Your data matches 25 different statistics following compositions of up to 3 maps.
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Matching statistic: St001279
Mapping
St001279: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
 => 2
[1,1]
 => 0
[3]
 => 3
[2,1]
 => 2
[1,1,1]
 => 0
[4]
 => 4
[3,1]
 => 3
[2,2]
 => 4
[2,1,1]
 => 2
[1,1,1,1]
 => 0
[5]
 => 5
[4,1]
 => 4
[3,2]
 => 5
[3,1,1]
 => 3
[2,2,1]
 => 4
[2,1,1,1]
 => 2
[1,1,1,1,1]
 => 0
[6]
 => 6
[5,1]
 => 5
[4,2]
 => 6
[4,1,1]
 => 4
[3,3]
 => 6
[3,2,1]
 => 5
[3,1,1,1]
 => 3
[2,2,2]
 => 6
[2,2,1,1]
 => 4
[2,1,1,1,1]
 => 2
[1,1,1,1,1,1]
 => 0
[1,1,1,1,1,1,1]
 => 0
Description
The sum of the parts of an integer partition that are at least two.
Matching statistic: St000673
Mapping
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000673: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
 => [[1,2]]
 => {{1,2}}
 => [2,1] => 2
[1,1]
 => [[1],[2]]
 => {{1},{2}}
 => [1,2] => 0
[3]
 => [[1,2,3]]
 => {{1,2,3}}
 => [2,3,1] => 3
[2,1]
 => [[1,3],[2]]
 => {{1,3},{2}}
 => [3,2,1] => 2
[1,1,1]
 => [[1],[2],[3]]
 => {{1},{2},{3}}
 => [1,2,3] => 0
[4]
 => [[1,2,3,4]]
 => {{1,2,3,4}}
 => [2,3,4,1] => 4
[3,1]
 => [[1,3,4],[2]]
 => {{1,3,4},{2}}
 => [3,2,4,1] => 3
[2,2]
 => [[1,2],[3,4]]
 => {{1,2},{3,4}}
 => [2,1,4,3] => 4
[2,1,1]
 => [[1,4],[2],[3]]
 => {{1,4},{2},{3}}
 => [4,2,3,1] => 2
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => {{1},{2},{3},{4}}
 => [1,2,3,4] => 0
[5]
 => [[1,2,3,4,5]]
 => {{1,2,3,4,5}}
 => [2,3,4,5,1] => 5
[4,1]
 => [[1,3,4,5],[2]]
 => {{1,3,4,5},{2}}
 => [3,2,4,5,1] => 4
[3,2]
 => [[1,2,5],[3,4]]
 => {{1,2,5},{3,4}}
 => [2,5,4,3,1] => 5
[3,1,1]
 => [[1,4,5],[2],[3]]
 => {{1,4,5},{2},{3}}
 => [4,2,3,5,1] => 3
[2,2,1]
 => [[1,3],[2,5],[4]]
 => {{1,3},{2,5},{4}}
 => [3,5,1,4,2] => 4
[2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => {{1,5},{2},{3},{4}}
 => [5,2,3,4,1] => 2
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => {{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => 0
[6]
 => [[1,2,3,4,5,6]]
 => {{1,2,3,4,5,6}}
 => [2,3,4,5,6,1] => 6
[5,1]
 => [[1,3,4,5,6],[2]]
 => {{1,3,4,5,6},{2}}
 => [3,2,4,5,6,1] => 5
[4,2]
 => [[1,2,5,6],[3,4]]
 => {{1,2,5,6},{3,4}}
 => [2,5,4,3,6,1] => 6
[4,1,1]
 => [[1,4,5,6],[2],[3]]
 => {{1,4,5,6},{2},{3}}
 => [4,2,3,5,6,1] => 4
[3,3]
 => [[1,2,3],[4,5,6]]
 => {{1,2,3},{4,5,6}}
 => [2,3,1,5,6,4] => 6
[3,2,1]
 => [[1,3,6],[2,5],[4]]
 => {{1,3,6},{2,5},{4}}
 => [3,5,6,4,2,1] => 5
[3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => {{1,5,6},{2},{3},{4}}
 => [5,2,3,4,6,1] => 3
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => {{1,2},{3,4},{5,6}}
 => [2,1,4,3,6,5] => 6
[2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => {{1,4},{2,6},{3},{5}}
 => [4,6,3,1,5,2] => 4
[2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => {{1,6},{2},{3},{4},{5}}
 => [6,2,3,4,5,1] => 2
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => {{1},{2},{3},{4},{5},{6}}
 => [1,2,3,4,5,6] => 0
[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] => 0
Description
The number of non-fixed points of a permutation. In other words, this statistic is $n$ minus the number of fixed points ([[St000022]]) of $\pi$.
Matching statistic: St001255
Mapping
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001255: Dyck paths ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[2]
 => [[1,2]]
 => [1,2] => [1,0,1,0]
 => 3 = 2 + 1
[1,1]
 => [[1],[2]]
 => [2,1] => [1,1,0,0]
 => 1 = 0 + 1
[3]
 => [[1,2,3]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 4 = 3 + 1
[2,1]
 => [[1,3],[2]]
 => [2,1,3] => [1,1,0,0,1,0]
 => 3 = 2 + 1
[1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => [1,1,1,0,0,0]
 => 1 = 0 + 1
[4]
 => [[1,2,3,4]]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 4 = 3 + 1
[2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 5 = 4 + 1
[2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => 6 = 5 + 1
[4,1]
 => [[1,3,4,5],[2]]
 => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[3,2]
 => [[1,2,5],[3,4]]
 => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
 => 6 = 5 + 1
[3,1,1]
 => [[1,4,5],[2],[3]]
 => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => 4 = 3 + 1
[2,2,1]
 => [[1,3],[2,5],[4]]
 => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
 => 5 = 4 + 1
[2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => 3 = 2 + 1
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[6]
 => [[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
[5,1]
 => [[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 6 = 5 + 1
[4,2]
 => [[1,2,5,6],[3,4]]
 => [3,4,1,2,5,6] => [1,1,1,0,1,0,0,0,1,0,1,0]
 => 7 = 6 + 1
[4,1,1]
 => [[1,4,5,6],[2],[3]]
 => [3,2,1,4,5,6] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[3,3]
 => [[1,2,3],[4,5,6]]
 => [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 7 = 6 + 1
[3,2,1]
 => [[1,3,6],[2,5],[4]]
 => [4,2,5,1,3,6] => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 6 = 5 + 1
[3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => [4,3,2,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 4 = 3 + 1
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [5,6,3,4,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 7 = 6 + 1
[2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [5,3,2,6,1,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 5 = 4 + 1
[2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 3 = 2 + 1
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1 = 0 + 1
[1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0 + 1
Description
The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J.
Matching statistic: St000259
Mapping
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: 17% ●values known / values provided: 21%●distinct values known / distinct values provided: 17%
Values
[2]
 => [[1,2]]
 => [2] => ([],2)
 => ? = 2 + 1
[1,1]
 => [[1],[2]]
 => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
[3]
 => [[1,2,3]]
 => [3] => ([],3)
 => ? = 3 + 1
[2,1]
 => [[1,3],[2]]
 => [1,2] => ([(1,2)],3)
 => ? = 2 + 1
[1,1,1]
 => [[1],[2],[3]]
 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1 = 0 + 1
[4]
 => [[1,2,3,4]]
 => [4] => ([],4)
 => ? = 4 + 1
[3,1]
 => [[1,3,4],[2]]
 => [1,3] => ([(2,3)],4)
 => ? = 3 + 1
[2,2]
 => [[1,2],[3,4]]
 => [2,2] => ([(1,3),(2,3)],4)
 => ? = 4 + 1
[2,1,1]
 => [[1,4],[2],[3]]
 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 2 + 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)
 => 1 = 0 + 1
[5]
 => [[1,2,3,4,5]]
 => [5] => ([],5)
 => ? = 5 + 1
[4,1]
 => [[1,3,4,5],[2]]
 => [1,4] => ([(3,4)],5)
 => ? = 4 + 1
[3,2]
 => [[1,2,5],[3,4]]
 => [2,3] => ([(2,4),(3,4)],5)
 => ? = 5 + 1
[3,1,1]
 => [[1,4,5],[2],[3]]
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[2,2,1]
 => [[1,3],[2,5],[4]]
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 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)
 => ? = 2 + 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)
 => 1 = 0 + 1
[6]
 => [[1,2,3,4,5,6]]
 => [6] => ([],6)
 => ? = 6 + 1
[5,1]
 => [[1,3,4,5,6],[2]]
 => [1,5] => ([(4,5)],6)
 => ? = 5 + 1
[4,2]
 => [[1,2,5,6],[3,4]]
 => [2,4] => ([(3,5),(4,5)],6)
 => ? = 6 + 1
[4,1,1]
 => [[1,4,5,6],[2],[3]]
 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ? = 4 + 1
[3,3]
 => [[1,2,3],[4,5,6]]
 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ? = 6 + 1
[3,2,1]
 => [[1,3,6],[2,5],[4]]
 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 + 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)
 => ? = 3 + 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)
 => ? = 6 + 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)
 => ? = 4 + 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)
 => ? = 2 + 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)
 => 1 = 0 + 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 = 0 + 1
Description
The diameter of a connected graph. This is the greatest distance between any pair of vertices.
Matching statistic: St000260
Mapping
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: 17% ●values known / values provided: 21%●distinct values known / distinct values provided: 17%
Values
[2]
 => [[1,2]]
 => [2] => ([],2)
 => ? = 2 + 1
[1,1]
 => [[1],[2]]
 => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
[3]
 => [[1,2,3]]
 => [3] => ([],3)
 => ? = 3 + 1
[2,1]
 => [[1,3],[2]]
 => [1,2] => ([(1,2)],3)
 => ? = 2 + 1
[1,1,1]
 => [[1],[2],[3]]
 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1 = 0 + 1
[4]
 => [[1,2,3,4]]
 => [4] => ([],4)
 => ? = 4 + 1
[3,1]
 => [[1,3,4],[2]]
 => [1,3] => ([(2,3)],4)
 => ? = 3 + 1
[2,2]
 => [[1,2],[3,4]]
 => [2,2] => ([(1,3),(2,3)],4)
 => ? = 4 + 1
[2,1,1]
 => [[1,4],[2],[3]]
 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 2 + 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)
 => 1 = 0 + 1
[5]
 => [[1,2,3,4,5]]
 => [5] => ([],5)
 => ? = 5 + 1
[4,1]
 => [[1,3,4,5],[2]]
 => [1,4] => ([(3,4)],5)
 => ? = 4 + 1
[3,2]
 => [[1,2,5],[3,4]]
 => [2,3] => ([(2,4),(3,4)],5)
 => ? = 5 + 1
[3,1,1]
 => [[1,4,5],[2],[3]]
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[2,2,1]
 => [[1,3],[2,5],[4]]
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 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)
 => ? = 2 + 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)
 => 1 = 0 + 1
[6]
 => [[1,2,3,4,5,6]]
 => [6] => ([],6)
 => ? = 6 + 1
[5,1]
 => [[1,3,4,5,6],[2]]
 => [1,5] => ([(4,5)],6)
 => ? = 5 + 1
[4,2]
 => [[1,2,5,6],[3,4]]
 => [2,4] => ([(3,5),(4,5)],6)
 => ? = 6 + 1
[4,1,1]
 => [[1,4,5,6],[2],[3]]
 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ? = 4 + 1
[3,3]
 => [[1,2,3],[4,5,6]]
 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ? = 6 + 1
[3,2,1]
 => [[1,3,6],[2,5],[4]]
 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 + 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)
 => ? = 3 + 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)
 => ? = 6 + 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)
 => ? = 4 + 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)
 => ? = 2 + 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)
 => 1 = 0 + 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 = 0 + 1
Description
The radius of a connected graph. This is the minimum eccentricity of any vertex.
Matching statistic: St000777
Mapping
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: 17% ●values known / values provided: 21%●distinct values known / distinct values provided: 17%
Values
[2]
 => [[1,2]]
 => [2] => ([],2)
 => ? = 2 + 2
[1,1]
 => [[1],[2]]
 => [1,1] => ([(0,1)],2)
 => 2 = 0 + 2
[3]
 => [[1,2,3]]
 => [3] => ([],3)
 => ? = 3 + 2
[2,1]
 => [[1,3],[2]]
 => [1,2] => ([(1,2)],3)
 => ? = 2 + 2
[1,1,1]
 => [[1],[2],[3]]
 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 0 + 2
[4]
 => [[1,2,3,4]]
 => [4] => ([],4)
 => ? = 4 + 2
[3,1]
 => [[1,3,4],[2]]
 => [1,3] => ([(2,3)],4)
 => ? = 3 + 2
[2,2]
 => [[1,2],[3,4]]
 => [2,2] => ([(1,3),(2,3)],4)
 => ? = 4 + 2
[2,1,1]
 => [[1,4],[2],[3]]
 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 2 + 2
[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 = 0 + 2
[5]
 => [[1,2,3,4,5]]
 => [5] => ([],5)
 => ? = 5 + 2
[4,1]
 => [[1,3,4,5],[2]]
 => [1,4] => ([(3,4)],5)
 => ? = 4 + 2
[3,2]
 => [[1,2,5],[3,4]]
 => [2,3] => ([(2,4),(3,4)],5)
 => ? = 5 + 2
[3,1,1]
 => [[1,4,5],[2],[3]]
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 3 + 2
[2,2,1]
 => [[1,3],[2,5],[4]]
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 2
[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)
 => ? = 2 + 2
[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 = 0 + 2
[6]
 => [[1,2,3,4,5,6]]
 => [6] => ([],6)
 => ? = 6 + 2
[5,1]
 => [[1,3,4,5,6],[2]]
 => [1,5] => ([(4,5)],6)
 => ? = 5 + 2
[4,2]
 => [[1,2,5,6],[3,4]]
 => [2,4] => ([(3,5),(4,5)],6)
 => ? = 6 + 2
[4,1,1]
 => [[1,4,5,6],[2],[3]]
 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ? = 4 + 2
[3,3]
 => [[1,2,3],[4,5,6]]
 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ? = 6 + 2
[3,2,1]
 => [[1,3,6],[2,5],[4]]
 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 + 2
[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)
 => ? = 3 + 2
[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)
 => ? = 6 + 2
[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)
 => ? = 4 + 2
[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)
 => ? = 2 + 2
[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 = 0 + 2
[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 = 0 + 2
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Matching statistic: St001713
(load all 7 compositions to match this statistic)
Mapping
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St001713: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 50%
Values
[2]
 => [[1,2]]
 => [[2,0],[1]]
 => 2
[1,1]
 => [[1],[2]]
 => [[1,1],[1]]
 => 0
[3]
 => [[1,2,3]]
 => [[3,0,0],[2,0],[1]]
 => 3
[2,1]
 => [[1,2],[3]]
 => [[2,1,0],[2,0],[1]]
 => 2
[1,1,1]
 => [[1],[2],[3]]
 => [[1,1,1],[1,1],[1]]
 => 0
[4]
 => [[1,2,3,4]]
 => [[4,0,0,0],[3,0,0],[2,0],[1]]
 => ? = 4
[3,1]
 => [[1,2,3],[4]]
 => [[3,1,0,0],[3,0,0],[2,0],[1]]
 => ? = 3
[2,2]
 => [[1,2],[3,4]]
 => [[2,2,0,0],[2,1,0],[2,0],[1]]
 => ? = 4
[2,1,1]
 => [[1,2],[3],[4]]
 => [[2,1,1,0],[2,1,0],[2,0],[1]]
 => ? = 2
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,1,1,1],[1,1,1],[1,1],[1]]
 => ? = 0
[5]
 => [[1,2,3,4,5]]
 => [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => ? = 5
[4,1]
 => [[1,2,3,4],[5]]
 => [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => ? = 4
[3,2]
 => [[1,2,3],[4,5]]
 => [[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
 => ? = 5
[3,1,1]
 => [[1,2,3],[4],[5]]
 => [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
 => ? = 3
[2,2,1]
 => [[1,2],[3,4],[5]]
 => [[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
 => ? = 4
[2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
 => ? = 2
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
 => ? = 0
[6]
 => [[1,2,3,4,5,6]]
 => [[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => ? = 6
[5,1]
 => [[1,2,3,4,5],[6]]
 => [[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => ? = 5
[4,2]
 => [[1,2,3,4],[5,6]]
 => [[4,2,0,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => ? = 6
[4,1,1]
 => [[1,2,3,4],[5],[6]]
 => [[4,1,1,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => ? = 4
[3,3]
 => [[1,2,3],[4,5,6]]
 => [[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
 => ? = 6
[3,2,1]
 => [[1,2,3],[4,5],[6]]
 => [[3,2,1,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
 => ? = 5
[3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => [[3,1,1,1,0,0],[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
 => ? = 3
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [[2,2,2,0,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
 => ? = 6
[2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => [[2,2,1,1,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
 => ? = 4
[2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => [[2,1,1,1,1,0],[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
 => ? = 2
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [[1,1,1,1,1,1],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
 => ? = 0
[1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => [[1,1,1,1,1,1,1],[1,1,1,1,1,1],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
 => ? = 0
Description
The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern.
Matching statistic: St001232
(load all 3 compositions to match this statistic)
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 83%
Values
[2]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 2
[1,1]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => ? = 0
[3]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 3
[2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => ? = 2
[1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => ? = 0
[4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => ? = 3
[2,2]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? = 4
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => ? = 2
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => ? = 0
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 5
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => ? = 4
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => ? = 5
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => ? = 3
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => ? = 4
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => ? = 2
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 0
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => 6
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 5
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => ? = 6
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => ? = 4
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => ? = 6
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => ? = 5
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 3
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => ? = 6
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 4
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 2
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 0
[1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 0
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001645
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 83%
Values
[2]
 => [1,0,1,0]
 => [1,1] => ([(0,1)],2)
 => 2
[1,1]
 => [1,1,0,0]
 => [2] => ([],2)
 => ? = 0
[3]
 => [1,0,1,0,1,0]
 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[2,1]
 => [1,0,1,1,0,0]
 => [1,2] => ([(1,2)],3)
 => ? = 2
[1,1,1]
 => [1,1,0,1,0,0]
 => [3] => ([],3)
 => ? = 0
[4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 3
[2,2]
 => [1,1,1,0,0,0]
 => [3] => ([],3)
 => ? = 4
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3] => ([(2,3)],4)
 => ? = 2
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [4] => ([],4)
 => ? = 0
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [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)
 => 5
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,3] => ([(2,3)],4)
 => ? = 5
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 3
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [4] => ([],4)
 => ? = 4
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,4] => ([(3,4)],5)
 => ? = 2
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [5] => ([],5)
 => ? = 0
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [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)
 => 6
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [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)
 => ? = 5
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 6
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [4] => ([],4)
 => ? = 6
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4] => ([(3,4)],5)
 => ? = 5
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ? = 3
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [4] => ([],4)
 => ? = 6
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [5] => ([],5)
 => ? = 4
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,5] => ([(4,5)],6)
 => ? = 2
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [6] => ([],6)
 => ? = 0
[1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [7] => ([],7)
 => ? = 0
Description
The pebbling number of a connected graph.
Matching statistic: St000771
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 83%
Values
[2]
 => [1,0,1,0]
 => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[1,1]
 => [1,1,0,0]
 => [2] => ([],2)
 => ? = 0 - 1
[3]
 => [1,0,1,0,1,0]
 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[2,1]
 => [1,0,1,1,0,0]
 => [1,2] => ([(1,2)],3)
 => ? = 2 - 1
[1,1,1]
 => [1,1,0,1,0,0]
 => [3] => ([],3)
 => ? = 0 - 1
[4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 3 - 1
[2,2]
 => [1,1,1,0,0,0]
 => [3] => ([],3)
 => ? = 4 - 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3] => ([(2,3)],4)
 => ? = 2 - 1
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [4] => ([],4)
 => ? = 0 - 1
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [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)
 => 4 = 5 - 1
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 1
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,3] => ([(2,3)],4)
 => ? = 5 - 1
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 3 - 1
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [4] => ([],4)
 => ? = 4 - 1
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,4] => ([(3,4)],5)
 => ? = 2 - 1
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [5] => ([],5)
 => ? = 0 - 1
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [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)
 => 5 = 6 - 1
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [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)
 => ? = 5 - 1
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 6 - 1
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 1
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [4] => ([],4)
 => ? = 6 - 1
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4] => ([(3,4)],5)
 => ? = 5 - 1
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ? = 3 - 1
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [4] => ([],4)
 => ? = 6 - 1
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [5] => ([],5)
 => ? = 4 - 1
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,5] => ([(4,5)],6)
 => ? = 2 - 1
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [6] => ([],6)
 => ? = 0 - 1
[1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [7] => ([],7)
 => ? = 0 - 1
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 $1$.
The following 15 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000753The Grundy value for the game of Kayles on a binary word. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001060The distinguishing index of a graph. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000740The last entry of a permutation. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000355The number of occurrences of the pattern 21-3. St001288The number of primes obtained by multiplying preimage and image of a permutation and adding one. St000400The path length of an ordered tree.