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Your data matches 27 different statistics following compositions of up to 3 maps.
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Matching statistic: St001200
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2,3]]
=> [1,2,3] => [1,2] => [1,0,1,0]
=> 2
[[1,2],[3]]
=> [3,1,2] => [1,2] => [1,0,1,0]
=> 2
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3] => [1,0,1,0,1,0]
=> 3
[[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3] => [1,1,0,0,1,0]
=> 2
[[1,2,3],[4]]
=> [4,1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 3
[[1,3],[2,4]]
=> [2,4,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 2
[[1,3],[2],[4]]
=> [4,2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 2
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 2
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 2
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 3
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 3
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 3
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> 3
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> 3
[[1,3,4,6],[2,5]]
=> [2,5,1,3,4,6] => [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> 2
[[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> 2
[[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[[1,3,4,5],[2,6]]
=> [2,6,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 3
[[1,2,4,5],[3,6]]
=> [3,6,1,2,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[[1,2,3,5],[4,6]]
=> [4,6,1,2,3,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[[1,3,5,6],[2],[4]]
=> [4,2,1,3,5,6] => [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[[1,2,5,6],[3],[4]]
=> [4,3,1,2,5,6] => [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[[1,3,4,5],[2],[6]]
=> [6,2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 3
[[1,2,4,5],[3],[6]]
=> [6,3,1,2,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[[1,2,3,5],[4],[6]]
=> [6,4,1,2,3,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> 3
[[1,2,5],[3,4,6]]
=> [3,4,6,1,2,5] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> 3
[[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> 2
[[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> 2
[[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[[1,4,6],[2,5],[3]]
=> [3,2,5,1,4,6] => [3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
=> 2
[[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[[1,2,6],[3,5],[4]]
=> [4,3,5,1,2,6] => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[[1,4,5],[2,6],[3]]
=> [3,2,6,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 3
Description
The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St000260
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 50%
Mp00252: Permutations —restriction⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 50%
Values
[[1,2,3]]
=> [1,2,3] => [1,2] => ([],2)
=> ? = 2
[[1,2],[3]]
=> [3,1,2] => [1,2] => ([],2)
=> ? = 2
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3] => ([],3)
=> ? = 3
[[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3] => ([(1,2)],3)
=> ? = 2
[[1,2,3],[4]]
=> [4,1,2,3] => [1,2,3] => ([],3)
=> ? = 3
[[1,3],[2,4]]
=> [2,4,1,3] => [2,1,3] => ([(1,2)],3)
=> ? = 2
[[1,3],[2],[4]]
=> [4,2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? = 2
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4] => ([],4)
=> ? = 3
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4] => ([(2,3)],4)
=> ? = 3
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? = 2
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,2,3,4] => ([],4)
=> ? = 3
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? = 3
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? = 2
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? = 2
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? = 3
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? = 2
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? = 2
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? = 2
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5] => ([],5)
=> ? = 3
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5] => ([(3,4)],5)
=> ? = 3
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ? = 3
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? = 3
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ? = 3
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 3
[[1,3,4,6],[2,5]]
=> [2,5,1,3,4,6] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[[1,3,4,5],[2,6]]
=> [2,6,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> ? = 3
[[1,2,4,5],[3,6]]
=> [3,6,1,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ? = 3
[[1,2,3,5],[4,6]]
=> [4,6,1,2,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2
[[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ? = 3
[[1,3,5,6],[2],[4]]
=> [4,2,1,3,5,6] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[[1,2,5,6],[3],[4]]
=> [4,3,1,2,5,6] => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[[1,3,4,5],[2],[6]]
=> [6,2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> ? = 3
[[1,2,4,5],[3],[6]]
=> [6,3,1,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ? = 3
[[1,2,3,5],[4],[6]]
=> [6,4,1,2,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2
[[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ? = 3
[[1,2,5],[3,4,6]]
=> [3,4,6,1,2,5] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 3
[[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[[1,4,6],[2,5],[3]]
=> [3,2,5,1,4,6] => [3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[[1,2,6],[3,5],[4]]
=> [4,3,5,1,2,6] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[[1,4,5],[2,6],[3]]
=> [3,2,6,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ? = 3
[[1,3,5],[2,6],[4]]
=> [4,2,6,1,3,5] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[[1,2,5],[3,6],[4]]
=> [4,3,6,1,2,5] => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[[1,3,5],[2,4],[6]]
=> [6,2,4,1,3,5] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ? = 3
[[1,2,5],[3,4],[6]]
=> [6,3,4,1,2,5] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 3
[[1,3,4],[2,5],[6]]
=> [6,2,5,1,3,4] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[[1,2,4],[3,5],[6]]
=> [6,3,5,1,2,4] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[[1,4,5],[2],[3],[6]]
=> [6,3,2,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ? = 3
[[1,3,5],[2],[4],[6]]
=> [6,4,2,1,3,5] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[[1,2,5],[3],[4],[6]]
=> [6,4,3,1,2,5] => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[[1,4],[2,5],[3,6]]
=> [3,6,2,5,1,4] => [3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[[1,3],[2,5],[4,6]]
=> [4,6,2,5,1,3] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[[1,2],[3,5],[4,6]]
=> [4,6,3,5,1,2] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[[1,5],[2,6],[3],[4]]
=> [4,3,2,6,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[[1,4],[2,5],[3],[6]]
=> [6,3,2,5,1,4] => [3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[[1,2],[3,5],[4],[6]]
=> [6,4,3,5,1,2] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[[1,5],[2],[3],[4],[6]]
=> [6,4,3,2,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St000259
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00226: Standard tableaux —row-to-column-descents⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 50%
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 50%
Values
[[1,2,3]]
=> [[1,2,3]]
=> [3] => ([],3)
=> ? = 2
[[1,2],[3]]
=> [[1,3],[2]]
=> [1,2] => ([(1,2)],3)
=> ? = 2
[[1,2,3,4]]
=> [[1,2,3,4]]
=> [4] => ([],4)
=> ? = 3
[[1,3,4],[2]]
=> [[1,2,4],[3]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 2
[[1,2,3],[4]]
=> [[1,3,4],[2]]
=> [1,3] => ([(2,3)],4)
=> ? = 3
[[1,3],[2,4]]
=> [[1,2],[3,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 2
[[1,3],[2],[4]]
=> [[1,2],[3],[4]]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[1,2,3,4,5]]
=> [[1,2,3,4,5]]
=> [5] => ([],5)
=> ? = 3
[[1,3,4,5],[2]]
=> [[1,2,4,5],[3]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 3
[[1,2,4,5],[3]]
=> [[1,2,3,5],[4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2
[[1,2,3,4],[5]]
=> [[1,3,4,5],[2]]
=> [1,4] => ([(3,4)],5)
=> ? = 3
[[1,3,5],[2,4]]
=> [[1,2,4],[3,5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[1,2,5],[3,4]]
=> [[1,3,4],[2,5]]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[1,3,4],[2,5]]
=> [[1,2,5],[3,4]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 3
[[1,2,4],[3,5]]
=> [[1,2,3],[4,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2
[[1,4,5],[2],[3]]
=> [[1,3,5],[2],[4]]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[[1,3,4],[2],[5]]
=> [[1,2,5],[3],[4]]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[[1,2,4],[3],[5]]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[1,4],[2,5],[3]]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[[1,3],[2,4],[5]]
=> [[1,2],[3,5],[4]]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[[1,2],[3,4],[5]]
=> [[1,4],[2,5],[3]]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[1,4],[2],[3],[5]]
=> [[1,3],[2],[4],[5]]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[1,2,3,4,5,6]]
=> [[1,2,3,4,5,6]]
=> [6] => ([],6)
=> ? = 3
[[1,3,4,5,6],[2]]
=> [[1,2,4,5,6],[3]]
=> [2,4] => ([(3,5),(4,5)],6)
=> ? = 3
[[1,2,4,5,6],[3]]
=> [[1,2,3,5,6],[4]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 3
[[1,2,3,5,6],[4]]
=> [[1,2,3,4,6],[5]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 2
[[1,2,3,4,5],[6]]
=> [[1,3,4,5,6],[2]]
=> [1,5] => ([(4,5)],6)
=> ? = 3
[[1,3,5,6],[2,4]]
=> [[1,2,4,6],[3,5]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[[1,2,5,6],[3,4]]
=> [[1,3,4,6],[2,5]]
=> [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[[1,3,4,6],[2,5]]
=> [[1,2,4,5],[3,6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2,4,6],[3,5]]
=> [[1,2,3,5],[4,6]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2,3,6],[4,5]]
=> [[1,3,4,5],[2,6]]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,3,4,5],[2,6]]
=> [[1,2,5,6],[3,4]]
=> [2,4] => ([(3,5),(4,5)],6)
=> ? = 3
[[1,2,4,5],[3,6]]
=> [[1,2,3,6],[4,5]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 3
[[1,2,3,5],[4,6]]
=> [[1,2,3,4],[5,6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 2
[[1,4,5,6],[2],[3]]
=> [[1,3,5,6],[2],[4]]
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[[1,3,5,6],[2],[4]]
=> [[1,2,4,6],[3],[5]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[[1,2,5,6],[3],[4]]
=> [[1,3,4,6],[2],[5]]
=> [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[[1,3,4,5],[2],[6]]
=> [[1,2,5,6],[3],[4]]
=> [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[[1,2,4,5],[3],[6]]
=> [[1,2,3,6],[4],[5]]
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[[1,2,3,5],[4],[6]]
=> [[1,2,3,4],[5],[6]]
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,3,5],[2,4,6]]
=> [[1,2,4],[3,5,6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[[1,2,5],[3,4,6]]
=> [[1,3,4],[2,5,6]]
=> [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[[1,3,4],[2,5,6]]
=> [[1,2,5],[3,4,6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2,4],[3,5,6]]
=> [[1,2,3],[4,5,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 2
[[1,2,3],[4,5,6]]
=> [[1,3,5],[2,4,6]]
=> [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,4,6],[2,5],[3]]
=> [[1,3,5],[2,6],[4]]
=> [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,3,6],[2,5],[4]]
=> [[1,2,4],[3,5],[6]]
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2,6],[3,5],[4]]
=> [[1,3,4],[2,5],[6]]
=> [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,4,5],[2,6],[3]]
=> [[1,3,6],[2,5],[4]]
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[[1,3,5],[2,6],[4]]
=> [[1,2,6],[3,4],[5]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[[1,2,5],[3,6],[4]]
=> [[1,3,6],[2,4],[5]]
=> [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[[1,3,5],[2,4],[6]]
=> [[1,2,4],[3,6],[5]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[[1,2,5],[3,4],[6]]
=> [[1,3,4],[2,6],[5]]
=> [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[[1,3,4],[2,5],[6]]
=> [[1,2,6],[3,5],[4]]
=> [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[[1,2,4],[3,5],[6]]
=> [[1,2,3],[4,6],[5]]
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[[1,2,3],[4,5],[6]]
=> [[1,4,6],[2,5],[3]]
=> [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[[1,5,6],[2],[3],[4]]
=> [[1,4,6],[2],[3],[5]]
=> [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[[1,4,5],[2],[3],[6]]
=> [[1,3,6],[2],[4],[5]]
=> [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[[1,3,5],[2],[4],[6]]
=> [[1,2,4],[3],[5],[6]]
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2,5],[3],[4],[6]]
=> [[1,3,4],[2],[5],[6]]
=> [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,4],[2,5],[3,6]]
=> [[1,3],[2,5],[4,6]]
=> [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,3],[2,5],[4,6]]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[[1,2],[3,5],[4,6]]
=> [[1,3],[2,4],[5,6]]
=> [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[[1,5],[2,6],[3],[4]]
=> [[1,4],[2,6],[3],[5]]
=> [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[[1,4],[2,5],[3],[6]]
=> [[1,3],[2,6],[4],[5]]
=> [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[[1,3],[2,5],[4],[6]]
=> [[1,2],[3,4],[5],[6]]
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2],[3,5],[4],[6]]
=> [[1,3],[2,4],[5],[6]]
=> [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,5],[2],[3],[4],[6]]
=> [[1,4],[2],[3],[5],[6]]
=> [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St001616
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001616: Lattices ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 50%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001616: Lattices ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 50%
Values
[[1,2,3]]
=> [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2
[[1,2],[3]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 3
[[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 2
[[1,2,3],[4]]
=> [4,1,2,3] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 3
[[1,3],[2,4]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 2
[[1,3],[2],[4]]
=> [4,2,1,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 2
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 3
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [3,2,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 2
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 3
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [4,3,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? = 2
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> 2
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [5,4,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [3,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 2
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [2,3,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 2
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [2,5,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 3
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [5,4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 2
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [2,5,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 2
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [2,5,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 2
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 2
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [3,2,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 2
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,15),(2,14),(3,19),(3,21),(4,20),(4,21),(5,14),(5,19),(6,15),(6,20),(8,10),(9,11),(10,12),(11,13),(12,7),(13,7),(14,8),(15,9),(16,10),(16,18),(17,11),(17,18),(18,12),(18,13),(19,8),(19,16),(20,9),(20,17),(21,16),(21,17)],22)
=> ? = 3
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,14),(3,13),(4,12),(4,16),(5,13),(5,17),(6,16),(6,17),(8,10),(9,11),(10,7),(11,7),(12,8),(13,9),(14,12),(15,10),(15,11),(16,8),(16,15),(17,9),(17,15)],18)
=> ? = 3
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [3,2,1,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,12),(4,13),(4,14),(5,11),(5,15),(6,12),(6,15),(8,11),(9,7),(10,7),(11,9),(12,10),(13,8),(14,8),(15,9),(15,10)],16)
=> ? = 3
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [4,3,2,1,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,12),(4,11),(4,14),(5,12),(5,15),(6,13),(6,15),(8,11),(9,8),(10,8),(11,7),(12,9),(13,10),(14,7),(15,9),(15,10)],16)
=> ? = 2
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,15),(2,14),(3,19),(3,21),(4,20),(4,21),(5,14),(5,19),(6,15),(6,20),(8,10),(9,11),(10,12),(11,13),(12,7),(13,7),(14,8),(15,9),(16,10),(16,18),(17,11),(17,18),(18,12),(18,13),(19,8),(19,16),(20,9),(20,17),(21,16),(21,17)],22)
=> ? = 3
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [4,3,1,2,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,13),(3,13),(4,11),(5,10),(5,12),(6,9),(6,11),(8,10),(9,8),(10,7),(11,8),(12,7),(13,9)],14)
=> ? = 3
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [3,1,4,2,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,10),(5,7),(6,7),(6,8),(7,9),(8,9),(10,8)],11)
=> ? = 3
[[1,3,4,6],[2,5]]
=> [2,5,1,3,4,6] => [5,4,3,1,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,7),(5,11),(5,14),(6,12),(6,14),(8,10),(9,10),(10,7),(11,8),(12,9),(13,11),(14,8),(14,9)],15)
=> ? = 2
[[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => [3,1,5,4,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 2
[[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => [5,3,1,4,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,9),(3,9),(4,9),(5,9),(6,7),(7,8),(9,7)],10)
=> ? = 2
[[1,3,4,5],[2,6]]
=> [2,6,1,3,4,5] => [6,5,4,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,14),(3,13),(4,12),(4,16),(5,13),(5,17),(6,16),(6,17),(8,10),(9,11),(10,7),(11,7),(12,8),(13,9),(14,12),(15,10),(15,11),(16,8),(16,15),(17,9),(17,15)],18)
=> ? = 3
[[1,2,4,5],[3,6]]
=> [3,6,1,2,4,5] => [3,1,6,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,8),(5,7),(6,7),(6,8),(7,9),(8,9),(9,10)],11)
=> ? = 3
[[1,2,3,5],[4,6]]
=> [4,6,1,2,3,5] => [6,5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,10),(5,7),(6,7),(6,8),(7,9),(8,9),(10,8)],11)
=> ? = 2
[[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [2,3,1,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,13),(3,13),(4,11),(5,12),(5,14),(6,10),(6,14),(8,7),(9,7),(10,9),(11,10),(12,8),(13,11),(14,8),(14,9)],15)
=> ? = 3
[[1,3,5,6],[2],[4]]
=> [4,2,1,3,5,6] => [2,4,3,1,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,10),(5,11),(6,7),(6,9),(7,12),(8,11),(9,12),(10,9),(11,10)],13)
=> ? = 2
[[1,2,5,6],[3],[4]]
=> [4,3,1,2,5,6] => [4,2,3,1,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,12),(4,13),(5,13),(6,9),(6,12),(8,9),(9,7),(10,8),(11,8),(12,7),(13,10),(13,11)],14)
=> ? = 2
[[1,3,4,5],[2],[6]]
=> [6,2,1,3,4,5] => [2,6,5,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,13),(3,12),(4,7),(5,12),(5,14),(6,13),(6,14),(8,11),(9,8),(10,8),(11,7),(12,9),(13,10),(14,9),(14,10)],15)
=> ? = 3
[[1,2,4,5],[3],[6]]
=> [6,3,1,2,4,5] => [6,5,4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,17),(3,17),(4,12),(5,15),(5,16),(6,13),(6,16),(8,10),(9,11),(10,7),(11,7),(12,9),(13,8),(14,10),(14,11),(15,9),(15,14),(16,8),(16,14),(17,12),(17,15)],18)
=> ? = 3
[[1,2,3,5],[4],[6]]
=> [6,4,1,2,3,5] => [4,2,6,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 2
[[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => [4,1,2,6,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,9),(8,9)],10)
=> ? = 3
[[1,2,5],[3,4,6]]
=> [3,4,6,1,2,5] => [6,5,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,10),(5,7),(6,7),(6,8),(7,9),(8,9),(10,8)],11)
=> ? = 3
[[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => [6,4,1,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 2
[[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => [5,2,6,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2
[[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [4,1,5,2,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2
[[1,4,6],[2,5],[3]]
=> [3,2,5,1,4,6] => [2,5,4,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 2
[[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => [2,4,1,5,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 2
[[1,2,6],[3,5],[4]]
=> [4,3,5,1,2,6] => [4,1,5,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 2
[[1,4,5],[2,6],[3]]
=> [3,2,6,1,4,5] => [2,6,5,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,8),(5,7),(6,7),(6,8),(7,9),(8,9),(9,10)],11)
=> ? = 3
[[1,3,5],[2,6],[4]]
=> [4,2,6,1,3,5] => [2,4,1,6,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2
[[1,2,5],[3,6],[4]]
=> [4,3,6,1,2,5] => [4,1,6,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,9),(8,9)],10)
=> ? = 2
[[1,3,5],[2,4],[6]]
=> [6,2,4,1,3,5] => [2,6,5,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,9),(5,11),(6,7),(6,10),(7,12),(8,10),(10,12),(11,9),(12,11)],13)
=> ? = 3
[[1,2,5],[3,4],[6]]
=> [6,3,4,1,2,5] => [6,5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,12),(3,11),(4,10),(5,12),(5,13),(6,11),(6,15),(8,7),(9,7),(10,9),(11,8),(12,14),(13,14),(14,10),(14,15),(15,8),(15,9)],16)
=> ? = 3
[[1,3,4],[2,5],[6]]
=> [6,2,5,1,3,4] => [2,5,3,6,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,9),(3,9),(4,9),(5,9),(6,7),(7,8),(9,7)],10)
=> ? = 2
[[1,2,4],[3,5],[6]]
=> [6,3,5,1,2,4] => [5,2,3,6,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 2
[[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => [6,3,5,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,10),(5,8),(6,7),(7,9),(8,9),(10,7),(10,8)],11)
=> ? = 2
[[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [3,2,4,1,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,10),(5,11),(6,7),(6,9),(7,12),(8,11),(9,12),(10,9),(11,10)],13)
=> ? = 2
[[1,4,5],[2],[3],[6]]
=> [6,3,2,1,4,5] => [3,2,6,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
=> ? = 3
[[1,3,5],[2],[4],[6]]
=> [6,4,2,1,3,5] => [6,5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,14),(3,12),(4,12),(5,10),(6,11),(6,13),(8,7),(9,7),(10,9),(11,8),(12,14),(13,8),(13,9),(14,10),(14,13)],15)
=> ? = 2
[[1,2,5],[3],[4],[6]]
=> [6,4,3,1,2,5] => [3,6,5,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 2
[[1,4],[2,5],[3,6]]
=> [3,6,2,5,1,4] => [6,4,5,1,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,10),(3,7),(4,7),(5,8),(6,8),(7,10),(8,9),(8,11),(9,12),(10,11),(11,12)],13)
=> ? = 2
[[1,3],[2,5],[4,6]]
=> [4,6,2,5,1,3] => [5,1,4,6,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2
[[1,2],[3,5],[4,6]]
=> [4,6,3,5,1,2] => [3,5,1,4,6,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2
Description
The number of neutral elements in a lattice.
An element $e$ of the lattice $L$ is neutral if the sublattice generated by $e$, $x$ and $y$ is distributive for all $x, y \in L$.
Matching statistic: St001720
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001720: Lattices ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 50%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001720: Lattices ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 50%
Values
[[1,2,3]]
=> [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2
[[1,2],[3]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 3
[[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 2
[[1,2,3],[4]]
=> [4,1,2,3] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 3
[[1,3],[2,4]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 2
[[1,3],[2],[4]]
=> [4,2,1,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 2
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 3
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [3,2,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 2
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 3
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [4,3,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? = 2
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> 2
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [5,4,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [3,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 2
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [2,3,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 2
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [2,5,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 3
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [5,4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 2
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [2,5,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 2
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [2,5,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 2
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 2
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [3,2,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 2
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,15),(2,14),(3,19),(3,21),(4,20),(4,21),(5,14),(5,19),(6,15),(6,20),(8,10),(9,11),(10,12),(11,13),(12,7),(13,7),(14,8),(15,9),(16,10),(16,18),(17,11),(17,18),(18,12),(18,13),(19,8),(19,16),(20,9),(20,17),(21,16),(21,17)],22)
=> ? = 3
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,14),(3,13),(4,12),(4,16),(5,13),(5,17),(6,16),(6,17),(8,10),(9,11),(10,7),(11,7),(12,8),(13,9),(14,12),(15,10),(15,11),(16,8),(16,15),(17,9),(17,15)],18)
=> ? = 3
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [3,2,1,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,12),(4,13),(4,14),(5,11),(5,15),(6,12),(6,15),(8,11),(9,7),(10,7),(11,9),(12,10),(13,8),(14,8),(15,9),(15,10)],16)
=> ? = 3
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [4,3,2,1,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,12),(4,11),(4,14),(5,12),(5,15),(6,13),(6,15),(8,11),(9,8),(10,8),(11,7),(12,9),(13,10),(14,7),(15,9),(15,10)],16)
=> ? = 2
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,15),(2,14),(3,19),(3,21),(4,20),(4,21),(5,14),(5,19),(6,15),(6,20),(8,10),(9,11),(10,12),(11,13),(12,7),(13,7),(14,8),(15,9),(16,10),(16,18),(17,11),(17,18),(18,12),(18,13),(19,8),(19,16),(20,9),(20,17),(21,16),(21,17)],22)
=> ? = 3
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [4,3,1,2,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,13),(3,13),(4,11),(5,10),(5,12),(6,9),(6,11),(8,10),(9,8),(10,7),(11,8),(12,7),(13,9)],14)
=> ? = 3
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [3,1,4,2,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,10),(5,7),(6,7),(6,8),(7,9),(8,9),(10,8)],11)
=> ? = 3
[[1,3,4,6],[2,5]]
=> [2,5,1,3,4,6] => [5,4,3,1,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,7),(5,11),(5,14),(6,12),(6,14),(8,10),(9,10),(10,7),(11,8),(12,9),(13,11),(14,8),(14,9)],15)
=> ? = 2
[[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => [3,1,5,4,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 2
[[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => [5,3,1,4,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,9),(3,9),(4,9),(5,9),(6,7),(7,8),(9,7)],10)
=> ? = 2
[[1,3,4,5],[2,6]]
=> [2,6,1,3,4,5] => [6,5,4,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,14),(3,13),(4,12),(4,16),(5,13),(5,17),(6,16),(6,17),(8,10),(9,11),(10,7),(11,7),(12,8),(13,9),(14,12),(15,10),(15,11),(16,8),(16,15),(17,9),(17,15)],18)
=> ? = 3
[[1,2,4,5],[3,6]]
=> [3,6,1,2,4,5] => [3,1,6,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,8),(5,7),(6,7),(6,8),(7,9),(8,9),(9,10)],11)
=> ? = 3
[[1,2,3,5],[4,6]]
=> [4,6,1,2,3,5] => [6,5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,10),(5,7),(6,7),(6,8),(7,9),(8,9),(10,8)],11)
=> ? = 2
[[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [2,3,1,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,13),(3,13),(4,11),(5,12),(5,14),(6,10),(6,14),(8,7),(9,7),(10,9),(11,10),(12,8),(13,11),(14,8),(14,9)],15)
=> ? = 3
[[1,3,5,6],[2],[4]]
=> [4,2,1,3,5,6] => [2,4,3,1,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,10),(5,11),(6,7),(6,9),(7,12),(8,11),(9,12),(10,9),(11,10)],13)
=> ? = 2
[[1,2,5,6],[3],[4]]
=> [4,3,1,2,5,6] => [4,2,3,1,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,12),(4,13),(5,13),(6,9),(6,12),(8,9),(9,7),(10,8),(11,8),(12,7),(13,10),(13,11)],14)
=> ? = 2
[[1,3,4,5],[2],[6]]
=> [6,2,1,3,4,5] => [2,6,5,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,13),(3,12),(4,7),(5,12),(5,14),(6,13),(6,14),(8,11),(9,8),(10,8),(11,7),(12,9),(13,10),(14,9),(14,10)],15)
=> ? = 3
[[1,2,4,5],[3],[6]]
=> [6,3,1,2,4,5] => [6,5,4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,17),(3,17),(4,12),(5,15),(5,16),(6,13),(6,16),(8,10),(9,11),(10,7),(11,7),(12,9),(13,8),(14,10),(14,11),(15,9),(15,14),(16,8),(16,14),(17,12),(17,15)],18)
=> ? = 3
[[1,2,3,5],[4],[6]]
=> [6,4,1,2,3,5] => [4,2,6,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 2
[[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => [4,1,2,6,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,9),(8,9)],10)
=> ? = 3
[[1,2,5],[3,4,6]]
=> [3,4,6,1,2,5] => [6,5,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,10),(5,7),(6,7),(6,8),(7,9),(8,9),(10,8)],11)
=> ? = 3
[[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => [6,4,1,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 2
[[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => [5,2,6,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2
[[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [4,1,5,2,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2
[[1,4,6],[2,5],[3]]
=> [3,2,5,1,4,6] => [2,5,4,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 2
[[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => [2,4,1,5,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 2
[[1,2,6],[3,5],[4]]
=> [4,3,5,1,2,6] => [4,1,5,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 2
[[1,4,5],[2,6],[3]]
=> [3,2,6,1,4,5] => [2,6,5,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,8),(5,7),(6,7),(6,8),(7,9),(8,9),(9,10)],11)
=> ? = 3
[[1,3,5],[2,6],[4]]
=> [4,2,6,1,3,5] => [2,4,1,6,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2
[[1,2,5],[3,6],[4]]
=> [4,3,6,1,2,5] => [4,1,6,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,9),(8,9)],10)
=> ? = 2
[[1,3,5],[2,4],[6]]
=> [6,2,4,1,3,5] => [2,6,5,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,9),(5,11),(6,7),(6,10),(7,12),(8,10),(10,12),(11,9),(12,11)],13)
=> ? = 3
[[1,2,5],[3,4],[6]]
=> [6,3,4,1,2,5] => [6,5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,12),(3,11),(4,10),(5,12),(5,13),(6,11),(6,15),(8,7),(9,7),(10,9),(11,8),(12,14),(13,14),(14,10),(14,15),(15,8),(15,9)],16)
=> ? = 3
[[1,3,4],[2,5],[6]]
=> [6,2,5,1,3,4] => [2,5,3,6,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,9),(3,9),(4,9),(5,9),(6,7),(7,8),(9,7)],10)
=> ? = 2
[[1,2,4],[3,5],[6]]
=> [6,3,5,1,2,4] => [5,2,3,6,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 2
[[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => [6,3,5,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,10),(5,8),(6,7),(7,9),(8,9),(10,7),(10,8)],11)
=> ? = 2
[[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [3,2,4,1,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,10),(5,11),(6,7),(6,9),(7,12),(8,11),(9,12),(10,9),(11,10)],13)
=> ? = 2
[[1,4,5],[2],[3],[6]]
=> [6,3,2,1,4,5] => [3,2,6,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
=> ? = 3
[[1,3,5],[2],[4],[6]]
=> [6,4,2,1,3,5] => [6,5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,14),(3,12),(4,12),(5,10),(6,11),(6,13),(8,7),(9,7),(10,9),(11,8),(12,14),(13,8),(13,9),(14,10),(14,13)],15)
=> ? = 2
[[1,2,5],[3],[4],[6]]
=> [6,4,3,1,2,5] => [3,6,5,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 2
[[1,4],[2,5],[3,6]]
=> [3,6,2,5,1,4] => [6,4,5,1,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,10),(3,7),(4,7),(5,8),(6,8),(7,10),(8,9),(8,11),(9,12),(10,11),(11,12)],13)
=> ? = 2
[[1,3],[2,5],[4,6]]
=> [4,6,2,5,1,3] => [5,1,4,6,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2
[[1,2],[3,5],[4,6]]
=> [4,6,3,5,1,2] => [3,5,1,4,6,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2
Description
The minimal length of a chain of small intervals in a lattice.
An interval $[a, b]$ is small if $b$ is a join of elements covering $a$.
Matching statistic: St001613
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001613: Lattices ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 50%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001613: Lattices ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 50%
Values
[[1,2,3]]
=> [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[[1,2],[3]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 3 - 1
[[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 1 = 2 - 1
[[1,2,3],[4]]
=> [4,1,2,3] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 3 - 1
[[1,3],[2,4]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 1 = 2 - 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 1 = 2 - 1
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 3 - 1
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3 - 1
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [3,2,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 2 - 1
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 3 - 1
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [4,3,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? = 2 - 1
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> 1 = 2 - 1
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [5,4,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3 - 1
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [3,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 1 = 2 - 1
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [2,3,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 2 - 1
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [2,5,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 3 - 1
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [5,4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 2 - 1
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [2,5,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 1 = 2 - 1
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [2,5,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 2 - 1
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 2 - 1
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [3,2,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 2 - 1
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,15),(2,14),(3,19),(3,21),(4,20),(4,21),(5,14),(5,19),(6,15),(6,20),(8,10),(9,11),(10,12),(11,13),(12,7),(13,7),(14,8),(15,9),(16,10),(16,18),(17,11),(17,18),(18,12),(18,13),(19,8),(19,16),(20,9),(20,17),(21,16),(21,17)],22)
=> ? = 3 - 1
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,14),(3,13),(4,12),(4,16),(5,13),(5,17),(6,16),(6,17),(8,10),(9,11),(10,7),(11,7),(12,8),(13,9),(14,12),(15,10),(15,11),(16,8),(16,15),(17,9),(17,15)],18)
=> ? = 3 - 1
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [3,2,1,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,12),(4,13),(4,14),(5,11),(5,15),(6,12),(6,15),(8,11),(9,7),(10,7),(11,9),(12,10),(13,8),(14,8),(15,9),(15,10)],16)
=> ? = 3 - 1
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [4,3,2,1,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,12),(4,11),(4,14),(5,12),(5,15),(6,13),(6,15),(8,11),(9,8),(10,8),(11,7),(12,9),(13,10),(14,7),(15,9),(15,10)],16)
=> ? = 2 - 1
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,15),(2,14),(3,19),(3,21),(4,20),(4,21),(5,14),(5,19),(6,15),(6,20),(8,10),(9,11),(10,12),(11,13),(12,7),(13,7),(14,8),(15,9),(16,10),(16,18),(17,11),(17,18),(18,12),(18,13),(19,8),(19,16),(20,9),(20,17),(21,16),(21,17)],22)
=> ? = 3 - 1
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [4,3,1,2,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,13),(3,13),(4,11),(5,10),(5,12),(6,9),(6,11),(8,10),(9,8),(10,7),(11,8),(12,7),(13,9)],14)
=> ? = 3 - 1
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [3,1,4,2,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,10),(5,7),(6,7),(6,8),(7,9),(8,9),(10,8)],11)
=> ? = 3 - 1
[[1,3,4,6],[2,5]]
=> [2,5,1,3,4,6] => [5,4,3,1,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,7),(5,11),(5,14),(6,12),(6,14),(8,10),(9,10),(10,7),(11,8),(12,9),(13,11),(14,8),(14,9)],15)
=> ? = 2 - 1
[[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => [3,1,5,4,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 2 - 1
[[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => [5,3,1,4,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,9),(3,9),(4,9),(5,9),(6,7),(7,8),(9,7)],10)
=> ? = 2 - 1
[[1,3,4,5],[2,6]]
=> [2,6,1,3,4,5] => [6,5,4,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,14),(3,13),(4,12),(4,16),(5,13),(5,17),(6,16),(6,17),(8,10),(9,11),(10,7),(11,7),(12,8),(13,9),(14,12),(15,10),(15,11),(16,8),(16,15),(17,9),(17,15)],18)
=> ? = 3 - 1
[[1,2,4,5],[3,6]]
=> [3,6,1,2,4,5] => [3,1,6,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,8),(5,7),(6,7),(6,8),(7,9),(8,9),(9,10)],11)
=> ? = 3 - 1
[[1,2,3,5],[4,6]]
=> [4,6,1,2,3,5] => [6,5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,10),(5,7),(6,7),(6,8),(7,9),(8,9),(10,8)],11)
=> ? = 2 - 1
[[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [2,3,1,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,13),(3,13),(4,11),(5,12),(5,14),(6,10),(6,14),(8,7),(9,7),(10,9),(11,10),(12,8),(13,11),(14,8),(14,9)],15)
=> ? = 3 - 1
[[1,3,5,6],[2],[4]]
=> [4,2,1,3,5,6] => [2,4,3,1,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,10),(5,11),(6,7),(6,9),(7,12),(8,11),(9,12),(10,9),(11,10)],13)
=> ? = 2 - 1
[[1,2,5,6],[3],[4]]
=> [4,3,1,2,5,6] => [4,2,3,1,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,12),(4,13),(5,13),(6,9),(6,12),(8,9),(9,7),(10,8),(11,8),(12,7),(13,10),(13,11)],14)
=> ? = 2 - 1
[[1,3,4,5],[2],[6]]
=> [6,2,1,3,4,5] => [2,6,5,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,13),(3,12),(4,7),(5,12),(5,14),(6,13),(6,14),(8,11),(9,8),(10,8),(11,7),(12,9),(13,10),(14,9),(14,10)],15)
=> ? = 3 - 1
[[1,2,4,5],[3],[6]]
=> [6,3,1,2,4,5] => [6,5,4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,17),(3,17),(4,12),(5,15),(5,16),(6,13),(6,16),(8,10),(9,11),(10,7),(11,7),(12,9),(13,8),(14,10),(14,11),(15,9),(15,14),(16,8),(16,14),(17,12),(17,15)],18)
=> ? = 3 - 1
[[1,2,3,5],[4],[6]]
=> [6,4,1,2,3,5] => [4,2,6,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 2 - 1
[[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => [4,1,2,6,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,9),(8,9)],10)
=> ? = 3 - 1
[[1,2,5],[3,4,6]]
=> [3,4,6,1,2,5] => [6,5,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,10),(5,7),(6,7),(6,8),(7,9),(8,9),(10,8)],11)
=> ? = 3 - 1
[[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => [6,4,1,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 2 - 1
[[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => [5,2,6,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
[[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [4,1,5,2,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
[[1,4,6],[2,5],[3]]
=> [3,2,5,1,4,6] => [2,5,4,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 2 - 1
[[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => [2,4,1,5,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 2 - 1
[[1,2,6],[3,5],[4]]
=> [4,3,5,1,2,6] => [4,1,5,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 2 - 1
[[1,4,5],[2,6],[3]]
=> [3,2,6,1,4,5] => [2,6,5,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,8),(5,7),(6,7),(6,8),(7,9),(8,9),(9,10)],11)
=> ? = 3 - 1
[[1,3,5],[2,6],[4]]
=> [4,2,6,1,3,5] => [2,4,1,6,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
[[1,2,5],[3,6],[4]]
=> [4,3,6,1,2,5] => [4,1,6,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,9),(8,9)],10)
=> ? = 2 - 1
[[1,3,5],[2,4],[6]]
=> [6,2,4,1,3,5] => [2,6,5,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,9),(5,11),(6,7),(6,10),(7,12),(8,10),(10,12),(11,9),(12,11)],13)
=> ? = 3 - 1
[[1,2,5],[3,4],[6]]
=> [6,3,4,1,2,5] => [6,5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,12),(3,11),(4,10),(5,12),(5,13),(6,11),(6,15),(8,7),(9,7),(10,9),(11,8),(12,14),(13,14),(14,10),(14,15),(15,8),(15,9)],16)
=> ? = 3 - 1
[[1,3,4],[2,5],[6]]
=> [6,2,5,1,3,4] => [2,5,3,6,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,9),(3,9),(4,9),(5,9),(6,7),(7,8),(9,7)],10)
=> ? = 2 - 1
[[1,2,4],[3,5],[6]]
=> [6,3,5,1,2,4] => [5,2,3,6,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 2 - 1
[[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => [6,3,5,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,10),(5,8),(6,7),(7,9),(8,9),(10,7),(10,8)],11)
=> ? = 2 - 1
[[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [3,2,4,1,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,10),(5,11),(6,7),(6,9),(7,12),(8,11),(9,12),(10,9),(11,10)],13)
=> ? = 2 - 1
[[1,4,5],[2],[3],[6]]
=> [6,3,2,1,4,5] => [3,2,6,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
=> ? = 3 - 1
[[1,3,5],[2],[4],[6]]
=> [6,4,2,1,3,5] => [6,5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,14),(3,12),(4,12),(5,10),(6,11),(6,13),(8,7),(9,7),(10,9),(11,8),(12,14),(13,8),(13,9),(14,10),(14,13)],15)
=> ? = 2 - 1
[[1,2,5],[3],[4],[6]]
=> [6,4,3,1,2,5] => [3,6,5,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 2 - 1
[[1,4],[2,5],[3,6]]
=> [3,6,2,5,1,4] => [6,4,5,1,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,10),(3,7),(4,7),(5,8),(6,8),(7,10),(8,9),(8,11),(9,12),(10,11),(11,12)],13)
=> ? = 2 - 1
[[1,3],[2,5],[4,6]]
=> [4,6,2,5,1,3] => [5,1,4,6,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
[[1,2],[3,5],[4,6]]
=> [4,6,3,5,1,2] => [3,5,1,4,6,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
Description
The binary logarithm of the size of the center of a lattice.
An element of a lattice is central if it is neutral and has a complement. The subposet induced by central elements is a Boolean lattice.
Matching statistic: St001719
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001719: Lattices ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 50%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001719: Lattices ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 50%
Values
[[1,2,3]]
=> [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[[1,2],[3]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 3 - 1
[[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 1 = 2 - 1
[[1,2,3],[4]]
=> [4,1,2,3] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 3 - 1
[[1,3],[2,4]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 1 = 2 - 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 1 = 2 - 1
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 3 - 1
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3 - 1
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [3,2,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 2 - 1
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 3 - 1
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [4,3,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? = 2 - 1
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> 1 = 2 - 1
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [5,4,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3 - 1
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [3,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 1 = 2 - 1
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [2,3,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 2 - 1
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [2,5,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 3 - 1
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [5,4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 2 - 1
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [2,5,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 1 = 2 - 1
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [2,5,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 2 - 1
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 2 - 1
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [3,2,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 2 - 1
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,15),(2,14),(3,19),(3,21),(4,20),(4,21),(5,14),(5,19),(6,15),(6,20),(8,10),(9,11),(10,12),(11,13),(12,7),(13,7),(14,8),(15,9),(16,10),(16,18),(17,11),(17,18),(18,12),(18,13),(19,8),(19,16),(20,9),(20,17),(21,16),(21,17)],22)
=> ? = 3 - 1
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,14),(3,13),(4,12),(4,16),(5,13),(5,17),(6,16),(6,17),(8,10),(9,11),(10,7),(11,7),(12,8),(13,9),(14,12),(15,10),(15,11),(16,8),(16,15),(17,9),(17,15)],18)
=> ? = 3 - 1
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [3,2,1,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,12),(4,13),(4,14),(5,11),(5,15),(6,12),(6,15),(8,11),(9,7),(10,7),(11,9),(12,10),(13,8),(14,8),(15,9),(15,10)],16)
=> ? = 3 - 1
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [4,3,2,1,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,12),(4,11),(4,14),(5,12),(5,15),(6,13),(6,15),(8,11),(9,8),(10,8),(11,7),(12,9),(13,10),(14,7),(15,9),(15,10)],16)
=> ? = 2 - 1
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,15),(2,14),(3,19),(3,21),(4,20),(4,21),(5,14),(5,19),(6,15),(6,20),(8,10),(9,11),(10,12),(11,13),(12,7),(13,7),(14,8),(15,9),(16,10),(16,18),(17,11),(17,18),(18,12),(18,13),(19,8),(19,16),(20,9),(20,17),(21,16),(21,17)],22)
=> ? = 3 - 1
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [4,3,1,2,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,13),(3,13),(4,11),(5,10),(5,12),(6,9),(6,11),(8,10),(9,8),(10,7),(11,8),(12,7),(13,9)],14)
=> ? = 3 - 1
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [3,1,4,2,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,10),(5,7),(6,7),(6,8),(7,9),(8,9),(10,8)],11)
=> ? = 3 - 1
[[1,3,4,6],[2,5]]
=> [2,5,1,3,4,6] => [5,4,3,1,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,7),(5,11),(5,14),(6,12),(6,14),(8,10),(9,10),(10,7),(11,8),(12,9),(13,11),(14,8),(14,9)],15)
=> ? = 2 - 1
[[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => [3,1,5,4,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 2 - 1
[[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => [5,3,1,4,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,9),(3,9),(4,9),(5,9),(6,7),(7,8),(9,7)],10)
=> ? = 2 - 1
[[1,3,4,5],[2,6]]
=> [2,6,1,3,4,5] => [6,5,4,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,14),(3,13),(4,12),(4,16),(5,13),(5,17),(6,16),(6,17),(8,10),(9,11),(10,7),(11,7),(12,8),(13,9),(14,12),(15,10),(15,11),(16,8),(16,15),(17,9),(17,15)],18)
=> ? = 3 - 1
[[1,2,4,5],[3,6]]
=> [3,6,1,2,4,5] => [3,1,6,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,8),(5,7),(6,7),(6,8),(7,9),(8,9),(9,10)],11)
=> ? = 3 - 1
[[1,2,3,5],[4,6]]
=> [4,6,1,2,3,5] => [6,5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,10),(5,7),(6,7),(6,8),(7,9),(8,9),(10,8)],11)
=> ? = 2 - 1
[[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [2,3,1,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,13),(3,13),(4,11),(5,12),(5,14),(6,10),(6,14),(8,7),(9,7),(10,9),(11,10),(12,8),(13,11),(14,8),(14,9)],15)
=> ? = 3 - 1
[[1,3,5,6],[2],[4]]
=> [4,2,1,3,5,6] => [2,4,3,1,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,10),(5,11),(6,7),(6,9),(7,12),(8,11),(9,12),(10,9),(11,10)],13)
=> ? = 2 - 1
[[1,2,5,6],[3],[4]]
=> [4,3,1,2,5,6] => [4,2,3,1,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,12),(4,13),(5,13),(6,9),(6,12),(8,9),(9,7),(10,8),(11,8),(12,7),(13,10),(13,11)],14)
=> ? = 2 - 1
[[1,3,4,5],[2],[6]]
=> [6,2,1,3,4,5] => [2,6,5,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,13),(3,12),(4,7),(5,12),(5,14),(6,13),(6,14),(8,11),(9,8),(10,8),(11,7),(12,9),(13,10),(14,9),(14,10)],15)
=> ? = 3 - 1
[[1,2,4,5],[3],[6]]
=> [6,3,1,2,4,5] => [6,5,4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,17),(3,17),(4,12),(5,15),(5,16),(6,13),(6,16),(8,10),(9,11),(10,7),(11,7),(12,9),(13,8),(14,10),(14,11),(15,9),(15,14),(16,8),(16,14),(17,12),(17,15)],18)
=> ? = 3 - 1
[[1,2,3,5],[4],[6]]
=> [6,4,1,2,3,5] => [4,2,6,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 2 - 1
[[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => [4,1,2,6,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,9),(8,9)],10)
=> ? = 3 - 1
[[1,2,5],[3,4,6]]
=> [3,4,6,1,2,5] => [6,5,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,10),(5,7),(6,7),(6,8),(7,9),(8,9),(10,8)],11)
=> ? = 3 - 1
[[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => [6,4,1,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 2 - 1
[[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => [5,2,6,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
[[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [4,1,5,2,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
[[1,4,6],[2,5],[3]]
=> [3,2,5,1,4,6] => [2,5,4,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 2 - 1
[[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => [2,4,1,5,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 2 - 1
[[1,2,6],[3,5],[4]]
=> [4,3,5,1,2,6] => [4,1,5,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 2 - 1
[[1,4,5],[2,6],[3]]
=> [3,2,6,1,4,5] => [2,6,5,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,8),(5,7),(6,7),(6,8),(7,9),(8,9),(9,10)],11)
=> ? = 3 - 1
[[1,3,5],[2,6],[4]]
=> [4,2,6,1,3,5] => [2,4,1,6,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
[[1,2,5],[3,6],[4]]
=> [4,3,6,1,2,5] => [4,1,6,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,9),(8,9)],10)
=> ? = 2 - 1
[[1,3,5],[2,4],[6]]
=> [6,2,4,1,3,5] => [2,6,5,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,9),(5,11),(6,7),(6,10),(7,12),(8,10),(10,12),(11,9),(12,11)],13)
=> ? = 3 - 1
[[1,2,5],[3,4],[6]]
=> [6,3,4,1,2,5] => [6,5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,12),(3,11),(4,10),(5,12),(5,13),(6,11),(6,15),(8,7),(9,7),(10,9),(11,8),(12,14),(13,14),(14,10),(14,15),(15,8),(15,9)],16)
=> ? = 3 - 1
[[1,3,4],[2,5],[6]]
=> [6,2,5,1,3,4] => [2,5,3,6,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,9),(3,9),(4,9),(5,9),(6,7),(7,8),(9,7)],10)
=> ? = 2 - 1
[[1,2,4],[3,5],[6]]
=> [6,3,5,1,2,4] => [5,2,3,6,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 2 - 1
[[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => [6,3,5,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,10),(5,8),(6,7),(7,9),(8,9),(10,7),(10,8)],11)
=> ? = 2 - 1
[[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [3,2,4,1,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,10),(5,11),(6,7),(6,9),(7,12),(8,11),(9,12),(10,9),(11,10)],13)
=> ? = 2 - 1
[[1,4,5],[2],[3],[6]]
=> [6,3,2,1,4,5] => [3,2,6,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
=> ? = 3 - 1
[[1,3,5],[2],[4],[6]]
=> [6,4,2,1,3,5] => [6,5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,14),(3,12),(4,12),(5,10),(6,11),(6,13),(8,7),(9,7),(10,9),(11,8),(12,14),(13,8),(13,9),(14,10),(14,13)],15)
=> ? = 2 - 1
[[1,2,5],[3],[4],[6]]
=> [6,4,3,1,2,5] => [3,6,5,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 2 - 1
[[1,4],[2,5],[3,6]]
=> [3,6,2,5,1,4] => [6,4,5,1,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,10),(3,7),(4,7),(5,8),(6,8),(7,10),(8,9),(8,11),(9,12),(10,11),(11,12)],13)
=> ? = 2 - 1
[[1,3],[2,5],[4,6]]
=> [4,6,2,5,1,3] => [5,1,4,6,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
[[1,2],[3,5],[4,6]]
=> [4,6,3,5,1,2] => [3,5,1,4,6,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
Description
The number of shortest chains of small intervals from the bottom to the top in a lattice.
An interval $[a, b]$ in a lattice is small if $b$ is a join of elements covering $a$.
Matching statistic: St001881
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001881: Lattices ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 50%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001881: Lattices ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 50%
Values
[[1,2,3]]
=> [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[[1,2],[3]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 3 - 1
[[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 1 = 2 - 1
[[1,2,3],[4]]
=> [4,1,2,3] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 3 - 1
[[1,3],[2,4]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 1 = 2 - 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 1 = 2 - 1
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 3 - 1
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3 - 1
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [3,2,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 2 - 1
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 3 - 1
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [4,3,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? = 2 - 1
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> 1 = 2 - 1
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [5,4,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3 - 1
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [3,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 1 = 2 - 1
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [2,3,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 2 - 1
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [2,5,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 3 - 1
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [5,4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 2 - 1
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [2,5,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 1 = 2 - 1
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [2,5,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 2 - 1
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 2 - 1
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [3,2,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 2 - 1
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,15),(2,14),(3,19),(3,21),(4,20),(4,21),(5,14),(5,19),(6,15),(6,20),(8,10),(9,11),(10,12),(11,13),(12,7),(13,7),(14,8),(15,9),(16,10),(16,18),(17,11),(17,18),(18,12),(18,13),(19,8),(19,16),(20,9),(20,17),(21,16),(21,17)],22)
=> ? = 3 - 1
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,14),(3,13),(4,12),(4,16),(5,13),(5,17),(6,16),(6,17),(8,10),(9,11),(10,7),(11,7),(12,8),(13,9),(14,12),(15,10),(15,11),(16,8),(16,15),(17,9),(17,15)],18)
=> ? = 3 - 1
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [3,2,1,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,12),(4,13),(4,14),(5,11),(5,15),(6,12),(6,15),(8,11),(9,7),(10,7),(11,9),(12,10),(13,8),(14,8),(15,9),(15,10)],16)
=> ? = 3 - 1
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [4,3,2,1,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,12),(4,11),(4,14),(5,12),(5,15),(6,13),(6,15),(8,11),(9,8),(10,8),(11,7),(12,9),(13,10),(14,7),(15,9),(15,10)],16)
=> ? = 2 - 1
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,15),(2,14),(3,19),(3,21),(4,20),(4,21),(5,14),(5,19),(6,15),(6,20),(8,10),(9,11),(10,12),(11,13),(12,7),(13,7),(14,8),(15,9),(16,10),(16,18),(17,11),(17,18),(18,12),(18,13),(19,8),(19,16),(20,9),(20,17),(21,16),(21,17)],22)
=> ? = 3 - 1
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [4,3,1,2,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,13),(3,13),(4,11),(5,10),(5,12),(6,9),(6,11),(8,10),(9,8),(10,7),(11,8),(12,7),(13,9)],14)
=> ? = 3 - 1
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [3,1,4,2,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,10),(5,7),(6,7),(6,8),(7,9),(8,9),(10,8)],11)
=> ? = 3 - 1
[[1,3,4,6],[2,5]]
=> [2,5,1,3,4,6] => [5,4,3,1,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,7),(5,11),(5,14),(6,12),(6,14),(8,10),(9,10),(10,7),(11,8),(12,9),(13,11),(14,8),(14,9)],15)
=> ? = 2 - 1
[[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => [3,1,5,4,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 2 - 1
[[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => [5,3,1,4,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,9),(3,9),(4,9),(5,9),(6,7),(7,8),(9,7)],10)
=> ? = 2 - 1
[[1,3,4,5],[2,6]]
=> [2,6,1,3,4,5] => [6,5,4,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,14),(3,13),(4,12),(4,16),(5,13),(5,17),(6,16),(6,17),(8,10),(9,11),(10,7),(11,7),(12,8),(13,9),(14,12),(15,10),(15,11),(16,8),(16,15),(17,9),(17,15)],18)
=> ? = 3 - 1
[[1,2,4,5],[3,6]]
=> [3,6,1,2,4,5] => [3,1,6,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,8),(5,7),(6,7),(6,8),(7,9),(8,9),(9,10)],11)
=> ? = 3 - 1
[[1,2,3,5],[4,6]]
=> [4,6,1,2,3,5] => [6,5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,10),(5,7),(6,7),(6,8),(7,9),(8,9),(10,8)],11)
=> ? = 2 - 1
[[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [2,3,1,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,13),(3,13),(4,11),(5,12),(5,14),(6,10),(6,14),(8,7),(9,7),(10,9),(11,10),(12,8),(13,11),(14,8),(14,9)],15)
=> ? = 3 - 1
[[1,3,5,6],[2],[4]]
=> [4,2,1,3,5,6] => [2,4,3,1,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,10),(5,11),(6,7),(6,9),(7,12),(8,11),(9,12),(10,9),(11,10)],13)
=> ? = 2 - 1
[[1,2,5,6],[3],[4]]
=> [4,3,1,2,5,6] => [4,2,3,1,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,12),(4,13),(5,13),(6,9),(6,12),(8,9),(9,7),(10,8),(11,8),(12,7),(13,10),(13,11)],14)
=> ? = 2 - 1
[[1,3,4,5],[2],[6]]
=> [6,2,1,3,4,5] => [2,6,5,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,13),(3,12),(4,7),(5,12),(5,14),(6,13),(6,14),(8,11),(9,8),(10,8),(11,7),(12,9),(13,10),(14,9),(14,10)],15)
=> ? = 3 - 1
[[1,2,4,5],[3],[6]]
=> [6,3,1,2,4,5] => [6,5,4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,17),(3,17),(4,12),(5,15),(5,16),(6,13),(6,16),(8,10),(9,11),(10,7),(11,7),(12,9),(13,8),(14,10),(14,11),(15,9),(15,14),(16,8),(16,14),(17,12),(17,15)],18)
=> ? = 3 - 1
[[1,2,3,5],[4],[6]]
=> [6,4,1,2,3,5] => [4,2,6,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 2 - 1
[[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => [4,1,2,6,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,9),(8,9)],10)
=> ? = 3 - 1
[[1,2,5],[3,4,6]]
=> [3,4,6,1,2,5] => [6,5,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,10),(5,7),(6,7),(6,8),(7,9),(8,9),(10,8)],11)
=> ? = 3 - 1
[[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => [6,4,1,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 2 - 1
[[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => [5,2,6,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
[[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [4,1,5,2,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
[[1,4,6],[2,5],[3]]
=> [3,2,5,1,4,6] => [2,5,4,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 2 - 1
[[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => [2,4,1,5,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 2 - 1
[[1,2,6],[3,5],[4]]
=> [4,3,5,1,2,6] => [4,1,5,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 2 - 1
[[1,4,5],[2,6],[3]]
=> [3,2,6,1,4,5] => [2,6,5,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,8),(5,7),(6,7),(6,8),(7,9),(8,9),(9,10)],11)
=> ? = 3 - 1
[[1,3,5],[2,6],[4]]
=> [4,2,6,1,3,5] => [2,4,1,6,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
[[1,2,5],[3,6],[4]]
=> [4,3,6,1,2,5] => [4,1,6,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,9),(8,9)],10)
=> ? = 2 - 1
[[1,3,5],[2,4],[6]]
=> [6,2,4,1,3,5] => [2,6,5,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,9),(5,11),(6,7),(6,10),(7,12),(8,10),(10,12),(11,9),(12,11)],13)
=> ? = 3 - 1
[[1,2,5],[3,4],[6]]
=> [6,3,4,1,2,5] => [6,5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,12),(3,11),(4,10),(5,12),(5,13),(6,11),(6,15),(8,7),(9,7),(10,9),(11,8),(12,14),(13,14),(14,10),(14,15),(15,8),(15,9)],16)
=> ? = 3 - 1
[[1,3,4],[2,5],[6]]
=> [6,2,5,1,3,4] => [2,5,3,6,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,9),(3,9),(4,9),(5,9),(6,7),(7,8),(9,7)],10)
=> ? = 2 - 1
[[1,2,4],[3,5],[6]]
=> [6,3,5,1,2,4] => [5,2,3,6,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 2 - 1
[[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => [6,3,5,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,10),(5,8),(6,7),(7,9),(8,9),(10,7),(10,8)],11)
=> ? = 2 - 1
[[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [3,2,4,1,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,10),(5,11),(6,7),(6,9),(7,12),(8,11),(9,12),(10,9),(11,10)],13)
=> ? = 2 - 1
[[1,4,5],[2],[3],[6]]
=> [6,3,2,1,4,5] => [3,2,6,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
=> ? = 3 - 1
[[1,3,5],[2],[4],[6]]
=> [6,4,2,1,3,5] => [6,5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,14),(3,12),(4,12),(5,10),(6,11),(6,13),(8,7),(9,7),(10,9),(11,8),(12,14),(13,8),(13,9),(14,10),(14,13)],15)
=> ? = 2 - 1
[[1,2,5],[3],[4],[6]]
=> [6,4,3,1,2,5] => [3,6,5,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 2 - 1
[[1,4],[2,5],[3,6]]
=> [3,6,2,5,1,4] => [6,4,5,1,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,10),(3,7),(4,7),(5,8),(6,8),(7,10),(8,9),(8,11),(9,12),(10,11),(11,12)],13)
=> ? = 2 - 1
[[1,3],[2,5],[4,6]]
=> [4,6,2,5,1,3] => [5,1,4,6,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
[[1,2],[3,5],[4,6]]
=> [4,6,3,5,1,2] => [3,5,1,4,6,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
Description
The number of factors of a lattice as a Cartesian product of lattices.
Since the cardinality of a lattice is the product of the cardinalities of its factors, this statistic is one whenever the cardinality of the lattice is prime.
Matching statistic: St001545
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001545: Graphs ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 50%
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001545: Graphs ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 50%
Values
[[1,2,3]]
=> [[1,2,3]]
=> [3] => ([],3)
=> ? = 2 - 2
[[1,2],[3]]
=> [[1,2,3]]
=> [3] => ([],3)
=> ? = 2 - 2
[[1,2,3,4]]
=> [[1,2,3,4]]
=> [4] => ([],4)
=> ? = 3 - 2
[[1,3,4],[2]]
=> [[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[[1,2,3],[4]]
=> [[1,2,3,4]]
=> [4] => ([],4)
=> ? = 3 - 2
[[1,3],[2,4]]
=> [[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[[1,2,3,4,5]]
=> [[1,2,3,4,5]]
=> [5] => ([],5)
=> ? = 3 - 2
[[1,3,4,5],[2]]
=> [[1,2,4,5],[3]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 3 - 2
[[1,2,4,5],[3]]
=> [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[[1,2,3,4],[5]]
=> [[1,2,3,4,5]]
=> [5] => ([],5)
=> ? = 3 - 2
[[1,3,5],[2,4]]
=> [[1,2,4],[3,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 - 2
[[1,2,5],[3,4]]
=> [[1,2,3,4],[5]]
=> [5] => ([],5)
=> ? = 2 - 2
[[1,3,4],[2,5]]
=> [[1,2,4,5],[3]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 3 - 2
[[1,2,4],[3,5]]
=> [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[[1,4,5],[2],[3]]
=> [[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[[1,3,4],[2],[5]]
=> [[1,2,4,5],[3]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 3 - 2
[[1,2,4],[3],[5]]
=> [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[[1,4],[2,5],[3]]
=> [[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[[1,3],[2,4],[5]]
=> [[1,2,4],[3,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 - 2
[[1,2],[3,4],[5]]
=> [[1,2,3,4],[5]]
=> [5] => ([],5)
=> ? = 2 - 2
[[1,4],[2],[3],[5]]
=> [[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[[1,2,3,4,5,6]]
=> [[1,2,3,4,5,6]]
=> [6] => ([],6)
=> ? = 3 - 2
[[1,3,4,5,6],[2]]
=> [[1,2,4,5,6],[3]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 3 - 2
[[1,2,4,5,6],[3]]
=> [[1,2,3,5,6],[4]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 3 - 2
[[1,2,3,5,6],[4]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 2 - 2
[[1,2,3,4,5],[6]]
=> [[1,2,3,4,5,6]]
=> [6] => ([],6)
=> ? = 3 - 2
[[1,3,5,6],[2,4]]
=> [[1,2,4,6],[3,5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 2
[[1,2,5,6],[3,4]]
=> [[1,2,3,4],[5,6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 3 - 2
[[1,3,4,6],[2,5]]
=> [[1,2,4,5],[3,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 2 - 2
[[1,2,4,6],[3,5]]
=> [[1,2,3,5],[4,6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 2 - 2
[[1,2,3,6],[4,5]]
=> [[1,2,3,4,5],[6]]
=> [6] => ([],6)
=> ? = 2 - 2
[[1,3,4,5],[2,6]]
=> [[1,2,4,5,6],[3]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 3 - 2
[[1,2,4,5],[3,6]]
=> [[1,2,3,5,6],[4]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 3 - 2
[[1,2,3,5],[4,6]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 2 - 2
[[1,4,5,6],[2],[3]]
=> [[1,2,5,6],[3],[4]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 3 - 2
[[1,3,5,6],[2],[4]]
=> [[1,2,4,6],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[[1,2,5,6],[3],[4]]
=> [[1,2,3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 2 - 2
[[1,3,4,5],[2],[6]]
=> [[1,2,4,5,6],[3]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 3 - 2
[[1,2,4,5],[3],[6]]
=> [[1,2,3,5,6],[4]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 3 - 2
[[1,2,3,5],[4],[6]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 2 - 2
[[1,3,5],[2,4,6]]
=> [[1,2,4,6],[3,5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 2
[[1,2,5],[3,4,6]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 3 - 2
[[1,3,4],[2,5,6]]
=> [[1,2,4,5,6],[3]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 2 - 2
[[1,2,4],[3,5,6]]
=> [[1,2,3,5,6],[4]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 2 - 2
[[1,2,3],[4,5,6]]
=> [[1,2,3,4,5,6]]
=> [6] => ([],6)
=> ? = 2 - 2
[[1,4,6],[2,5],[3]]
=> [[1,2,5],[3,6],[4]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 2 - 2
[[1,3,6],[2,5],[4]]
=> [[1,2,4,5],[3],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 2 - 2
[[1,2,6],[3,5],[4]]
=> [[1,2,3,5],[4],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 2 - 2
[[1,4,5],[2,6],[3]]
=> [[1,2,5,6],[3],[4]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 3 - 2
[[1,3,5],[2,6],[4]]
=> [[1,2,4,6],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[[1,2,5],[3,6],[4]]
=> [[1,2,3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 2 - 2
[[1,3,5],[2,4],[6]]
=> [[1,2,4,6],[3,5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 2
[[1,2,5],[3,4],[6]]
=> [[1,2,3,4],[5,6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 3 - 2
[[1,3,4],[2,5],[6]]
=> [[1,2,4,5],[3,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 2 - 2
[[1,2,4],[3,5],[6]]
=> [[1,2,3,5],[4,6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 2 - 2
[[1,2,3],[4,5],[6]]
=> [[1,2,3,4,5],[6]]
=> [6] => ([],6)
=> ? = 2 - 2
[[1,5,6],[2],[3],[4]]
=> [[1,2,6],[3],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 2 - 2
[[1,4,5],[2],[3],[6]]
=> [[1,2,5,6],[3],[4]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 3 - 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.
Matching statistic: St000181
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00158: Binary words —alternating inverse⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000181: Posets ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 50%
Mp00158: Binary words —alternating inverse⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000181: Posets ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 50%
Values
[[1,2,3]]
=> 00 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[1,2],[3]]
=> 01 => 00 => ([(0,2),(2,1)],3)
=> 1 = 2 - 1
[[1,2,3,4]]
=> 000 => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 3 - 1
[[1,3,4],[2]]
=> 100 => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2 - 1
[[1,2,3],[4]]
=> 001 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 3 - 1
[[1,3],[2,4]]
=> 101 => 111 => ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[[1,3],[2],[4]]
=> 101 => 111 => ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[[1,2,3,4,5]]
=> 0000 => 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 3 - 1
[[1,3,4,5],[2]]
=> 1000 => 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 3 - 1
[[1,2,4,5],[3]]
=> 0100 => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2 - 1
[[1,2,3,4],[5]]
=> 0001 => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 3 - 1
[[1,3,5],[2,4]]
=> 1010 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 2 - 1
[[1,2,5],[3,4]]
=> 0100 => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2 - 1
[[1,3,4],[2,5]]
=> 1001 => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3 - 1
[[1,2,4],[3,5]]
=> 0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 2 - 1
[[1,4,5],[2],[3]]
=> 1100 => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 2 - 1
[[1,3,4],[2],[5]]
=> 1001 => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3 - 1
[[1,2,4],[3],[5]]
=> 0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 2 - 1
[[1,4],[2,5],[3]]
=> 1101 => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2 - 1
[[1,3],[2,4],[5]]
=> 1011 => 1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2 - 1
[[1,2],[3,4],[5]]
=> 0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 2 - 1
[[1,4],[2],[3],[5]]
=> 1101 => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2 - 1
[[1,2,3,4,5,6]]
=> 00000 => 01010 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ? = 3 - 1
[[1,3,4,5,6],[2]]
=> 10000 => 11010 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ? = 3 - 1
[[1,2,4,5,6],[3]]
=> 01000 => 00010 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 3 - 1
[[1,2,3,5,6],[4]]
=> 00100 => 01110 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ? = 2 - 1
[[1,2,3,4,5],[6]]
=> 00001 => 01011 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ? = 3 - 1
[[1,3,5,6],[2,4]]
=> 10100 => 11110 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 3 - 1
[[1,2,5,6],[3,4]]
=> 01000 => 00010 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 3 - 1
[[1,3,4,6],[2,5]]
=> 10010 => 11000 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 2 - 1
[[1,2,4,6],[3,5]]
=> 01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 2 - 1
[[1,2,3,6],[4,5]]
=> 00100 => 01110 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ? = 2 - 1
[[1,3,4,5],[2,6]]
=> 10001 => 11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ? = 3 - 1
[[1,2,4,5],[3,6]]
=> 01001 => 00011 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 3 - 1
[[1,2,3,5],[4,6]]
=> 00101 => 01111 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 2 - 1
[[1,4,5,6],[2],[3]]
=> 11000 => 10010 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ? = 3 - 1
[[1,3,5,6],[2],[4]]
=> 10100 => 11110 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 2 - 1
[[1,2,5,6],[3],[4]]
=> 01100 => 00110 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 2 - 1
[[1,3,4,5],[2],[6]]
=> 10001 => 11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ? = 3 - 1
[[1,2,4,5],[3],[6]]
=> 01001 => 00011 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 3 - 1
[[1,2,3,5],[4],[6]]
=> 00101 => 01111 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 2 - 1
[[1,3,5],[2,4,6]]
=> 10101 => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 3 - 1
[[1,2,5],[3,4,6]]
=> 01001 => 00011 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 3 - 1
[[1,3,4],[2,5,6]]
=> 10010 => 11000 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 2 - 1
[[1,2,4],[3,5,6]]
=> 01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 2 - 1
[[1,2,3],[4,5,6]]
=> 00100 => 01110 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ? = 2 - 1
[[1,4,6],[2,5],[3]]
=> 11010 => 10000 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 2 - 1
[[1,3,6],[2,5],[4]]
=> 10100 => 11110 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 2 - 1
[[1,2,6],[3,5],[4]]
=> 01100 => 00110 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 2 - 1
[[1,4,5],[2,6],[3]]
=> 11001 => 10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 3 - 1
[[1,3,5],[2,6],[4]]
=> 10101 => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 2 - 1
[[1,2,5],[3,6],[4]]
=> 01101 => 00111 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 2 - 1
[[1,3,5],[2,4],[6]]
=> 10101 => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 3 - 1
[[1,2,5],[3,4],[6]]
=> 01001 => 00011 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 3 - 1
[[1,3,4],[2,5],[6]]
=> 10011 => 11001 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 2 - 1
[[1,2,4],[3,5],[6]]
=> 01011 => 00001 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 2 - 1
[[1,2,3],[4,5],[6]]
=> 00101 => 01111 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 2 - 1
[[1,5,6],[2],[3],[4]]
=> 11100 => 10110 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ? = 2 - 1
Description
The number of connected components of the Hasse diagram for the poset.
The following 17 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001890The maximum magnitude of the Möbius function of a poset. St000173The segment statistic of a semistandard tableau. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001404The number of distinct entries in a Gelfand Tsetlin pattern. St000942The number of critical left to right maxima of the parking functions. St001209The pmaj statistic of a parking function. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001935The number of ascents in a parking function. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001822The number of alignments of a signed permutation. St001823The Stasinski-Voll length of a signed permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons.
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