searching the database
Your data matches 58 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000182
St000182: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> 1
[1,1]
=> 1
[3]
=> 2
[2,1]
=> 3
[1,1,1]
=> 1
[4]
=> 6
[3,1]
=> 8
[2,2]
=> 3
[2,1,1]
=> 6
[1,1,1,1]
=> 1
[5]
=> 24
[4,1]
=> 30
[3,2]
=> 20
[3,1,1]
=> 20
[2,2,1]
=> 15
[2,1,1,1]
=> 10
[1,1,1,1,1]
=> 1
[6]
=> 120
[5,1]
=> 144
[4,2]
=> 90
[4,1,1]
=> 90
[3,3]
=> 40
[3,2,1]
=> 120
[3,1,1,1]
=> 40
[2,2,2]
=> 15
[2,2,1,1]
=> 45
[2,1,1,1,1]
=> 15
[1,1,1,1,1,1]
=> 1
[1,1,1,1,1,1,1]
=> 1
Description
The number of permutations whose cycle type is the given integer partition.
This number is given by
$$\{ \pi \in \mathfrak{S}_n : \text{type}(\pi) = \lambda\} = \frac{n!}{\lambda_1 \cdots \lambda_k \mu_1(\lambda)! \cdots \mu_n(\lambda)!}$$
where $\mu_j(\lambda)$ denotes the number of parts of $\lambda$ equal to $j$.
All permutations with the same cycle type form a [[wikipedia:Conjugacy class]].
Matching statistic: St000690
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000690: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00284: Standard tableaux —rows⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000690: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> [[1,2]]
=> {{1,2}}
=> [2,1] => 1
[1,1]
=> [[1],[2]]
=> {{1},{2}}
=> [1,2] => 1
[3]
=> [[1,2,3]]
=> {{1,2,3}}
=> [2,3,1] => 2
[2,1]
=> [[1,3],[2]]
=> {{1,3},{2}}
=> [3,2,1] => 3
[1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> [1,2,3] => 1
[4]
=> [[1,2,3,4]]
=> {{1,2,3,4}}
=> [2,3,4,1] => 6
[3,1]
=> [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> [3,2,4,1] => 8
[2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> [2,1,4,3] => 3
[2,1,1]
=> [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> [4,2,3,1] => 6
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> [1,2,3,4] => 1
[5]
=> [[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> [2,3,4,5,1] => 24
[4,1]
=> [[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> [3,2,4,5,1] => 30
[3,2]
=> [[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> [2,5,4,3,1] => 20
[3,1,1]
=> [[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> [4,2,3,5,1] => 20
[2,2,1]
=> [[1,3],[2,5],[4]]
=> {{1,3},{2,5},{4}}
=> [3,5,1,4,2] => 15
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> {{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => 10
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => 1
[6]
=> [[1,2,3,4,5,6]]
=> {{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => 120
[5,1]
=> [[1,3,4,5,6],[2]]
=> {{1,3,4,5,6},{2}}
=> [3,2,4,5,6,1] => 144
[4,2]
=> [[1,2,5,6],[3,4]]
=> {{1,2,5,6},{3,4}}
=> [2,5,4,3,6,1] => 90
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> {{1,4,5,6},{2},{3}}
=> [4,2,3,5,6,1] => 90
[3,3]
=> [[1,2,3],[4,5,6]]
=> {{1,2,3},{4,5,6}}
=> [2,3,1,5,6,4] => 40
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> {{1,3,6},{2,5},{4}}
=> [3,5,6,4,2,1] => 120
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> {{1,5,6},{2},{3},{4}}
=> [5,2,3,4,6,1] => 40
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> {{1,2},{3,4},{5,6}}
=> [2,1,4,3,6,5] => 15
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> {{1,4},{2,6},{3},{5}}
=> [4,6,3,1,5,2] => 45
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> {{1,6},{2},{3},{4},{5}}
=> [6,2,3,4,5,1] => 15
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> {{1},{2},{3},{4},{5},{6}}
=> [1,2,3,4,5,6] => 1
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> {{1},{2},{3},{4},{5},{6},{7}}
=> [1,2,3,4,5,6,7] => 1
Description
The size of the conjugacy class of a permutation.
Two permutations are conjugate if and only if they have the same cycle type, this statistic is then computed as described in [[St000182]].
Matching statistic: St000259
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 21%●distinct values known / distinct values provided: 7%
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 21%●distinct values known / distinct values provided: 7%
Values
[2]
=> [[1,2]]
=> [2] => ([],2)
=> ? = 1
[1,1]
=> [[1],[2]]
=> [1,1] => ([(0,1)],2)
=> 1
[3]
=> [[1,2,3]]
=> [3] => ([],3)
=> ? = 2
[2,1]
=> [[1,3],[2]]
=> [1,2] => ([(1,2)],3)
=> ? = 3
[1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[4]
=> [[1,2,3,4]]
=> [4] => ([],4)
=> ? = 6
[3,1]
=> [[1,3,4],[2]]
=> [1,3] => ([(2,3)],4)
=> ? = 8
[2,2]
=> [[1,2],[3,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 3
[2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 6
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[5]
=> [[1,2,3,4,5]]
=> [5] => ([],5)
=> ? = 24
[4,1]
=> [[1,3,4,5],[2]]
=> [1,4] => ([(3,4)],5)
=> ? = 30
[3,2]
=> [[1,2,5],[3,4]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 20
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? = 20
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 15
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 10
[1,1,1,1,1]
=> [[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)
=> 1
[6]
=> [[1,2,3,4,5,6]]
=> [6] => ([],6)
=> ? = 120
[5,1]
=> [[1,3,4,5,6],[2]]
=> [1,5] => ([(4,5)],6)
=> ? = 144
[4,2]
=> [[1,2,5,6],[3,4]]
=> [2,4] => ([(3,5),(4,5)],6)
=> ? = 90
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ? = 90
[3,3]
=> [[1,2,3],[4,5,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 40
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 120
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 40
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 15
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 45
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 15
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => ([(0,1),(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)
=> 1
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St000260
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 21%●distinct values known / distinct values provided: 7%
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 21%●distinct values known / distinct values provided: 7%
Values
[2]
=> [[1,2]]
=> [2] => ([],2)
=> ? = 1
[1,1]
=> [[1],[2]]
=> [1,1] => ([(0,1)],2)
=> 1
[3]
=> [[1,2,3]]
=> [3] => ([],3)
=> ? = 2
[2,1]
=> [[1,3],[2]]
=> [1,2] => ([(1,2)],3)
=> ? = 3
[1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[4]
=> [[1,2,3,4]]
=> [4] => ([],4)
=> ? = 6
[3,1]
=> [[1,3,4],[2]]
=> [1,3] => ([(2,3)],4)
=> ? = 8
[2,2]
=> [[1,2],[3,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 3
[2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 6
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[5]
=> [[1,2,3,4,5]]
=> [5] => ([],5)
=> ? = 24
[4,1]
=> [[1,3,4,5],[2]]
=> [1,4] => ([(3,4)],5)
=> ? = 30
[3,2]
=> [[1,2,5],[3,4]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 20
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? = 20
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 15
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 10
[1,1,1,1,1]
=> [[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)
=> 1
[6]
=> [[1,2,3,4,5,6]]
=> [6] => ([],6)
=> ? = 120
[5,1]
=> [[1,3,4,5,6],[2]]
=> [1,5] => ([(4,5)],6)
=> ? = 144
[4,2]
=> [[1,2,5,6],[3,4]]
=> [2,4] => ([(3,5),(4,5)],6)
=> ? = 90
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ? = 90
[3,3]
=> [[1,2,3],[4,5,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 40
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 120
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 40
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 15
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 45
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 15
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => ([(0,1),(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)
=> 1
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St000527
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000527: Posets ⟶ ℤResult quality: 13% ●values known / values provided: 21%●distinct values known / distinct values provided: 13%
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000527: Posets ⟶ ℤResult quality: 13% ●values known / values provided: 21%●distinct values known / distinct values provided: 13%
Values
[2]
=> 100 => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
[1,1]
=> 110 => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[3]
=> 1000 => 0010 => ([(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)
=> 3 = 2 + 1
[2,1]
=> 1010 => 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)
=> ? = 3 + 1
[1,1,1]
=> 1110 => 1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 2 = 1 + 1
[4]
=> 10000 => 00010 => ([(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)
=> ? = 6 + 1
[3,1]
=> 10010 => 00110 => ([(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)
=> ? = 8 + 1
[2,2]
=> 1100 => 1010 => ([(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
[2,1,1]
=> 10110 => 01110 => ([(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)
=> ? = 6 + 1
[1,1,1,1]
=> 11110 => 11110 => ([(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)
=> 2 = 1 + 1
[5]
=> 100000 => 000010 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ? = 24 + 1
[4,1]
=> 100010 => 000110 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ? = 30 + 1
[3,2]
=> 10100 => 10010 => ([(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)
=> ? = 20 + 1
[3,1,1]
=> 100110 => 001110 => ([(0,4),(0,5),(1,12),(2,3),(2,13),(2,16),(3,8),(3,14),(4,1),(4,9),(4,15),(5,2),(5,9),(5,15),(7,10),(8,11),(9,13),(10,6),(11,6),(12,7),(13,8),(14,10),(14,11),(15,12),(15,16),(16,7),(16,14)],17)
=> ? = 20 + 1
[2,2,1]
=> 11010 => 10110 => ([(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)
=> ? = 15 + 1
[2,1,1,1]
=> 101110 => 011110 => ([(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)
=> ? = 10 + 1
[1,1,1,1,1]
=> 111110 => 111110 => ([(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)
=> 2 = 1 + 1
[6]
=> 1000000 => 0000010 => ([(0,5),(0,6),(1,4),(1,14),(2,13),(3,2),(3,16),(4,3),(4,17),(5,12),(5,15),(6,1),(6,12),(6,15),(8,11),(9,10),(10,8),(11,7),(12,9),(13,7),(14,10),(14,17),(15,9),(15,14),(16,11),(16,13),(17,8),(17,16)],18)
=> ? = 120 + 1
[5,1]
=> 1000010 => 0000110 => ([(0,5),(0,6),(1,3),(1,15),(2,14),(3,2),(3,20),(4,8),(4,16),(5,1),(5,13),(5,19),(6,4),(6,13),(6,19),(8,9),(9,10),(10,11),(11,7),(12,7),(13,8),(14,12),(15,18),(15,20),(16,9),(16,18),(17,11),(17,12),(18,10),(18,17),(19,15),(19,16),(20,14),(20,17)],21)
=> ? = 144 + 1
[4,2]
=> 100100 => 100010 => ([(0,3),(0,4),(1,2),(1,11),(1,15),(2,7),(2,12),(3,13),(3,14),(4,1),(4,13),(4,14),(6,9),(7,10),(8,6),(9,5),(10,5),(11,7),(12,9),(12,10),(13,8),(13,15),(14,8),(14,11),(15,6),(15,12)],16)
=> ? = 90 + 1
[4,1,1]
=> 1000110 => 0001110 => ([(0,5),(0,6),(1,2),(1,20),(2,8),(3,4),(3,15),(3,21),(4,9),(4,17),(5,1),(5,10),(5,18),(6,3),(6,10),(6,18),(8,11),(9,12),(10,15),(11,13),(12,14),(13,7),(14,7),(15,9),(16,11),(16,19),(17,12),(17,19),(18,20),(18,21),(19,13),(19,14),(20,8),(20,16),(21,16),(21,17)],22)
=> ? = 90 + 1
[3,3]
=> 11000 => 01010 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ? = 40 + 1
[3,2,1]
=> 101010 => 100110 => ([(0,3),(0,4),(1,11),(1,16),(2,10),(2,15),(3,2),(3,13),(3,14),(4,1),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,15),(13,16),(14,10),(14,11),(15,6),(15,12),(16,7),(16,12)],17)
=> ? = 120 + 1
[3,1,1,1]
=> 1001110 => 0011110 => ([(0,5),(0,6),(1,14),(2,4),(2,9),(2,18),(3,16),(3,20),(4,3),(4,15),(4,19),(5,2),(5,10),(5,17),(6,1),(6,10),(6,17),(8,12),(9,15),(10,9),(11,8),(12,7),(13,7),(14,11),(15,16),(16,13),(17,14),(17,18),(18,11),(18,19),(19,8),(19,20),(20,12),(20,13)],21)
=> ? = 40 + 1
[2,2,2]
=> 11100 => 11010 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ? = 15 + 1
[2,2,1,1]
=> 110110 => 101110 => ([(0,3),(0,4),(1,2),(1,11),(1,15),(2,7),(2,12),(3,13),(3,14),(4,1),(4,13),(4,14),(6,9),(7,10),(8,6),(9,5),(10,5),(11,7),(12,9),(12,10),(13,8),(13,15),(14,8),(14,11),(15,6),(15,12)],16)
=> ? = 45 + 1
[2,1,1,1,1]
=> 1011110 => 0111110 => ([(0,5),(0,6),(1,4),(1,8),(1,9),(2,16),(2,17),(3,2),(3,12),(3,13),(4,3),(4,14),(4,15),(5,10),(5,11),(6,1),(6,10),(6,11),(8,15),(9,14),(10,9),(11,8),(12,16),(13,17),(14,12),(15,13),(16,7),(17,7)],18)
=> ? = 15 + 1
[1,1,1,1,1,1]
=> 1111110 => 1111110 => ([(0,2),(0,7),(1,9),(2,8),(3,4),(3,11),(4,6),(4,10),(5,3),(5,13),(6,1),(6,12),(7,5),(7,8),(8,13),(10,12),(11,10),(12,9),(13,11)],14)
=> ? = 1 + 1
[1,1,1,1,1,1,1]
=> 11111110 => 11111110 => ([(0,2),(0,8),(1,10),(2,9),(3,5),(3,11),(4,3),(4,13),(5,7),(5,12),(6,4),(6,15),(7,1),(7,14),(8,6),(8,9),(9,15),(11,12),(12,14),(13,11),(14,10),(15,13)],16)
=> ? = 1 + 1
Description
The width of the poset.
This is the size of the poset's longest antichain, also called Dilworth number.
Matching statistic: St000777
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 21%●distinct values known / distinct values provided: 7%
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 21%●distinct values known / distinct values provided: 7%
Values
[2]
=> [[1,2]]
=> [2] => ([],2)
=> ? = 1 + 1
[1,1]
=> [[1],[2]]
=> [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[3]
=> [[1,2,3]]
=> [3] => ([],3)
=> ? = 2 + 1
[2,1]
=> [[1,3],[2]]
=> [1,2] => ([(1,2)],3)
=> ? = 3 + 1
[1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[4]
=> [[1,2,3,4]]
=> [4] => ([],4)
=> ? = 6 + 1
[3,1]
=> [[1,3,4],[2]]
=> [1,3] => ([(2,3)],4)
=> ? = 8 + 1
[2,2]
=> [[1,2],[3,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 3 + 1
[2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 6 + 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[5]
=> [[1,2,3,4,5]]
=> [5] => ([],5)
=> ? = 24 + 1
[4,1]
=> [[1,3,4,5],[2]]
=> [1,4] => ([(3,4)],5)
=> ? = 30 + 1
[3,2]
=> [[1,2,5],[3,4]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 20 + 1
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? = 20 + 1
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 15 + 1
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 10 + 1
[1,1,1,1,1]
=> [[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)
=> 2 = 1 + 1
[6]
=> [[1,2,3,4,5,6]]
=> [6] => ([],6)
=> ? = 120 + 1
[5,1]
=> [[1,3,4,5,6],[2]]
=> [1,5] => ([(4,5)],6)
=> ? = 144 + 1
[4,2]
=> [[1,2,5,6],[3,4]]
=> [2,4] => ([(3,5),(4,5)],6)
=> ? = 90 + 1
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ? = 90 + 1
[3,3]
=> [[1,2,3],[4,5,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 40 + 1
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 120 + 1
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 40 + 1
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 15 + 1
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 45 + 1
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 15 + 1
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => ([(0,1),(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)
=> 2 = 1 + 1
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 1 + 1
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Matching statistic: St000068
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000068: Posets ⟶ ℤResult quality: 7% ●values known / values provided: 14%●distinct values known / distinct values provided: 7%
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000068: Posets ⟶ ℤResult quality: 7% ●values known / values provided: 14%●distinct values known / distinct values provided: 7%
Values
[2]
=> 100 => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[1,1]
=> 110 => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[3]
=> 1000 => 0010 => ([(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
[2,1]
=> 1010 => 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)
=> ? = 3
[1,1,1]
=> 1110 => 1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1
[4]
=> 10000 => 00010 => ([(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)
=> ? = 6
[3,1]
=> 10010 => 00110 => ([(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)
=> ? = 8
[2,2]
=> 1100 => 1010 => ([(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
[2,1,1]
=> 10110 => 01110 => ([(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)
=> ? = 6
[1,1,1,1]
=> 11110 => 11110 => ([(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)
=> 1
[5]
=> 100000 => 000010 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ? = 24
[4,1]
=> 100010 => 000110 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ? = 30
[3,2]
=> 10100 => 10010 => ([(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)
=> ? = 20
[3,1,1]
=> 100110 => 001110 => ([(0,4),(0,5),(1,12),(2,3),(2,13),(2,16),(3,8),(3,14),(4,1),(4,9),(4,15),(5,2),(5,9),(5,15),(7,10),(8,11),(9,13),(10,6),(11,6),(12,7),(13,8),(14,10),(14,11),(15,12),(15,16),(16,7),(16,14)],17)
=> ? = 20
[2,2,1]
=> 11010 => 10110 => ([(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)
=> ? = 15
[2,1,1,1]
=> 101110 => 011110 => ([(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)
=> ? = 10
[1,1,1,1,1]
=> 111110 => 111110 => ([(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)
=> ? = 1
[6]
=> 1000000 => 0000010 => ([(0,5),(0,6),(1,4),(1,14),(2,13),(3,2),(3,16),(4,3),(4,17),(5,12),(5,15),(6,1),(6,12),(6,15),(8,11),(9,10),(10,8),(11,7),(12,9),(13,7),(14,10),(14,17),(15,9),(15,14),(16,11),(16,13),(17,8),(17,16)],18)
=> ? = 120
[5,1]
=> 1000010 => 0000110 => ([(0,5),(0,6),(1,3),(1,15),(2,14),(3,2),(3,20),(4,8),(4,16),(5,1),(5,13),(5,19),(6,4),(6,13),(6,19),(8,9),(9,10),(10,11),(11,7),(12,7),(13,8),(14,12),(15,18),(15,20),(16,9),(16,18),(17,11),(17,12),(18,10),(18,17),(19,15),(19,16),(20,14),(20,17)],21)
=> ? = 144
[4,2]
=> 100100 => 100010 => ([(0,3),(0,4),(1,2),(1,11),(1,15),(2,7),(2,12),(3,13),(3,14),(4,1),(4,13),(4,14),(6,9),(7,10),(8,6),(9,5),(10,5),(11,7),(12,9),(12,10),(13,8),(13,15),(14,8),(14,11),(15,6),(15,12)],16)
=> ? = 90
[4,1,1]
=> 1000110 => 0001110 => ([(0,5),(0,6),(1,2),(1,20),(2,8),(3,4),(3,15),(3,21),(4,9),(4,17),(5,1),(5,10),(5,18),(6,3),(6,10),(6,18),(8,11),(9,12),(10,15),(11,13),(12,14),(13,7),(14,7),(15,9),(16,11),(16,19),(17,12),(17,19),(18,20),(18,21),(19,13),(19,14),(20,8),(20,16),(21,16),(21,17)],22)
=> ? = 90
[3,3]
=> 11000 => 01010 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ? = 40
[3,2,1]
=> 101010 => 100110 => ([(0,3),(0,4),(1,11),(1,16),(2,10),(2,15),(3,2),(3,13),(3,14),(4,1),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,15),(13,16),(14,10),(14,11),(15,6),(15,12),(16,7),(16,12)],17)
=> ? = 120
[3,1,1,1]
=> 1001110 => 0011110 => ([(0,5),(0,6),(1,14),(2,4),(2,9),(2,18),(3,16),(3,20),(4,3),(4,15),(4,19),(5,2),(5,10),(5,17),(6,1),(6,10),(6,17),(8,12),(9,15),(10,9),(11,8),(12,7),(13,7),(14,11),(15,16),(16,13),(17,14),(17,18),(18,11),(18,19),(19,8),(19,20),(20,12),(20,13)],21)
=> ? = 40
[2,2,2]
=> 11100 => 11010 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ? = 15
[2,2,1,1]
=> 110110 => 101110 => ([(0,3),(0,4),(1,2),(1,11),(1,15),(2,7),(2,12),(3,13),(3,14),(4,1),(4,13),(4,14),(6,9),(7,10),(8,6),(9,5),(10,5),(11,7),(12,9),(12,10),(13,8),(13,15),(14,8),(14,11),(15,6),(15,12)],16)
=> ? = 45
[2,1,1,1,1]
=> 1011110 => 0111110 => ([(0,5),(0,6),(1,4),(1,8),(1,9),(2,16),(2,17),(3,2),(3,12),(3,13),(4,3),(4,14),(4,15),(5,10),(5,11),(6,1),(6,10),(6,11),(8,15),(9,14),(10,9),(11,8),(12,16),(13,17),(14,12),(15,13),(16,7),(17,7)],18)
=> ? = 15
[1,1,1,1,1,1]
=> 1111110 => 1111110 => ([(0,2),(0,7),(1,9),(2,8),(3,4),(3,11),(4,6),(4,10),(5,3),(5,13),(6,1),(6,12),(7,5),(7,8),(8,13),(10,12),(11,10),(12,9),(13,11)],14)
=> ? = 1
[1,1,1,1,1,1,1]
=> 11111110 => 11111110 => ([(0,2),(0,8),(1,10),(2,9),(3,5),(3,11),(4,3),(4,13),(5,7),(5,12),(6,4),(6,15),(7,1),(7,14),(8,6),(8,9),(9,15),(11,12),(12,14),(13,11),(14,10),(15,13)],16)
=> ? = 1
Description
The number of minimal elements in a poset.
Matching statistic: St000232
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000232: Set partitions ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 13%
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000232: Set partitions ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 13%
Values
[2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> {{1,2,5},{3,4,6}}
=> 1
[1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> {{1,3,4},{2,5,6}}
=> 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> {{1,2,3,7},{4,5,6,8}}
=> ? = 2
[2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> {{1,3,5},{2,4,6}}
=> 3
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> {{1,3,4,5},{2,6,7,8}}
=> ? = 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> {{1,2,3,4,9},{5,6,7,8,10}}
=> ? = 6
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> {{1,2,4,7},{3,5,6,8}}
=> ? = 8
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> {{1,2,5,6},{3,4,7,8}}
=> ? = 3
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> {{1,3,4,6},{2,5,7,8}}
=> ? = 6
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> {{1,3,4,5,6},{2,7,8,9,10}}
=> ? = 1
[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]]
=> {{1,2,3,4,5,11},{6,7,8,9,10,12}}
=> ? = 24
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> {{1,2,3,5,9},{4,6,7,8,10}}
=> ? = 30
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> {{1,2,5,7},{3,4,6,8}}
=> ? = 20
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> {{1,3,4,7},{2,5,6,8}}
=> ? = 20
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> {{1,3,5,6},{2,4,7,8}}
=> ? = 15
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> {{1,3,4,5,7},{2,6,8,9,10}}
=> ? = 10
[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]]
=> {{1,3,4,5,6,7},{2,8,9,10,11,12}}
=> ? = 1
[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]]
=> {{1,2,3,4,5,6,13},{7,8,9,10,11,12,14}}
=> ? = 120
[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]]
=> {{1,2,3,4,6,11},{5,7,8,9,10,12}}
=> ? = 144
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> {{1,2,3,6,9},{4,5,7,8,10}}
=> ? = 90
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> {{1,2,4,5,9},{3,6,7,8,10}}
=> ? = 90
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> {{1,2,3,7,8},{4,5,6,9,10}}
=> ? = 40
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> {{1,3,5,7},{2,4,6,8}}
=> ? = 120
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> {{1,3,4,5,8},{2,6,7,9,10}}
=> ? = 40
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[1,2,5,6,7],[3,4,8,9,10]]
=> {{1,2,5,6,7},{3,4,8,9,10}}
=> ? = 15
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[1,3,4,6,7],[2,5,8,9,10]]
=> {{1,3,4,6,7},{2,5,8,9,10}}
=> ? = 45
[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]]
=> {{1,3,4,5,6,8},{2,7,9,10,11,12}}
=> ? = 15
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8],[2,9,10,11,12,13,14]]
=> {{1,3,4,5,6,7,8},{2,9,10,11,12,13,14}}
=> ? = 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8,9],[2,10,11,12,13,14,15,16]]
=> {{1,3,4,5,6,7,8,9},{2,10,11,12,13,14,15,16}}
=> ? = 1
Description
The number of crossings of a set partition.
This is given by the number of $i < i' < j < j'$ such that $i,j$ are two consecutive entries on one block, and $i',j'$ are consecutive entries in another block.
Matching statistic: St000236
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St000236: Permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 13%
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St000236: Permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 13%
Values
[2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => 1
[1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => ? = 2
[2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 3
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,6,7,8,5,4,3] => ? = 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> [5,6,7,8,4,3,2,1,10,9] => ? = 6
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => ? = 8
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [3,4,2,1,7,8,6,5] => ? = 3
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => ? = 6
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [2,1,7,8,9,10,6,5,4,3] => ? = 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
=> [6,7,8,9,10,5,4,3,2,1,12,11] => ? = 24
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> [4,6,7,3,8,5,2,1,10,9] => ? = 30
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => ? = 20
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => ? = 20
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,7,8,6,5] => ? = 15
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> [2,1,6,8,9,5,10,7,4,3] => ? = 10
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> [2,1,8,9,10,11,12,7,6,5,4,3] => ? = 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14)]
=> [7,8,9,10,11,12,6,5,4,3,2,1,14,13] => ? = 120
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7),(11,12)]
=> [5,7,8,9,4,10,6,3,2,1,12,11] => ? = 144
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> [4,5,7,3,2,8,6,1,10,9] => ? = 90
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> [3,6,2,7,8,5,4,1,10,9] => ? = 90
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> [4,5,6,3,2,1,9,10,8,7] => ? = 40
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => ? = 120
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [2,1,6,7,9,5,4,10,8,3] => ? = 40
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [3,4,2,1,8,9,10,7,6,5] => ? = 15
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> [2,1,5,8,4,9,10,7,6,3] => ? = 45
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
=> [2,1,7,9,10,11,6,12,8,5,4,3] => ? = 15
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> [2,1,9,10,11,12,13,14,8,7,6,5,4,3] => ? = 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [(1,2),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> [2,1,10,11,12,13,14,15,16,9,8,7,6,5,4,3] => ? = 1
Description
The number of cyclical small weak excedances.
A cyclical small weak excedance is an index $i$ such that $\pi_i \in \{ i,i+1 \}$ considered cyclically.
Matching statistic: St000239
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St000239: Permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 13%
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St000239: Permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 13%
Values
[2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => 1
[1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => ? = 2
[2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 3
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,6,7,8,5,4,3] => ? = 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> [5,6,7,8,4,3,2,1,10,9] => ? = 6
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => ? = 8
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [3,4,2,1,7,8,6,5] => ? = 3
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => ? = 6
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [2,1,7,8,9,10,6,5,4,3] => ? = 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
=> [6,7,8,9,10,5,4,3,2,1,12,11] => ? = 24
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> [4,6,7,3,8,5,2,1,10,9] => ? = 30
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => ? = 20
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => ? = 20
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,7,8,6,5] => ? = 15
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> [2,1,6,8,9,5,10,7,4,3] => ? = 10
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> [2,1,8,9,10,11,12,7,6,5,4,3] => ? = 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14)]
=> [7,8,9,10,11,12,6,5,4,3,2,1,14,13] => ? = 120
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7),(11,12)]
=> [5,7,8,9,4,10,6,3,2,1,12,11] => ? = 144
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> [4,5,7,3,2,8,6,1,10,9] => ? = 90
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> [3,6,2,7,8,5,4,1,10,9] => ? = 90
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> [4,5,6,3,2,1,9,10,8,7] => ? = 40
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => ? = 120
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [2,1,6,7,9,5,4,10,8,3] => ? = 40
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [3,4,2,1,8,9,10,7,6,5] => ? = 15
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> [2,1,5,8,4,9,10,7,6,3] => ? = 45
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
=> [2,1,7,9,10,11,6,12,8,5,4,3] => ? = 15
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> [2,1,9,10,11,12,13,14,8,7,6,5,4,3] => ? = 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [(1,2),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> [2,1,10,11,12,13,14,15,16,9,8,7,6,5,4,3] => ? = 1
Description
The number of small weak excedances.
A small weak excedance is an index $i$ such that $\pi_i \in \{i,i+1\}$.
This is the sum of [[St000022]] and [[St000237]].
The following 48 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000241The number of cyclical small excedances. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000782The indicator function of whether a given perfect matching is an L & P matching. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001625The Möbius invariant of a lattice. St001722The number of minimal chains with small intervals between a binary word and the top element. St000357The number of occurrences of the pattern 12-3. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000422The energy of a graph, if it is integral. St000646The number of big ascents of a permutation. St000650The number of 3-rises of a permutation. St000837The number of ascents of distance 2 of a permutation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001298The number of repeated entries in the Lehmer code of a permutation. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001760The number of prefix or suffix reversals needed to sort a permutation. St000037The sign of a permutation. St000348The non-inversion sum of a binary word. St000490The intertwining number of a set partition. St000691The number of changes of a binary word. St000717The number of ordinal summands of a poset. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St001313The number of Dyck paths above the lattice path given by a binary word. St001388The number of non-attacking neighbors of a permutation. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001641The number of ascent tops in the flattened set partition such that all smaller elements appear before. St000355The number of occurrences of the pattern 21-3. St000498The lcs statistic of a set partition. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000638The number of up-down runs of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001377The major index minus the number of inversions of a permutation. St001439The number of even weak deficiencies and of odd weak exceedences. St001536The number of cyclic misalignments of a permutation. St001821The sorting index of a signed permutation. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000520The number of patterns in a permutation. St001639The number of alternating subsets such that applying the permutation does not yield an alternating subset. St001168The vector space dimension of the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!