Your data matches 47 different statistics following compositions of up to 3 maps.
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Matching statistic: St000173
Mapping
St000173: Semistandard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => 1
[[2,2]]
 => 1
[[1],[2]]
 => 0
[[1,3]]
 => 1
[[2,3]]
 => 2
[[3,3]]
 => 1
[[1],[3]]
 => 1
[[2],[3]]
 => 2
[[1,1,2]]
 => 1
[[1,2,2]]
 => 1
[[2,2,2]]
 => 1
[[1,1],[2]]
 => 0
[[1,2],[2]]
 => 1
[[1,4]]
 => 1
[[2,4]]
 => 2
[[3,4]]
 => 2
[[4,4]]
 => 1
[[1],[4]]
 => 1
[[2],[4]]
 => 2
[[3],[4]]
 => 2
[[1,1,3]]
 => 1
[[1,2,3]]
 => 2
[[1,3,3]]
 => 1
[[2,2,3]]
 => 2
[[2,3,3]]
 => 2
[[3,3,3]]
 => 1
[[1,1],[3]]
 => 1
[[1,2],[3]]
 => 2
[[1,3],[2]]
 => 1
[[1,3],[3]]
 => 2
[[2,2],[3]]
 => 2
[[2,3],[3]]
 => 3
[[1],[2],[3]]
 => 0
[[1,1,1,2]]
 => 1
[[1,1,2,2]]
 => 1
[[1,2,2,2]]
 => 1
[[2,2,2,2]]
 => 1
[[1,1,1],[2]]
 => 0
[[1,1,2],[2]]
 => 1
[[1,2,2],[2]]
 => 1
[[1,1],[2,2]]
 => 0
[[1,1,4]]
 => 1
[[1,2,4]]
 => 2
[[1,3,4]]
 => 2
[[1,4,4]]
 => 1
[[2,2,4]]
 => 2
[[2,3,4]]
 => 3
[[2,4,4]]
 => 2
[[3,3,4]]
 => 2
[[3,4,4]]
 => 2
Description
The segment statistic of a semistandard tableau. Let ''T'' be a tableau. A ''k''-segment of ''T'' (in the ''i''th row) is defined to be a maximal consecutive sequence of ''k''-boxes in the ith row. Note that the possible ''i''-boxes in the ''i''th row are not considered to be ''i''-segments. Then seg(''T'') is the total number of ''k''-segments in ''T'' as ''k'' varies over all possible values.
Matching statistic: St000264
(load all 5 compositions to match this statistic)
Mapping
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 29%
Values
[[1,2]]
 => [1,2] => ([],2)
 => ([],1)
 => ? ∊ {0,1,1}
[[2,2]]
 => [1,2] => ([],2)
 => ([],1)
 => ? ∊ {0,1,1}
[[1],[2]]
 => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => ? ∊ {0,1,1}
[[1,3]]
 => [1,2] => ([],2)
 => ([],1)
 => ? ∊ {1,1,1,2,2}
[[2,3]]
 => [1,2] => ([],2)
 => ([],1)
 => ? ∊ {1,1,1,2,2}
[[3,3]]
 => [1,2] => ([],2)
 => ([],1)
 => ? ∊ {1,1,1,2,2}
[[1],[3]]
 => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,2,2}
[[2],[3]]
 => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,2,2}
[[1,1,2]]
 => [1,2,3] => ([],3)
 => ([],1)
 => ? ∊ {0,1,1,1,1}
[[1,2,2]]
 => [1,2,3] => ([],3)
 => ([],1)
 => ? ∊ {0,1,1,1,1}
[[2,2,2]]
 => [1,2,3] => ([],3)
 => ([],1)
 => ? ∊ {0,1,1,1,1}
[[1,1],[2]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => ? ∊ {0,1,1,1,1}
[[1,2],[2]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {0,1,1,1,1}
[[1,4]]
 => [1,2] => ([],2)
 => ([],1)
 => ? ∊ {1,1,1,2,2,2,2}
[[2,4]]
 => [1,2] => ([],2)
 => ([],1)
 => ? ∊ {1,1,1,2,2,2,2}
[[3,4]]
 => [1,2] => ([],2)
 => ([],1)
 => ? ∊ {1,1,1,2,2,2,2}
[[4,4]]
 => [1,2] => ([],2)
 => ([],1)
 => ? ∊ {1,1,1,2,2,2,2}
[[1],[4]]
 => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,2,2,2,2}
[[2],[4]]
 => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,2,2,2,2}
[[3],[4]]
 => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,2,2,2,2}
[[1,1,3]]
 => [1,2,3] => ([],3)
 => ([],1)
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2}
[[1,2,3]]
 => [1,2,3] => ([],3)
 => ([],1)
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2}
[[1,3,3]]
 => [1,2,3] => ([],3)
 => ([],1)
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2}
[[2,2,3]]
 => [1,2,3] => ([],3)
 => ([],1)
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2}
[[2,3,3]]
 => [1,2,3] => ([],3)
 => ([],1)
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2}
[[3,3,3]]
 => [1,2,3] => ([],3)
 => ([],1)
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2}
[[1,1],[3]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2}
[[1,2],[3]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2}
[[1,3],[2]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2}
[[1,3],[3]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2}
[[2,2],[3]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2}
[[2,3],[3]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2}
[[1],[2],[3]]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,1,1,2]]
 => [1,2,3,4] => ([],4)
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,1,2,2]]
 => [1,2,3,4] => ([],4)
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,2,2,2]]
 => [1,2,3,4] => ([],4)
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[2,2,2,2]]
 => [1,2,3,4] => ([],4)
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,1,1],[2]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,1,2],[2]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,2,2],[2]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,1],[2,2]]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,1,4]]
 => [1,2,3] => ([],3)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[[1,2,4]]
 => [1,2,3] => ([],3)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[[1,3,4]]
 => [1,2,3] => ([],3)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[[1,4,4]]
 => [1,2,3] => ([],3)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[[2,2,4]]
 => [1,2,3] => ([],3)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[[2,3,4]]
 => [1,2,3] => ([],3)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[[2,4,4]]
 => [1,2,3] => ([],3)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[[3,3,4]]
 => [1,2,3] => ([],3)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[[3,4,4]]
 => [1,2,3] => ([],3)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[[4,4,4]]
 => [1,2,3] => ([],3)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[[1],[2],[4]]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1],[3],[4]]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[2],[3],[4]]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,1],[2],[3]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,2],[2],[3]]
 => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1,3],[2],[3]]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[[1,1],[2],[4]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,1],[3],[4]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,2],[2],[4]]
 => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1,2],[3],[4]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,3],[2],[4]]
 => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1,4],[2],[3]]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[[1,4],[2],[4]]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[[1,3],[3],[4]]
 => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1,4],[3],[4]]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[[2,2],[3],[4]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[2,3],[3],[4]]
 => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[2,4],[3],[4]]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[[1],[2],[3],[4]]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1,1,1],[2],[3]]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,1,2],[2],[3]]
 => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1,1,3],[2],[3]]
 => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[[1,2,2],[2],[3]]
 => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1,2,3],[2],[3]]
 => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,3,3],[2],[3]]
 => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[[1,1],[2,2],[3]]
 => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,1],[2,3],[3]]
 => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1,2],[2,3],[3]]
 => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 3
[[1,1,1],[2],[4]]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,1,1],[3],[4]]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,1,2],[2],[4]]
 => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1,1,2],[3],[4]]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,1,3],[2],[4]]
 => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1,1,4],[2],[3]]
 => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[[1,1,4],[2],[4]]
 => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[[1,1,3],[3],[4]]
 => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1,1,4],[3],[4]]
 => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[[1,2,2],[2],[4]]
 => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1,2,2],[3],[4]]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,2,3],[2],[4]]
 => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1,2,4],[2],[3]]
 => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,4],[2],[4]]
 => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,3],[3],[4]]
 => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1,3,3],[2],[4]]
 => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1,3,4],[2],[3]]
 => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[[1,2,4],[3],[4]]
 => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[[1,3,4],[2],[4]]
 => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,4,4],[2],[3]]
 => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[[1,4,4],[2],[4]]
 => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
Description
The girth of a graph, which is not a tree. This is the length of the shortest cycle in the graph.
Matching statistic: St000806
Mapping
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000806: Integer compositions ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 29%
Values
[[1,2]]
 => [1,2] => [2] => [1] => ? ∊ {0,1,1}
[[2,2]]
 => [1,2] => [2] => [1] => ? ∊ {0,1,1}
[[1],[2]]
 => [2,1] => [2] => [1] => ? ∊ {0,1,1}
[[1,3]]
 => [1,2] => [2] => [1] => ? ∊ {1,1,1,2,2}
[[2,3]]
 => [1,2] => [2] => [1] => ? ∊ {1,1,1,2,2}
[[3,3]]
 => [1,2] => [2] => [1] => ? ∊ {1,1,1,2,2}
[[1],[3]]
 => [2,1] => [2] => [1] => ? ∊ {1,1,1,2,2}
[[2],[3]]
 => [2,1] => [2] => [1] => ? ∊ {1,1,1,2,2}
[[1,1,2]]
 => [1,2,3] => [3] => [1] => ? ∊ {0,1,1,1,1}
[[1,2,2]]
 => [1,2,3] => [3] => [1] => ? ∊ {0,1,1,1,1}
[[2,2,2]]
 => [1,2,3] => [3] => [1] => ? ∊ {0,1,1,1,1}
[[1,1],[2]]
 => [3,1,2] => [3] => [1] => ? ∊ {0,1,1,1,1}
[[1,2],[2]]
 => [2,1,3] => [3] => [1] => ? ∊ {0,1,1,1,1}
[[1,4]]
 => [1,2] => [2] => [1] => ? ∊ {1,1,1,2,2,2,2}
[[2,4]]
 => [1,2] => [2] => [1] => ? ∊ {1,1,1,2,2,2,2}
[[3,4]]
 => [1,2] => [2] => [1] => ? ∊ {1,1,1,2,2,2,2}
[[4,4]]
 => [1,2] => [2] => [1] => ? ∊ {1,1,1,2,2,2,2}
[[1],[4]]
 => [2,1] => [2] => [1] => ? ∊ {1,1,1,2,2,2,2}
[[2],[4]]
 => [2,1] => [2] => [1] => ? ∊ {1,1,1,2,2,2,2}
[[3],[4]]
 => [2,1] => [2] => [1] => ? ∊ {1,1,1,2,2,2,2}
[[1,1,3]]
 => [1,2,3] => [3] => [1] => ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2}
[[1,2,3]]
 => [1,2,3] => [3] => [1] => ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2}
[[1,3,3]]
 => [1,2,3] => [3] => [1] => ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2}
[[2,2,3]]
 => [1,2,3] => [3] => [1] => ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2}
[[2,3,3]]
 => [1,2,3] => [3] => [1] => ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2}
[[3,3,3]]
 => [1,2,3] => [3] => [1] => ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2}
[[1,1],[3]]
 => [3,1,2] => [3] => [1] => ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2}
[[1,2],[3]]
 => [3,1,2] => [3] => [1] => ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2}
[[1,3],[2]]
 => [2,1,3] => [3] => [1] => ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2}
[[1,3],[3]]
 => [2,1,3] => [3] => [1] => ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2}
[[2,2],[3]]
 => [3,1,2] => [3] => [1] => ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2}
[[2,3],[3]]
 => [2,1,3] => [3] => [1] => ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2}
[[1],[2],[3]]
 => [3,2,1] => [2,1] => [1,1] => 3
[[1,1,1,2]]
 => [1,2,3,4] => [4] => [1] => ? ∊ {0,0,1,1,1,1,1,1}
[[1,1,2,2]]
 => [1,2,3,4] => [4] => [1] => ? ∊ {0,0,1,1,1,1,1,1}
[[1,2,2,2]]
 => [1,2,3,4] => [4] => [1] => ? ∊ {0,0,1,1,1,1,1,1}
[[2,2,2,2]]
 => [1,2,3,4] => [4] => [1] => ? ∊ {0,0,1,1,1,1,1,1}
[[1,1,1],[2]]
 => [4,1,2,3] => [4] => [1] => ? ∊ {0,0,1,1,1,1,1,1}
[[1,1,2],[2]]
 => [3,1,2,4] => [4] => [1] => ? ∊ {0,0,1,1,1,1,1,1}
[[1,2,2],[2]]
 => [2,1,3,4] => [4] => [1] => ? ∊ {0,0,1,1,1,1,1,1}
[[1,1],[2,2]]
 => [3,4,1,2] => [4] => [1] => ? ∊ {0,0,1,1,1,1,1,1}
[[1,1,4]]
 => [1,2,3] => [3] => [1] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[[1,2,4]]
 => [1,2,3] => [3] => [1] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[[1,3,4]]
 => [1,2,3] => [3] => [1] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[[1,4,4]]
 => [1,2,3] => [3] => [1] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[[2,2,4]]
 => [1,2,3] => [3] => [1] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[[2,3,4]]
 => [1,2,3] => [3] => [1] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[[2,4,4]]
 => [1,2,3] => [3] => [1] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[[3,3,4]]
 => [1,2,3] => [3] => [1] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[[3,4,4]]
 => [1,2,3] => [3] => [1] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[[4,4,4]]
 => [1,2,3] => [3] => [1] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[[1],[2],[4]]
 => [3,2,1] => [2,1] => [1,1] => 3
[[1],[3],[4]]
 => [3,2,1] => [2,1] => [1,1] => 3
[[2],[3],[4]]
 => [3,2,1] => [2,1] => [1,1] => 3
[[1,1],[2],[3]]
 => [4,3,1,2] => [1,3] => [1,1] => 3
[[1,2],[2],[3]]
 => [4,2,1,3] => [2,2] => [2] => 3
[[1,3],[2],[3]]
 => [3,2,1,4] => [2,2] => [2] => 3
[[1,1],[2],[4]]
 => [4,3,1,2] => [1,3] => [1,1] => 3
[[1,1],[3],[4]]
 => [4,3,1,2] => [1,3] => [1,1] => 3
[[1,2],[2],[4]]
 => [4,2,1,3] => [2,2] => [2] => 3
[[1,2],[3],[4]]
 => [4,3,1,2] => [1,3] => [1,1] => 3
[[1,3],[2],[4]]
 => [4,2,1,3] => [2,2] => [2] => 3
[[1,4],[2],[3]]
 => [3,2,1,4] => [2,2] => [2] => 3
[[1,4],[2],[4]]
 => [3,2,1,4] => [2,2] => [2] => 3
[[1,3],[3],[4]]
 => [4,2,1,3] => [2,2] => [2] => 3
[[1,4],[3],[4]]
 => [3,2,1,4] => [2,2] => [2] => 3
[[2,2],[3],[4]]
 => [4,3,1,2] => [1,3] => [1,1] => 3
[[2,3],[3],[4]]
 => [4,2,1,3] => [2,2] => [2] => 3
[[2,4],[3],[4]]
 => [3,2,1,4] => [2,2] => [2] => 3
[[1],[2],[3],[4]]
 => [4,3,2,1] => [1,2,1] => [1,1,1] => 4
[[1,1,1],[2],[3]]
 => [5,4,1,2,3] => [1,4] => [1,1] => 3
[[1,1,2],[2],[3]]
 => [5,3,1,2,4] => [1,4] => [1,1] => 3
[[1,1,3],[2],[3]]
 => [4,3,1,2,5] => [1,4] => [1,1] => 3
[[1,2,2],[2],[3]]
 => [5,2,1,3,4] => [2,3] => [1,1] => 3
[[1,2,3],[2],[3]]
 => [4,2,1,3,5] => [2,3] => [1,1] => 3
[[1,3,3],[2],[3]]
 => [3,2,1,4,5] => [2,3] => [1,1] => 3
[[1,1],[2,2],[3]]
 => [5,3,4,1,2] => [1,4] => [1,1] => 3
[[1,1],[2,3],[3]]
 => [4,3,5,1,2] => [1,4] => [1,1] => 3
[[1,2],[2,3],[3]]
 => [4,2,5,1,3] => [2,3] => [1,1] => 3
[[1,1,1],[2],[4]]
 => [5,4,1,2,3] => [1,4] => [1,1] => 3
[[1,1,1],[3],[4]]
 => [5,4,1,2,3] => [1,4] => [1,1] => 3
[[1,1,2],[2],[4]]
 => [5,3,1,2,4] => [1,4] => [1,1] => 3
[[1,1,2],[3],[4]]
 => [5,4,1,2,3] => [1,4] => [1,1] => 3
[[1,1,3],[2],[4]]
 => [5,3,1,2,4] => [1,4] => [1,1] => 3
[[1,1,4],[2],[3]]
 => [4,3,1,2,5] => [1,4] => [1,1] => 3
[[1,1,4],[2],[4]]
 => [4,3,1,2,5] => [1,4] => [1,1] => 3
[[1,1,3],[3],[4]]
 => [5,3,1,2,4] => [1,4] => [1,1] => 3
[[1,1,4],[3],[4]]
 => [4,3,1,2,5] => [1,4] => [1,1] => 3
[[1,2,2],[2],[4]]
 => [5,2,1,3,4] => [2,3] => [1,1] => 3
[[1,2,2],[3],[4]]
 => [5,4,1,2,3] => [1,4] => [1,1] => 3
[[1,2,3],[2],[4]]
 => [5,2,1,3,4] => [2,3] => [1,1] => 3
[[1,2,4],[2],[3]]
 => [4,2,1,3,5] => [2,3] => [1,1] => 3
[[1,2,4],[2],[4]]
 => [4,2,1,3,5] => [2,3] => [1,1] => 3
[[1,2,3],[3],[4]]
 => [5,3,1,2,4] => [1,4] => [1,1] => 3
[[1,3,3],[2],[4]]
 => [5,2,1,3,4] => [2,3] => [1,1] => 3
[[1,3,4],[2],[3]]
 => [3,2,1,4,5] => [2,3] => [1,1] => 3
[[1,2,4],[3],[4]]
 => [4,3,1,2,5] => [1,4] => [1,1] => 3
[[1,3,4],[2],[4]]
 => [4,2,1,3,5] => [2,3] => [1,1] => 3
[[1,4,4],[2],[3]]
 => [3,2,1,4,5] => [2,3] => [1,1] => 3
[[1,4,4],[2],[4]]
 => [3,2,1,4,5] => [2,3] => [1,1] => 3
Description
The semiperimeter of the associated bargraph. Interpret the composition as the sequence of heights of the bars of a bargraph. This statistic is the semiperimeter of the polygon determined by the axis and the bargraph. Put differently, it is the sum of the number of up steps and the number of horizontal steps when regarding the bargraph as a path with up, horizontal and down steps.
Matching statistic: St001060
Mapping
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 14% ●values known / values provided: 24%●distinct values known / distinct values provided: 14%
Values
[[1,2]]
 => [1,2] => ([],2)
 => ([],1)
 => ? ∊ {0,1,1}
[[2,2]]
 => [1,2] => ([],2)
 => ([],1)
 => ? ∊ {0,1,1}
[[1],[2]]
 => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => ? ∊ {0,1,1}
[[1,3]]
 => [1,2] => ([],2)
 => ([],1)
 => ? ∊ {1,1,1,2,2}
[[2,3]]
 => [1,2] => ([],2)
 => ([],1)
 => ? ∊ {1,1,1,2,2}
[[3,3]]
 => [1,2] => ([],2)
 => ([],1)
 => ? ∊ {1,1,1,2,2}
[[1],[3]]
 => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,2,2}
[[2],[3]]
 => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,2,2}
[[1,1,2]]
 => [1,2,3] => ([],3)
 => ([],1)
 => ? ∊ {0,1,1,1,1}
[[1,2,2]]
 => [1,2,3] => ([],3)
 => ([],1)
 => ? ∊ {0,1,1,1,1}
[[2,2,2]]
 => [1,2,3] => ([],3)
 => ([],1)
 => ? ∊ {0,1,1,1,1}
[[1,1],[2]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => ? ∊ {0,1,1,1,1}
[[1,2],[2]]
 => [2,1,3] => ([(1,2)],3)
 => ([(0,1)],2)
 => ? ∊ {0,1,1,1,1}
[[1,4]]
 => [1,2] => ([],2)
 => ([],1)
 => ? ∊ {1,1,1,2,2,2,2}
[[2,4]]
 => [1,2] => ([],2)
 => ([],1)
 => ? ∊ {1,1,1,2,2,2,2}
[[3,4]]
 => [1,2] => ([],2)
 => ([],1)
 => ? ∊ {1,1,1,2,2,2,2}
[[4,4]]
 => [1,2] => ([],2)
 => ([],1)
 => ? ∊ {1,1,1,2,2,2,2}
[[1],[4]]
 => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,2,2,2,2}
[[2],[4]]
 => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,2,2,2,2}
[[3],[4]]
 => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,2,2,2,2}
[[1,1,3]]
 => [1,2,3] => ([],3)
 => ([],1)
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2}
[[1,2,3]]
 => [1,2,3] => ([],3)
 => ([],1)
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2}
[[1,3,3]]
 => [1,2,3] => ([],3)
 => ([],1)
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2}
[[2,2,3]]
 => [1,2,3] => ([],3)
 => ([],1)
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2}
[[2,3,3]]
 => [1,2,3] => ([],3)
 => ([],1)
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2}
[[3,3,3]]
 => [1,2,3] => ([],3)
 => ([],1)
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2}
[[1,1],[3]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2}
[[1,2],[3]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2}
[[1,3],[2]]
 => [2,1,3] => ([(1,2)],3)
 => ([(0,1)],2)
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2}
[[1,3],[3]]
 => [2,1,3] => ([(1,2)],3)
 => ([(0,1)],2)
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2}
[[2,2],[3]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2}
[[2,3],[3]]
 => [2,1,3] => ([(1,2)],3)
 => ([(0,1)],2)
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2}
[[1],[2],[3]]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,1,1,2]]
 => [1,2,3,4] => ([],4)
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,1,2,2]]
 => [1,2,3,4] => ([],4)
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,2,2,2]]
 => [1,2,3,4] => ([],4)
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[2,2,2,2]]
 => [1,2,3,4] => ([],4)
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,1,1],[2]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,1,2],[2]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,2,2],[2]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(0,1)],2)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,1],[2,2]]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,1,4]]
 => [1,2,3] => ([],3)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[[1,2,4]]
 => [1,2,3] => ([],3)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[[1,3,4]]
 => [1,2,3] => ([],3)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[[1,4,4]]
 => [1,2,3] => ([],3)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[[2,2,4]]
 => [1,2,3] => ([],3)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[[2,3,4]]
 => [1,2,3] => ([],3)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[[2,4,4]]
 => [1,2,3] => ([],3)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[[3,3,4]]
 => [1,2,3] => ([],3)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[[3,4,4]]
 => [1,2,3] => ([],3)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[[4,4,4]]
 => [1,2,3] => ([],3)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[[1],[2],[4]]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1],[3],[4]]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[2],[3],[4]]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,1],[2],[3]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,2],[2],[3]]
 => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,3],[2],[3]]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,1],[2],[4]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,1],[3],[4]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,2],[2],[4]]
 => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,2],[3],[4]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,3],[2],[4]]
 => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,4],[2],[3]]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,4],[2],[4]]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,3],[3],[4]]
 => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,4],[3],[4]]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[2,2],[3],[4]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[2,3],[3],[4]]
 => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[2,4],[3],[4]]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1],[2],[3],[4]]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1,1,1],[2],[3]]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,1,2],[2],[3]]
 => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,1,3],[2],[3]]
 => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,2,2],[2],[3]]
 => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,2,3],[2],[3]]
 => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,3,3],[2],[3]]
 => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,1],[2,2],[3]]
 => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,1],[2,3],[3]]
 => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,2],[2,3],[3]]
 => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,1,1],[2],[4]]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,1,1],[3],[4]]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,1,2],[2],[4]]
 => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,1,2],[3],[4]]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,1,3],[2],[4]]
 => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,1,4],[2],[3]]
 => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,1,4],[2],[4]]
 => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,1,3],[3],[4]]
 => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,1,4],[3],[4]]
 => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,2,2],[2],[4]]
 => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,2,2],[3],[4]]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,2,3],[2],[4]]
 => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,2,4],[2],[3]]
 => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,2,4],[2],[4]]
 => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,2,3],[3],[4]]
 => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,3,3],[2],[4]]
 => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,3,4],[2],[3]]
 => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,2,4],[3],[4]]
 => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,3,4],[2],[4]]
 => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,4,4],[2],[3]]
 => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,4,4],[2],[4]]
 => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
Description
The distinguishing index of a graph. This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism. If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.
Matching statistic: St000259
(load all 5 compositions to match this statistic)
Mapping
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 29%
Values
[[1,2]]
 => [[1,2]]
 => [1,2] => ([],2)
 => ? ∊ {0,1,1}
[[2,2]]
 => [[2,2]]
 => [1,2] => ([],2)
 => ? ∊ {0,1,1}
[[1],[2]]
 => [[1,2]]
 => [1,2] => ([],2)
 => ? ∊ {0,1,1}
[[1,3]]
 => [[1,3]]
 => [1,2] => ([],2)
 => ? ∊ {1,1,1,2,2}
[[2,3]]
 => [[2,3]]
 => [1,2] => ([],2)
 => ? ∊ {1,1,1,2,2}
[[3,3]]
 => [[3,3]]
 => [1,2] => ([],2)
 => ? ∊ {1,1,1,2,2}
[[1],[3]]
 => [[1,3]]
 => [1,2] => ([],2)
 => ? ∊ {1,1,1,2,2}
[[2],[3]]
 => [[2,3]]
 => [1,2] => ([],2)
 => ? ∊ {1,1,1,2,2}
[[1,1,2]]
 => [[1,1,2]]
 => [1,2,3] => ([],3)
 => ? ∊ {0,1,1,1,1}
[[1,2,2]]
 => [[1,2,2]]
 => [1,2,3] => ([],3)
 => ? ∊ {0,1,1,1,1}
[[2,2,2]]
 => [[2,2,2]]
 => [1,2,3] => ([],3)
 => ? ∊ {0,1,1,1,1}
[[1,1],[2]]
 => [[1,1,2]]
 => [1,2,3] => ([],3)
 => ? ∊ {0,1,1,1,1}
[[1,2],[2]]
 => [[1,2,2]]
 => [1,2,3] => ([],3)
 => ? ∊ {0,1,1,1,1}
[[1,4]]
 => [[1,4]]
 => [1,2] => ([],2)
 => ? ∊ {1,1,1,2,2,2,2}
[[2,4]]
 => [[2,4]]
 => [1,2] => ([],2)
 => ? ∊ {1,1,1,2,2,2,2}
[[3,4]]
 => [[3,4]]
 => [1,2] => ([],2)
 => ? ∊ {1,1,1,2,2,2,2}
[[4,4]]
 => [[4,4]]
 => [1,2] => ([],2)
 => ? ∊ {1,1,1,2,2,2,2}
[[1],[4]]
 => [[1,4]]
 => [1,2] => ([],2)
 => ? ∊ {1,1,1,2,2,2,2}
[[2],[4]]
 => [[2,4]]
 => [1,2] => ([],2)
 => ? ∊ {1,1,1,2,2,2,2}
[[3],[4]]
 => [[3,4]]
 => [1,2] => ([],2)
 => ? ∊ {1,1,1,2,2,2,2}
[[1,1,3]]
 => [[1,1,3]]
 => [1,2,3] => ([],3)
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,3}
[[1,2,3]]
 => [[1,2,3]]
 => [1,2,3] => ([],3)
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,3}
[[1,3,3]]
 => [[1,3,3]]
 => [1,2,3] => ([],3)
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,3}
[[2,2,3]]
 => [[2,2,3]]
 => [1,2,3] => ([],3)
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,3}
[[2,3,3]]
 => [[2,3,3]]
 => [1,2,3] => ([],3)
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,3}
[[3,3,3]]
 => [[3,3,3]]
 => [1,2,3] => ([],3)
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,3}
[[1,1],[3]]
 => [[1,1,3]]
 => [1,2,3] => ([],3)
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,3}
[[1,2],[3]]
 => [[1,2,3]]
 => [1,2,3] => ([],3)
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,3}
[[1,3],[2]]
 => [[1,2],[3]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 2
[[1,3],[3]]
 => [[1,3,3]]
 => [1,2,3] => ([],3)
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,3}
[[2,2],[3]]
 => [[2,2,3]]
 => [1,2,3] => ([],3)
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,3}
[[2,3],[3]]
 => [[2,3,3]]
 => [1,2,3] => ([],3)
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,3}
[[1],[2],[3]]
 => [[1,2],[3]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 2
[[1,1,1,2]]
 => [[1,1,1,2]]
 => [1,2,3,4] => ([],4)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,1,2,2]]
 => [[1,1,2,2]]
 => [1,2,3,4] => ([],4)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,2,2,2]]
 => [[1,2,2,2]]
 => [1,2,3,4] => ([],4)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[2,2,2,2]]
 => [[2,2,2,2]]
 => [1,2,3,4] => ([],4)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,1,1],[2]]
 => [[1,1,1,2]]
 => [1,2,3,4] => ([],4)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,1,2],[2]]
 => [[1,1,2,2]]
 => [1,2,3,4] => ([],4)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,2,2],[2]]
 => [[1,2,2,2]]
 => [1,2,3,4] => ([],4)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,1],[2,2]]
 => [[1,1,2,2]]
 => [1,2,3,4] => ([],4)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,1,4]]
 => [[1,1,4]]
 => [1,2,3] => ([],3)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[1,2,4]]
 => [[1,2,4]]
 => [1,2,3] => ([],3)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[1,3,4]]
 => [[1,3,4]]
 => [1,2,3] => ([],3)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[1,4,4]]
 => [[1,4,4]]
 => [1,2,3] => ([],3)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[2,2,4]]
 => [[2,2,4]]
 => [1,2,3] => ([],3)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[2,3,4]]
 => [[2,3,4]]
 => [1,2,3] => ([],3)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[2,4,4]]
 => [[2,4,4]]
 => [1,2,3] => ([],3)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[3,3,4]]
 => [[3,3,4]]
 => [1,2,3] => ([],3)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[3,4,4]]
 => [[3,4,4]]
 => [1,2,3] => ([],3)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[4,4,4]]
 => [[4,4,4]]
 => [1,2,3] => ([],3)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[1,1],[4]]
 => [[1,1,4]]
 => [1,2,3] => ([],3)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[1,4],[2]]
 => [[1,2],[4]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 2
[[1,4],[3]]
 => [[1,3],[4]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 2
[[2,4],[3]]
 => [[2,3],[4]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 2
[[1],[2],[4]]
 => [[1,2],[4]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 2
[[1],[3],[4]]
 => [[1,3],[4]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 2
[[2],[3],[4]]
 => [[2,3],[4]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 2
[[1,1,3],[2]]
 => [[1,1,2],[3]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[1,2,3],[2]]
 => [[1,2,2],[3]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[1,1],[2],[3]]
 => [[1,1,2],[3]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[1,2],[2],[3]]
 => [[1,2,2],[3]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[1,1,4],[2]]
 => [[1,1,2],[4]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[1,1,4],[3]]
 => [[1,1,3],[4]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[1,2,4],[2]]
 => [[1,2,2],[4]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[1,2,4],[3]]
 => [[1,2,3],[4]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[1,3,4],[3]]
 => [[1,3,3],[4]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[2,2,4],[3]]
 => [[2,2,3],[4]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[2,3,4],[3]]
 => [[2,3,3],[4]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[1,1],[2],[4]]
 => [[1,1,2],[4]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[1,1],[3],[4]]
 => [[1,1,3],[4]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[1,2],[2],[4]]
 => [[1,2,2],[4]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[1,2],[3],[4]]
 => [[1,2,3],[4]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[1,4],[2],[3]]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[[1,3],[3],[4]]
 => [[1,3,3],[4]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[2,2],[3],[4]]
 => [[2,2,3],[4]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[2,3],[3],[4]]
 => [[2,3,3],[4]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[[1,1,1,3],[2]]
 => [[1,1,1,2],[3]]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[[1,1,2,3],[2]]
 => [[1,1,2,2],[3]]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[[1,2,2,3],[2]]
 => [[1,2,2,2],[3]]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[[1,1,3],[2,2]]
 => [[1,1,2,2],[3]]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[[1,1,1],[2],[3]]
 => [[1,1,1,2],[3]]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[[1,1,2],[2],[3]]
 => [[1,1,2,2],[3]]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[[1,2,2],[2],[3]]
 => [[1,2,2,2],[3]]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[[1,1],[2,2],[3]]
 => [[1,1,2,2],[3]]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[[1,1,1,4],[2]]
 => [[1,1,1,2],[4]]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[[1,1,1,4],[3]]
 => [[1,1,1,3],[4]]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[[1,1,2,4],[2]]
 => [[1,1,2,2],[4]]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[[1,1,2,4],[3]]
 => [[1,1,2,3],[4]]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[[1,1,3,4],[3]]
 => [[1,1,3,3],[4]]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[[1,2,2,4],[2]]
 => [[1,2,2,2],[4]]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[[1,2,2,4],[3]]
 => [[1,2,2,3],[4]]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[[1,2,3,4],[3]]
 => [[1,2,3,3],[4]]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[[1,3,3,4],[3]]
 => [[1,3,3,3],[4]]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[[2,2,2,4],[3]]
 => [[2,2,2,3],[4]]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[[2,2,3,4],[3]]
 => [[2,2,3,3],[4]]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[[2,3,3,4],[3]]
 => [[2,3,3,3],[4]]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[[1,1,4],[2,2]]
 => [[1,1,2,2],[4]]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[[1,1,4],[2,3]]
 => [[1,1,2,3],[4]]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
Description
The diameter of a connected graph. This is the greatest distance between any pair of vertices.
Matching statistic: St000456
Mapping
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 14%
Values
[[1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ([],1)
 => ? ∊ {0,1,1}
[[2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {0,1,1}
[[1],[2]]
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {0,1,1}
[[1,3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {1,2,2}
[[2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1],[3]]
 => ([(0,1)],2)
 => ([],2)
 => ([],1)
 => ? ∊ {1,2,2}
[[2],[3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {1,2,2}
[[1,1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ([],1)
 => ? ∊ {0,1,1,1,1}
[[1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {0,1,1,1,1}
[[2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => ? ∊ {0,1,1,1,1}
[[1,1],[2]]
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {0,1,1,1,1}
[[1,2],[2]]
 => ([(0,1)],2)
 => ([],2)
 => ([],1)
 => ? ∊ {0,1,1,1,1}
[[1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => ? ∊ {1,2,2,2,2}
[[2,4]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => ?
 => ? ∊ {1,2,2,2,2}
[[3,4]]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
 => ?
 => ? ∊ {1,2,2,2,2}
[[4,4]]
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
 => ?
 => ? ∊ {1,2,2,2,2}
[[1],[4]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {1,2,2,2,2}
[[2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,1,3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {0,1,1,1,2,2,2,2,2,2,3}
[[1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[2,2,3]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => ?
 => ? ∊ {0,1,1,1,2,2,2,2,2,2,3}
[[2,3,3]]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
 => ?
 => ? ∊ {0,1,1,1,2,2,2,2,2,2,3}
[[3,3,3]]
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
 => ?
 => ? ∊ {0,1,1,1,2,2,2,2,2,2,3}
[[1,1],[3]]
 => ([(0,1)],2)
 => ([],2)
 => ([],1)
 => ? ∊ {0,1,1,1,2,2,2,2,2,2,3}
[[1,2],[3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {0,1,1,1,2,2,2,2,2,2,3}
[[1,3],[2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {0,1,1,1,2,2,2,2,2,2,3}
[[1,3],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => ? ∊ {0,1,1,1,2,2,2,2,2,2,3}
[[2,2],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => ? ∊ {0,1,1,1,2,2,2,2,2,2,3}
[[2,3],[3]]
 => ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
 => ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
 => ?
 => ? ∊ {0,1,1,1,2,2,2,2,2,2,3}
[[1],[2],[3]]
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {0,1,1,1,2,2,2,2,2,2,3}
[[1,1,1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,1,1],[2]]
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,1,2],[2]]
 => ([(0,1)],2)
 => ([],2)
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,1],[2,2]]
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[1,2,4]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => ?
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[1,3,4]]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
 => ?
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[1,4,4]]
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
 => ?
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[2,2,4]]
 => ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
 => ([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
 => ?
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[2,3,4]]
 => ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
 => ([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
 => ?
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[2,4,4]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => ([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
 => ?
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[3,3,4]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => ([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
 => ?
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[3,4,4]]
 => ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
 => ([(3,4),(3,16),(4,15),(5,6),(5,17),(6,18),(7,17),(7,18),(8,15),(8,16),(9,12),(9,13),(9,14),(9,15),(9,16),(10,11),(10,13),(10,14),(10,15),(10,18),(11,12),(11,14),(11,16),(11,17),(12,13),(12,15),(12,18),(13,16),(13,17),(14,17),(14,18),(15,16),(15,17),(16,18),(17,18)],19)
 => ?
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[4,4,4]]
 => ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
 => ([(4,5),(4,17),(5,16),(6,7),(6,18),(7,19),(8,18),(8,19),(9,16),(9,17),(10,13),(10,14),(10,15),(10,16),(10,17),(11,12),(11,14),(11,15),(11,16),(11,19),(12,13),(12,15),(12,17),(12,18),(13,14),(13,16),(13,19),(14,17),(14,18),(15,18),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
 => ?
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[1,1],[4]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[1,2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[1,4],[2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[1,3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,4],[3]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,4],[4]]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ([(5,6)],7)
 => ?
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[2,2],[4]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => ?
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[2,3],[4]]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
 => ?
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[1,1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[1,1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,2,3],[2]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[1,3,3],[2]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,2],[3,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[2,2],[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,1,2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[1,1,3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,1,4],[3]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,1],[3,4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[1,1],[4,4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,2],[2,4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[1,3],[2,4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,2],[2],[4]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ([(0,1)],2)
 => 1
[[1,3],[2],[4]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[1,3],[3],[4]]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,1,1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[1,1,1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,1,2,3],[2]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[1,1,3,3],[2]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,1,2],[3,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[1,2,2],[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,2,3],[2],[3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[1,3,3],[2],[3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,1,1,2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[1,1,1,3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,1,1,4],[3]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,1,1],[3,4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[1,1,1],[4,4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,1,2],[2,4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[1,1,3],[2,4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,1,4],[2,3]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,1,2],[2],[4]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ([(0,1)],2)
 => 1
[[1,1,3],[2],[4]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[1,1,3],[3],[4]]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,2,2],[2],[4]]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,2],[2,4],[3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[1,2],[3,3],[4]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,3],[2,4],[3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,1,1,1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[1,1,1,1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
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.
Matching statistic: St000464
Mapping
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St000464: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 14%
Values
[[1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ([],1)
 => ? ∊ {0,1,1}
[[2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {0,1,1}
[[1],[2]]
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {0,1,1}
[[1,3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {1,1,1}
[[2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 2
[[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 2
[[1],[3]]
 => ([(0,1)],2)
 => ([],2)
 => ([],1)
 => ? ∊ {1,1,1}
[[2],[3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {1,1,1}
[[1,1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ([],1)
 => ? ∊ {0,1,1,1,1}
[[1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {0,1,1,1,1}
[[2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => ? ∊ {0,1,1,1,1}
[[1,1],[2]]
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {0,1,1,1,1}
[[1,2],[2]]
 => ([(0,1)],2)
 => ([],2)
 => ([],1)
 => ? ∊ {0,1,1,1,1}
[[1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => ? ∊ {1,1,1,2,2}
[[2,4]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => ?
 => ? ∊ {1,1,1,2,2}
[[3,4]]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
 => ?
 => ? ∊ {1,1,1,2,2}
[[4,4]]
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
 => ?
 => ? ∊ {1,1,1,2,2}
[[1],[4]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {1,1,1,2,2}
[[2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 2
[[3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 2
[[1,1,3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,3}
[[1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 2
[[1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 2
[[2,2,3]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => ?
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,3}
[[2,3,3]]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
 => ?
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,3}
[[3,3,3]]
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
 => ?
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,3}
[[1,1],[3]]
 => ([(0,1)],2)
 => ([],2)
 => ([],1)
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,3}
[[1,2],[3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,3}
[[1,3],[2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,3}
[[1,3],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,3}
[[2,2],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,3}
[[2,3],[3]]
 => ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
 => ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
 => ?
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,3}
[[1],[2],[3]]
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,3}
[[1,1,1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,1,1],[2]]
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,1,2],[2]]
 => ([(0,1)],2)
 => ([],2)
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,1],[2,2]]
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[1,2,4]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => ?
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[1,3,4]]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
 => ?
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[1,4,4]]
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
 => ?
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[2,2,4]]
 => ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
 => ([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
 => ?
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[2,3,4]]
 => ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
 => ([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
 => ?
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[2,4,4]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => ([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
 => ?
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[3,3,4]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => ([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
 => ?
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[3,4,4]]
 => ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
 => ([(3,4),(3,16),(4,15),(5,6),(5,17),(6,18),(7,17),(7,18),(8,15),(8,16),(9,12),(9,13),(9,14),(9,15),(9,16),(10,11),(10,13),(10,14),(10,15),(10,18),(11,12),(11,14),(11,16),(11,17),(12,13),(12,15),(12,18),(13,16),(13,17),(14,17),(14,18),(15,16),(15,17),(16,18),(17,18)],19)
 => ?
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[4,4,4]]
 => ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
 => ([(4,5),(4,17),(5,16),(6,7),(6,18),(7,19),(8,18),(8,19),(9,16),(9,17),(10,13),(10,14),(10,15),(10,16),(10,17),(11,12),(11,14),(11,15),(11,16),(11,19),(12,13),(12,15),(12,17),(12,18),(13,14),(13,16),(13,19),(14,17),(14,18),(15,18),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
 => ?
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[1,1],[4]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[1,2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 2
[[1,4],[2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[1,3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 2
[[1,4],[3]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 2
[[1,4],[4]]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ([(5,6)],7)
 => ?
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[2,2],[4]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => ?
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[2,3],[4]]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
 => ?
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[1,1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 2
[[1,1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 2
[[1,2,3],[2]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 2
[[1,3,3],[2]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 2
[[1,2],[3,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 2
[[2,2],[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 2
[[1,1,2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 2
[[1,1,3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 2
[[1,1,4],[3]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 2
[[1,1],[3,4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 2
[[1,1],[4,4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 2
[[1,2],[2,4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 2
[[1,3],[2,4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 2
[[1,2],[2],[4]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ([(0,1)],2)
 => 2
[[1,3],[2],[4]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 2
[[1,3],[3],[4]]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 2
[[1,1,1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 2
[[1,1,1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 2
[[1,1,2,3],[2]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 2
[[1,1,3,3],[2]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 2
[[1,1,2],[3,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 2
[[1,2,2],[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 2
[[1,2,3],[2],[3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 2
[[1,3,3],[2],[3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 2
[[1,1,1,2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 2
[[1,1,1,3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 2
[[1,1,1,4],[3]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 2
[[1,1,1],[3,4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 2
[[1,1,1],[4,4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 2
[[1,1,2],[2,4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 2
[[1,1,3],[2,4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 2
[[1,1,4],[2,3]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 2
[[1,1,2],[2],[4]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ([(0,1)],2)
 => 2
[[1,1,3],[2],[4]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 2
[[1,1,3],[3],[4]]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 2
[[1,2,2],[2],[4]]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => 2
[[1,2],[2,4],[3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 2
[[1,2],[3,3],[4]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 2
[[1,3],[2,4],[3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 2
[[1,1,1,1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 2
[[1,1,1,1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 2
Description
The Schultz index of a connected graph. This is $$\sum_{\{u,v\}\subseteq V} (d(u)+d(v))d(u,v)$$ where $d(u)$ is the degree of vertex $u$ and $d(u,v)$ is the distance between vertices $u$ and $v$. For trees on $n$ vertices, the Schultz index is related to the Wiener index via $S(T)=4W(T)-n(n-1)$ [2].
Matching statistic: St001118
(load all 2 compositions to match this statistic)
Mapping
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001118: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 14%
Values
[[1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ([],1)
 => ? ∊ {0,1,1}
[[2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {0,1,1}
[[1],[2]]
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {0,1,1}
[[1,3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {1,2,2}
[[2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1],[3]]
 => ([(0,1)],2)
 => ([],2)
 => ([],1)
 => ? ∊ {1,2,2}
[[2],[3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {1,2,2}
[[1,1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ([],1)
 => ? ∊ {0,1,1,1,1}
[[1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {0,1,1,1,1}
[[2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => ? ∊ {0,1,1,1,1}
[[1,1],[2]]
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {0,1,1,1,1}
[[1,2],[2]]
 => ([(0,1)],2)
 => ([],2)
 => ([],1)
 => ? ∊ {0,1,1,1,1}
[[1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => ? ∊ {1,2,2,2,2}
[[2,4]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => ?
 => ? ∊ {1,2,2,2,2}
[[3,4]]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
 => ?
 => ? ∊ {1,2,2,2,2}
[[4,4]]
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
 => ?
 => ? ∊ {1,2,2,2,2}
[[1],[4]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {1,2,2,2,2}
[[2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,1,3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {0,1,1,1,2,2,2,2,2,2,3}
[[1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[2,2,3]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => ?
 => ? ∊ {0,1,1,1,2,2,2,2,2,2,3}
[[2,3,3]]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
 => ?
 => ? ∊ {0,1,1,1,2,2,2,2,2,2,3}
[[3,3,3]]
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
 => ?
 => ? ∊ {0,1,1,1,2,2,2,2,2,2,3}
[[1,1],[3]]
 => ([(0,1)],2)
 => ([],2)
 => ([],1)
 => ? ∊ {0,1,1,1,2,2,2,2,2,2,3}
[[1,2],[3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {0,1,1,1,2,2,2,2,2,2,3}
[[1,3],[2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {0,1,1,1,2,2,2,2,2,2,3}
[[1,3],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => ? ∊ {0,1,1,1,2,2,2,2,2,2,3}
[[2,2],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => ? ∊ {0,1,1,1,2,2,2,2,2,2,3}
[[2,3],[3]]
 => ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
 => ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
 => ?
 => ? ∊ {0,1,1,1,2,2,2,2,2,2,3}
[[1],[2],[3]]
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {0,1,1,1,2,2,2,2,2,2,3}
[[1,1,1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,1,1],[2]]
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,1,2],[2]]
 => ([(0,1)],2)
 => ([],2)
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,1],[2,2]]
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[1,2,4]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => ?
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[1,3,4]]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
 => ?
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[1,4,4]]
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
 => ?
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[2,2,4]]
 => ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
 => ([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
 => ?
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[2,3,4]]
 => ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
 => ([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
 => ?
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[2,4,4]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => ([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
 => ?
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[3,3,4]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => ([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
 => ?
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[3,4,4]]
 => ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
 => ([(3,4),(3,16),(4,15),(5,6),(5,17),(6,18),(7,17),(7,18),(8,15),(8,16),(9,12),(9,13),(9,14),(9,15),(9,16),(10,11),(10,13),(10,14),(10,15),(10,18),(11,12),(11,14),(11,16),(11,17),(12,13),(12,15),(12,18),(13,16),(13,17),(14,17),(14,18),(15,16),(15,17),(16,18),(17,18)],19)
 => ?
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[4,4,4]]
 => ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
 => ([(4,5),(4,17),(5,16),(6,7),(6,18),(7,19),(8,18),(8,19),(9,16),(9,17),(10,13),(10,14),(10,15),(10,16),(10,17),(11,12),(11,14),(11,15),(11,16),(11,19),(12,13),(12,15),(12,17),(12,18),(13,14),(13,16),(13,19),(14,17),(14,18),(15,18),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
 => ?
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[1,1],[4]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[1,2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[1,4],[2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[1,3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,4],[3]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,4],[4]]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ([(5,6)],7)
 => ?
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[2,2],[4]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => ?
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[2,3],[4]]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
 => ?
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[1,1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[1,1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,2,3],[2]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[1,3,3],[2]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,2],[3,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[2,2],[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,1,2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[1,1,3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,1,4],[3]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,1],[3,4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[1,1],[4,4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,2],[2,4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[1,3],[2,4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,2],[2],[4]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ([(0,1)],2)
 => 1
[[1,3],[2],[4]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[1,3],[3],[4]]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,1,1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[1,1,1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,1,2,3],[2]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[1,1,3,3],[2]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,1,2],[3,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[1,2,2],[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,2,3],[2],[3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[1,3,3],[2],[3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,1,1,2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[1,1,1,3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,1,1,4],[3]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,1,1],[3,4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[1,1,1],[4,4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,1,2],[2,4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[1,1,3],[2,4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,1,4],[2,3]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,1,2],[2],[4]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ([(0,1)],2)
 => 1
[[1,1,3],[2],[4]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[1,1,3],[3],[4]]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,2,2],[2],[4]]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,2],[2,4],[3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[1,2],[3,3],[4]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,3],[2,4],[3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,1,1,1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[1,1,1,1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
Description
The acyclic chromatic index of a graph. An acyclic edge coloring of a graph is a proper colouring of the edges of a graph such that the union of the edges colored with any two given colours is a forest. The smallest number of colours such that such a colouring exists is the acyclic chromatic index.
Matching statistic: St001281
Mapping
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001281: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 14%
Values
[[1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ([],1)
 => ? ∊ {0,1,1}
[[2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {0,1,1}
[[1],[2]]
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {0,1,1}
[[1,3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {1,2,2}
[[2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1],[3]]
 => ([(0,1)],2)
 => ([],2)
 => ([],1)
 => ? ∊ {1,2,2}
[[2],[3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {1,2,2}
[[1,1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ([],1)
 => ? ∊ {0,1,1,1,1}
[[1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {0,1,1,1,1}
[[2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => ? ∊ {0,1,1,1,1}
[[1,1],[2]]
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {0,1,1,1,1}
[[1,2],[2]]
 => ([(0,1)],2)
 => ([],2)
 => ([],1)
 => ? ∊ {0,1,1,1,1}
[[1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => ? ∊ {1,2,2,2,2}
[[2,4]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => ?
 => ? ∊ {1,2,2,2,2}
[[3,4]]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
 => ?
 => ? ∊ {1,2,2,2,2}
[[4,4]]
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
 => ?
 => ? ∊ {1,2,2,2,2}
[[1],[4]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {1,2,2,2,2}
[[2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,1,3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {0,1,1,1,2,2,2,2,2,2,3}
[[1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[2,2,3]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => ?
 => ? ∊ {0,1,1,1,2,2,2,2,2,2,3}
[[2,3,3]]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
 => ?
 => ? ∊ {0,1,1,1,2,2,2,2,2,2,3}
[[3,3,3]]
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
 => ?
 => ? ∊ {0,1,1,1,2,2,2,2,2,2,3}
[[1,1],[3]]
 => ([(0,1)],2)
 => ([],2)
 => ([],1)
 => ? ∊ {0,1,1,1,2,2,2,2,2,2,3}
[[1,2],[3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {0,1,1,1,2,2,2,2,2,2,3}
[[1,3],[2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {0,1,1,1,2,2,2,2,2,2,3}
[[1,3],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => ? ∊ {0,1,1,1,2,2,2,2,2,2,3}
[[2,2],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => ? ∊ {0,1,1,1,2,2,2,2,2,2,3}
[[2,3],[3]]
 => ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
 => ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
 => ?
 => ? ∊ {0,1,1,1,2,2,2,2,2,2,3}
[[1],[2],[3]]
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {0,1,1,1,2,2,2,2,2,2,3}
[[1,1,1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,1,1],[2]]
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,1,2],[2]]
 => ([(0,1)],2)
 => ([],2)
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,1],[2,2]]
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[1,2,4]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => ?
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[1,3,4]]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
 => ?
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[1,4,4]]
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
 => ?
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[2,2,4]]
 => ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
 => ([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
 => ?
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[2,3,4]]
 => ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
 => ([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
 => ?
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[2,4,4]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => ([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
 => ?
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[3,3,4]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => ([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
 => ?
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[3,4,4]]
 => ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
 => ([(3,4),(3,16),(4,15),(5,6),(5,17),(6,18),(7,17),(7,18),(8,15),(8,16),(9,12),(9,13),(9,14),(9,15),(9,16),(10,11),(10,13),(10,14),(10,15),(10,18),(11,12),(11,14),(11,16),(11,17),(12,13),(12,15),(12,18),(13,16),(13,17),(14,17),(14,18),(15,16),(15,17),(16,18),(17,18)],19)
 => ?
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[4,4,4]]
 => ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
 => ([(4,5),(4,17),(5,16),(6,7),(6,18),(7,19),(8,18),(8,19),(9,16),(9,17),(10,13),(10,14),(10,15),(10,16),(10,17),(11,12),(11,14),(11,15),(11,16),(11,19),(12,13),(12,15),(12,17),(12,18),(13,14),(13,16),(13,19),(14,17),(14,18),(15,18),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
 => ?
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[1,1],[4]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[1,2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[1,4],[2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[1,3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,4],[3]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,4],[4]]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ([(5,6)],7)
 => ?
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[2,2],[4]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => ?
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[2,3],[4]]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
 => ?
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[1,1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[1,1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,2,3],[2]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[1,3,3],[2]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,2],[3,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[2,2],[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,1,2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[1,1,3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,1,4],[3]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,1],[3,4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[1,1],[4,4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,2],[2,4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[1,3],[2,4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,2],[2],[4]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ([(0,1)],2)
 => 1
[[1,3],[2],[4]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[1,3],[3],[4]]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,1,1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[1,1,1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,1,2,3],[2]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[1,1,3,3],[2]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,1,2],[3,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[1,2,2],[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,2,3],[2],[3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[1,3,3],[2],[3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,1,1,2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[1,1,1,3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,1,1,4],[3]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,1,1],[3,4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[1,1,1],[4,4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,1,2],[2,4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[1,1,3],[2,4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,1,4],[2,3]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,1,2],[2],[4]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ([(0,1)],2)
 => 1
[[1,1,3],[2],[4]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[1,1,3],[3],[4]]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,2,2],[2],[4]]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,2],[2,4],[3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[1,2],[3,3],[4]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,3],[2,4],[3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
[[1,1,1,1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 1
[[1,1,1,1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 1
Description
The normalized isoperimetric number of a graph. The isoperimetric number, or Cheeger constant, of a graph $G$ is $$ i(G) = \min\left\{\frac{|\partial A|}{|A|}\ : \ A\subseteq V(G), 0 < |A|\leq |V(G)|/2\right\}, $$ where $$ \partial A := \{(x, y)\in E(G)\ : \ x\in A, y\in V(G)\setminus A \}. $$ This statistic is $i(G)\cdot\lfloor n/2\rfloor$.
Matching statistic: St001545
Mapping
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001545: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 14%
Values
[[1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ([],1)
 => ? ∊ {0,1,1}
[[2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {0,1,1}
[[1],[2]]
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {0,1,1}
[[1,3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {1,1,1}
[[2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 2
[[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 2
[[1],[3]]
 => ([(0,1)],2)
 => ([],2)
 => ([],1)
 => ? ∊ {1,1,1}
[[2],[3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {1,1,1}
[[1,1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ([],1)
 => ? ∊ {0,1,1,1,1}
[[1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {0,1,1,1,1}
[[2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => ? ∊ {0,1,1,1,1}
[[1,1],[2]]
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {0,1,1,1,1}
[[1,2],[2]]
 => ([(0,1)],2)
 => ([],2)
 => ([],1)
 => ? ∊ {0,1,1,1,1}
[[1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => ? ∊ {1,1,1,2,2}
[[2,4]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => ?
 => ? ∊ {1,1,1,2,2}
[[3,4]]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
 => ?
 => ? ∊ {1,1,1,2,2}
[[4,4]]
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
 => ?
 => ? ∊ {1,1,1,2,2}
[[1],[4]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {1,1,1,2,2}
[[2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 2
[[3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 2
[[1,1,3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,3}
[[1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 2
[[1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 2
[[2,2,3]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => ?
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,3}
[[2,3,3]]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
 => ?
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,3}
[[3,3,3]]
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
 => ?
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,3}
[[1,1],[3]]
 => ([(0,1)],2)
 => ([],2)
 => ([],1)
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,3}
[[1,2],[3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,3}
[[1,3],[2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,3}
[[1,3],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,3}
[[2,2],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,3}
[[2,3],[3]]
 => ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
 => ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
 => ?
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,3}
[[1],[2],[3]]
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {0,1,1,1,1,1,2,2,2,2,3}
[[1,1,1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,1,1],[2]]
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,1,2],[2]]
 => ([(0,1)],2)
 => ([],2)
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,1],[2,2]]
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1}
[[1,1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[1,2,4]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => ?
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[1,3,4]]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
 => ?
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[1,4,4]]
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
 => ?
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[2,2,4]]
 => ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
 => ([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
 => ?
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[2,3,4]]
 => ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
 => ([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
 => ?
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[2,4,4]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => ([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
 => ?
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[3,3,4]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => ([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
 => ?
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[3,4,4]]
 => ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
 => ([(3,4),(3,16),(4,15),(5,6),(5,17),(6,18),(7,17),(7,18),(8,15),(8,16),(9,12),(9,13),(9,14),(9,15),(9,16),(10,11),(10,13),(10,14),(10,15),(10,18),(11,12),(11,14),(11,16),(11,17),(12,13),(12,15),(12,18),(13,16),(13,17),(14,17),(14,18),(15,16),(15,17),(16,18),(17,18)],19)
 => ?
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[4,4,4]]
 => ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
 => ([(4,5),(4,17),(5,16),(6,7),(6,18),(7,19),(8,18),(8,19),(9,16),(9,17),(10,13),(10,14),(10,15),(10,16),(10,17),(11,12),(11,14),(11,15),(11,16),(11,19),(12,13),(12,15),(12,17),(12,18),(13,14),(13,16),(13,19),(14,17),(14,18),(15,18),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
 => ?
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[1,1],[4]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[1,2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 2
[[1,4],[2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[1,3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 2
[[1,4],[3]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 2
[[1,4],[4]]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ([(5,6)],7)
 => ?
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[2,2],[4]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => ?
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[2,3],[4]]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
 => ?
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[1,1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 2
[[1,1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 2
[[1,2,3],[2]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 2
[[1,3,3],[2]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 2
[[1,2],[3,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 2
[[2,2],[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 2
[[1,1,2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 2
[[1,1,3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 2
[[1,1,4],[3]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 2
[[1,1],[3,4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 2
[[1,1],[4,4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 2
[[1,2],[2,4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 2
[[1,3],[2,4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 2
[[1,2],[2],[4]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ([(0,1)],2)
 => 2
[[1,3],[2],[4]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 2
[[1,3],[3],[4]]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 2
[[1,1,1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 2
[[1,1,1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 2
[[1,1,2,3],[2]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 2
[[1,1,3,3],[2]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 2
[[1,1,2],[3,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 2
[[1,2,2],[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 2
[[1,2,3],[2],[3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 2
[[1,3,3],[2],[3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 2
[[1,1,1,2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 2
[[1,1,1,3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 2
[[1,1,1,4],[3]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 2
[[1,1,1],[3,4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 2
[[1,1,1],[4,4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 2
[[1,1,2],[2,4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 2
[[1,1,3],[2,4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 2
[[1,1,4],[2,3]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 2
[[1,1,2],[2],[4]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ([(0,1)],2)
 => 2
[[1,1,3],[2],[4]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 2
[[1,1,3],[3],[4]]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 2
[[1,2,2],[2],[4]]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => 2
[[1,2],[2,4],[3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 2
[[1,2],[3,3],[4]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 2
[[1,3],[2,4],[3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 2
[[1,1,1,1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 2
[[1,1,1,1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 2
Description
The second Elser number of a connected graph. For a connected graph $G$ the $k$-th Elser number is $$ els_k(G) = (-1)^{|V(G)|+1} \sum_N (-1)^{|E(N)|} |V(N)|^k $$ where the sum is over all nuclei of $G$, that is, the connected subgraphs of $G$ whose vertex set is a vertex cover of $G$. It is clear that this number is even. It was shown in [1] that it is non-negative.
The following 37 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001592The maximal number of simple paths between any two different vertices of a graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000668The least common multiple of the parts of the partition. St000706The product of the factorials of the multiplicities of an integer partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000815The number of semistandard Young tableaux of partition weight of given shape. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St001877Number of indecomposable injective modules with projective dimension 2. St000454The largest eigenvalue of a graph if it is integral. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St000260The radius of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph.