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Your data matches 4 different statistics following compositions of up to 3 maps.
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Matching statistic: St001673
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Mp00295: Standard tableaux —valley composition⟶ Integer compositions
St001673: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001673: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => 0
[[1,2]]
=> [2] => 0
[[1],[2]]
=> [2] => 0
[[1,2,3]]
=> [3] => 0
[[1,3],[2]]
=> [2,1] => 1
[[1,2],[3]]
=> [3] => 0
[[1],[2],[3]]
=> [3] => 0
[[1,2,3,4]]
=> [4] => 0
[[1,3,4],[2]]
=> [2,2] => 0
[[1,2,4],[3]]
=> [3,1] => 1
[[1,2,3],[4]]
=> [4] => 0
[[1,3],[2,4]]
=> [2,2] => 0
[[1,2],[3,4]]
=> [3,1] => 1
[[1,4],[2],[3]]
=> [3,1] => 1
[[1,3],[2],[4]]
=> [2,2] => 0
[[1,2],[3],[4]]
=> [4] => 0
[[1],[2],[3],[4]]
=> [4] => 0
[[1,2,3,4,5]]
=> [5] => 0
[[1,3,4,5],[2]]
=> [2,3] => 1
[[1,2,4,5],[3]]
=> [3,2] => 1
[[1,2,3,5],[4]]
=> [4,1] => 1
[[1,2,3,4],[5]]
=> [5] => 0
[[1,3,5],[2,4]]
=> [2,2,1] => 1
[[1,2,5],[3,4]]
=> [3,2] => 1
[[1,3,4],[2,5]]
=> [2,3] => 1
[[1,2,4],[3,5]]
=> [3,2] => 1
[[1,2,3],[4,5]]
=> [4,1] => 1
[[1,4,5],[2],[3]]
=> [3,2] => 1
[[1,3,5],[2],[4]]
=> [2,2,1] => 1
[[1,2,5],[3],[4]]
=> [4,1] => 1
[[1,3,4],[2],[5]]
=> [2,3] => 1
[[1,2,4],[3],[5]]
=> [3,2] => 1
[[1,2,3],[4],[5]]
=> [5] => 0
[[1,4],[2,5],[3]]
=> [3,2] => 1
[[1,3],[2,5],[4]]
=> [2,2,1] => 1
[[1,2],[3,5],[4]]
=> [4,1] => 1
[[1,3],[2,4],[5]]
=> [2,3] => 1
[[1,2],[3,4],[5]]
=> [3,2] => 1
[[1,5],[2],[3],[4]]
=> [4,1] => 1
[[1,4],[2],[3],[5]]
=> [3,2] => 1
[[1,3],[2],[4],[5]]
=> [2,3] => 1
[[1,2],[3],[4],[5]]
=> [5] => 0
[[1],[2],[3],[4],[5]]
=> [5] => 0
[[1,2,3,4,5,6]]
=> [6] => 0
[[1,3,4,5,6],[2]]
=> [2,4] => 1
[[1,2,4,5,6],[3]]
=> [3,3] => 0
[[1,2,3,5,6],[4]]
=> [4,2] => 1
[[1,2,3,4,6],[5]]
=> [5,1] => 1
[[1,2,3,4,5],[6]]
=> [6] => 0
[[1,3,5,6],[2,4]]
=> [2,2,2] => 0
Description
The degree of asymmetry of an integer composition.
This is the number of pairs of symmetrically positioned distinct entries.
Matching statistic: St000526
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000526: Posets ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 67%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000526: Posets ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 67%
Values
[[1]]
=> [1] => [[1],[]]
=> ([],1)
=> ? = 0 + 1
[[1,2]]
=> [2] => [[2],[]]
=> ([(0,1)],2)
=> 1 = 0 + 1
[[1],[2]]
=> [2] => [[2],[]]
=> ([(0,1)],2)
=> 1 = 0 + 1
[[1,2,3]]
=> [3] => [[3],[]]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[1,3],[2]]
=> [2,1] => [[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1,2],[3]]
=> [3] => [[3],[]]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[1],[2],[3]]
=> [3] => [[3],[]]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[1,2,3,4]]
=> [4] => [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[1,3,4],[2]]
=> [2,2] => [[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> 1 = 0 + 1
[[1,2,4],[3]]
=> [3,1] => [[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[[1,2,3],[4]]
=> [4] => [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[1,3],[2,4]]
=> [2,2] => [[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> 1 = 0 + 1
[[1,2],[3,4]]
=> [3,1] => [[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[[1,4],[2],[3]]
=> [3,1] => [[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[[1,3],[2],[4]]
=> [2,2] => [[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> 1 = 0 + 1
[[1,2],[3],[4]]
=> [4] => [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[1],[2],[3],[4]]
=> [4] => [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[1,2,3,4,5]]
=> [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[1,3,4,5],[2]]
=> [2,3] => [[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> 2 = 1 + 1
[[1,2,4,5],[3]]
=> [3,2] => [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> 2 = 1 + 1
[[1,2,3,5],[4]]
=> [4,1] => [[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2 = 1 + 1
[[1,2,3,4],[5]]
=> [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[1,3,5],[2,4]]
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[[1,2,5],[3,4]]
=> [3,2] => [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> 2 = 1 + 1
[[1,3,4],[2,5]]
=> [2,3] => [[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> 2 = 1 + 1
[[1,2,4],[3,5]]
=> [3,2] => [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> 2 = 1 + 1
[[1,2,3],[4,5]]
=> [4,1] => [[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2 = 1 + 1
[[1,4,5],[2],[3]]
=> [3,2] => [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> 2 = 1 + 1
[[1,3,5],[2],[4]]
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[[1,2,5],[3],[4]]
=> [4,1] => [[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2 = 1 + 1
[[1,3,4],[2],[5]]
=> [2,3] => [[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> 2 = 1 + 1
[[1,2,4],[3],[5]]
=> [3,2] => [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> 2 = 1 + 1
[[1,2,3],[4],[5]]
=> [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[1,4],[2,5],[3]]
=> [3,2] => [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> 2 = 1 + 1
[[1,3],[2,5],[4]]
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[[1,2],[3,5],[4]]
=> [4,1] => [[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2 = 1 + 1
[[1,3],[2,4],[5]]
=> [2,3] => [[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> 2 = 1 + 1
[[1,2],[3,4],[5]]
=> [3,2] => [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> 2 = 1 + 1
[[1,5],[2],[3],[4]]
=> [4,1] => [[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2 = 1 + 1
[[1,4],[2],[3],[5]]
=> [3,2] => [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> 2 = 1 + 1
[[1,3],[2],[4],[5]]
=> [2,3] => [[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> 2 = 1 + 1
[[1,2],[3],[4],[5]]
=> [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[1],[2],[3],[4],[5]]
=> [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[1,2,3,4,5,6]]
=> [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[[1,3,4,5,6],[2]]
=> [2,4] => [[5,2],[1]]
=> ([(0,5),(1,4),(1,5),(3,2),(4,3)],6)
=> 2 = 1 + 1
[[1,2,4,5,6],[3]]
=> [3,3] => [[5,3],[2]]
=> ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
=> 1 = 0 + 1
[[1,2,3,5,6],[4]]
=> [4,2] => [[5,4],[3]]
=> ([(0,4),(1,3),(1,5),(2,5),(4,2)],6)
=> 2 = 1 + 1
[[1,2,3,4,6],[5]]
=> [5,1] => [[5,5],[4]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 2 = 1 + 1
[[1,2,3,4,5],[6]]
=> [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[[1,3,5,6],[2,4]]
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
=> 1 = 0 + 1
[[1,2,5,6],[3,4]]
=> [3,3] => [[5,3],[2]]
=> ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
=> 1 = 0 + 1
[[1,2,3,4,5,6,7]]
=> [7] => [[7],[]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 0 + 1
[[1,3,4,5,6,7],[2]]
=> [2,5] => [[6,2],[1]]
=> ([(0,6),(1,5),(1,6),(3,4),(4,2),(5,3)],7)
=> ? = 1 + 1
[[1,2,4,5,6,7],[3]]
=> [3,4] => [[6,3],[2]]
=> ([(0,3),(1,5),(1,6),(3,6),(4,2),(5,4)],7)
=> ? = 1 + 1
[[1,2,3,5,6,7],[4]]
=> [4,3] => [[6,4],[3]]
=> ([(0,4),(1,5),(1,6),(3,6),(4,3),(5,2)],7)
=> ? = 1 + 1
[[1,2,3,4,6,7],[5]]
=> [5,2] => [[6,5],[4]]
=> ([(0,5),(1,3),(1,6),(2,6),(4,2),(5,4)],7)
=> ? = 1 + 1
[[1,2,3,4,5,7],[6]]
=> [6,1] => [[6,6],[5]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ? = 1 + 1
[[1,2,3,4,5,6],[7]]
=> [7] => [[7],[]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 0 + 1
[[1,3,5,6,7],[2,4]]
=> [2,2,3] => [[5,3,2],[2,1]]
=> ([(0,5),(1,5),(1,6),(2,3),(2,6),(3,4)],7)
=> ? = 1 + 1
[[1,2,5,6,7],[3,4]]
=> [3,4] => [[6,3],[2]]
=> ([(0,3),(1,5),(1,6),(3,6),(4,2),(5,4)],7)
=> ? = 1 + 1
[[1,3,4,6,7],[2,5]]
=> [2,3,2] => [[5,4,2],[3,1]]
=> ([(0,5),(1,3),(1,6),(2,4),(2,5),(4,6)],7)
=> ? = 0 + 1
[[1,2,4,6,7],[3,5]]
=> [3,2,2] => [[5,4,3],[3,2]]
=> ([(0,5),(0,6),(1,4),(2,3),(2,5),(4,6)],7)
=> ? = 1 + 1
[[1,2,3,6,7],[4,5]]
=> [4,3] => [[6,4],[3]]
=> ([(0,4),(1,5),(1,6),(3,6),(4,3),(5,2)],7)
=> ? = 1 + 1
[[1,3,4,5,7],[2,6]]
=> [2,4,1] => [[5,5,2],[4,1]]
=> ([(0,5),(1,6),(2,3),(2,5),(3,4),(4,6)],7)
=> ? = 1 + 1
[[1,2,4,5,7],[3,6]]
=> [3,3,1] => [[5,5,3],[4,2]]
=> ([(0,6),(1,3),(2,4),(2,5),(3,5),(4,6)],7)
=> ? = 1 + 1
[[1,2,3,5,7],[4,6]]
=> [4,2,1] => [[5,5,4],[4,3]]
=> ([(0,5),(1,5),(1,6),(2,3),(3,4),(4,6)],7)
=> ? = 1 + 1
[[1,2,3,4,7],[5,6]]
=> [5,2] => [[6,5],[4]]
=> ([(0,5),(1,3),(1,6),(2,6),(4,2),(5,4)],7)
=> ? = 1 + 1
[[1,3,4,5,6],[2,7]]
=> [2,5] => [[6,2],[1]]
=> ([(0,6),(1,5),(1,6),(3,4),(4,2),(5,3)],7)
=> ? = 1 + 1
[[1,2,4,5,6],[3,7]]
=> [3,4] => [[6,3],[2]]
=> ([(0,3),(1,5),(1,6),(3,6),(4,2),(5,4)],7)
=> ? = 1 + 1
[[1,2,3,5,6],[4,7]]
=> [4,3] => [[6,4],[3]]
=> ([(0,4),(1,5),(1,6),(3,6),(4,3),(5,2)],7)
=> ? = 1 + 1
[[1,2,3,4,6],[5,7]]
=> [5,2] => [[6,5],[4]]
=> ([(0,5),(1,3),(1,6),(2,6),(4,2),(5,4)],7)
=> ? = 1 + 1
[[1,2,3,4,5],[6,7]]
=> [6,1] => [[6,6],[5]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ? = 1 + 1
[[1,4,5,6,7],[2],[3]]
=> [3,4] => [[6,3],[2]]
=> ([(0,3),(1,5),(1,6),(3,6),(4,2),(5,4)],7)
=> ? = 1 + 1
[[1,3,5,6,7],[2],[4]]
=> [2,2,3] => [[5,3,2],[2,1]]
=> ([(0,5),(1,5),(1,6),(2,3),(2,6),(3,4)],7)
=> ? = 1 + 1
[[1,2,5,6,7],[3],[4]]
=> [4,3] => [[6,4],[3]]
=> ([(0,4),(1,5),(1,6),(3,6),(4,3),(5,2)],7)
=> ? = 1 + 1
[[1,3,4,6,7],[2],[5]]
=> [2,3,2] => [[5,4,2],[3,1]]
=> ([(0,5),(1,3),(1,6),(2,4),(2,5),(4,6)],7)
=> ? = 0 + 1
[[1,2,4,6,7],[3],[5]]
=> [3,2,2] => [[5,4,3],[3,2]]
=> ([(0,5),(0,6),(1,4),(2,3),(2,5),(4,6)],7)
=> ? = 1 + 1
[[1,2,3,6,7],[4],[5]]
=> [5,2] => [[6,5],[4]]
=> ([(0,5),(1,3),(1,6),(2,6),(4,2),(5,4)],7)
=> ? = 1 + 1
[[1,3,4,5,7],[2],[6]]
=> [2,4,1] => [[5,5,2],[4,1]]
=> ([(0,5),(1,6),(2,3),(2,5),(3,4),(4,6)],7)
=> ? = 1 + 1
[[1,2,4,5,7],[3],[6]]
=> [3,3,1] => [[5,5,3],[4,2]]
=> ([(0,6),(1,3),(2,4),(2,5),(3,5),(4,6)],7)
=> ? = 1 + 1
[[1,2,3,5,7],[4],[6]]
=> [4,2,1] => [[5,5,4],[4,3]]
=> ([(0,5),(1,5),(1,6),(2,3),(3,4),(4,6)],7)
=> ? = 1 + 1
[[1,2,3,4,7],[5],[6]]
=> [6,1] => [[6,6],[5]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ? = 1 + 1
[[1,3,4,5,6],[2],[7]]
=> [2,5] => [[6,2],[1]]
=> ([(0,6),(1,5),(1,6),(3,4),(4,2),(5,3)],7)
=> ? = 1 + 1
[[1,2,4,5,6],[3],[7]]
=> [3,4] => [[6,3],[2]]
=> ([(0,3),(1,5),(1,6),(3,6),(4,2),(5,4)],7)
=> ? = 1 + 1
[[1,2,3,5,6],[4],[7]]
=> [4,3] => [[6,4],[3]]
=> ([(0,4),(1,5),(1,6),(3,6),(4,3),(5,2)],7)
=> ? = 1 + 1
[[1,2,3,4,6],[5],[7]]
=> [5,2] => [[6,5],[4]]
=> ([(0,5),(1,3),(1,6),(2,6),(4,2),(5,4)],7)
=> ? = 1 + 1
[[1,2,3,4,5],[6],[7]]
=> [7] => [[7],[]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 0 + 1
[[1,2,5,7],[3,4,6]]
=> [3,3,1] => [[5,5,3],[4,2]]
=> ([(0,6),(1,3),(2,4),(2,5),(3,5),(4,6)],7)
=> ? = 1 + 1
[[1,3,4,7],[2,5,6]]
=> [2,3,2] => [[5,4,2],[3,1]]
=> ([(0,5),(1,3),(1,6),(2,4),(2,5),(4,6)],7)
=> ? = 0 + 1
[[1,2,4,7],[3,5,6]]
=> [3,2,2] => [[5,4,3],[3,2]]
=> ([(0,5),(0,6),(1,4),(2,3),(2,5),(4,6)],7)
=> ? = 1 + 1
[[1,2,3,7],[4,5,6]]
=> [4,3] => [[6,4],[3]]
=> ([(0,4),(1,5),(1,6),(3,6),(4,3),(5,2)],7)
=> ? = 1 + 1
[[1,3,5,6],[2,4,7]]
=> [2,2,3] => [[5,3,2],[2,1]]
=> ([(0,5),(1,5),(1,6),(2,3),(2,6),(3,4)],7)
=> ? = 1 + 1
[[1,2,5,6],[3,4,7]]
=> [3,4] => [[6,3],[2]]
=> ([(0,3),(1,5),(1,6),(3,6),(4,2),(5,4)],7)
=> ? = 1 + 1
[[1,3,4,6],[2,5,7]]
=> [2,3,2] => [[5,4,2],[3,1]]
=> ([(0,5),(1,3),(1,6),(2,4),(2,5),(4,6)],7)
=> ? = 0 + 1
[[1,2,4,6],[3,5,7]]
=> [3,2,2] => [[5,4,3],[3,2]]
=> ([(0,5),(0,6),(1,4),(2,3),(2,5),(4,6)],7)
=> ? = 1 + 1
[[1,2,3,6],[4,5,7]]
=> [4,3] => [[6,4],[3]]
=> ([(0,4),(1,5),(1,6),(3,6),(4,3),(5,2)],7)
=> ? = 1 + 1
[[1,3,4,5],[2,6,7]]
=> [2,4,1] => [[5,5,2],[4,1]]
=> ([(0,5),(1,6),(2,3),(2,5),(3,4),(4,6)],7)
=> ? = 1 + 1
[[1,2,4,5],[3,6,7]]
=> [3,3,1] => [[5,5,3],[4,2]]
=> ([(0,6),(1,3),(2,4),(2,5),(3,5),(4,6)],7)
=> ? = 1 + 1
[[1,2,3,5],[4,6,7]]
=> [4,2,1] => [[5,5,4],[4,3]]
=> ([(0,5),(1,5),(1,6),(2,3),(3,4),(4,6)],7)
=> ? = 1 + 1
[[1,2,3,4],[5,6,7]]
=> [5,2] => [[6,5],[4]]
=> ([(0,5),(1,3),(1,6),(2,6),(4,2),(5,4)],7)
=> ? = 1 + 1
Description
The number of posets with combinatorially isomorphic order polytopes.
Matching statistic: St000260
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 67%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 67%
Values
[[1]]
=> [1] => ([],1)
=> 0
[[1,2]]
=> [2] => ([],2)
=> ? = 0
[[1],[2]]
=> [2] => ([],2)
=> ? = 0
[[1,2,3]]
=> [3] => ([],3)
=> ? = 0
[[1,3],[2]]
=> [2,1] => ([(0,2),(1,2)],3)
=> 1
[[1,2],[3]]
=> [3] => ([],3)
=> ? = 0
[[1],[2],[3]]
=> [3] => ([],3)
=> ? = 0
[[1,2,3,4]]
=> [4] => ([],4)
=> ? = 0
[[1,3,4],[2]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 0
[[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[[1,2,3],[4]]
=> [4] => ([],4)
=> ? = 0
[[1,3],[2,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 0
[[1,2],[3,4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[[1,4],[2],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[[1,3],[2],[4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 0
[[1,2],[3],[4]]
=> [4] => ([],4)
=> ? = 0
[[1],[2],[3],[4]]
=> [4] => ([],4)
=> ? = 0
[[1,2,3,4,5]]
=> [5] => ([],5)
=> ? = 0
[[1,3,4,5],[2]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 1
[[1,2,4,5],[3]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1
[[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[[1,2,3,4],[5]]
=> [5] => ([],5)
=> ? = 0
[[1,3,5],[2,4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[1,2,5],[3,4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1
[[1,3,4],[2,5]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 1
[[1,2,4],[3,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1
[[1,2,3],[4,5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[[1,4,5],[2],[3]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1
[[1,3,5],[2],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[[1,3,4],[2],[5]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 1
[[1,2,4],[3],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1
[[1,2,3],[4],[5]]
=> [5] => ([],5)
=> ? = 0
[[1,4],[2,5],[3]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1
[[1,3],[2,5],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[1,2],[3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[[1,3],[2,4],[5]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 1
[[1,2],[3,4],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1
[[1,5],[2],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[[1,4],[2],[3],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1
[[1,3],[2],[4],[5]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 1
[[1,2],[3],[4],[5]]
=> [5] => ([],5)
=> ? = 0
[[1],[2],[3],[4],[5]]
=> [5] => ([],5)
=> ? = 0
[[1,2,3,4,5,6]]
=> [6] => ([],6)
=> ? = 0
[[1,3,4,5,6],[2]]
=> [2,4] => ([(3,5),(4,5)],6)
=> ? = 1
[[1,2,4,5,6],[3]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 0
[[1,2,3,5,6],[4]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1
[[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,2,3,4,5],[6]]
=> [6] => ([],6)
=> ? = 0
[[1,3,5,6],[2,4]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[[1,2,5,6],[3,4]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 0
[[1,3,4,6],[2,5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[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)
=> 1
[[1,2,3,6],[4,5]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1
[[1,3,4,5],[2,6]]
=> [2,4] => ([(3,5),(4,5)],6)
=> ? = 1
[[1,2,4,5],[3,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 0
[[1,2,3,5],[4,6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1
[[1,2,3,4],[5,6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,4,5,6],[2],[3]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 0
[[1,3,5,6],[2],[4]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[[1,2,5,6],[3],[4]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1
[[1,3,4,6],[2],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[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)
=> 1
[[1,2,3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,3,4,5],[2],[6]]
=> [2,4] => ([(3,5),(4,5)],6)
=> ? = 1
[[1,2,4,5],[3],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 0
[[1,2,3,5],[4],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1
[[1,2,3,4],[5],[6]]
=> [6] => ([],6)
=> ? = 0
[[1,3,5],[2,4,6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[[1,2,5],[3,4,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 0
[[1,3,4],[2,5,6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,2,4],[3,5,6]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,4,6],[2,5],[3]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,3,6],[2,4],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,2,6],[3,4],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,3,4],[2,6],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,2,4],[3,6],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,2,3],[4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,4,6],[2],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,3,6],[2],[4],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,2,6],[3],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,3],[2,4],[5,6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,2],[3,4],[5,6]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,4],[2,6],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,3],[2,6],[4],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,2],[3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,6],[2],[3],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,2,3,4,5,7],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[[1,3,4,5,7],[2,6]]
=> [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[[1,2,4,5,7],[3,6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[[1,2,3,5,7],[4,6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[[1,2,3,4,5],[6,7]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[[1,3,4,5,7],[2],[6]]
=> [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[[1,2,4,5,7],[3],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[[1,2,3,5,7],[4],[6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[[1,2,3,4,7],[5],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[[1,3,5,7],[2,4,6]]
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[[1,2,5,7],[3,4,6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[[1,3,4,5],[2,6,7]]
=> [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[[1,2,4,5],[3,6,7]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St000455
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 67%
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 67%
Values
[[1]]
=> [1] => [1] => ([],1)
=> ? = 0 - 1
[[1,2]]
=> [2] => [1,1] => ([(0,1)],2)
=> -1 = 0 - 1
[[1],[2]]
=> [2] => [1,1] => ([(0,1)],2)
=> -1 = 0 - 1
[[1,2,3]]
=> [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> -1 = 0 - 1
[[1,3],[2]]
=> [2,1] => [1,2] => ([(1,2)],3)
=> 0 = 1 - 1
[[1,2],[3]]
=> [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> -1 = 0 - 1
[[1],[2],[3]]
=> [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> -1 = 0 - 1
[[1,2,3,4]]
=> [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> -1 = 0 - 1
[[1,3,4],[2]]
=> [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 - 1
[[1,2,4],[3]]
=> [3,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[1,2,3],[4]]
=> [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> -1 = 0 - 1
[[1,3],[2,4]]
=> [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 - 1
[[1,2],[3,4]]
=> [3,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[1,4],[2],[3]]
=> [3,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[1,3],[2],[4]]
=> [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 - 1
[[1,2],[3],[4]]
=> [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> -1 = 0 - 1
[[1],[2],[3],[4]]
=> [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> -1 = 0 - 1
[[1,2,3,4,5]]
=> [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> -1 = 0 - 1
[[1,3,4,5],[2]]
=> [2,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[[1,2,4,5],[3]]
=> [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[[1,2,3,5],[4]]
=> [4,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[1,2,3,4],[5]]
=> [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> -1 = 0 - 1
[[1,3,5],[2,4]]
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[[1,2,5],[3,4]]
=> [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[[1,3,4],[2,5]]
=> [2,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[[1,2,4],[3,5]]
=> [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[[1,2,3],[4,5]]
=> [4,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[1,4,5],[2],[3]]
=> [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[[1,3,5],[2],[4]]
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[[1,2,5],[3],[4]]
=> [4,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[1,3,4],[2],[5]]
=> [2,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[[1,2,4],[3],[5]]
=> [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[[1,2,3],[4],[5]]
=> [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> -1 = 0 - 1
[[1,4],[2,5],[3]]
=> [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[[1,3],[2,5],[4]]
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[[1,2],[3,5],[4]]
=> [4,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[1,3],[2,4],[5]]
=> [2,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[[1,2],[3,4],[5]]
=> [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[[1,5],[2],[3],[4]]
=> [4,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[1,4],[2],[3],[5]]
=> [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[[1,3],[2],[4],[5]]
=> [2,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[[1,2],[3],[4],[5]]
=> [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> -1 = 0 - 1
[[1],[2],[3],[4],[5]]
=> [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> -1 = 0 - 1
[[1,2,3,4,5,6]]
=> [6] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> -1 = 0 - 1
[[1,3,4,5,6],[2]]
=> [2,4] => [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)
=> ? = 1 - 1
[[1,2,4,5,6],[3]]
=> [3,3] => [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)
=> ? = 0 - 1
[[1,2,3,5,6],[4]]
=> [4,2] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
[[1,2,3,4,6],[5]]
=> [5,1] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,2,3,4,5],[6]]
=> [6] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> -1 = 0 - 1
[[1,3,5,6],[2,4]]
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 - 1
[[1,2,5,6],[3,4]]
=> [3,3] => [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)
=> ? = 0 - 1
[[1,3,4,6],[2,5]]
=> [2,3,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
[[1,2,4,6],[3,5]]
=> [3,2,1] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
[[1,2,3,6],[4,5]]
=> [4,2] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
[[1,3,4,5],[2,6]]
=> [2,4] => [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)
=> ? = 1 - 1
[[1,2,4,5],[3,6]]
=> [3,3] => [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)
=> ? = 0 - 1
[[1,2,3,5],[4,6]]
=> [4,2] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
[[1,2,3,4],[5,6]]
=> [5,1] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,4,5,6],[2],[3]]
=> [3,3] => [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)
=> ? = 0 - 1
[[1,3,5,6],[2],[4]]
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 - 1
[[1,2,5,6],[3],[4]]
=> [4,2] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
[[1,3,4,6],[2],[5]]
=> [2,3,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
[[1,2,4,6],[3],[5]]
=> [3,2,1] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
[[1,2,3,6],[4],[5]]
=> [5,1] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3,4,5],[2],[6]]
=> [2,4] => [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)
=> ? = 1 - 1
[[1,2,4,5],[3],[6]]
=> [3,3] => [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)
=> ? = 0 - 1
[[1,2,3,5],[4],[6]]
=> [4,2] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
[[1,2,3,4],[5],[6]]
=> [6] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> -1 = 0 - 1
[[1,3,5],[2,4,6]]
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 - 1
[[1,2,5],[3,4,6]]
=> [3,3] => [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)
=> ? = 0 - 1
[[1,3,4],[2,5,6]]
=> [2,3,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
[[1,2,4],[3,5,6]]
=> [3,2,1] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
[[1,2,3],[4,5,6]]
=> [4,2] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
[[1,4,6],[2,5],[3]]
=> [3,2,1] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
[[1,3,6],[2,5],[4]]
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 - 1
[[1,2,6],[3,5],[4]]
=> [4,2] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
[[1,3,6],[2,4],[5]]
=> [2,3,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
[[1,2,6],[3,4],[5]]
=> [3,2,1] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
[[1,4,5],[2,6],[3]]
=> [3,3] => [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)
=> ? = 0 - 1
[[1,2,3],[4,6],[5]]
=> [5,1] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,2,6],[3],[4],[5]]
=> [5,1] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,2,3],[4],[5],[6]]
=> [6] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> -1 = 0 - 1
[[1,2],[3,6],[4],[5]]
=> [5,1] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,6],[2],[3],[4],[5]]
=> [5,1] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,2],[3],[4],[5],[6]]
=> [6] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> -1 = 0 - 1
[[1],[2],[3],[4],[5],[6]]
=> [6] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> -1 = 0 - 1
[[1,2,3,4,5,6,7]]
=> [7] => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> -1 = 0 - 1
[[1,2,3,4,5,7],[6]]
=> [6,1] => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,2,3,4,5,6],[7]]
=> [7] => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> -1 = 0 - 1
[[1,2,3,4,5],[6,7]]
=> [6,1] => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,2,3,4,7],[5],[6]]
=> [6,1] => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,2,3,4,5],[6],[7]]
=> [7] => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> -1 = 0 - 1
[[1,2,3,4],[5,7],[6]]
=> [6,1] => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,2,3,7],[4],[5],[6]]
=> [6,1] => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,2,3,4],[5],[6],[7]]
=> [7] => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> -1 = 0 - 1
[[1,2,3],[4,7],[5],[6]]
=> [6,1] => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,2,7],[3],[4],[5],[6]]
=> [6,1] => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,2,3],[4],[5],[6],[7]]
=> [7] => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> -1 = 0 - 1
[[1,2],[3,7],[4],[5],[6]]
=> [6,1] => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,7],[2],[3],[4],[5],[6]]
=> [6,1] => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
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