]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
Description
The interval resolution global dimension of a poset.
This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.
Matching statistic: St001232
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 20%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 20%
Values
[1]
=> []
=> []
=> ? = 1
[2]
=> []
=> []
=> ? = 1
[3]
=> []
=> []
=> ? = 1
[2,1]
=> [1]
=> [1,0,1,0]
=> 1
[4]
=> []
=> []
=> ? = 1
[5]
=> []
=> []
=> ? = 1
[4,1]
=> [1]
=> [1,0,1,0]
=> 1
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> ? = 1
[6]
=> []
=> []
=> ? = 1
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> ? = 2
[7]
=> []
=> []
=> ? = 1
[6,1]
=> [1]
=> [1,0,1,0]
=> 1
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> ? = 1
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> ? = 2
[8]
=> []
=> []
=> ? = 1
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> ? = 2
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
[3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> ? = 1
[9]
=> []
=> []
=> ? = 1
[8,1]
=> [1]
=> [1,0,1,0]
=> 1
[7,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[6,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[6,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> ? = 1
[5,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[4,4,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 1
[4,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 2
[4,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> ? = 2
[3,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> ? = 2
[2,2,2,2,1]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> ? = 3
[10]
=> []
=> []
=> ? = 1
[7,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> ? = 2
[5,4,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 2
[5,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
[5,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> ? = 1
[4,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> ? = 3
[3,2,2,2,1]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> ? = 2
[11]
=> []
=> []
=> ? = 1
[10,1]
=> [1]
=> [1,0,1,0]
=> 1
[9,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[8,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[8,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> ? = 1
[7,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[6,5]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[6,4,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 1
[6,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 2
[6,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> ? = 2
[5,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> ? = 2
[4,4,3]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 1
[4,4,2,1]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? = 2
[4,2,2,2,1]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> ? = 3
[3,3,3,2]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> ? = 2
[3,3,2,2,1]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[2,2,2,2,2,1]
=> [2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 4
[12]
=> []
=> []
=> ? = 1
[9,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> ? = 2
[7,4,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 2
[7,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
[7,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> ? = 1
[6,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> ? = 3
[5,5,2]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 1
[5,4,3]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 2
[5,4,2,1]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? = 1
[12,1]
=> [1]
=> [1,0,1,0]
=> 1
[11,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[10,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[9,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[8,5]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[7,6]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1
[14,1]
=> [1]
=> [1,0,1,0]
=> 1
[13,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[12,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[11,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[10,5]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[9,6]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1
[16,1]
=> [1]
=> [1,0,1,0]
=> 1
[15,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[14,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[13,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[12,5]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[11,6]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000022
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 20%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 20%
Values
[1]
=> [1]
=> [1,0]
=> [2,1] => 0 = 1 - 1
[2]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 0 = 1 - 1
[3]
=> [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 0 = 1 - 1
[2,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 0 = 1 - 1
[4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 0 = 1 - 1
[5]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => 0 = 1 - 1
[4,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => 0 = 1 - 1
[3,2]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 0 = 1 - 1
[2,2,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 0 = 1 - 1
[6]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 0 = 1 - 1
[3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 2 - 1
[7]
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,1,2,3,4,5,6,7] => 0 = 1 - 1
[6,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => 0 = 1 - 1
[5,2]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [8,1,2,3,4,7,5,6] => ? = 1 - 1
[4,3]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [7,1,2,8,3,4,5,6] => ? = 1 - 1
[4,2,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,7,1,2,3,4,5,6] => 0 = 1 - 1
[2,2,2,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [8,1,2,3,7,4,5,6] => ? = 2 - 1
[8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,9,1,2,3,4,5,6,7] => 0 = 1 - 1
[5,2,1]
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [9,1,2,3,4,8,5,6,7] => ? = 2 - 1
[3,3,2]
=> [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [9,1,2,3,4,5,8,6,7] => ? = 1 - 1
[3,2,2,1]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => ? = 1 - 1
[9]
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [10,1,2,3,4,5,6,7,8,9] => 0 = 1 - 1
[8,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [10,9,1,2,3,4,5,6,7,8] => 0 = 1 - 1
[7,2]
=> [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [10,1,2,3,4,5,6,9,7,8] => ? = 1 - 1
[6,3]
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 0 = 1 - 1
[6,2,1]
=> [3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [8,3,7,1,2,4,5,6] => ? = 1 - 1
[5,4]
=> [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [9,1,2,3,4,10,5,6,7,8] => ? = 1 - 1
[4,4,1]
=> [8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [9,1,2,3,4,5,6,7,10,8] => 0 = 1 - 1
[4,3,2]
=> [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [10,1,2,9,3,4,5,6,7,8] => ? = 2 - 1
[4,2,2,1]
=> [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [10,1,2,3,9,4,5,6,7,8] => ? = 2 - 1
[3,3,2,1]
=> [6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [10,1,2,3,4,5,9,6,7,8] => ? = 2 - 1
[2,2,2,2,1]
=> [4,4,1]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [6,7,4,1,2,3,5] => ? = 3 - 1
[10]
=> [5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [6,7,5,1,2,3,4] => ? = 1 - 1
[7,2,1]
=> [7,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [11,1,2,3,4,5,6,10,7,8,9] => ? = 2 - 1
[5,4,1]
=> [5,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [11,1,2,3,4,10,5,6,7,8,9] => ? = 2 - 1
[5,3,2]
=> [5,3,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [8,1,2,5,9,3,4,6,7] => ? = 1 - 1
[5,2,2,1]
=> [5,4,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [7,1,8,5,2,3,4,6] => ? = 1 - 1
[4,3,2,1]
=> [3,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [10,1,2,11,3,4,5,6,7,8,9] => ? = 3 - 1
[3,2,2,2,1]
=> [4,3,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [9,1,4,8,2,3,5,6,7] => ? = 2 - 1
[11]
=> [11]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [12,1,2,3,4,5,6,7,8,9,10,11] => ? = 1 - 1
[10,1]
=> [5,5,1]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [8,7,5,1,2,3,4,6] => ? = 1 - 1
[9,2]
=> [9,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [12,1,2,3,4,5,6,7,8,11,9,10] => ? = 1 - 1
[8,3]
=> [3,1,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [12,1,2,11,3,4,5,6,7,8,9,10] => ? = 1 - 1
[8,2,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [11,12,1,2,3,4,5,6,7,8,9,10] => ? = 1 - 1
[7,4]
=> [7,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [11,1,2,3,4,5,6,12,7,8,9,10] => ? = 1 - 1
[6,5]
=> [5,3,3]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [4,1,2,5,6,7,8,3] => ? = 1 - 1
[6,4,1]
=> [3,3,1,1,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [9,3,10,1,2,4,5,6,7,8] => ? = 1 - 1
[6,3,2]
=> [3,3,3,1,1]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [2,3,4,8,7,1,5,6] => ? = 2 - 1
[6,2,2,1]
=> [4,3,3,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [3,1,4,5,8,7,2,6] => ? = 2 - 1
[5,3,2,1]
=> [5,3,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [10,1,2,5,9,3,4,6,7,8] => ? = 2 - 1
[4,4,3]
=> [8,3]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [10,1,2,3,4,5,8,9,6,7] => ? = 1 - 1
[4,4,2,1]
=> [8,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [12,1,2,3,4,5,6,7,11,8,9,10] => ? = 2 - 1
[4,2,2,2,1]
=> [4,1,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [11,1,2,3,12,4,5,6,7,8,9,10] => ? = 3 - 1
[3,3,3,2]
=> [6,3,1,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [9,1,2,3,6,10,4,5,7,8] => ? = 2 - 1
[3,3,2,2,1]
=> [6,4,1]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [8,1,2,9,6,3,4,5,7] => ? = 1 - 1
[2,2,2,2,2,1]
=> [4,4,1,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [9,8,4,1,2,3,5,6,7] => ? = 4 - 1
[12]
=> [3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 1 - 1
[9,2,1]
=> [9,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? => ? = 2 - 1
[7,4,1]
=> [7,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ? = 2 - 1
[7,3,2]
=> [7,3,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [10,1,2,3,4,7,11,5,6,8,9] => ? = 1 - 1
[7,2,2,1]
=> [7,4,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [9,1,2,3,10,7,4,5,6,8] => ? = 1 - 1
[6,3,2,1]
=> [3,3,3,1,1,1]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> [2,3,4,8,9,1,5,6,7] => ? = 3 - 1
[5,5,2]
=> [10,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [13,1,2,3,4,5,6,7,8,9,12,10,11] => ? = 1 - 1
[5,4,3]
=> [5,3,1,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [11,1,2,5,10,3,4,6,7,8,9] => ? = 2 - 1
[5,4,2,1]
=> [5,1,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ? = 1 - 1
[5,2,2,2,1]
=> [5,4,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [10,1,9,5,2,3,4,6,7,8] => ? = 2 - 1
[4,3,3,2]
=> [6,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ? = 1 - 1
[4,3,2,2,1]
=> [4,3,1,1,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [10,1,4,11,2,3,5,6,7,8,9] => ? = 2 - 1
[3,3,3,2,1]
=> [6,3,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [11,1,2,3,6,10,4,5,7,8,9] => ? = 2 - 1
[14]
=> [7,7]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [9,7,8,1,2,3,4,5,6] => 0 = 1 - 1
[10,5]
=> [5,5,5]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [8,7,4,5,6,1,2,3] => 0 = 1 - 1
Description
The number of fixed points of a permutation.
Matching statistic: St000405
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000405: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 20%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000405: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 20%
Values
[1]
=> [1]
=> [1,0]
=> [2,1] => 0 = 1 - 1
[2]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 0 = 1 - 1
[3]
=> [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 0 = 1 - 1
[2,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 0 = 1 - 1
[4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 0 = 1 - 1
[5]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => 0 = 1 - 1
[4,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => 0 = 1 - 1
[3,2]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 0 = 1 - 1
[2,2,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 0 = 1 - 1
[6]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 0 = 1 - 1
[3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 2 - 1
[7]
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,1,2,3,4,5,6,7] => 0 = 1 - 1
[6,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => 0 = 1 - 1
[5,2]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [8,1,2,3,4,7,5,6] => ? = 1 - 1
[4,3]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [7,1,2,8,3,4,5,6] => ? = 1 - 1
[4,2,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,7,1,2,3,4,5,6] => 0 = 1 - 1
[2,2,2,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [8,1,2,3,7,4,5,6] => ? = 2 - 1
[8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,9,1,2,3,4,5,6,7] => 0 = 1 - 1
[5,2,1]
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [9,1,2,3,4,8,5,6,7] => ? = 2 - 1
[3,3,2]
=> [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [9,1,2,3,4,5,8,6,7] => ? = 1 - 1
[3,2,2,1]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => ? = 1 - 1
[9]
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [10,1,2,3,4,5,6,7,8,9] => 0 = 1 - 1
[8,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [10,9,1,2,3,4,5,6,7,8] => 0 = 1 - 1
[7,2]
=> [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [10,1,2,3,4,5,6,9,7,8] => ? = 1 - 1
[6,3]
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 0 = 1 - 1
[6,2,1]
=> [3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [8,3,7,1,2,4,5,6] => ? = 1 - 1
[5,4]
=> [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [9,1,2,3,4,10,5,6,7,8] => ? = 1 - 1
[4,4,1]
=> [8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [9,1,2,3,4,5,6,7,10,8] => 0 = 1 - 1
[4,3,2]
=> [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [10,1,2,9,3,4,5,6,7,8] => ? = 2 - 1
[4,2,2,1]
=> [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [10,1,2,3,9,4,5,6,7,8] => ? = 2 - 1
[3,3,2,1]
=> [6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [10,1,2,3,4,5,9,6,7,8] => ? = 2 - 1
[2,2,2,2,1]
=> [4,4,1]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [6,7,4,1,2,3,5] => ? = 3 - 1
[10]
=> [5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [6,7,5,1,2,3,4] => ? = 1 - 1
[7,2,1]
=> [7,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [11,1,2,3,4,5,6,10,7,8,9] => ? = 2 - 1
[5,4,1]
=> [5,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [11,1,2,3,4,10,5,6,7,8,9] => ? = 2 - 1
[5,3,2]
=> [5,3,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [8,1,2,5,9,3,4,6,7] => ? = 1 - 1
[5,2,2,1]
=> [5,4,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [7,1,8,5,2,3,4,6] => ? = 1 - 1
[4,3,2,1]
=> [3,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [10,1,2,11,3,4,5,6,7,8,9] => ? = 3 - 1
[3,2,2,2,1]
=> [4,3,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [9,1,4,8,2,3,5,6,7] => ? = 2 - 1
[11]
=> [11]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [12,1,2,3,4,5,6,7,8,9,10,11] => ? = 1 - 1
[10,1]
=> [5,5,1]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [8,7,5,1,2,3,4,6] => ? = 1 - 1
[9,2]
=> [9,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [12,1,2,3,4,5,6,7,8,11,9,10] => ? = 1 - 1
[8,3]
=> [3,1,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [12,1,2,11,3,4,5,6,7,8,9,10] => ? = 1 - 1
[8,2,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [11,12,1,2,3,4,5,6,7,8,9,10] => ? = 1 - 1
[7,4]
=> [7,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [11,1,2,3,4,5,6,12,7,8,9,10] => ? = 1 - 1
[6,5]
=> [5,3,3]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [4,1,2,5,6,7,8,3] => ? = 1 - 1
[6,4,1]
=> [3,3,1,1,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [9,3,10,1,2,4,5,6,7,8] => ? = 1 - 1
[6,3,2]
=> [3,3,3,1,1]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [2,3,4,8,7,1,5,6] => ? = 2 - 1
[6,2,2,1]
=> [4,3,3,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [3,1,4,5,8,7,2,6] => ? = 2 - 1
[5,3,2,1]
=> [5,3,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [10,1,2,5,9,3,4,6,7,8] => ? = 2 - 1
[4,4,3]
=> [8,3]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [10,1,2,3,4,5,8,9,6,7] => ? = 1 - 1
[4,4,2,1]
=> [8,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [12,1,2,3,4,5,6,7,11,8,9,10] => ? = 2 - 1
[4,2,2,2,1]
=> [4,1,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [11,1,2,3,12,4,5,6,7,8,9,10] => ? = 3 - 1
[3,3,3,2]
=> [6,3,1,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [9,1,2,3,6,10,4,5,7,8] => ? = 2 - 1
[3,3,2,2,1]
=> [6,4,1]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [8,1,2,9,6,3,4,5,7] => ? = 1 - 1
[2,2,2,2,2,1]
=> [4,4,1,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [9,8,4,1,2,3,5,6,7] => ? = 4 - 1
[12]
=> [3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 1 - 1
[9,2,1]
=> [9,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? => ? = 2 - 1
[7,4,1]
=> [7,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ? = 2 - 1
[7,3,2]
=> [7,3,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [10,1,2,3,4,7,11,5,6,8,9] => ? = 1 - 1
[7,2,2,1]
=> [7,4,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [9,1,2,3,10,7,4,5,6,8] => ? = 1 - 1
[6,3,2,1]
=> [3,3,3,1,1,1]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> [2,3,4,8,9,1,5,6,7] => ? = 3 - 1
[5,5,2]
=> [10,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [13,1,2,3,4,5,6,7,8,9,12,10,11] => ? = 1 - 1
[5,4,3]
=> [5,3,1,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [11,1,2,5,10,3,4,6,7,8,9] => ? = 2 - 1
[5,4,2,1]
=> [5,1,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ? = 1 - 1
[5,2,2,2,1]
=> [5,4,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [10,1,9,5,2,3,4,6,7,8] => ? = 2 - 1
[4,3,3,2]
=> [6,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ? = 1 - 1
[4,3,2,2,1]
=> [4,3,1,1,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [10,1,4,11,2,3,5,6,7,8,9] => ? = 2 - 1
[3,3,3,2,1]
=> [6,3,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [11,1,2,3,6,10,4,5,7,8,9] => ? = 2 - 1
[14]
=> [7,7]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [9,7,8,1,2,3,4,5,6] => 0 = 1 - 1
[10,5]
=> [5,5,5]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [8,7,4,5,6,1,2,3] => 0 = 1 - 1
Description
The number of occurrences of the pattern 1324 in a permutation.
There is no explicit formula known for the number of permutations avoiding this pattern (denoted by $S_n(1324)$), but it is shown in [1], improving bounds in [2] and [3] that
$$\lim_{n \rightarrow \infty} \sqrt[n]{S_n(1324)} \leq 13.73718.$$
Matching statistic: St000842
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000842: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 20%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000842: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 20%
Values
[1]
=> [1]
=> [1,0]
=> [2,1] => 2 = 1 + 1
[2]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 1 + 1
[3]
=> [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 2 = 1 + 1
[2,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 2 = 1 + 1
[4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 2 = 1 + 1
[5]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => 2 = 1 + 1
[4,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => 2 = 1 + 1
[3,2]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 2 = 1 + 1
[2,2,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 2 = 1 + 1
[6]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 2 = 1 + 1
[3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 2 + 1
[7]
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,1,2,3,4,5,6,7] => 2 = 1 + 1
[6,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => 2 = 1 + 1
[5,2]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [8,1,2,3,4,7,5,6] => ? = 1 + 1
[4,3]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [7,1,2,8,3,4,5,6] => ? = 1 + 1
[4,2,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,7,1,2,3,4,5,6] => 2 = 1 + 1
[2,2,2,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [8,1,2,3,7,4,5,6] => ? = 2 + 1
[8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,9,1,2,3,4,5,6,7] => 2 = 1 + 1
[5,2,1]
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [9,1,2,3,4,8,5,6,7] => ? = 2 + 1
[3,3,2]
=> [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [9,1,2,3,4,5,8,6,7] => ? = 1 + 1
[3,2,2,1]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => ? = 1 + 1
[9]
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [10,1,2,3,4,5,6,7,8,9] => 2 = 1 + 1
[8,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [10,9,1,2,3,4,5,6,7,8] => 2 = 1 + 1
[7,2]
=> [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [10,1,2,3,4,5,6,9,7,8] => ? = 1 + 1
[6,3]
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 2 = 1 + 1
[6,2,1]
=> [3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [8,3,7,1,2,4,5,6] => ? = 1 + 1
[5,4]
=> [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [9,1,2,3,4,10,5,6,7,8] => ? = 1 + 1
[4,4,1]
=> [8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [9,1,2,3,4,5,6,7,10,8] => 2 = 1 + 1
[4,3,2]
=> [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [10,1,2,9,3,4,5,6,7,8] => ? = 2 + 1
[4,2,2,1]
=> [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [10,1,2,3,9,4,5,6,7,8] => ? = 2 + 1
[3,3,2,1]
=> [6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [10,1,2,3,4,5,9,6,7,8] => ? = 2 + 1
[2,2,2,2,1]
=> [4,4,1]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [6,7,4,1,2,3,5] => ? = 3 + 1
[10]
=> [5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [6,7,5,1,2,3,4] => ? = 1 + 1
[7,2,1]
=> [7,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [11,1,2,3,4,5,6,10,7,8,9] => ? = 2 + 1
[5,4,1]
=> [5,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [11,1,2,3,4,10,5,6,7,8,9] => ? = 2 + 1
[5,3,2]
=> [5,3,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [8,1,2,5,9,3,4,6,7] => ? = 1 + 1
[5,2,2,1]
=> [5,4,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [7,1,8,5,2,3,4,6] => ? = 1 + 1
[4,3,2,1]
=> [3,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [10,1,2,11,3,4,5,6,7,8,9] => ? = 3 + 1
[3,2,2,2,1]
=> [4,3,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [9,1,4,8,2,3,5,6,7] => ? = 2 + 1
[11]
=> [11]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [12,1,2,3,4,5,6,7,8,9,10,11] => ? = 1 + 1
[10,1]
=> [5,5,1]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [8,7,5,1,2,3,4,6] => ? = 1 + 1
[9,2]
=> [9,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [12,1,2,3,4,5,6,7,8,11,9,10] => ? = 1 + 1
[8,3]
=> [3,1,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [12,1,2,11,3,4,5,6,7,8,9,10] => ? = 1 + 1
[8,2,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [11,12,1,2,3,4,5,6,7,8,9,10] => ? = 1 + 1
[7,4]
=> [7,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [11,1,2,3,4,5,6,12,7,8,9,10] => ? = 1 + 1
[6,5]
=> [5,3,3]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [4,1,2,5,6,7,8,3] => ? = 1 + 1
[6,4,1]
=> [3,3,1,1,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [9,3,10,1,2,4,5,6,7,8] => ? = 1 + 1
[6,3,2]
=> [3,3,3,1,1]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [2,3,4,8,7,1,5,6] => ? = 2 + 1
[6,2,2,1]
=> [4,3,3,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [3,1,4,5,8,7,2,6] => ? = 2 + 1
[5,3,2,1]
=> [5,3,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [10,1,2,5,9,3,4,6,7,8] => ? = 2 + 1
[4,4,3]
=> [8,3]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [10,1,2,3,4,5,8,9,6,7] => ? = 1 + 1
[4,4,2,1]
=> [8,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [12,1,2,3,4,5,6,7,11,8,9,10] => ? = 2 + 1
[4,2,2,2,1]
=> [4,1,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [11,1,2,3,12,4,5,6,7,8,9,10] => ? = 3 + 1
[3,3,3,2]
=> [6,3,1,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [9,1,2,3,6,10,4,5,7,8] => ? = 2 + 1
[3,3,2,2,1]
=> [6,4,1]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [8,1,2,9,6,3,4,5,7] => ? = 1 + 1
[2,2,2,2,2,1]
=> [4,4,1,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [9,8,4,1,2,3,5,6,7] => ? = 4 + 1
[12]
=> [3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 2 = 1 + 1
[9,2,1]
=> [9,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? => ? = 2 + 1
[7,4,1]
=> [7,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ? = 2 + 1
[7,3,2]
=> [7,3,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [10,1,2,3,4,7,11,5,6,8,9] => ? = 1 + 1
[7,2,2,1]
=> [7,4,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [9,1,2,3,10,7,4,5,6,8] => ? = 1 + 1
[6,3,2,1]
=> [3,3,3,1,1,1]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> [2,3,4,8,9,1,5,6,7] => ? = 3 + 1
[5,5,2]
=> [10,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [13,1,2,3,4,5,6,7,8,9,12,10,11] => ? = 1 + 1
[5,4,3]
=> [5,3,1,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [11,1,2,5,10,3,4,6,7,8,9] => ? = 2 + 1
[5,4,2,1]
=> [5,1,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ? = 1 + 1
[5,2,2,2,1]
=> [5,4,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [10,1,9,5,2,3,4,6,7,8] => ? = 2 + 1
[4,3,3,2]
=> [6,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ? = 1 + 1
[4,3,2,2,1]
=> [4,3,1,1,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [10,1,4,11,2,3,5,6,7,8,9] => ? = 2 + 1
[3,3,3,2,1]
=> [6,3,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [11,1,2,3,6,10,4,5,7,8,9] => ? = 2 + 1
[14]
=> [7,7]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [9,7,8,1,2,3,4,5,6] => 2 = 1 + 1
[10,5]
=> [5,5,5]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [8,7,4,5,6,1,2,3] => 2 = 1 + 1
Description
The breadth of a permutation.
According to [1, Def.1.6], this is the minimal Manhattan distance between two ones in the permutation matrix of $\pi$: $$\min\{|i-j|+|\pi(i)-\pi(j)|: i\neq j\}.$$
According to [1, Def.1.3], a permutation $\pi$ is $k$-prolific, if the set of permutations obtained from $\pi$ by deleting any $k$ elements and standardising has maximal cardinality, i.e., $\binom{n}{k}$.
By [1, Thm.2.22], a permutation is $k$-prolific if and only if its breath is at least $k+2$.
By [1, Cor.4.3], the smallest permutations that are $k$-prolific have size $\lceil k^2+2k+1\rceil$, and by [1, Thm.4.4], there are $k$-prolific permutations of any size larger than this.
According to [2] the proportion of $k$-prolific permutations in the set of all permutations is asymptotically equal to $\exp(-k^2-k)$.
Matching statistic: St000223
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000223: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 40%
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000223: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 40%
Values
[1]
=> [[1]]
=> [[1]]
=> [1] => 0 = 1 - 1
[2]
=> [[1,2]]
=> [[1,2]]
=> [1,2] => 0 = 1 - 1
[3]
=> [[1,2,3]]
=> [[1,2,3]]
=> [1,2,3] => 0 = 1 - 1
[2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> [3,1,2] => 0 = 1 - 1
[4]
=> [[1,2,3,4]]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0 = 1 - 1
[5]
=> [[1,2,3,4,5]]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 0 = 1 - 1
[4,1]
=> [[1,3,4,5],[2]]
=> [[1,2,4,5],[3]]
=> [3,1,2,4,5] => 0 = 1 - 1
[3,2]
=> [[1,2,5],[3,4]]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 0 = 1 - 1
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [[1,2,4,5],[3]]
=> [3,1,2,4,5] => 0 = 1 - 1
[6]
=> [[1,2,3,4,5,6]]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => 0 = 1 - 1
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [[1,2,4,5],[3],[6]]
=> [6,3,1,2,4,5] => 1 = 2 - 1
[7]
=> [[1,2,3,4,5,6,7]]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => 0 = 1 - 1
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> [[1,2,4,5,6,7],[3]]
=> [3,1,2,4,5,6,7] => 0 = 1 - 1
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> [[1,2,3,4,7],[5,6]]
=> [5,6,1,2,3,4,7] => 0 = 1 - 1
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => 0 = 1 - 1
[4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [[1,2,4,5],[3,7],[6]]
=> [6,3,7,1,2,4,5] => ? = 1 - 1
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [[1,2,4,5],[3,7],[6]]
=> [6,3,7,1,2,4,5] => ? = 2 - 1
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => 0 = 1 - 1
[5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> [[1,2,4,5,8],[3,7],[6]]
=> [6,3,7,1,2,4,5,8] => ? = 2 - 1
[3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> [[1,2,3,4,7,8],[5,6]]
=> [5,6,1,2,3,4,7,8] => ? = 1 - 1
[3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> [[1,2,4,5],[3,7],[6],[8]]
=> [8,6,3,7,1,2,4,5] => ? = 1 - 1
[9]
=> [[1,2,3,4,5,6,7,8,9]]
=> [[1,2,3,4,5,6,7,8,9]]
=> [1,2,3,4,5,6,7,8,9] => 0 = 1 - 1
[8,1]
=> [[1,3,4,5,6,7,8,9],[2]]
=> [[1,2,4,5,6,7,8,9],[3]]
=> [3,1,2,4,5,6,7,8,9] => ? = 1 - 1
[7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [[1,2,3,4,7,8,9],[5,6]]
=> [5,6,1,2,3,4,7,8,9] => ? = 1 - 1
[6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> [[1,2,3,4,5,6],[7,8,9]]
=> [7,8,9,1,2,3,4,5,6] => 0 = 1 - 1
[6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> [[1,2,4,5,8,9],[3,7],[6]]
=> [6,3,7,1,2,4,5,8,9] => ? = 1 - 1
[5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [[1,2,3,4,5,6,7,8],[9]]
=> [9,1,2,3,4,5,6,7,8] => 0 = 1 - 1
[4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> [[1,2,4,5,6,7,8,9],[3]]
=> [3,1,2,4,5,6,7,8,9] => ? = 1 - 1
[4,3,2]
=> [[1,2,5,9],[3,4,8],[6,7]]
=> [[1,2,3,4,7,8],[5,6],[9]]
=> [9,5,6,1,2,3,4,7,8] => ? = 2 - 1
[4,2,2,1]
=> [[1,3,8,9],[2,5],[4,7],[6]]
=> [[1,2,4,5],[3,7],[6,9],[8]]
=> [8,6,9,3,7,1,2,4,5] => ? = 2 - 1
[3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8],[7]]
=> [[1,2,4,5,8,9],[3,7],[6]]
=> [6,3,7,1,2,4,5,8,9] => ? = 2 - 1
[2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> [[1,2,4,5],[3,7],[6,9],[8]]
=> [8,6,9,3,7,1,2,4,5] => ? = 3 - 1
[10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> [1,2,3,4,5,6,7,8,9,10] => 0 = 1 - 1
[7,2,1]
=> [[1,3,6,7,8,9,10],[2,5],[4]]
=> [[1,2,4,5,8,9,10],[3,7],[6]]
=> [6,3,7,1,2,4,5,8,9,10] => ? = 2 - 1
[5,4,1]
=> [[1,3,4,5,10],[2,7,8,9],[6]]
=> [[1,2,4,5,6,7,8,9],[3],[10]]
=> [10,3,1,2,4,5,6,7,8,9] => ? = 2 - 1
[5,3,2]
=> [[1,2,5,9,10],[3,4,8],[6,7]]
=> [[1,2,3,4,7,8],[5,6],[9,10]]
=> [9,10,5,6,1,2,3,4,7,8] => ? = 1 - 1
[5,2,2,1]
=> [[1,3,8,9,10],[2,5],[4,7],[6]]
=> [[1,2,4,5,10],[3,7],[6,9],[8]]
=> [8,6,9,3,7,1,2,4,5,10] => ? = 1 - 1
[4,3,2,1]
=> [[1,3,6,10],[2,5,9],[4,8],[7]]
=> [[1,2,4,5,8,9],[3,7],[6],[10]]
=> [10,6,3,7,1,2,4,5,8,9] => ? = 3 - 1
[3,2,2,2,1]
=> [[1,3,10],[2,5],[4,7],[6,9],[8]]
=> [[1,2,4,5],[3,7],[6,9],[8],[10]]
=> [10,8,6,9,3,7,1,2,4,5] => ? = 2 - 1
[11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ?
=> ? => ? = 1 - 1
[10,1]
=> [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ?
=> ? => ? = 1 - 1
[9,2]
=> [[1,2,5,6,7,8,9,10,11],[3,4]]
=> ?
=> ? => ? = 1 - 1
[8,3]
=> [[1,2,3,7,8,9,10,11],[4,5,6]]
=> ?
=> ? => ? = 1 - 1
[8,2,1]
=> [[1,3,6,7,8,9,10,11],[2,5],[4]]
=> ?
=> ? => ? = 1 - 1
[7,4]
=> [[1,2,3,4,9,10,11],[5,6,7,8]]
=> ?
=> ? => ? = 1 - 1
[6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> ?
=> ? => ? = 1 - 1
[6,4,1]
=> [[1,3,4,5,10,11],[2,7,8,9],[6]]
=> ?
=> ? => ? = 1 - 1
[6,3,2]
=> [[1,2,5,9,10,11],[3,4,8],[6,7]]
=> ?
=> ? => ? = 2 - 1
[6,2,2,1]
=> [[1,3,8,9,10,11],[2,5],[4,7],[6]]
=> ?
=> ? => ? = 2 - 1
[5,3,2,1]
=> [[1,3,6,10,11],[2,5,9],[4,8],[7]]
=> [[1,2,4,5,8,9],[3,7],[6,11],[10]]
=> ? => ? = 2 - 1
[4,4,3]
=> [[1,2,3,7],[4,5,6,11],[8,9,10]]
=> [[1,2,3,4,5,6,10,11],[7,8,9]]
=> ? => ? = 1 - 1
[4,4,2,1]
=> [[1,3,6,7],[2,5,10,11],[4,9],[8]]
=> [[1,2,4,5,8,9,10,11],[3,7],[6]]
=> ? => ? = 2 - 1
[4,2,2,2,1]
=> [[1,3,10,11],[2,5],[4,7],[6,9],[8]]
=> [[1,2,4,5],[3,7],[6,9],[8,11],[10]]
=> ? => ? = 3 - 1
[3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10]]
=> [[1,2,3,4,7,8],[5,6,11],[9,10]]
=> ? => ? = 2 - 1
[3,3,2,2,1]
=> [[1,3,8],[2,5,11],[4,7],[6,10],[9]]
=> [[1,2,4,5,10,11],[3,7],[6,9],[8]]
=> ? => ? = 1 - 1
[2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> ?
=> ? => ? = 4 - 1
[12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [1,2,3,4,5,6,7,8,9,10,11,12] => ? = 1 - 1
[9,2,1]
=> [[1,3,6,7,8,9,10,11,12],[2,5],[4]]
=> ?
=> ? => ? = 2 - 1
[7,4,1]
=> [[1,3,4,5,10,11,12],[2,7,8,9],[6]]
=> ?
=> ? => ? = 2 - 1
[7,3,2]
=> [[1,2,5,9,10,11,12],[3,4,8],[6,7]]
=> ?
=> ? => ? = 1 - 1
[7,2,2,1]
=> [[1,3,8,9,10,11,12],[2,5],[4,7],[6]]
=> ?
=> ? => ? = 1 - 1
[6,3,2,1]
=> [[1,3,6,10,11,12],[2,5,9],[4,8],[7]]
=> ?
=> ? => ? = 3 - 1
[5,5,2]
=> [[1,2,5,6,7],[3,4,10,11,12],[8,9]]
=> ?
=> ? => ? = 1 - 1
[5,4,3]
=> [[1,2,3,7,12],[4,5,6,11],[8,9,10]]
=> [[1,2,3,4,5,6,10,11],[7,8,9],[12]]
=> ? => ? = 2 - 1
[5,4,2,1]
=> [[1,3,6,7,12],[2,5,10,11],[4,9],[8]]
=> [[1,2,4,5,8,9,10,11],[3,7],[6],[12]]
=> ? => ? = 1 - 1
[5,2,2,2,1]
=> [[1,3,10,11,12],[2,5],[4,7],[6,9],[8]]
=> [[1,2,4,5,12],[3,7],[6,9],[8,11],[10]]
=> ? => ? = 2 - 1
[4,3,3,2]
=> [[1,2,5,12],[3,4,8],[6,7,11],[9,10]]
=> [[1,2,3,4,7,8],[5,6,11],[9,10],[12]]
=> ? => ? = 1 - 1
[4,3,2,2,1]
=> [[1,3,8,12],[2,5,11],[4,7],[6,10],[9]]
=> [[1,2,4,5,10,11],[3,7],[6,9],[8],[12]]
=> ? => ? = 2 - 1
[3,3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8,12],[7,11],[10]]
=> [[1,2,4,5,8,9],[3,7,12],[6,11],[10]]
=> ? => ? = 2 - 1
[3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> ?
=> ? => ? = 3 - 1
Description
The number of nestings in the permutation.
The following 4 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000237The number of small exceedances. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$.
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