Your data matches 8 different statistics following compositions of up to 3 maps.
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Matching statistic: St001805
Mapping
St001805: Semistandard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => 0
[[2,2]]
 => 1
[[1],[2]]
 => 2
[[1,3]]
 => 0
[[2,3]]
 => 0
[[3,3]]
 => 1
[[1],[3]]
 => 2
[[2],[3]]
 => 2
[[1,1,2]]
 => 0
[[1,2,2]]
 => 0
[[2,2,2]]
 => 1
[[1,1],[2]]
 => 1
[[1,2],[2]]
 => 1
[[1,4]]
 => 0
[[2,4]]
 => 0
[[3,4]]
 => 0
[[4,4]]
 => 1
[[1],[4]]
 => 2
[[2],[4]]
 => 2
[[3],[4]]
 => 2
[[1,1,3]]
 => 0
[[1,2,3]]
 => 0
[[1,3,3]]
 => 0
[[2,2,3]]
 => 0
[[2,3,3]]
 => 0
[[3,3,3]]
 => 1
[[1,1],[3]]
 => 1
[[1,2],[3]]
 => 1
[[1,3],[2]]
 => 0
[[1,3],[3]]
 => 1
[[2,2],[3]]
 => 1
[[2,3],[3]]
 => 1
[[1],[2],[3]]
 => 3
[[1,1,1,2]]
 => 0
[[1,1,2,2]]
 => 0
[[1,2,2,2]]
 => 0
[[2,2,2,2]]
 => 1
[[1,1,1],[2]]
 => 1
[[1,1,2],[2]]
 => 1
[[1,2,2],[2]]
 => 1
[[1,1],[2,2]]
 => 2
[[1,5]]
 => 0
[[2,5]]
 => 0
[[3,5]]
 => 0
[[4,5]]
 => 0
[[5,5]]
 => 1
[[1],[5]]
 => 2
[[2],[5]]
 => 2
[[3],[5]]
 => 2
[[4],[5]]
 => 2
Description
The maximal overlap of a cylindrical tableau associated with a semistandard tableau. A cylindrical tableau associated with a semistandard Young tableau $T$ is the skew semistandard tableau obtained by gluing two copies of $T$ such that the inner shape is a rectangle. The overlap, recorded in this statistic, equals $\max_C\big(2\ell(T) - \ell(C)\big)$, where $\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux.
Matching statistic: St001645
(load all 2 compositions to match this statistic)
Mapping
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 60%
Values
[[1,2]]
 => [1,2] => [2] => ([],2)
 => ? = 0
[[2,2]]
 => [1,2] => [2] => ([],2)
 => ? = 1
[[1],[2]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 2
[[1,3]]
 => [1,2] => [2] => ([],2)
 => ? = 0
[[2,3]]
 => [1,2] => [2] => ([],2)
 => ? = 0
[[3,3]]
 => [1,2] => [2] => ([],2)
 => ? = 1
[[1],[3]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 2
[[2],[3]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 2
[[1,1,2]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0
[[1,2,2]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0
[[2,2,2]]
 => [1,2,3] => [3] => ([],3)
 => ? = 1
[[1,1],[2]]
 => [3,1,2] => [1,2] => ([(1,2)],3)
 => ? = 1
[[1,2],[2]]
 => [2,1,3] => [1,2] => ([(1,2)],3)
 => ? = 1
[[1,4]]
 => [1,2] => [2] => ([],2)
 => ? = 0
[[2,4]]
 => [1,2] => [2] => ([],2)
 => ? = 0
[[3,4]]
 => [1,2] => [2] => ([],2)
 => ? = 0
[[4,4]]
 => [1,2] => [2] => ([],2)
 => ? = 1
[[1],[4]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 2
[[2],[4]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 2
[[3],[4]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 2
[[1,1,3]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0
[[1,2,3]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0
[[1,3,3]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0
[[2,2,3]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0
[[2,3,3]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0
[[3,3,3]]
 => [1,2,3] => [3] => ([],3)
 => ? = 1
[[1,1],[3]]
 => [3,1,2] => [1,2] => ([(1,2)],3)
 => ? = 1
[[1,2],[3]]
 => [3,1,2] => [1,2] => ([(1,2)],3)
 => ? = 1
[[1,3],[2]]
 => [2,1,3] => [1,2] => ([(1,2)],3)
 => ? = 0
[[1,3],[3]]
 => [2,1,3] => [1,2] => ([(1,2)],3)
 => ? = 1
[[2,2],[3]]
 => [3,1,2] => [1,2] => ([(1,2)],3)
 => ? = 1
[[2,3],[3]]
 => [2,1,3] => [1,2] => ([(1,2)],3)
 => ? = 1
[[1],[2],[3]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,1,1,2]]
 => [1,2,3,4] => [4] => ([],4)
 => ? = 0
[[1,1,2,2]]
 => [1,2,3,4] => [4] => ([],4)
 => ? = 0
[[1,2,2,2]]
 => [1,2,3,4] => [4] => ([],4)
 => ? = 0
[[2,2,2,2]]
 => [1,2,3,4] => [4] => ([],4)
 => ? = 1
[[1,1,1],[2]]
 => [4,1,2,3] => [1,3] => ([(2,3)],4)
 => ? = 1
[[1,1,2],[2]]
 => [3,1,2,4] => [1,3] => ([(2,3)],4)
 => ? = 1
[[1,2,2],[2]]
 => [2,1,3,4] => [1,3] => ([(2,3)],4)
 => ? = 1
[[1,1],[2,2]]
 => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 2
[[1,5]]
 => [1,2] => [2] => ([],2)
 => ? = 0
[[2,5]]
 => [1,2] => [2] => ([],2)
 => ? = 0
[[3,5]]
 => [1,2] => [2] => ([],2)
 => ? = 0
[[4,5]]
 => [1,2] => [2] => ([],2)
 => ? = 0
[[5,5]]
 => [1,2] => [2] => ([],2)
 => ? = 1
[[1],[5]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 2
[[2],[5]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 2
[[3],[5]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 2
[[4],[5]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 2
[[1,1,4]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0
[[1,2,4]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0
[[1,3,4]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0
[[1,4,4]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0
[[2,2,4]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0
[[2,3,4]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0
[[2,4,4]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0
[[3,3,4]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0
[[3,4,4]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0
[[4,4,4]]
 => [1,2,3] => [3] => ([],3)
 => ? = 1
[[1,1],[4]]
 => [3,1,2] => [1,2] => ([(1,2)],3)
 => ? = 1
[[1],[2],[4]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1],[3],[4]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[[2],[3],[4]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1],[6]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 2
[[2],[6]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 2
[[3],[6]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 2
[[4],[6]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 2
[[5],[6]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 2
[[1],[2],[5]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1],[3],[5]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1],[4],[5]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[[2],[3],[5]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[[2],[4],[5]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[[3],[4],[5]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1],[2],[3],[4]]
 => [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[[1],[7]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 2
[[2],[7]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 2
[[3],[7]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 2
[[4],[7]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 2
[[5],[7]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 2
[[6],[7]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 2
[[1],[2],[6]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1],[3],[6]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1],[4],[6]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1],[5],[6]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[[2],[3],[6]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[[2],[4],[6]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[[2],[5],[6]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[[3],[4],[6]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[[3],[5],[6]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[[4],[5],[6]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1],[2],[3],[5]]
 => [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[[1],[2],[4],[5]]
 => [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[[1],[3],[4],[5]]
 => [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[[2],[3],[4],[5]]
 => [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[[1],[8]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 2
[[2],[8]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 2
[[3],[8]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 2
[[4],[8]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 2
Description
The pebbling number of a connected graph.
Matching statistic: St000771
Mapping
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 60%
Values
[[1,2]]
 => [1,2] => [2] => ([],2)
 => ? = 0 - 1
[[2,2]]
 => [1,2] => [2] => ([],2)
 => ? = 1 - 1
[[1],[2]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[[1,3]]
 => [1,2] => [2] => ([],2)
 => ? = 0 - 1
[[2,3]]
 => [1,2] => [2] => ([],2)
 => ? = 0 - 1
[[3,3]]
 => [1,2] => [2] => ([],2)
 => ? = 1 - 1
[[1],[3]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[[2],[3]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[[1,1,2]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0 - 1
[[1,2,2]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0 - 1
[[2,2,2]]
 => [1,2,3] => [3] => ([],3)
 => ? = 1 - 1
[[1,1],[2]]
 => [3,1,2] => [1,2] => ([(1,2)],3)
 => ? = 1 - 1
[[1,2],[2]]
 => [2,1,3] => [1,2] => ([(1,2)],3)
 => ? = 1 - 1
[[1,4]]
 => [1,2] => [2] => ([],2)
 => ? = 0 - 1
[[2,4]]
 => [1,2] => [2] => ([],2)
 => ? = 0 - 1
[[3,4]]
 => [1,2] => [2] => ([],2)
 => ? = 0 - 1
[[4,4]]
 => [1,2] => [2] => ([],2)
 => ? = 1 - 1
[[1],[4]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[[2],[4]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[[3],[4]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[[1,1,3]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0 - 1
[[1,2,3]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0 - 1
[[1,3,3]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0 - 1
[[2,2,3]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0 - 1
[[2,3,3]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0 - 1
[[3,3,3]]
 => [1,2,3] => [3] => ([],3)
 => ? = 1 - 1
[[1,1],[3]]
 => [3,1,2] => [1,2] => ([(1,2)],3)
 => ? = 1 - 1
[[1,2],[3]]
 => [3,1,2] => [1,2] => ([(1,2)],3)
 => ? = 1 - 1
[[1,3],[2]]
 => [2,1,3] => [1,2] => ([(1,2)],3)
 => ? = 0 - 1
[[1,3],[3]]
 => [2,1,3] => [1,2] => ([(1,2)],3)
 => ? = 1 - 1
[[2,2],[3]]
 => [3,1,2] => [1,2] => ([(1,2)],3)
 => ? = 1 - 1
[[2,3],[3]]
 => [2,1,3] => [1,2] => ([(1,2)],3)
 => ? = 1 - 1
[[1],[2],[3]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[[1,1,1,2]]
 => [1,2,3,4] => [4] => ([],4)
 => ? = 0 - 1
[[1,1,2,2]]
 => [1,2,3,4] => [4] => ([],4)
 => ? = 0 - 1
[[1,2,2,2]]
 => [1,2,3,4] => [4] => ([],4)
 => ? = 0 - 1
[[2,2,2,2]]
 => [1,2,3,4] => [4] => ([],4)
 => ? = 1 - 1
[[1,1,1],[2]]
 => [4,1,2,3] => [1,3] => ([(2,3)],4)
 => ? = 1 - 1
[[1,1,2],[2]]
 => [3,1,2,4] => [1,3] => ([(2,3)],4)
 => ? = 1 - 1
[[1,2,2],[2]]
 => [2,1,3,4] => [1,3] => ([(2,3)],4)
 => ? = 1 - 1
[[1,1],[2,2]]
 => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 2 - 1
[[1,5]]
 => [1,2] => [2] => ([],2)
 => ? = 0 - 1
[[2,5]]
 => [1,2] => [2] => ([],2)
 => ? = 0 - 1
[[3,5]]
 => [1,2] => [2] => ([],2)
 => ? = 0 - 1
[[4,5]]
 => [1,2] => [2] => ([],2)
 => ? = 0 - 1
[[5,5]]
 => [1,2] => [2] => ([],2)
 => ? = 1 - 1
[[1],[5]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[[2],[5]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[[3],[5]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[[4],[5]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[[1,1,4]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0 - 1
[[1,2,4]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0 - 1
[[1,3,4]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0 - 1
[[1,4,4]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0 - 1
[[2,2,4]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0 - 1
[[2,3,4]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0 - 1
[[2,4,4]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0 - 1
[[3,3,4]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0 - 1
[[3,4,4]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0 - 1
[[4,4,4]]
 => [1,2,3] => [3] => ([],3)
 => ? = 1 - 1
[[1,1],[4]]
 => [3,1,2] => [1,2] => ([(1,2)],3)
 => ? = 1 - 1
[[1],[2],[4]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[[1],[3],[4]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[[2],[3],[4]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[[1],[6]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[[2],[6]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[[3],[6]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[[4],[6]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[[5],[6]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[[1],[2],[5]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[[1],[3],[5]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[[1],[4],[5]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[[2],[3],[5]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[[2],[4],[5]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[[3],[4],[5]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[[1],[2],[3],[4]]
 => [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[[1],[7]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[[2],[7]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[[3],[7]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[[4],[7]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[[5],[7]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[[6],[7]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[[1],[2],[6]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[[1],[3],[6]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[[1],[4],[6]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[[1],[5],[6]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[[2],[3],[6]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[[2],[4],[6]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[[2],[5],[6]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[[3],[4],[6]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[[3],[5],[6]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[[4],[5],[6]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[[1],[2],[3],[5]]
 => [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[[1],[2],[4],[5]]
 => [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[[1],[3],[4],[5]]
 => [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[[2],[3],[4],[5]]
 => [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[[1],[8]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[[2],[8]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[[3],[8]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[[4],[8]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 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 $1$.
Matching statistic: St000772
Mapping
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 60%
Values
[[1,2]]
 => [1,2] => [2] => ([],2)
 => ? = 0 - 1
[[2,2]]
 => [1,2] => [2] => ([],2)
 => ? = 1 - 1
[[1],[2]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[[1,3]]
 => [1,2] => [2] => ([],2)
 => ? = 0 - 1
[[2,3]]
 => [1,2] => [2] => ([],2)
 => ? = 0 - 1
[[3,3]]
 => [1,2] => [2] => ([],2)
 => ? = 1 - 1
[[1],[3]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[[2],[3]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[[1,1,2]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0 - 1
[[1,2,2]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0 - 1
[[2,2,2]]
 => [1,2,3] => [3] => ([],3)
 => ? = 1 - 1
[[1,1],[2]]
 => [3,1,2] => [1,2] => ([(1,2)],3)
 => ? = 1 - 1
[[1,2],[2]]
 => [2,1,3] => [1,2] => ([(1,2)],3)
 => ? = 1 - 1
[[1,4]]
 => [1,2] => [2] => ([],2)
 => ? = 0 - 1
[[2,4]]
 => [1,2] => [2] => ([],2)
 => ? = 0 - 1
[[3,4]]
 => [1,2] => [2] => ([],2)
 => ? = 0 - 1
[[4,4]]
 => [1,2] => [2] => ([],2)
 => ? = 1 - 1
[[1],[4]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[[2],[4]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[[3],[4]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[[1,1,3]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0 - 1
[[1,2,3]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0 - 1
[[1,3,3]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0 - 1
[[2,2,3]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0 - 1
[[2,3,3]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0 - 1
[[3,3,3]]
 => [1,2,3] => [3] => ([],3)
 => ? = 1 - 1
[[1,1],[3]]
 => [3,1,2] => [1,2] => ([(1,2)],3)
 => ? = 1 - 1
[[1,2],[3]]
 => [3,1,2] => [1,2] => ([(1,2)],3)
 => ? = 1 - 1
[[1,3],[2]]
 => [2,1,3] => [1,2] => ([(1,2)],3)
 => ? = 0 - 1
[[1,3],[3]]
 => [2,1,3] => [1,2] => ([(1,2)],3)
 => ? = 1 - 1
[[2,2],[3]]
 => [3,1,2] => [1,2] => ([(1,2)],3)
 => ? = 1 - 1
[[2,3],[3]]
 => [2,1,3] => [1,2] => ([(1,2)],3)
 => ? = 1 - 1
[[1],[2],[3]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[[1,1,1,2]]
 => [1,2,3,4] => [4] => ([],4)
 => ? = 0 - 1
[[1,1,2,2]]
 => [1,2,3,4] => [4] => ([],4)
 => ? = 0 - 1
[[1,2,2,2]]
 => [1,2,3,4] => [4] => ([],4)
 => ? = 0 - 1
[[2,2,2,2]]
 => [1,2,3,4] => [4] => ([],4)
 => ? = 1 - 1
[[1,1,1],[2]]
 => [4,1,2,3] => [1,3] => ([(2,3)],4)
 => ? = 1 - 1
[[1,1,2],[2]]
 => [3,1,2,4] => [1,3] => ([(2,3)],4)
 => ? = 1 - 1
[[1,2,2],[2]]
 => [2,1,3,4] => [1,3] => ([(2,3)],4)
 => ? = 1 - 1
[[1,1],[2,2]]
 => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 2 - 1
[[1,5]]
 => [1,2] => [2] => ([],2)
 => ? = 0 - 1
[[2,5]]
 => [1,2] => [2] => ([],2)
 => ? = 0 - 1
[[3,5]]
 => [1,2] => [2] => ([],2)
 => ? = 0 - 1
[[4,5]]
 => [1,2] => [2] => ([],2)
 => ? = 0 - 1
[[5,5]]
 => [1,2] => [2] => ([],2)
 => ? = 1 - 1
[[1],[5]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[[2],[5]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[[3],[5]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[[4],[5]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[[1,1,4]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0 - 1
[[1,2,4]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0 - 1
[[1,3,4]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0 - 1
[[1,4,4]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0 - 1
[[2,2,4]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0 - 1
[[2,3,4]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0 - 1
[[2,4,4]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0 - 1
[[3,3,4]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0 - 1
[[3,4,4]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0 - 1
[[4,4,4]]
 => [1,2,3] => [3] => ([],3)
 => ? = 1 - 1
[[1,1],[4]]
 => [3,1,2] => [1,2] => ([(1,2)],3)
 => ? = 1 - 1
[[1],[2],[4]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[[1],[3],[4]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[[2],[3],[4]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[[1],[6]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[[2],[6]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[[3],[6]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[[4],[6]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[[5],[6]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[[1],[2],[5]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[[1],[3],[5]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[[1],[4],[5]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[[2],[3],[5]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[[2],[4],[5]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[[3],[4],[5]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[[1],[2],[3],[4]]
 => [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[[1],[7]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[[2],[7]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[[3],[7]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[[4],[7]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[[5],[7]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[[6],[7]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[[1],[2],[6]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[[1],[3],[6]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[[1],[4],[6]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[[1],[5],[6]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[[2],[3],[6]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[[2],[4],[6]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[[2],[5],[6]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[[3],[4],[6]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[[3],[5],[6]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[[4],[5],[6]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[[1],[2],[3],[5]]
 => [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[[1],[2],[4],[5]]
 => [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[[1],[3],[4],[5]]
 => [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[[2],[3],[4],[5]]
 => [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[[1],[8]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[[2],[8]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[[3],[8]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[[4],[8]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
Description
The multiplicity of the largest 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 $1$. The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore also statistic $1$. The graphs with statistic $n-1$, $n-2$ and $n-3$ have been characterised, see [1].
Matching statistic: St000777
Mapping
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 40%
Values
[[1,2]]
 => [1,2] => [2] => ([],2)
 => ? = 0
[[2,2]]
 => [1,2] => [2] => ([],2)
 => ? = 1
[[1],[2]]
 => [2,1] => [2] => ([],2)
 => ? = 2
[[1,3]]
 => [1,2] => [2] => ([],2)
 => ? = 0
[[2,3]]
 => [1,2] => [2] => ([],2)
 => ? = 0
[[3,3]]
 => [1,2] => [2] => ([],2)
 => ? = 1
[[1],[3]]
 => [2,1] => [2] => ([],2)
 => ? = 2
[[2],[3]]
 => [2,1] => [2] => ([],2)
 => ? = 2
[[1,1,2]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0
[[1,2,2]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0
[[2,2,2]]
 => [1,2,3] => [3] => ([],3)
 => ? = 1
[[1,1],[2]]
 => [3,1,2] => [3] => ([],3)
 => ? = 1
[[1,2],[2]]
 => [2,1,3] => [3] => ([],3)
 => ? = 1
[[1,4]]
 => [1,2] => [2] => ([],2)
 => ? = 0
[[2,4]]
 => [1,2] => [2] => ([],2)
 => ? = 0
[[3,4]]
 => [1,2] => [2] => ([],2)
 => ? = 0
[[4,4]]
 => [1,2] => [2] => ([],2)
 => ? = 1
[[1],[4]]
 => [2,1] => [2] => ([],2)
 => ? = 2
[[2],[4]]
 => [2,1] => [2] => ([],2)
 => ? = 2
[[3],[4]]
 => [2,1] => [2] => ([],2)
 => ? = 2
[[1,1,3]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0
[[1,2,3]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0
[[1,3,3]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0
[[2,2,3]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0
[[2,3,3]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0
[[3,3,3]]
 => [1,2,3] => [3] => ([],3)
 => ? = 1
[[1,1],[3]]
 => [3,1,2] => [3] => ([],3)
 => ? = 1
[[1,2],[3]]
 => [3,1,2] => [3] => ([],3)
 => ? = 1
[[1,3],[2]]
 => [2,1,3] => [3] => ([],3)
 => ? = 0
[[1,3],[3]]
 => [2,1,3] => [3] => ([],3)
 => ? = 1
[[2,2],[3]]
 => [3,1,2] => [3] => ([],3)
 => ? = 1
[[2,3],[3]]
 => [2,1,3] => [3] => ([],3)
 => ? = 1
[[1],[2],[3]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[[1,1,1,2]]
 => [1,2,3,4] => [4] => ([],4)
 => ? = 0
[[1,1,2,2]]
 => [1,2,3,4] => [4] => ([],4)
 => ? = 0
[[1,2,2,2]]
 => [1,2,3,4] => [4] => ([],4)
 => ? = 0
[[2,2,2,2]]
 => [1,2,3,4] => [4] => ([],4)
 => ? = 1
[[1,1,1],[2]]
 => [4,1,2,3] => [4] => ([],4)
 => ? = 1
[[1,1,2],[2]]
 => [3,1,2,4] => [4] => ([],4)
 => ? = 1
[[1,2,2],[2]]
 => [2,1,3,4] => [4] => ([],4)
 => ? = 1
[[1,1],[2,2]]
 => [3,4,1,2] => [4] => ([],4)
 => ? = 2
[[1,5]]
 => [1,2] => [2] => ([],2)
 => ? = 0
[[2,5]]
 => [1,2] => [2] => ([],2)
 => ? = 0
[[3,5]]
 => [1,2] => [2] => ([],2)
 => ? = 0
[[4,5]]
 => [1,2] => [2] => ([],2)
 => ? = 0
[[5,5]]
 => [1,2] => [2] => ([],2)
 => ? = 1
[[1],[5]]
 => [2,1] => [2] => ([],2)
 => ? = 2
[[2],[5]]
 => [2,1] => [2] => ([],2)
 => ? = 2
[[3],[5]]
 => [2,1] => [2] => ([],2)
 => ? = 2
[[4],[5]]
 => [2,1] => [2] => ([],2)
 => ? = 2
[[1,1,4]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0
[[1],[2],[4]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[[1],[3],[4]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[[2],[3],[4]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[[1],[2],[5]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[[1],[3],[5]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[[1],[4],[5]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[[2],[3],[5]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[[2],[4],[5]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[[3],[4],[5]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[[1],[2],[3],[4]]
 => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[[1],[2],[6]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[[1],[3],[6]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[[1],[4],[6]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[[1],[5],[6]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[[2],[3],[6]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[[2],[4],[6]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[[2],[5],[6]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[[3],[4],[6]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[[3],[5],[6]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[[4],[5],[6]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[[1],[2],[3],[5]]
 => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[[1],[2],[4],[5]]
 => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[[1],[3],[4],[5]]
 => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[[2],[3],[4],[5]]
 => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[[1],[2],[7]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[[1],[3],[7]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[[1],[4],[7]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[[1],[5],[7]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[[1],[6],[7]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[[2],[3],[7]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[[2],[4],[7]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[[2],[5],[7]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[[2],[6],[7]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[[3],[4],[7]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[[3],[5],[7]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[[3],[6],[7]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[[4],[5],[7]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[[4],[6],[7]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[[5],[6],[7]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Matching statistic: St001880
(load all 4 compositions to match this statistic)
Mapping
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 40%
Values
[[1,2]]
 => [1,2] => [1,2] => ([(0,1)],2)
 => ? = 0
[[2,2]]
 => [1,2] => [1,2] => ([(0,1)],2)
 => ? = 1
[[1],[2]]
 => [2,1] => [2,1] => ([],2)
 => ? = 2
[[1,3]]
 => [1,2] => [1,2] => ([(0,1)],2)
 => ? = 0
[[2,3]]
 => [1,2] => [1,2] => ([(0,1)],2)
 => ? = 0
[[3,3]]
 => [1,2] => [1,2] => ([(0,1)],2)
 => ? = 1
[[1],[3]]
 => [2,1] => [2,1] => ([],2)
 => ? = 2
[[2],[3]]
 => [2,1] => [2,1] => ([],2)
 => ? = 2
[[1,1,2]]
 => [1,2,3] => [3,2,1] => ([],3)
 => ? = 0
[[1,2,2]]
 => [1,2,3] => [3,2,1] => ([],3)
 => ? = 0
[[2,2,2]]
 => [1,2,3] => [3,2,1] => ([],3)
 => ? = 1
[[1,1],[2]]
 => [3,1,2] => [2,1,3] => ([(0,2),(1,2)],3)
 => ? = 1
[[1,2],[2]]
 => [2,1,3] => [3,1,2] => ([(1,2)],3)
 => ? = 1
[[1,4]]
 => [1,2] => [1,2] => ([(0,1)],2)
 => ? = 0
[[2,4]]
 => [1,2] => [1,2] => ([(0,1)],2)
 => ? = 0
[[3,4]]
 => [1,2] => [1,2] => ([(0,1)],2)
 => ? = 0
[[4,4]]
 => [1,2] => [1,2] => ([(0,1)],2)
 => ? = 1
[[1],[4]]
 => [2,1] => [2,1] => ([],2)
 => ? = 2
[[2],[4]]
 => [2,1] => [2,1] => ([],2)
 => ? = 2
[[3],[4]]
 => [2,1] => [2,1] => ([],2)
 => ? = 2
[[1,1,3]]
 => [1,2,3] => [3,2,1] => ([],3)
 => ? = 0
[[1,2,3]]
 => [1,2,3] => [3,2,1] => ([],3)
 => ? = 0
[[1,3,3]]
 => [1,2,3] => [3,2,1] => ([],3)
 => ? = 0
[[2,2,3]]
 => [1,2,3] => [3,2,1] => ([],3)
 => ? = 0
[[2,3,3]]
 => [1,2,3] => [3,2,1] => ([],3)
 => ? = 0
[[3,3,3]]
 => [1,2,3] => [3,2,1] => ([],3)
 => ? = 1
[[1,1],[3]]
 => [3,1,2] => [2,1,3] => ([(0,2),(1,2)],3)
 => ? = 1
[[1,2],[3]]
 => [3,1,2] => [2,1,3] => ([(0,2),(1,2)],3)
 => ? = 1
[[1,3],[2]]
 => [2,1,3] => [3,1,2] => ([(1,2)],3)
 => ? = 0
[[1,3],[3]]
 => [2,1,3] => [3,1,2] => ([(1,2)],3)
 => ? = 1
[[2,2],[3]]
 => [3,1,2] => [2,1,3] => ([(0,2),(1,2)],3)
 => ? = 1
[[2,3],[3]]
 => [2,1,3] => [3,1,2] => ([(1,2)],3)
 => ? = 1
[[1],[2],[3]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[[1,1,1,2]]
 => [1,2,3,4] => [4,2,3,1] => ([(2,3)],4)
 => ? = 0
[[1,1,2,2]]
 => [1,2,3,4] => [4,2,3,1] => ([(2,3)],4)
 => ? = 0
[[1,2,2,2]]
 => [1,2,3,4] => [4,2,3,1] => ([(2,3)],4)
 => ? = 0
[[2,2,2,2]]
 => [1,2,3,4] => [4,2,3,1] => ([(2,3)],4)
 => ? = 1
[[1,1,1],[2]]
 => [4,1,2,3] => [3,4,2,1] => ([(2,3)],4)
 => ? = 1
[[1,1,2],[2]]
 => [3,1,2,4] => [2,4,3,1] => ([(1,2),(1,3)],4)
 => ? = 1
[[1,2,2],[2]]
 => [2,1,3,4] => [4,1,2,3] => ([(1,2),(2,3)],4)
 => ? = 1
[[1,1],[2,2]]
 => [3,4,1,2] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ? = 2
[[1,5]]
 => [1,2] => [1,2] => ([(0,1)],2)
 => ? = 0
[[2,5]]
 => [1,2] => [1,2] => ([(0,1)],2)
 => ? = 0
[[3,5]]
 => [1,2] => [1,2] => ([(0,1)],2)
 => ? = 0
[[4,5]]
 => [1,2] => [1,2] => ([(0,1)],2)
 => ? = 0
[[5,5]]
 => [1,2] => [1,2] => ([(0,1)],2)
 => ? = 1
[[1],[5]]
 => [2,1] => [2,1] => ([],2)
 => ? = 2
[[2],[5]]
 => [2,1] => [2,1] => ([],2)
 => ? = 2
[[3],[5]]
 => [2,1] => [2,1] => ([],2)
 => ? = 2
[[4],[5]]
 => [2,1] => [2,1] => ([],2)
 => ? = 2
[[1,1,4]]
 => [1,2,3] => [3,2,1] => ([],3)
 => ? = 0
[[1],[2],[4]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[[1],[3],[4]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[[2],[3],[4]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[[1],[2],[5]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[[1],[3],[5]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[[1],[4],[5]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[[2],[3],[5]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[[2],[4],[5]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[[3],[4],[5]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[[1],[2],[3],[4]]
 => [4,3,2,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 4
[[1],[2],[6]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[[1],[3],[6]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[[1],[4],[6]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[[1],[5],[6]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[[2],[3],[6]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[[2],[4],[6]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[[2],[5],[6]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[[3],[4],[6]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[[3],[5],[6]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[[4],[5],[6]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[[1],[2],[3],[5]]
 => [4,3,2,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 4
[[1],[2],[4],[5]]
 => [4,3,2,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 4
[[1],[3],[4],[5]]
 => [4,3,2,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 4
[[2],[3],[4],[5]]
 => [4,3,2,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 4
[[1],[2],[7]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[[1],[3],[7]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[[1],[4],[7]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[[1],[5],[7]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[[1],[6],[7]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[[2],[3],[7]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[[2],[4],[7]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[[2],[5],[7]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[[2],[6],[7]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[[3],[4],[7]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[[3],[5],[7]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[[3],[6],[7]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[[4],[5],[7]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[[4],[6],[7]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[[5],[6],[7]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St001879
(load all 3 compositions to match this statistic)
Mapping
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 40%
Values
[[1,2]]
 => [1,2] => [2,1] => ([],2)
 => ? = 0 - 1
[[2,2]]
 => [1,2] => [2,1] => ([],2)
 => ? = 1 - 1
[[1],[2]]
 => [2,1] => [1,2] => ([(0,1)],2)
 => ? = 2 - 1
[[1,3]]
 => [1,2] => [2,1] => ([],2)
 => ? = 0 - 1
[[2,3]]
 => [1,2] => [2,1] => ([],2)
 => ? = 0 - 1
[[3,3]]
 => [1,2] => [2,1] => ([],2)
 => ? = 1 - 1
[[1],[3]]
 => [2,1] => [1,2] => ([(0,1)],2)
 => ? = 2 - 1
[[2],[3]]
 => [2,1] => [1,2] => ([(0,1)],2)
 => ? = 2 - 1
[[1,1,2]]
 => [1,2,3] => [3,2,1] => ([],3)
 => ? = 0 - 1
[[1,2,2]]
 => [1,2,3] => [3,2,1] => ([],3)
 => ? = 0 - 1
[[2,2,2]]
 => [1,2,3] => [3,2,1] => ([],3)
 => ? = 1 - 1
[[1,1],[2]]
 => [3,1,2] => [1,3,2] => ([(0,1),(0,2)],3)
 => ? = 1 - 1
[[1,2],[2]]
 => [2,1,3] => [2,3,1] => ([(1,2)],3)
 => ? = 1 - 1
[[1,4]]
 => [1,2] => [2,1] => ([],2)
 => ? = 0 - 1
[[2,4]]
 => [1,2] => [2,1] => ([],2)
 => ? = 0 - 1
[[3,4]]
 => [1,2] => [2,1] => ([],2)
 => ? = 0 - 1
[[4,4]]
 => [1,2] => [2,1] => ([],2)
 => ? = 1 - 1
[[1],[4]]
 => [2,1] => [1,2] => ([(0,1)],2)
 => ? = 2 - 1
[[2],[4]]
 => [2,1] => [1,2] => ([(0,1)],2)
 => ? = 2 - 1
[[3],[4]]
 => [2,1] => [1,2] => ([(0,1)],2)
 => ? = 2 - 1
[[1,1,3]]
 => [1,2,3] => [3,2,1] => ([],3)
 => ? = 0 - 1
[[1,2,3]]
 => [1,2,3] => [3,2,1] => ([],3)
 => ? = 0 - 1
[[1,3,3]]
 => [1,2,3] => [3,2,1] => ([],3)
 => ? = 0 - 1
[[2,2,3]]
 => [1,2,3] => [3,2,1] => ([],3)
 => ? = 0 - 1
[[2,3,3]]
 => [1,2,3] => [3,2,1] => ([],3)
 => ? = 0 - 1
[[3,3,3]]
 => [1,2,3] => [3,2,1] => ([],3)
 => ? = 1 - 1
[[1,1],[3]]
 => [3,1,2] => [1,3,2] => ([(0,1),(0,2)],3)
 => ? = 1 - 1
[[1,2],[3]]
 => [3,1,2] => [1,3,2] => ([(0,1),(0,2)],3)
 => ? = 1 - 1
[[1,3],[2]]
 => [2,1,3] => [2,3,1] => ([(1,2)],3)
 => ? = 0 - 1
[[1,3],[3]]
 => [2,1,3] => [2,3,1] => ([(1,2)],3)
 => ? = 1 - 1
[[2,2],[3]]
 => [3,1,2] => [1,3,2] => ([(0,1),(0,2)],3)
 => ? = 1 - 1
[[2,3],[3]]
 => [2,1,3] => [2,3,1] => ([(1,2)],3)
 => ? = 1 - 1
[[1],[2],[3]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[[1,1,1,2]]
 => [1,2,3,4] => [4,3,2,1] => ([],4)
 => ? = 0 - 1
[[1,1,2,2]]
 => [1,2,3,4] => [4,3,2,1] => ([],4)
 => ? = 0 - 1
[[1,2,2,2]]
 => [1,2,3,4] => [4,3,2,1] => ([],4)
 => ? = 0 - 1
[[2,2,2,2]]
 => [1,2,3,4] => [4,3,2,1] => ([],4)
 => ? = 1 - 1
[[1,1,1],[2]]
 => [4,1,2,3] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ? = 1 - 1
[[1,1,2],[2]]
 => [3,1,2,4] => [2,4,3,1] => ([(1,2),(1,3)],4)
 => ? = 1 - 1
[[1,2,2],[2]]
 => [2,1,3,4] => [3,4,2,1] => ([(2,3)],4)
 => ? = 1 - 1
[[1,1],[2,2]]
 => [3,4,1,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ? = 2 - 1
[[1,5]]
 => [1,2] => [2,1] => ([],2)
 => ? = 0 - 1
[[2,5]]
 => [1,2] => [2,1] => ([],2)
 => ? = 0 - 1
[[3,5]]
 => [1,2] => [2,1] => ([],2)
 => ? = 0 - 1
[[4,5]]
 => [1,2] => [2,1] => ([],2)
 => ? = 0 - 1
[[5,5]]
 => [1,2] => [2,1] => ([],2)
 => ? = 1 - 1
[[1],[5]]
 => [2,1] => [1,2] => ([(0,1)],2)
 => ? = 2 - 1
[[2],[5]]
 => [2,1] => [1,2] => ([(0,1)],2)
 => ? = 2 - 1
[[3],[5]]
 => [2,1] => [1,2] => ([(0,1)],2)
 => ? = 2 - 1
[[4],[5]]
 => [2,1] => [1,2] => ([(0,1)],2)
 => ? = 2 - 1
[[1,1,4]]
 => [1,2,3] => [3,2,1] => ([],3)
 => ? = 0 - 1
[[1],[2],[4]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[[1],[3],[4]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[[2],[3],[4]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[[1],[2],[5]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[[1],[3],[5]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[[1],[4],[5]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[[2],[3],[5]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[[2],[4],[5]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[[3],[4],[5]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[[1],[2],[3],[4]]
 => [4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[[1],[2],[6]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[[1],[3],[6]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[[1],[4],[6]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[[1],[5],[6]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[[2],[3],[6]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[[2],[4],[6]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[[2],[5],[6]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[[3],[4],[6]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[[3],[5],[6]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[[4],[5],[6]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[[1],[2],[3],[5]]
 => [4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[[1],[2],[4],[5]]
 => [4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[[1],[3],[4],[5]]
 => [4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[[2],[3],[4],[5]]
 => [4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[[1],[2],[7]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[[1],[3],[7]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[[1],[4],[7]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[[1],[5],[7]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[[1],[6],[7]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[[2],[3],[7]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[[2],[4],[7]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[[2],[5],[7]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[[2],[6],[7]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[[3],[4],[7]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[[3],[5],[7]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[[3],[6],[7]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[[4],[5],[7]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[[4],[6],[7]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[[5],[6],[7]]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
Description
The 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.
Matching statistic: St000456
Mapping
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 40%
Values
[[1,2]]
 => [1,2] => [2] => ([],2)
 => ? = 0 - 2
[[2,2]]
 => [1,2] => [2] => ([],2)
 => ? = 1 - 2
[[1],[2]]
 => [2,1] => [2] => ([],2)
 => ? = 2 - 2
[[1,3]]
 => [1,2] => [2] => ([],2)
 => ? = 0 - 2
[[2,3]]
 => [1,2] => [2] => ([],2)
 => ? = 0 - 2
[[3,3]]
 => [1,2] => [2] => ([],2)
 => ? = 1 - 2
[[1],[3]]
 => [2,1] => [2] => ([],2)
 => ? = 2 - 2
[[2],[3]]
 => [2,1] => [2] => ([],2)
 => ? = 2 - 2
[[1,1,2]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0 - 2
[[1,2,2]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0 - 2
[[2,2,2]]
 => [1,2,3] => [3] => ([],3)
 => ? = 1 - 2
[[1,1],[2]]
 => [3,1,2] => [3] => ([],3)
 => ? = 1 - 2
[[1,2],[2]]
 => [2,1,3] => [3] => ([],3)
 => ? = 1 - 2
[[1,4]]
 => [1,2] => [2] => ([],2)
 => ? = 0 - 2
[[2,4]]
 => [1,2] => [2] => ([],2)
 => ? = 0 - 2
[[3,4]]
 => [1,2] => [2] => ([],2)
 => ? = 0 - 2
[[4,4]]
 => [1,2] => [2] => ([],2)
 => ? = 1 - 2
[[1],[4]]
 => [2,1] => [2] => ([],2)
 => ? = 2 - 2
[[2],[4]]
 => [2,1] => [2] => ([],2)
 => ? = 2 - 2
[[3],[4]]
 => [2,1] => [2] => ([],2)
 => ? = 2 - 2
[[1,1,3]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0 - 2
[[1,2,3]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0 - 2
[[1,3,3]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0 - 2
[[2,2,3]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0 - 2
[[2,3,3]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0 - 2
[[3,3,3]]
 => [1,2,3] => [3] => ([],3)
 => ? = 1 - 2
[[1,1],[3]]
 => [3,1,2] => [3] => ([],3)
 => ? = 1 - 2
[[1,2],[3]]
 => [3,1,2] => [3] => ([],3)
 => ? = 1 - 2
[[1,3],[2]]
 => [2,1,3] => [3] => ([],3)
 => ? = 0 - 2
[[1,3],[3]]
 => [2,1,3] => [3] => ([],3)
 => ? = 1 - 2
[[2,2],[3]]
 => [3,1,2] => [3] => ([],3)
 => ? = 1 - 2
[[2,3],[3]]
 => [2,1,3] => [3] => ([],3)
 => ? = 1 - 2
[[1],[2],[3]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
[[1,1,1,2]]
 => [1,2,3,4] => [4] => ([],4)
 => ? = 0 - 2
[[1,1,2,2]]
 => [1,2,3,4] => [4] => ([],4)
 => ? = 0 - 2
[[1,2,2,2]]
 => [1,2,3,4] => [4] => ([],4)
 => ? = 0 - 2
[[2,2,2,2]]
 => [1,2,3,4] => [4] => ([],4)
 => ? = 1 - 2
[[1,1,1],[2]]
 => [4,1,2,3] => [4] => ([],4)
 => ? = 1 - 2
[[1,1,2],[2]]
 => [3,1,2,4] => [4] => ([],4)
 => ? = 1 - 2
[[1,2,2],[2]]
 => [2,1,3,4] => [4] => ([],4)
 => ? = 1 - 2
[[1,1],[2,2]]
 => [3,4,1,2] => [4] => ([],4)
 => ? = 2 - 2
[[1,5]]
 => [1,2] => [2] => ([],2)
 => ? = 0 - 2
[[2,5]]
 => [1,2] => [2] => ([],2)
 => ? = 0 - 2
[[3,5]]
 => [1,2] => [2] => ([],2)
 => ? = 0 - 2
[[4,5]]
 => [1,2] => [2] => ([],2)
 => ? = 0 - 2
[[5,5]]
 => [1,2] => [2] => ([],2)
 => ? = 1 - 2
[[1],[5]]
 => [2,1] => [2] => ([],2)
 => ? = 2 - 2
[[2],[5]]
 => [2,1] => [2] => ([],2)
 => ? = 2 - 2
[[3],[5]]
 => [2,1] => [2] => ([],2)
 => ? = 2 - 2
[[4],[5]]
 => [2,1] => [2] => ([],2)
 => ? = 2 - 2
[[1,1,4]]
 => [1,2,3] => [3] => ([],3)
 => ? = 0 - 2
[[1],[2],[4]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
[[1],[3],[4]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
[[2],[3],[4]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
[[1],[2],[5]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
[[1],[3],[5]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
[[1],[4],[5]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
[[2],[3],[5]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
[[2],[4],[5]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
[[3],[4],[5]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
[[1],[2],[3],[4]]
 => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[[1],[2],[6]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
[[1],[3],[6]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
[[1],[4],[6]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
[[1],[5],[6]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
[[2],[3],[6]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
[[2],[4],[6]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
[[2],[5],[6]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
[[3],[4],[6]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
[[3],[5],[6]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
[[4],[5],[6]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
[[1],[2],[3],[5]]
 => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[[1],[2],[4],[5]]
 => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[[1],[3],[4],[5]]
 => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[[2],[3],[4],[5]]
 => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[[1],[2],[7]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
[[1],[3],[7]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
[[1],[4],[7]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
[[1],[5],[7]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
[[1],[6],[7]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
[[2],[3],[7]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
[[2],[4],[7]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
[[2],[5],[7]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
[[2],[6],[7]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
[[3],[4],[7]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
[[3],[5],[7]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
[[3],[6],[7]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
[[4],[5],[7]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
[[4],[6],[7]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
[[5],[6],[7]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
Description
The monochromatic index of a connected graph. This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path. For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.