searching the database
Your data matches 22 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St001405
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001405: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001405: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => 0
[[1,2]]
=> [1,2] => 1
[[1],[2]]
=> [2,1] => 1
[[1,2,3]]
=> [1,2,3] => 2
[[1,3],[2]]
=> [2,1,3] => 1
[[1,2],[3]]
=> [3,1,2] => 1
[[1],[2],[3]]
=> [3,2,1] => 2
[[1,2,3,4]]
=> [1,2,3,4] => 3
[[1,3,4],[2]]
=> [2,1,3,4] => 2
[[1,2,4],[3]]
=> [3,1,2,4] => 1
[[1,2,3],[4]]
=> [4,1,2,3] => 2
[[1,3],[2,4]]
=> [2,4,1,3] => 0
[[1,2],[3,4]]
=> [3,4,1,2] => 2
[[1,4],[2],[3]]
=> [3,2,1,4] => 2
[[1,3],[2],[4]]
=> [4,2,1,3] => 1
[[1,2],[3],[4]]
=> [4,3,1,2] => 2
[[1],[2],[3],[4]]
=> [4,3,2,1] => 3
[[1,2,3,4,5]]
=> [1,2,3,4,5] => 4
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => 3
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => 2
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => 2
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => 3
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => 0
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => 2
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => 1
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => 1
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => 3
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 3
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => 1
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => 2
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => 2
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => 1
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 3
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => 1
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 0
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => 2
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => 0
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 3
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => 2
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => 2
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 3
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 4
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => 5
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => 4
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => 3
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => 3
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => 3
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => 4
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => 1
Description
The number of bonds in a permutation.
For a permutation $\pi$, the pair $(\pi_i, \pi_{i+1})$ is a bond if $|\pi_i-\pi_{i+1}| = 1$.
Matching statistic: St001630
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001630: Lattices ⟶ ℤResult quality: 33% ●values known / values provided: 41%●distinct values known / distinct values provided: 33%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001630: Lattices ⟶ ℤResult quality: 33% ●values known / values provided: 41%●distinct values known / distinct values provided: 33%
Values
[[1]]
=> [1] => ([],1)
=> ([],1)
=> ? = 0
[[1,2]]
=> [1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1}
[[1],[2]]
=> [2,1] => ([],2)
=> ([],1)
=> ? ∊ {1,1}
[[1,2,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[[1,3],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([],1)
=> ? ∊ {1,2,2}
[[1,2],[3]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {1,2,2}
[[1],[2],[3]]
=> [3,2,1] => ([],3)
=> ([],1)
=> ? ∊ {1,2,2}
[[1,2,3,4]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 2
[[1,3,4],[2]]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([],1)
=> ? ∊ {0,2,2,2,3,3}
[[1,2,4],[3]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,2,2,2,3,3}
[[1,2,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[[1,3],[2,4]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[[1,2],[3,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1,4],[2],[3]]
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> ? ∊ {0,2,2,2,3,3}
[[1,3],[2],[4]]
=> [4,2,1,3] => ([(1,3),(2,3)],4)
=> ([],1)
=> ? ∊ {0,2,2,2,3,3}
[[1,2],[3],[4]]
=> [4,3,1,2] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,2,2,2,3,3}
[[1],[2],[3],[4]]
=> [4,3,2,1] => ([],4)
=> ([],1)
=> ? ∊ {0,2,2,2,3,3}
[[1,2,3,4,5]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([],1)
=> ? ∊ {0,0,0,2,2,2,3,3,3,3,3,3,3,4,4}
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,2,2,2,3,3,3,3,3,3,3,4,4}
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 2
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 2
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([],1)
=> ? ∊ {0,0,0,2,2,2,3,3,3,3,3,3,3,4,4}
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([],1)
=> ? ∊ {0,0,0,2,2,2,3,3,3,3,3,3,3,4,4}
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,2,2,2,3,3,3,3,3,3,3,4,4}
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
=> ([],1)
=> ? ∊ {0,0,0,2,2,2,3,3,3,3,3,3,3,4,4}
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,2,2,2,3,3,3,3,3,3,3,4,4}
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ? ∊ {0,0,0,2,2,2,3,3,3,3,3,3,3,4,4}
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([],1)
=> ? ∊ {0,0,0,2,2,2,3,3,3,3,3,3,3,4,4}
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,2,2,2,3,3,3,3,3,3,3,4,4}
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => ([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ? ∊ {0,0,0,2,2,2,3,3,3,3,3,3,3,4,4}
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ? ∊ {0,0,0,2,2,2,3,3,3,3,3,3,3,4,4}
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => ([(2,4),(3,4)],5)
=> ([],1)
=> ? ∊ {0,0,0,2,2,2,3,3,3,3,3,3,3,4,4}
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,2,2,2,3,3,3,3,3,3,3,4,4}
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,2,2,2,3,3,3,3,3,3,3,4,4}
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,2),(2,1)],3)
=> 1
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 2
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ([(1,5),(3,4),(4,2),(5,3)],6)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => ([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
=> ([(0,2),(2,1)],3)
=> 1
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1,3,4,6],[2,5]]
=> [2,5,1,3,4,6] => ([(0,4),(1,2),(1,4),(2,5),(3,5),(4,3)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 2
[[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => ([(0,3),(1,2),(1,4),(2,5),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[[1,3,4,5],[2,6]]
=> [2,6,1,3,4,5] => ([(0,5),(1,3),(1,5),(4,2),(5,4)],6)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[1,2,4,5],[3,6]]
=> [3,6,1,2,4,5] => ([(0,4),(1,3),(1,5),(4,5),(5,2)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2
[[1,2,3,5],[4,6]]
=> [4,6,1,2,3,5] => ([(0,4),(1,3),(1,5),(2,5),(4,2)],6)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2
[[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => ([(0,5),(1,3),(4,2),(5,4)],6)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => ([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,3,5,6],[2],[4]]
=> [4,2,1,3,5,6] => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,2,5,6],[3],[4]]
=> [4,3,1,2,5,6] => ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,3,4,6],[2],[5]]
=> [5,2,1,3,4,6] => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,2,4,6],[3],[5]]
=> [5,3,1,2,4,6] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,2,3,6],[4],[5]]
=> [5,4,1,2,3,6] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 1
[[1,3,4,5],[2],[6]]
=> [6,2,1,3,4,5] => ([(1,5),(2,5),(3,4),(5,3)],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,2,4,5],[3],[6]]
=> [6,3,1,2,4,5] => ([(1,5),(2,3),(3,5),(5,4)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,2,3,5],[4],[6]]
=> [6,4,1,2,3,5] => ([(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 1
[[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => ([(2,3),(3,5),(5,4)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 2
[[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => ([(0,4),(1,2),(1,4),(2,3),(2,5),(4,5)],6)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2
[[1,2,5],[3,4,6]]
=> [3,4,6,1,2,5] => ([(0,3),(1,4),(3,5),(4,2),(4,5)],6)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2
[[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => ([(0,5),(1,4),(1,5),(4,2),(5,3)],6)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 2
[[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => ([(0,5),(1,4),(4,2),(5,3)],6)
=> ([(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)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,4,6],[2,5],[3]]
=> [3,2,5,1,4,6] => ([(0,5),(1,3),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,2,6],[3,5],[4]]
=> [4,3,5,1,2,6] => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,3,6],[2,4],[5]]
=> [5,2,4,1,3,6] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 1
[[1,2,6],[3,4],[5]]
=> [5,3,4,1,2,6] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1,4,5],[2,6],[3]]
=> [3,2,6,1,4,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(5,3)],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,3,5],[2,6],[4]]
=> [4,2,6,1,3,5] => ([(0,4),(1,3),(1,5),(2,3),(2,4),(4,5)],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,2,5],[3,6],[4]]
=> [4,3,6,1,2,5] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(3,5)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,3,4],[2,6],[5]]
=> [5,2,6,1,3,4] => ([(0,5),(1,4),(2,4),(2,5),(5,3)],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,2,4],[3,6],[5]]
=> [5,3,6,1,2,4] => ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,2,3],[4,6],[5]]
=> [5,4,6,1,2,3] => ([(0,5),(1,5),(2,3),(3,4)],6)
=> ([(0,2),(2,1)],3)
=> 1
[[1,3,5],[2,4],[6]]
=> [6,2,4,1,3,5] => ([(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 1
[[1,2,5],[3,4],[6]]
=> [6,3,4,1,2,5] => ([(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1,3,4],[2,5],[6]]
=> [6,2,5,1,3,4] => ([(1,5),(2,3),(2,5),(5,4)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 2
[[1,2,4],[3,5],[6]]
=> [6,3,5,1,2,4] => ([(1,4),(2,3),(2,5),(4,5)],6)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => ([(1,3),(2,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => ([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,4,6],[2],[3],[5]]
=> [5,3,2,1,4,6] => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,3,6],[2],[4],[5]]
=> [5,4,2,1,3,6] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => ([(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 1
[[1,4],[2,5],[3,6]]
=> [3,6,2,5,1,4] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 2
[[1,3],[2,5],[4,6]]
=> [4,6,2,5,1,3] => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 2
[[1,2],[3,5],[4,6]]
=> [4,6,3,5,1,2] => ([(0,5),(1,3),(2,4),(2,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[[1,3],[2,4],[5,6]]
=> [5,6,2,4,1,3] => ([(0,5),(1,3),(2,4),(2,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[[1,3],[2,4],[5],[6]]
=> [6,5,2,4,1,3] => ([(2,5),(3,4),(3,5)],6)
=> ([(0,2),(2,1)],3)
=> 1
[[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
Description
The global dimension of the incidence algebra of the lattice over the rational numbers.
Matching statistic: St000454
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 83%
Mp00252: Permutations —restriction⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 83%
Values
[[1]]
=> [1] => [] => ([],0)
=> ? = 0 - 1
[[1,2]]
=> [1,2] => [1] => ([],1)
=> 0 = 1 - 1
[[1],[2]]
=> [2,1] => [1] => ([],1)
=> 0 = 1 - 1
[[1,2,3]]
=> [1,2,3] => [1,2] => ([],2)
=> 0 = 1 - 1
[[1,3],[2]]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 1 = 2 - 1
[[1,2],[3]]
=> [3,1,2] => [1,2] => ([],2)
=> 0 = 1 - 1
[[1],[2],[3]]
=> [3,2,1] => [2,1] => ([(0,1)],2)
=> 1 = 2 - 1
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3] => ([],3)
=> 0 = 1 - 1
[[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3] => ([(1,2)],3)
=> 1 = 2 - 1
[[1,2,4],[3]]
=> [3,1,2,4] => [3,1,2] => ([(0,2),(1,2)],3)
=> ? ∊ {0,2,2} - 1
[[1,2,3],[4]]
=> [4,1,2,3] => [1,2,3] => ([],3)
=> 0 = 1 - 1
[[1,3],[2,4]]
=> [2,4,1,3] => [2,1,3] => ([(1,2)],3)
=> 1 = 2 - 1
[[1,2],[3,4]]
=> [3,4,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> ? ∊ {0,2,2} - 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [2,1,3] => ([(1,2)],3)
=> 1 = 2 - 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> ? ∊ {0,2,2} - 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4] => ([],4)
=> 0 = 1 - 1
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4] => ([(2,3)],4)
=> 1 = 2 - 1
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,2,3,3} - 1
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,2,3,3} - 1
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,2,3,4] => ([],4)
=> 0 = 1 - 1
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,2,3,3} - 1
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 3 - 1
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> 1 = 2 - 1
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,2,3,3} - 1
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,2,3,3} - 1
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,2,3,3} - 1
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,2,3,3} - 1
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> 1 = 2 - 1
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,2,3,3} - 1
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,2,3,3} - 1
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,2,3,3} - 1
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,2,3,3} - 1
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,2,3,3} - 1
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 3 - 1
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,2,3,3} - 1
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,2,3,3} - 1
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5] => ([],5)
=> 0 = 1 - 1
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5] => ([(3,4)],5)
=> 1 = 2 - 1
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4} - 1
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4} - 1
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0 = 1 - 1
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4} - 1
[[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)
=> 2 = 3 - 1
[[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)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4} - 1
[[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)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4} - 1
[[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)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4} - 1
[[1,3,4,5],[2,6]]
=> [2,6,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> 1 = 2 - 1
[[1,2,4,5],[3,6]]
=> [3,6,1,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4} - 1
[[1,2,3,5],[4,6]]
=> [4,6,1,2,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4} - 1
[[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[[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)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4} - 1
[[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)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4} - 1
[[1,3,4,6],[2],[5]]
=> [5,2,1,3,4,6] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4} - 1
[[1,2,4,6],[3],[5]]
=> [5,3,1,2,4,6] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4} - 1
[[1,2,3,6],[4],[5]]
=> [5,4,1,2,3,6] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[[1,3,4,5],[2],[6]]
=> [6,2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> 1 = 2 - 1
[[1,2,4,5],[3],[6]]
=> [6,3,1,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4} - 1
[[1,2,3,5],[4],[6]]
=> [6,4,1,2,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4} - 1
[[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4} - 1
[[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)
=> 2 = 3 - 1
[[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)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4} - 1
[[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)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4} - 1
[[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)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4} - 1
[[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)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4} - 1
[[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)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4} - 1
[[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)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4} - 1
[[1,3,6],[2,4],[5]]
=> [5,2,4,1,3,6] => [5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4} - 1
[[1,2,6],[3,4],[5]]
=> [5,3,4,1,2,6] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4} - 1
[[1,4,5],[2,6],[3]]
=> [3,2,6,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[[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)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4} - 1
[[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)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4} - 1
[[1,3,4],[2,6],[5]]
=> [5,2,6,1,3,4] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4} - 1
[[1,2,4],[3,6],[5]]
=> [5,3,6,1,2,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4} - 1
[[1,2,3],[4,6],[5]]
=> [5,4,6,1,2,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[[1,3,5],[2,4],[6]]
=> [6,2,4,1,3,5] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4} - 1
[[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)
=> 2 = 3 - 1
[[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)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4} - 1
[[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)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4} - 1
[[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)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4} - 1
[[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)
=> 3 = 4 - 1
[[1,4,6],[2],[3],[5]]
=> [5,3,2,1,4,6] => [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4} - 1
[[1,4,5],[2],[3],[6]]
=> [6,3,2,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[[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)
=> 3 = 4 - 1
[[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[[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)
=> 3 = 4 - 1
[[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
Description
The largest eigenvalue of a graph if it is integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St001632
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 50%
Mp00310: Permutations —toric promotion⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 50%
Values
[[1]]
=> [1] => [1] => ([],1)
=> ? = 0
[[1,2]]
=> [1,2] => [1,2] => ([(0,1)],2)
=> 1
[[1],[2]]
=> [2,1] => [2,1] => ([],2)
=> ? = 1
[[1,2,3]]
=> [1,2,3] => [3,2,1] => ([],3)
=> ? ∊ {2,2}
[[1,3],[2]]
=> [2,1,3] => [3,1,2] => ([(1,2)],3)
=> ? ∊ {2,2}
[[1,2],[3]]
=> [3,1,2] => [2,1,3] => ([(0,2),(1,2)],3)
=> 1
[[1],[2],[3]]
=> [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[1,2,3,4]]
=> [1,2,3,4] => [4,2,3,1] => ([(2,3)],4)
=> ? ∊ {2,2,2,3,3}
[[1,3,4],[2]]
=> [2,1,3,4] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? ∊ {2,2,2,3,3}
[[1,2,4],[3]]
=> [3,1,2,4] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> ? ∊ {2,2,2,3,3}
[[1,2,3],[4]]
=> [4,1,2,3] => [3,4,2,1] => ([(2,3)],4)
=> ? ∊ {2,2,2,3,3}
[[1,3],[2,4]]
=> [2,4,1,3] => [4,3,2,1] => ([],4)
=> ? ∊ {2,2,2,3,3}
[[1,2],[3,4]]
=> [3,4,1,2] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 2
[[1,2],[3],[4]]
=> [4,3,1,2] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 0
[[1],[2],[3],[4]]
=> [4,3,2,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ? ∊ {1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4}
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ? ∊ {1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4}
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
=> ? ∊ {1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4}
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
=> ? ∊ {1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4}
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [4,5,2,3,1] => ([(1,4),(2,3)],5)
=> ? ∊ {1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4}
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4}
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
=> ? ∊ {1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4}
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [5,4,2,3,1] => ([(3,4)],5)
=> ? ∊ {1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4}
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
=> ? ∊ {1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4}
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [3,4,5,2,1] => ([(2,3),(3,4)],5)
=> ? ∊ {1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4}
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> 1
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 2
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4}
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> 2
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
=> ? ∊ {1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4}
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [4,3,5,2,1] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4}
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> ? ∊ {1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4}
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [3,5,4,2,1] => ([(2,3),(2,4)],5)
=> ? ∊ {1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4}
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> 0
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [4,5,3,2,1] => ([(3,4)],5)
=> ? ∊ {1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4}
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 2
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> 1
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 0
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [6,2,3,4,5,1] => ([(2,3),(3,5),(5,4)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [6,1,2,3,4,5] => ([(1,5),(3,4),(4,2),(5,3)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [2,6,3,4,5,1] => ([(1,3),(1,5),(4,2),(5,4)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [3,6,2,4,5,1] => ([(1,5),(2,3),(2,5),(5,4)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [4,6,2,3,5,1] => ([(1,4),(2,3),(2,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [5,6,2,3,4,1] => ([(1,3),(2,4),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [6,3,2,4,5,1] => ([(2,5),(3,5),(5,4)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [2,3,6,4,5,1] => ([(1,5),(4,3),(5,2),(5,4)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,3,4,6],[2,5]]
=> [2,5,1,3,4,6] => [6,4,2,3,5,1] => ([(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => [2,4,6,3,5,1] => ([(1,3),(1,4),(3,5),(4,2),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => [3,4,6,2,5,1] => ([(1,5),(2,3),(3,4),(3,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,3,4,5],[2,6]]
=> [2,6,1,3,4,5] => [6,5,2,3,4,1] => ([(3,4),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,2,4,5],[3,6]]
=> [3,6,1,2,4,5] => [2,5,6,3,4,1] => ([(1,4),(1,5),(4,3),(5,2)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,2,3,5],[4,6]]
=> [4,6,1,2,3,5] => [3,5,6,2,4,1] => ([(1,5),(2,3),(2,5),(3,4)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => [4,5,6,2,3,1] => ([(1,3),(2,4),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [1,2,6,3,4,5] => ([(0,5),(3,4),(4,2),(5,1),(5,3)],6)
=> 1
[[1,3,5,6],[2],[4]]
=> [4,2,1,3,5,6] => [1,3,6,2,4,5] => ([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
=> 2
[[1,2,5,6],[3],[4]]
=> [4,3,1,2,5,6] => [3,2,6,4,5,1] => ([(1,4),(1,5),(2,4),(2,5),(5,3)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,3,4,6],[2],[5]]
=> [5,2,1,3,4,6] => [1,4,6,2,3,5] => ([(0,3),(0,4),(2,5),(3,2),(4,1),(4,5)],6)
=> 2
[[1,2,4,6],[3],[5]]
=> [5,3,1,2,4,6] => [4,2,6,3,5,1] => ([(1,4),(1,5),(2,3),(2,4),(3,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,2,3,6],[4],[5]]
=> [5,4,1,2,3,6] => [4,3,6,2,5,1] => ([(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,3,4,5],[2],[6]]
=> [6,2,1,3,4,5] => [1,5,6,2,3,4] => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 2
[[1,2,4,5],[3],[6]]
=> [6,3,1,2,4,5] => [5,2,6,3,4,1] => ([(1,5),(2,3),(2,5),(3,4)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,2,3,5],[4],[6]]
=> [6,4,1,2,3,5] => [5,3,6,2,4,1] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => [5,4,6,2,3,1] => ([(1,5),(2,5),(3,4)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => [6,3,5,2,4,1] => ([(2,5),(3,4),(3,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,2,5],[3,4,6]]
=> [3,4,6,1,2,5] => [2,3,5,6,4,1] => ([(1,5),(4,3),(5,2),(5,4)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => [6,4,5,2,3,1] => ([(2,5),(3,4)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => [2,4,5,6,3,1] => ([(1,3),(1,5),(4,2),(5,4)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [1,3,2,6,4,5] => ([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> 2
[[1,4,6],[2],[3],[5]]
=> [5,3,2,1,4,6] => [1,4,2,6,3,5] => ([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
=> 2
[[1,3,6],[2],[4],[5]]
=> [5,4,2,1,3,6] => [1,4,3,6,2,5] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 1
[[1,4,5],[2],[3],[6]]
=> [6,3,2,1,4,5] => [1,5,2,6,3,4] => ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 2
[[1,3,5],[2],[4],[6]]
=> [6,4,2,1,3,5] => [1,5,3,6,2,4] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
=> 2
[[1,3,4],[2],[5],[6]]
=> [6,5,2,1,3,4] => [1,5,4,6,2,3] => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> 1
[[1,2],[3,5],[4,6]]
=> [4,6,3,5,1,2] => [3,5,2,4,1,6] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 0
[[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [4,5,2,3,1,6] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 2
[[1,2],[3,6],[4],[5]]
=> [5,4,3,6,1,2] => [4,3,2,5,1,6] => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> 0
[[1,2],[3,5],[4],[6]]
=> [6,4,3,5,1,2] => [5,3,2,4,1,6] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 0
[[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => [5,4,2,3,1,6] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 1
[[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [1,4,3,2,6,5] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0
[[1,5],[2],[3],[4],[6]]
=> [6,4,3,2,1,5] => [1,5,3,2,6,4] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 1
[[1,4],[2],[3],[5],[6]]
=> [6,5,3,2,1,4] => [1,5,4,2,6,3] => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
=> 0
[[1,3],[2],[4],[5],[6]]
=> [6,5,4,2,1,3] => [1,5,4,3,6,2] => ([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
=> 1
[[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => [5,4,3,2,1,6] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [1,5,4,3,2,6] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 0
Description
The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset.
Matching statistic: St000259
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 100%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => [1] => ([],1)
=> 0
[[1,2]]
=> [1,2] => [1,2] => ([],2)
=> ? = 1
[[1],[2]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[1,2,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,2,2}
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {1,2,2}
[[1,2],[3]]
=> [3,1,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {1,2,2}
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,1,2,2,2,2,3}
[[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {0,1,2,2,2,2,3}
[[1,2,4],[3]]
=> [3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,1,2,2,2,2,3}
[[1,2,3],[4]]
=> [4,1,2,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,1,2,2,2,2,3}
[[1,3],[2,4]]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[1,2],[3,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 3
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,1,2,2,2,2,3}
[[1,3],[2],[4]]
=> [4,2,1,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {0,1,2,2,2,2,3}
[[1,2],[3],[4]]
=> [4,3,1,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,1,2,2,2,2,3}
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [5,2,1,4,3] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => ([(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [1,3,2,4,5,6] => ([(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [1,2,4,3,5,6] => ([(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [1,2,3,5,4,6] => ([(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [1,2,3,4,6,5] => ([(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [4,2,1,3,5,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [3,1,4,2,5,6] => ([(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,3,4,6],[2,5]]
=> [2,5,1,3,4,6] => [2,1,5,3,4,6] => ([(1,2),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => [1,5,3,2,4,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => [1,4,2,5,3,6] => ([(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,3,4,5],[2,6]]
=> [2,6,1,3,4,5] => [2,1,3,6,4,5] => ([(1,2),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,2,4,5],[3,6]]
=> [3,6,1,2,4,5] => [1,3,2,6,4,5] => ([(1,2),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,2,3,5],[4,6]]
=> [4,6,1,2,3,5] => [1,2,6,4,3,5] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => [1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [3,2,1,4,5,6] => ([(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,3,5,6],[2],[4]]
=> [4,2,1,3,5,6] => [2,1,4,3,5,6] => ([(2,5),(3,4)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,2,5,6],[3],[4]]
=> [4,3,1,2,5,6] => [1,4,3,2,5,6] => ([(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,3,4,6],[2],[5]]
=> [5,2,1,3,4,6] => [2,1,3,5,4,6] => ([(2,5),(3,4)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => [6,4,2,1,3,5] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2,5],[3,4,6]]
=> [3,4,6,1,2,5] => [6,3,1,4,2,5] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => [5,2,1,6,3,4] => ([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> 3
[[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => [5,1,6,3,2,4] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 3
[[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [4,1,5,2,6,3] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 3
[[1,4,5],[2,6],[3]]
=> [3,2,6,1,4,5] => [6,3,2,1,4,5] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,3,5],[2,6],[4]]
=> [4,2,6,1,3,5] => [4,2,1,6,3,5] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> 4
[[1,2,5],[3,6],[4]]
=> [4,3,6,1,2,5] => [4,1,6,3,2,5] => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[[1,3,4],[2,6],[5]]
=> [5,2,6,1,3,4] => [5,2,1,3,6,4] => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 3
[[1,2,4],[3,6],[5]]
=> [5,3,6,1,2,4] => [3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 5
[[1,4],[2,5],[3,6]]
=> [3,6,2,5,1,4] => [3,6,5,2,1,4] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,3],[2,5],[4,6]]
=> [4,6,2,5,1,3] => [4,6,2,1,5,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> 2
[[1,2],[3,5],[4,6]]
=> [4,6,3,5,1,2] => [6,4,3,1,5,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,3],[2,4],[5,6]]
=> [5,6,2,4,1,3] => [2,5,1,6,4,3] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> 4
[[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [5,3,1,6,4,2] => ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,5],[2,6],[3],[4]]
=> [4,3,2,6,1,5] => [4,3,6,2,1,5] => ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [3,2,5,1,6,4] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> 4
[[1,3],[2,6],[4],[5]]
=> [5,4,2,6,1,3] => [2,5,4,1,6,3] => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[[1,2],[3,6],[4],[5]]
=> [5,4,3,6,1,2] => [5,4,3,1,6,2] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,4],[2,5],[3],[6]]
=> [6,3,2,5,1,4] => [3,6,2,1,5,4] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 3
[[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => [2,4,1,6,5,3] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> 4
[[1,2],[3,5],[4],[6]]
=> [6,4,3,5,1,2] => [4,3,1,6,5,2] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 3
[[1,3],[2,4],[5],[6]]
=> [6,5,2,4,1,3] => [6,2,1,5,4,3] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => [3,1,6,5,4,2] => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(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
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St000260
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 67%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 67%
Values
[[1]]
=> [1] => [1] => ([],1)
=> 0
[[1,2]]
=> [1,2] => [1,2] => ([],2)
=> ? = 1
[[1],[2]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[1,2,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,2,2}
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {1,2,2}
[[1,2],[3]]
=> [3,1,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {1,2,2}
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,2,2,2,2,3,3}
[[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {0,2,2,2,2,3,3}
[[1,2,4],[3]]
=> [3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,2,2,2,2,3,3}
[[1,2,3],[4]]
=> [4,1,2,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,2,2,2,2,3,3}
[[1,3],[2,4]]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1,2],[3,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,2,2,2,2,3,3}
[[1,3],[2],[4]]
=> [4,2,1,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {0,2,2,2,2,3,3}
[[1,2],[3],[4]]
=> [4,3,1,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,2,2,2,2,3,3}
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4}
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4}
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4}
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4}
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4}
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4}
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4}
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4}
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4}
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4}
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4}
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4}
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4}
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4}
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4}
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4}
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [5,2,1,4,3] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4}
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4}
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4}
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4}
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => ([(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [1,3,2,4,5,6] => ([(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [1,2,4,3,5,6] => ([(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [1,2,3,5,4,6] => ([(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [1,2,3,4,6,5] => ([(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [4,2,1,3,5,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [3,1,4,2,5,6] => ([(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,3,4,6],[2,5]]
=> [2,5,1,3,4,6] => [2,1,5,3,4,6] => ([(1,2),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => [1,5,3,2,4,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => [1,4,2,5,3,6] => ([(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,3,4,5],[2,6]]
=> [2,6,1,3,4,5] => [2,1,3,6,4,5] => ([(1,2),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,2,4,5],[3,6]]
=> [3,6,1,2,4,5] => [1,3,2,6,4,5] => ([(1,2),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,2,3,5],[4,6]]
=> [4,6,1,2,3,5] => [1,2,6,4,3,5] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => [1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [3,2,1,4,5,6] => ([(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,3,5,6],[2],[4]]
=> [4,2,1,3,5,6] => [2,1,4,3,5,6] => ([(2,5),(3,4)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,2,5,6],[3],[4]]
=> [4,3,1,2,5,6] => [1,4,3,2,5,6] => ([(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,3,4,6],[2],[5]]
=> [5,2,1,3,4,6] => [2,1,3,5,4,6] => ([(2,5),(3,4)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
[[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => [6,4,2,1,3,5] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,2,5],[3,4,6]]
=> [3,4,6,1,2,5] => [6,3,1,4,2,5] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => [5,2,1,6,3,4] => ([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => [5,1,6,3,2,4] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [4,1,5,2,6,3] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 2
[[1,4,5],[2,6],[3]]
=> [3,2,6,1,4,5] => [6,3,2,1,4,5] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,3,5],[2,6],[4]]
=> [4,2,6,1,3,5] => [4,2,1,6,3,5] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> 2
[[1,2,5],[3,6],[4]]
=> [4,3,6,1,2,5] => [4,1,6,3,2,5] => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[[1,3,4],[2,6],[5]]
=> [5,2,6,1,3,4] => [5,2,1,3,6,4] => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 2
[[1,2,4],[3,6],[5]]
=> [5,3,6,1,2,4] => [3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 3
[[1,4],[2,5],[3,6]]
=> [3,6,2,5,1,4] => [3,6,5,2,1,4] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,3],[2,5],[4,6]]
=> [4,6,2,5,1,3] => [4,6,2,1,5,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> 2
[[1,2],[3,5],[4,6]]
=> [4,6,3,5,1,2] => [6,4,3,1,5,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,3],[2,4],[5,6]]
=> [5,6,2,4,1,3] => [2,5,1,6,4,3] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [5,3,1,6,4,2] => ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,5],[2,6],[3],[4]]
=> [4,3,2,6,1,5] => [4,3,6,2,1,5] => ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [3,2,5,1,6,4] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> 2
[[1,3],[2,6],[4],[5]]
=> [5,4,2,6,1,3] => [2,5,4,1,6,3] => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[[1,2],[3,6],[4],[5]]
=> [5,4,3,6,1,2] => [5,4,3,1,6,2] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,4],[2,5],[3],[6]]
=> [6,3,2,5,1,4] => [3,6,2,1,5,4] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 2
[[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => [2,4,1,6,5,3] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> 2
[[1,2],[3,5],[4],[6]]
=> [6,4,3,5,1,2] => [4,3,1,6,5,2] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 2
[[1,3],[2,4],[5],[6]]
=> [6,5,2,4,1,3] => [6,2,1,5,4,3] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => [3,1,6,5,4,2] => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(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
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St000939
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000939: Integer partitions ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 67%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000939: Integer partitions ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 67%
Values
[[1]]
=> [1]
=> []
=> ?
=> ? = 0
[[1,2]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1}
[[1],[2]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1}
[[1,2,3]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,2,2}
[[1,3],[2]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,2,2}
[[1,2],[3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,2,2}
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2}
[[1,2,3,4]]
=> [4]
=> []
=> ?
=> ? ∊ {0,1,1,2,2,2,2,3,3}
[[1,3,4],[2]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,1,1,2,2,2,2,3,3}
[[1,2,4],[3]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,1,1,2,2,2,2,3,3}
[[1,2,3],[4]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,1,1,2,2,2,2,3,3}
[[1,3],[2,4]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,1,1,2,2,2,2,3,3}
[[1,2],[3,4]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,1,1,2,2,2,2,3,3}
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,2,2,2,2,3,3}
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,2,2,2,2,3,3}
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,2,2,2,2,3,3}
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,2,3,4,5]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,3,4,5],[2]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2,4,5],[3]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2,3,5],[4]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2,3,4],[5]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,3,5],[2,4]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2,5],[3,4]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,3,4],[2,5]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2,4],[3,5]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2,3],[4,5]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,4,5],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,3,5],[2],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2,5],[3],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,3,4],[2],[5]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2,4],[3],[5]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,4],[2,5],[3]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,3],[2,5],[4]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2],[3,5],[4]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,3],[2,4],[5]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2],[3,4],[5]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,5],[2],[3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,4],[2],[3],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,3],[2],[4],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[[1,2,3,4,5,6]]
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[[1,3,4,5,6],[2]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[[1,2,4,5,6],[3]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[[1,2,3,5,6],[4]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[[1,2,3,4,6],[5]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[[1,2,3,4,5],[6]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[[1,3,5,6],[2,4]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[[1,2,5,6],[3,4]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[[1,3,4,6],[2,5]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[[1,2,4,6],[3,5]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[[1,2,3,6],[4,5]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[[1,3,4,5],[2,6]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[[1,2,4,5],[3,6]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[[1,5,6],[2],[3],[4]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,4,6],[2],[3],[5]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,3,6],[2],[4],[5]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,2,6],[3],[4],[5]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,4,5],[2],[3],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,3,5],[2],[4],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,2,5],[3],[4],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,3,4],[2],[5],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,2,4],[3],[5],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,2,3],[4],[5],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,4],[2,5],[3,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[[1,3],[2,5],[4,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[[1,2],[3,5],[4,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[[1,3],[2,4],[5,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[[1,2],[3,4],[5,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[[1,5],[2,6],[3],[4]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[[1,4],[2,6],[3],[5]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[[1,3],[2,6],[4],[5]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[[1,2],[3,6],[4],[5]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[[1,4],[2,5],[3],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[[1,3],[2,5],[4],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[[1,2],[3,5],[4],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[[1,3],[2,4],[5],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[[1,2],[3,4],[5],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[[1,6],[2],[3],[4],[5]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[[1,5],[2],[3],[4],[6]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[[1,4],[2],[3],[5],[6]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[[1,3],[2],[4],[5],[6]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 5
Description
The number of characters of the symmetric group whose value on the partition is positive.
Matching statistic: St000993
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 67%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 67%
Values
[[1]]
=> [1]
=> []
=> ?
=> ? = 0
[[1,2]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1}
[[1],[2]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1}
[[1,2,3]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,2,2}
[[1,3],[2]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,2,2}
[[1,2],[3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,2,2}
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2}
[[1,2,3,4]]
=> [4]
=> []
=> ?
=> ? ∊ {0,1,1,2,2,2,2,3,3}
[[1,3,4],[2]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,1,1,2,2,2,2,3,3}
[[1,2,4],[3]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,1,1,2,2,2,2,3,3}
[[1,2,3],[4]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,1,1,2,2,2,2,3,3}
[[1,3],[2,4]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,1,1,2,2,2,2,3,3}
[[1,2],[3,4]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,1,1,2,2,2,2,3,3}
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,2,2,2,2,3,3}
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,2,2,2,2,3,3}
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,2,2,2,2,3,3}
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,2,3,4,5]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,3,4,5],[2]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2,4,5],[3]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2,3,5],[4]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2,3,4],[5]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,3,5],[2,4]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2,5],[3,4]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,3,4],[2,5]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2,4],[3,5]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2,3],[4,5]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,4,5],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,3,5],[2],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2,5],[3],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,3,4],[2],[5]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2,4],[3],[5]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,4],[2,5],[3]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,3],[2,5],[4]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2],[3,5],[4]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,3],[2,4],[5]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2],[3,4],[5]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,5],[2],[3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,4],[2],[3],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,3],[2],[4],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[[1,2,3,4,5,6]]
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[[1,3,4,5,6],[2]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[[1,2,4,5,6],[3]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[[1,2,3,5,6],[4]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[[1,2,3,4,6],[5]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[[1,2,3,4,5],[6]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[[1,3,5,6],[2,4]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[[1,2,5,6],[3,4]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[[1,3,4,6],[2,5]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[[1,2,4,6],[3,5]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[[1,2,3,6],[4,5]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[[1,3,4,5],[2,6]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[[1,2,4,5],[3,6]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[[1,5,6],[2],[3],[4]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,4,6],[2],[3],[5]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,3,6],[2],[4],[5]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,2,6],[3],[4],[5]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,4,5],[2],[3],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,3,5],[2],[4],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,2,5],[3],[4],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,3,4],[2],[5],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,2,4],[3],[5],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,2,3],[4],[5],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,4],[2,5],[3,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[[1,3],[2,5],[4,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[[1,2],[3,5],[4,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[[1,3],[2,4],[5,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[[1,2],[3,4],[5,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[[1,5],[2,6],[3],[4]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[[1,4],[2,6],[3],[5]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[[1,3],[2,6],[4],[5]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[[1,2],[3,6],[4],[5]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[[1,4],[2,5],[3],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[[1,3],[2,5],[4],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[[1,2],[3,5],[4],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[[1,3],[2,4],[5],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[[1,2],[3,4],[5],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[[1,6],[2],[3],[4],[5]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[[1,5],[2],[3],[4],[6]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[[1,4],[2],[3],[5],[6]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[[1,3],[2],[4],[5],[6]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 4
Description
The multiplicity of the largest part of an integer partition.
Matching statistic: St001060
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 50%
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 50%
Values
[[1]]
=> [1] => [1] => ([],1)
=> ? = 0
[[1,2]]
=> [2] => [2] => ([],2)
=> ? ∊ {1,1}
[[1],[2]]
=> [2] => [2] => ([],2)
=> ? ∊ {1,1}
[[1,2,3]]
=> [3] => [3] => ([],3)
=> ? ∊ {1,1,2,2}
[[1,3],[2]]
=> [2,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,2,2}
[[1,2],[3]]
=> [3] => [3] => ([],3)
=> ? ∊ {1,1,2,2}
[[1],[2],[3]]
=> [3] => [3] => ([],3)
=> ? ∊ {1,1,2,2}
[[1,2,3,4]]
=> [4] => [4] => ([],4)
=> ? ∊ {0,1,1,2,2,3,3}
[[1,3,4],[2]]
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[[1,2,4],[3]]
=> [3,1] => [1,3] => ([(2,3)],4)
=> ? ∊ {0,1,1,2,2,3,3}
[[1,2,3],[4]]
=> [4] => [4] => ([],4)
=> ? ∊ {0,1,1,2,2,3,3}
[[1,3],[2,4]]
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[[1,2],[3,4]]
=> [3,1] => [1,3] => ([(2,3)],4)
=> ? ∊ {0,1,1,2,2,3,3}
[[1,4],[2],[3]]
=> [3,1] => [1,3] => ([(2,3)],4)
=> ? ∊ {0,1,1,2,2,3,3}
[[1,3],[2],[4]]
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[[1,2],[3],[4]]
=> [4] => [4] => ([],4)
=> ? ∊ {0,1,1,2,2,3,3}
[[1],[2],[3],[4]]
=> [4] => [4] => ([],4)
=> ? ∊ {0,1,1,2,2,3,3}
[[1,2,3,4,5]]
=> [5] => [5] => ([],5)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
[[1,3,4,5],[2]]
=> [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[[1,2,4,5],[3]]
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
[[1,2,3,5],[4]]
=> [4,1] => [1,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
[[1,2,3,4],[5]]
=> [5] => [5] => ([],5)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
[[1,3,5],[2,4]]
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[1,2,5],[3,4]]
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
[[1,3,4],[2,5]]
=> [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[[1,2,4],[3,5]]
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
[[1,2,3],[4,5]]
=> [4,1] => [1,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
[[1,4,5],[2],[3]]
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
[[1,3,5],[2],[4]]
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[1,2,5],[3],[4]]
=> [4,1] => [1,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
[[1,3,4],[2],[5]]
=> [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[[1,2,4],[3],[5]]
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
[[1,2,3],[4],[5]]
=> [5] => [5] => ([],5)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
[[1,4],[2,5],[3]]
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
[[1,3],[2,5],[4]]
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[1,2],[3,5],[4]]
=> [4,1] => [1,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
[[1,3],[2,4],[5]]
=> [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[[1,2],[3,4],[5]]
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
[[1,5],[2],[3],[4]]
=> [4,1] => [1,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
[[1,4],[2],[3],[5]]
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
[[1,3],[2],[4],[5]]
=> [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[[1,2],[3],[4],[5]]
=> [5] => [5] => ([],5)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
[[1],[2],[3],[4],[5]]
=> [5] => [5] => ([],5)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
[[1,2,3,4,5,6]]
=> [6] => [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,5}
[[1,3,4,5,6],[2]]
=> [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[[1,2,4,5,6],[3]]
=> [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,5}
[[1,2,3,5,6],[4]]
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,5}
[[1,2,3,4,6],[5]]
=> [5,1] => [1,5] => ([(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,5}
[[1,2,3,4,5],[6]]
=> [6] => [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,5}
[[1,3,5,6],[2,4]]
=> [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2,5,6],[3,4]]
=> [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,5}
[[1,3,4,6],[2,5]]
=> [2,3,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,5}
[[1,2,4,6],[3,5]]
=> [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2,3,6],[4,5]]
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,5}
[[1,3,4,5],[2,6]]
=> [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[[1,2,4,5],[3,6]]
=> [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,5}
[[1,2,3,5],[4,6]]
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,5}
[[1,2,3,4],[5,6]]
=> [5,1] => [1,5] => ([(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,5}
[[1,4,5,6],[2],[3]]
=> [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,5}
[[1,3,5,6],[2],[4]]
=> [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2,5,6],[3],[4]]
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,5}
[[1,3,4,6],[2],[5]]
=> [2,3,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,5}
[[1,2,4,6],[3],[5]]
=> [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2,3,6],[4],[5]]
=> [5,1] => [1,5] => ([(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,5}
[[1,3,4,5],[2],[6]]
=> [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[[1,2,4,5],[3],[6]]
=> [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,5}
[[1,2,3,5],[4],[6]]
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,5}
[[1,2,3,4],[5],[6]]
=> [6] => [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,5}
[[1,3,5],[2,4,6]]
=> [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2,4],[3,5,6]]
=> [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,4,6],[2,5],[3]]
=> [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,3,6],[2,5],[4]]
=> [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2,6],[3,4],[5]]
=> [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,3,5],[2,6],[4]]
=> [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2,4],[3,6],[5]]
=> [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,3,5],[2,4],[6]]
=> [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,3,4],[2,5],[6]]
=> [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[[1,4,6],[2],[3],[5]]
=> [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,3,5],[2],[4],[6]]
=> [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,3,4],[2],[5],[6]]
=> [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[[1,3],[2,5],[4,6]]
=> [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2],[3,4],[5,6]]
=> [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,4],[2,6],[3],[5]]
=> [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,3],[2,5],[4],[6]]
=> [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,3],[2,4],[5],[6]]
=> [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[[1,3],[2],[4],[5],[6]]
=> [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
Description
The distinguishing index of a graph.
This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism.
If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.
Matching statistic: St000456
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 50%
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 50%
Values
[[1]]
=> [[1]]
=> [1] => ([],1)
=> ? = 0
[[1,2]]
=> [[1,2]]
=> [2] => ([],2)
=> ? ∊ {1,1}
[[1],[2]]
=> [[1,2]]
=> [2] => ([],2)
=> ? ∊ {1,1}
[[1,2,3]]
=> [[1,2,3]]
=> [3] => ([],3)
=> ? ∊ {2,2}
[[1,3],[2]]
=> [[1,2],[3]]
=> [2,1] => ([(0,2),(1,2)],3)
=> 1
[[1,2],[3]]
=> [[1,2,3]]
=> [3] => ([],3)
=> ? ∊ {2,2}
[[1],[2],[3]]
=> [[1,2],[3]]
=> [2,1] => ([(0,2),(1,2)],3)
=> 1
[[1,2,3,4]]
=> [[1,2,3,4]]
=> [4] => ([],4)
=> ? ∊ {0,2,2,2,2,2,3,3}
[[1,3,4],[2]]
=> [[1,2,4],[3]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,2,2,2,2,2,3,3}
[[1,2,4],[3]]
=> [[1,2,3],[4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[[1,2,3],[4]]
=> [[1,2,3,4]]
=> [4] => ([],4)
=> ? ∊ {0,2,2,2,2,2,3,3}
[[1,3],[2,4]]
=> [[1,2,4],[3]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,2,2,2,2,2,3,3}
[[1,2],[3,4]]
=> [[1,2,3,4]]
=> [4] => ([],4)
=> ? ∊ {0,2,2,2,2,2,3,3}
[[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,2,2,2,2,2,3,3}
[[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,2,2,2,2,2,3,3}
[[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[[1],[2],[3],[4]]
=> [[1,2],[3],[4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,2,2,2,2,2,3,3}
[[1,2,3,4,5]]
=> [[1,2,3,4,5]]
=> [5] => ([],5)
=> ? ∊ {0,0,0,1,2,2,2,2,2,2,2,2,2,3,3,3,4,4}
[[1,3,4,5],[2]]
=> [[1,2,4,5],[3]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,2,2,2,2,2,2,2,2,2,3,3,3,4,4}
[[1,2,4,5],[3]]
=> [[1,2,3,5],[4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,2,2,2,2,2,2,2,2,2,3,3,3,4,4}
[[1,2,3,5],[4]]
=> [[1,2,3,4],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[[1,2,3,4],[5]]
=> [[1,2,3,4,5]]
=> [5] => ([],5)
=> ? ∊ {0,0,0,1,2,2,2,2,2,2,2,2,2,3,3,3,4,4}
[[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)
=> 3
[[1,2,5],[3,4]]
=> [[1,2,3,4],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[[1,3,4],[2,5]]
=> [[1,2,4,5],[3]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,2,2,2,2,2,2,2,2,2,3,3,3,4,4}
[[1,2,4],[3,5]]
=> [[1,2,3,5],[4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,2,2,2,2,2,2,2,2,2,3,3,3,4,4}
[[1,2,3],[4,5]]
=> [[1,2,3,4,5]]
=> [5] => ([],5)
=> ? ∊ {0,0,0,1,2,2,2,2,2,2,2,2,2,3,3,3,4,4}
[[1,4,5],[2],[3]]
=> [[1,2,5],[3],[4]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,2,2,2,2,2,2,2,2,2,3,3,3,4,4}
[[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)
=> 3
[[1,2,5],[3],[4]]
=> [[1,2,3],[4],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,2,2,2,2,2,2,2,2,2,3,3,3,4,4}
[[1,3,4],[2],[5]]
=> [[1,2,4,5],[3]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,2,2,2,2,2,2,2,2,2,3,3,3,4,4}
[[1,2,4],[3],[5]]
=> [[1,2,3,5],[4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,2,2,2,2,2,2,2,2,2,3,3,3,4,4}
[[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[[1,4],[2,5],[3]]
=> [[1,2,5],[3],[4]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,2,2,2,2,2,2,2,2,2,3,3,3,4,4}
[[1,3],[2,5],[4]]
=> [[1,2,4,5],[3]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,2,2,2,2,2,2,2,2,2,3,3,3,4,4}
[[1,2],[3,5],[4]]
=> [[1,2,3,5],[4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,2,2,2,2,2,2,2,2,2,3,3,3,4,4}
[[1,3],[2,4],[5]]
=> [[1,2,4],[3,5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,2],[3,4],[5]]
=> [[1,2,3,4],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[[1,5],[2],[3],[4]]
=> [[1,2],[3],[4],[5]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,2,2,2,2,2,2,2,2,2,3,3,3,4,4}
[[1,4],[2],[3],[5]]
=> [[1,2,5],[3],[4]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,2,2,2,2,2,2,2,2,2,3,3,3,4,4}
[[1,3],[2],[4],[5]]
=> [[1,2,4],[3],[5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,2],[3],[4],[5]]
=> [[1,2,3],[4],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,2,2,2,2,2,2,2,2,2,3,3,3,4,4}
[[1],[2],[3],[4],[5]]
=> [[1,2],[3],[4],[5]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,2,2,2,2,2,2,2,2,2,3,3,3,4,4}
[[1,2,3,4,5,6]]
=> [[1,2,3,4,5,6]]
=> [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,5,5}
[[1,3,4,5,6],[2]]
=> [[1,2,4,5,6],[3]]
=> [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,5,5}
[[1,2,4,5,6],[3]]
=> [[1,2,3,5,6],[4]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,5,5}
[[1,2,3,5,6],[4]]
=> [[1,2,3,4,6],[5]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,5,5}
[[1,2,3,4,6],[5]]
=> [[1,2,3,4,5],[6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,2,3,4,5],[6]]
=> [[1,2,3,4,5,6]]
=> [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,5,5}
[[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)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,5,5}
[[1,2,5,6],[3,4]]
=> [[1,2,3,4],[5,6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,5,5}
[[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)
=> 3
[[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)
=> 4
[[1,2,3,6],[4,5]]
=> [[1,2,3,4,5],[6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,3,4,5],[2,6]]
=> [[1,2,4,5,6],[3]]
=> [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,5,5}
[[1,2,4,5],[3,6]]
=> [[1,2,3,5,6],[4]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,5,5}
[[1,2,3,5],[4,6]]
=> [[1,2,3,4,6],[5]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,5,5}
[[1,2,3,4],[5,6]]
=> [[1,2,3,4,5,6]]
=> [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,5,5}
[[1,4,5,6],[2],[3]]
=> [[1,2,5,6],[3],[4]]
=> [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,5,5}
[[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)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,5,5}
[[1,2,5,6],[3],[4]]
=> [[1,2,3,6],[4],[5]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,5,5}
[[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)
=> 3
[[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)
=> 4
[[1,2,3,6],[4],[5]]
=> [[1,2,3,4],[5],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,5,5}
[[1,3,4,5],[2],[6]]
=> [[1,2,4,5,6],[3]]
=> [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,5,5}
[[1,2,4,5],[3],[6]]
=> [[1,2,3,5,6],[4]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,5,5}
[[1,2,3,5],[4],[6]]
=> [[1,2,3,4,6],[5]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,5,5}
[[1,2,3,4],[5],[6]]
=> [[1,2,3,4,5],[6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,3,5],[2,4,6]]
=> [[1,2,4,6],[3,5]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,5,5}
[[1,4,6],[2,5],[3]]
=> [[1,2,5],[3,6],[4]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,3,6],[2,5],[4]]
=> [[1,2,4,5],[3],[6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,2,6],[3,5],[4]]
=> [[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)
=> 4
[[1,3,4],[2,5],[6]]
=> [[1,2,4,5],[3,6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,2,4],[3,5],[6]]
=> [[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)
=> 4
[[1,2,3],[4,5],[6]]
=> [[1,2,3,4,5],[6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,4,6],[2],[3],[5]]
=> [[1,2,5],[3],[4],[6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,3,4],[2],[5],[6]]
=> [[1,2,4,5],[3],[6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,2,4],[3],[5],[6]]
=> [[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)
=> 4
[[1,4],[2,5],[3,6]]
=> [[1,2,5],[3,6],[4]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,3],[2,5],[4,6]]
=> [[1,2,4,5],[3,6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,2],[3,5],[4,6]]
=> [[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)
=> 4
[[1,4],[2,5],[3],[6]]
=> [[1,2,5],[3,6],[4]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,3],[2,5],[4],[6]]
=> [[1,2,4,5],[3],[6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,2],[3,5],[4],[6]]
=> [[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)
=> 4
[[1,4],[2],[3],[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)
=> 3
Description
The monochromatic index of a connected graph.
This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path.
For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.
The following 12 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000264The girth of a graph, which is not a tree. St000640The rank of the largest boolean interval in a poset. St000100The number of linear extensions of a poset. St000633The size of the automorphism group of a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!