Your data matches 6 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000498
Mapping
St000498: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => 0
{{1},{2}}
 => 1
{{1,2,3}}
 => 0
{{1,2},{3}}
 => 1
{{1,3},{2}}
 => 0
{{1},{2,3}}
 => 2
{{1},{2},{3}}
 => 3
{{1,2,3,4}}
 => 0
{{1,2,3},{4}}
 => 1
{{1,2,4},{3}}
 => 0
{{1,2},{3,4}}
 => 2
{{1,2},{3},{4}}
 => 3
{{1,3,4},{2}}
 => 0
{{1,3},{2,4}}
 => 1
{{1,3},{2},{4}}
 => 2
{{1,4},{2,3}}
 => 0
{{1},{2,3,4}}
 => 3
{{1},{2,3},{4}}
 => 4
{{1,4},{2},{3}}
 => 1
{{1},{2,4},{3}}
 => 3
{{1},{2},{3,4}}
 => 5
{{1},{2},{3},{4}}
 => 6
{{1,2,3,4,5}}
 => 0
{{1,2,3,4},{5}}
 => 1
{{1,2,3,5},{4}}
 => 0
{{1,2,3},{4,5}}
 => 2
{{1,2,3},{4},{5}}
 => 3
{{1,2,4,5},{3}}
 => 0
{{1,2,4},{3,5}}
 => 1
{{1,2,4},{3},{5}}
 => 2
{{1,2,5},{3,4}}
 => 0
{{1,2},{3,4,5}}
 => 3
{{1,2},{3,4},{5}}
 => 4
{{1,2,5},{3},{4}}
 => 1
{{1,2},{3,5},{4}}
 => 3
{{1,2},{3},{4,5}}
 => 5
{{1,2},{3},{4},{5}}
 => 6
{{1,3,4,5},{2}}
 => 0
{{1,3,4},{2,5}}
 => 1
{{1,3,4},{2},{5}}
 => 2
{{1,3,5},{2,4}}
 => 0
{{1,3},{2,4,5}}
 => 2
{{1,3},{2,4},{5}}
 => 3
{{1,3,5},{2},{4}}
 => 1
{{1,3},{2,5},{4}}
 => 2
{{1,3},{2},{4,5}}
 => 4
{{1,3},{2},{4},{5}}
 => 5
{{1,4,5},{2,3}}
 => 0
{{1,4},{2,3,5}}
 => 1
{{1,4},{2,3},{5}}
 => 2
Description
The lcs statistic of a set partition. Let $S = B_1,\ldots,B_k$ be a set partition with ordered blocks $B_i$ and with $\operatorname{min} B_a < \operatorname{min} B_b$ for $a < b$. According to [1, Definition 3], a '''lcs''' (left-closer-smaller) of $S$ is given by a pair $i > j$ such that $j = \operatorname{max} B_b$ and $i \in B_a$ for $a > b$.
Matching statistic: St000456
(load all 3 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 86%
Values
{{1,2}}
 => [2,1] => [2,1] => ([(0,1)],2)
 => 1
{{1},{2}}
 => [1,2] => [1,2] => ([],2)
 => ? = 0
{{1,2,3}}
 => [2,3,1] => [1,3,2] => ([(1,2)],3)
 => ? ∊ {0,0,2}
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => ([(1,2)],3)
 => ? ∊ {0,0,2}
{{1,3},{2}}
 => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
{{1},{2,3}}
 => [1,3,2] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => ([],3)
 => ? ∊ {0,0,2}
{{1,2,3,4}}
 => [2,3,4,1] => [1,2,4,3] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,3,3,3,5}
{{1,2,3},{4}}
 => [2,3,1,4] => [1,3,2,4] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,3,3,3,5}
{{1,2,4},{3}}
 => [2,4,3,1] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
{{1,2},{3,4}}
 => [2,1,4,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,3,3,3,5}
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,3,3,3,5}
{{1,3,4},{2}}
 => [3,2,4,1] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? ∊ {0,0,0,0,3,3,3,5}
{{1,3},{2,4}}
 => [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,3,3,3,5}
{{1,4},{2,3}}
 => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 6
{{1},{2,3,4}}
 => [1,3,4,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,3,3,3,5}
{{1,4},{2},{3}}
 => [4,2,3,1] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
{{1},{2,4},{3}}
 => [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
{{1},{2},{3,4}}
 => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? ∊ {0,0,0,0,3,3,3,5}
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [1,2,3,5,4] => ([(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,7,7,7,9}
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,7,7,7,9}
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,7,7,7,9}
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [1,3,2,4,5] => ([(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,7,7,7,9}
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,7,7,7,9}
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,7,7,7,9}
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [5,4,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 6
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,7,7,7,9}
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,7,7,7,9}
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 3
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,7,7,7,9}
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,7,7,7,9}
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,7,7,7,9}
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,7,7,7,9}
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,7,7,7,9}
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,7,7,7,9}
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,7,7,7,9}
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,7,7,7,9}
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,7,7,7,9}
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,7,7,7,9}
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,7,7,7,9}
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,7,7,7,9}
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,7,7,7,9}
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [4,2,1,5,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 2
{{1,4},{2,3},{5}}
 => [4,3,2,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,7,7,7,9}
{{1,5},{2,3,4}}
 => [5,3,4,2,1] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
{{1},{2,3,4,5}}
 => [1,3,4,5,2] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
{{1},{2,3,4},{5}}
 => [1,3,4,2,5] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,7,7,7,9}
{{1,5},{2,3},{4}}
 => [5,3,2,4,1] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 3
{{1},{2,3,5},{4}}
 => [1,3,5,4,2] => [5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
{{1},{2,3},{4,5}}
 => [1,3,2,5,4] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
{{1},{2,3},{4},{5}}
 => [1,3,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,7,7,7,9}
{{1,4,5},{2},{3}}
 => [4,2,3,5,1] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,7,7,7,9}
{{1,4},{2,5},{3}}
 => [4,5,3,1,2] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 3
{{1,4},{2},{3,5}}
 => [4,2,5,1,3] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
{{1,4},{2},{3},{5}}
 => [4,2,3,1,5] => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,7,7,7,9}
{{1,5},{2,4},{3}}
 => [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)
 => 10
{{1},{2,4,5},{3}}
 => [1,4,3,5,2] => [3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 2
{{1},{2,4},{3,5}}
 => [1,4,5,2,3] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 3
{{1},{2,4},{3},{5}}
 => [1,4,3,2,5] => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,7,7,7,9}
{{1,5},{2},{3,4}}
 => [5,2,4,3,1] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 6
{{1},{2,5},{3,4}}
 => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 8
{{1},{2},{3,4,5}}
 => [1,2,4,5,3] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2
{{1},{2},{3,4},{5}}
 => [1,2,4,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,7,7,7,9}
{{1,5},{2},{3},{4}}
 => [5,2,3,4,1] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
{{1},{2,5},{3},{4}}
 => [1,5,3,4,2] => [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 4
{{1},{2},{3,5},{4}}
 => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
{{1},{2},{3},{4,5}}
 => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,7,7,7,9}
{{1,2,3,4,5,6}}
 => [2,3,4,5,6,1] => [1,2,3,4,6,5] => ([(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,11,12,12,12,14}
{{1,2,3,4,5},{6}}
 => [2,3,4,5,1,6] => [1,2,3,5,4,6] => ([(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,11,12,12,12,14}
{{1,2,3,4,6},{5}}
 => [2,3,4,6,5,1] => [6,1,2,3,5,4] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
{{1,2,3,4},{5,6}}
 => [2,3,4,1,6,5] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,11,12,12,12,14}
{{1,2,3,4},{5},{6}}
 => [2,3,4,1,5,6] => [1,2,4,3,5,6] => ([(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,11,12,12,12,14}
{{1,2,3,5,6},{4}}
 => [2,3,5,4,6,1] => [4,1,2,3,6,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,11,12,12,12,14}
{{1,2,3,5},{4,6}}
 => [2,3,5,6,1,4] => [5,1,2,3,6,4] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 1
{{1,2,3,5},{4},{6}}
 => [2,3,5,4,1,6] => [5,1,2,4,3,6] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,11,12,12,12,14}
{{1,2,3,6},{4,5}}
 => [2,3,6,5,4,1] => [6,5,1,2,4,3] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 7
{{1,2,3},{4,5,6}}
 => [2,3,1,5,6,4] => [1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,11,12,12,12,14}
{{1,2,3},{4,5},{6}}
 => [2,3,1,5,4,6] => [1,2,5,3,4,6] => ([(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,11,12,12,12,14}
{{1,2,3,6},{4},{5}}
 => [2,3,6,4,5,1] => [4,6,1,2,5,3] => ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
 => 4
{{1,2,3},{4,6},{5}}
 => [2,3,1,6,5,4] => [6,1,2,5,3,4] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
{{1,2,3},{4},{5,6}}
 => [2,3,1,4,6,5] => [1,2,6,3,4,5] => ([(2,5),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,11,12,12,12,14}
{{1,2,4,5},{3,6}}
 => [2,4,6,5,1,3] => [2,5,6,1,4,3] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => 4
{{1,2,4,6},{3,5}}
 => [2,4,5,6,3,1] => [2,3,6,1,5,4] => ([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => 2
{{1,2,4},{3,5,6}}
 => [2,4,5,1,6,3] => [2,6,1,4,3,5] => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 2
{{1,2,4,6},{3},{5}}
 => [2,4,3,6,5,1] => [2,6,1,3,5,4] => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 2
{{1,2,4},{3,6},{5}}
 => [2,4,6,1,5,3] => [2,6,1,4,5,3] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
 => 3
{{1,2,5},{3,4,6}}
 => [2,5,4,6,1,3] => [5,3,1,2,6,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
 => 3
{{1,2,6},{3,4,5}}
 => [2,6,4,5,3,1] => [3,6,5,1,4,2] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 6
{{1,2,6},{3,4},{5}}
 => [2,6,4,3,5,1] => [4,3,6,1,5,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => 5
{{1,2},{3,4,6},{5}}
 => [2,1,4,6,5,3] => [6,1,3,5,2,4] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
{{1,2,5},{3,6},{4}}
 => [2,5,6,4,1,3] => [5,2,6,1,4,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => 5
{{1,2,5},{3},{4,6}}
 => [2,5,3,6,1,4] => [3,5,1,2,6,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => 2
{{1,2,6},{3,5},{4}}
 => [2,6,5,4,3,1] => [6,5,4,1,3,2] => ([(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)
 => 11
Description
The monochromatic index of a connected graph. This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path. For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.
Matching statistic: St000454
(load all 10 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 32%
Values
{{1,2}}
 => [2,1] => [2,1] => ([(0,1)],2)
 => 1
{{1},{2}}
 => [1,2] => [1,2] => ([],2)
 => 0
{{1,2,3}}
 => [2,3,1] => [2,1,3] => ([(1,2)],3)
 => 1
{{1,2},{3}}
 => [2,1,3] => [2,3,1] => ([(0,2),(1,2)],3)
 => ? ∊ {0,3}
{{1,3},{2}}
 => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
{{1},{2,3}}
 => [1,3,2] => [3,1,2] => ([(0,2),(1,2)],3)
 => ? ∊ {0,3}
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => ([],3)
 => 0
{{1,2,3,4}}
 => [2,3,4,1] => [2,1,3,4] => ([(2,3)],4)
 => 1
{{1,2,3},{4}}
 => [2,3,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,2,3,3,4,5,6}
{{1,2,4},{3}}
 => [2,4,3,1] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,2,3,3,4,5,6}
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,2,3,3,4,5,6}
{{1,3,4},{2}}
 => [3,2,4,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,2,3,3,4,5,6}
{{1,3},{2,4}}
 => [3,4,1,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
{{1,3},{2},{4}}
 => [3,2,1,4] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,2,3,3,4,5,6}
{{1,4},{2,3}}
 => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
{{1},{2,3,4}}
 => [1,3,4,2] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,2,3,3,4,5,6}
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
 => 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,2,3,3,4,5,6}
{{1},{2,4},{3}}
 => [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,2,3,3,4,5,6}
{{1},{2},{3,4}}
 => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,2,3,3,4,5,6}
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => ([],4)
 => 0
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [2,1,3,4,5] => ([(3,4)],5)
 => 1
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [5,2,1,4,3] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => 2
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 2
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,4},{2,3},{5}}
 => [4,3,2,1,5] => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,5},{2,3,4}}
 => [5,3,4,2,1] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1},{2,3,4,5}}
 => [1,3,4,5,2] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1},{2,3,4},{5}}
 => [1,3,4,2,5] => [1,3,2,4,5] => ([(3,4)],5)
 => 1
{{1,5},{2,3},{4}}
 => [5,3,2,4,1] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1},{2,3,5},{4}}
 => [1,3,5,4,2] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1},{2,3},{4,5}}
 => [1,3,2,5,4] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1},{2,3},{4},{5}}
 => [1,3,2,4,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,4,5},{2},{3}}
 => [4,2,3,5,1] => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,4},{2,5},{3}}
 => [4,5,3,1,2] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,4},{2},{3,5}}
 => [4,2,5,1,3] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,4},{2},{3},{5}}
 => [4,2,3,1,5] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,5},{2,4},{3}}
 => [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
{{1},{2,4,5},{3}}
 => [1,4,3,5,2] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1},{2,4},{3,5}}
 => [1,4,5,2,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1},{2,4},{3},{5}}
 => [1,4,3,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => 2
{{1,5},{2},{3,4}}
 => [5,2,4,3,1] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1},{2,5},{3,4}}
 => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1},{2},{3,4,5}}
 => [1,2,4,5,3] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1},{2},{3,4},{5}}
 => [1,2,4,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,5},{2},{3},{4}}
 => [5,2,3,4,1] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
{{1},{2,5},{3},{4}}
 => [1,5,3,4,2] => [5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1},{2},{3,5},{4}}
 => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
{{1},{2},{3},{4,5}}
 => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => 0
{{1,2,3,4,5,6}}
 => [2,3,4,5,6,1] => [2,1,3,4,5,6] => ([(4,5)],6)
 => 1
{{1,2,3},{4,5,6}}
 => [2,3,1,5,6,4] => [2,1,3,5,4,6] => ([(2,5),(3,4)],6)
 => 1
{{1,2,3},{4},{5},{6}}
 => [2,3,1,4,5,6] => [2,3,4,5,1,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2
{{1,2,4},{3,5,6}}
 => [2,4,5,1,6,3] => [2,1,4,3,5,6] => ([(2,5),(3,4)],6)
 => 1
{{1,2},{3,4},{5,6}}
 => [2,1,4,3,6,5] => [2,1,4,3,6,5] => ([(0,5),(1,4),(2,3)],6)
 => 1
{{1,2,5,6},{3},{4}}
 => [2,5,3,4,6,1] => [5,2,3,4,1,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
{{1,2},{3},{4},{5,6}}
 => [2,1,3,4,6,5] => [2,1,3,4,6,5] => ([(2,5),(3,4)],6)
 => 1
{{1,3,4,6},{2},{5}}
 => [3,2,4,6,5,1] => [3,2,1,4,6,5] => ([(1,2),(3,4),(3,5),(4,5)],6)
 => 2
{{1,3,5},{2,4,6}}
 => [3,4,5,6,1,2] => [3,4,1,2,5,6] => ([(2,4),(2,5),(3,4),(3,5)],6)
 => 2
{{1,3},{2},{4,6},{5}}
 => [3,2,1,6,5,4] => [3,2,1,6,5,4] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => 2
{{1},{2,3,4,5},{6}}
 => [1,3,4,5,2,6] => [1,3,2,4,5,6] => ([(4,5)],6)
 => 1
{{1},{2,3},{4,5},{6}}
 => [1,3,2,5,4,6] => [1,3,2,5,4,6] => ([(2,5),(3,4)],6)
 => 1
{{1,4},{2,5},{3,6}}
 => [4,5,6,1,2,3] => [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => 3
{{1},{2,4},{3,5},{6}}
 => [1,4,5,2,3,6] => [1,4,5,2,3,6] => ([(2,4),(2,5),(3,4),(3,5)],6)
 => 2
{{1,6},{2,5},{3,4}}
 => [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)
 => 5
{{1},{2,5},{3,4},{6}}
 => [1,5,4,3,2,6] => [1,5,4,3,2,6] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
{{1},{2},{3,4},{5},{6}}
 => [1,2,4,3,5,6] => [1,2,4,3,5,6] => ([(4,5)],6)
 => 1
{{1,5},{2,6},{3},{4}}
 => [5,6,3,4,1,2] => [5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 4
{{1},{2},{3},{4,5,6}}
 => [1,2,3,5,6,4] => [5,1,2,3,4,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2
{{1},{2},{3},{4},{5},{6}}
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
 => 0
{{1,2,3,4,5,6,7}}
 => [2,3,4,5,6,7,1] => [2,1,3,4,5,6,7] => ([(5,6)],7)
 => 1
{{1,2,3,4,5},{6,7}}
 => [2,3,4,5,1,7,6] => [2,1,7,3,4,5,6] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => 2
{{1,2,3,4},{5},{6},{7}}
 => [2,3,4,1,5,6,7] => [2,3,4,5,1,6,7] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 2
{{1,2,3,5,6,7},{4}}
 => [2,3,5,4,6,7,1] => [5,2,3,4,1,6,7] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3
{{1,2,3},{4},{5,6,7}}
 => [2,3,1,4,6,7,5] => [2,1,3,4,6,5,7] => ([(3,6),(4,5)],7)
 => 1
{{1,2},{3,4,5,6,7}}
 => [2,1,4,5,6,7,3] => [2,1,4,5,6,7,3] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => 2
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: St001645
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 41%
Values
{{1,2}}
 => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 2 = 1 + 1
{{1},{2}}
 => [1,2] => ([],2)
 => ([],1)
 => 1 = 0 + 1
{{1,2,3}}
 => [2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 2 = 1 + 1
{{1,2},{3}}
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {0,3} + 1
{{1,3},{2}}
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
{{1},{2,3}}
 => [1,3,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {0,3} + 1
{{1},{2},{3}}
 => [1,2,3] => ([],3)
 => ([],1)
 => 1 = 0 + 1
{{1,2,3,4}}
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 2 = 1 + 1
{{1,2,3},{4}}
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {0,0,0,1,2,3,3,4,5,6} + 1
{{1,2,4},{3}}
 => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,1,2,3,3,4,5,6} + 1
{{1,2},{3,4}}
 => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ? ∊ {0,0,0,1,2,3,3,4,5,6} + 1
{{1,2},{3},{4}}
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {0,0,0,1,2,3,3,4,5,6} + 1
{{1,3,4},{2}}
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,1,2,3,3,4,5,6} + 1
{{1,3},{2,4}}
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 2 = 1 + 1
{{1,3},{2},{4}}
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,1,2,3,3,4,5,6} + 1
{{1,4},{2,3}}
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
{{1},{2,3,4}}
 => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {0,0,0,1,2,3,3,4,5,6} + 1
{{1},{2,3},{4}}
 => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {0,0,0,1,2,3,3,4,5,6} + 1
{{1,4},{2},{3}}
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
{{1},{2,4},{3}}
 => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,1,2,3,3,4,5,6} + 1
{{1},{2},{3,4}}
 => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {0,0,0,1,2,3,3,4,5,6} + 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => ([],4)
 => ([],1)
 => 1 = 0 + 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2 = 1 + 1
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,8,9,10} + 1
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,8,9,10} + 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,8,9,10} + 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,8,9,10} + 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,8,9,10} + 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 8 = 7 + 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,8,9,10} + 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,8,9,10} + 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,8,9,10} + 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,8,9,10} + 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,8,9,10} + 1
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,8,9,10} + 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,8,9,10} + 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,8,9,10} + 1
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,8,9,10} + 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,8,9,10} + 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,8,9,10} + 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 8 = 7 + 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,8,9,10} + 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,8,9,10} + 1
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,8,9,10} + 1
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,8,9,10} + 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,8,9,10} + 1
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,8,9,10} + 1
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,8,9,10} + 1
{{1,4},{2,3},{5}}
 => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,8,9,10} + 1
{{1,5},{2,3,4}}
 => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
{{1},{2,3,4,5}}
 => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,8,9,10} + 1
{{1},{2,3,4},{5}}
 => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,8,9,10} + 1
{{1,5},{2,3},{4}}
 => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,8,9,10} + 1
{{1},{2,3,5},{4}}
 => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,8,9,10} + 1
{{1},{2,3},{4,5}}
 => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,8,9,10} + 1
{{1},{2,3},{4},{5}}
 => [1,3,2,4,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,8,9,10} + 1
{{1,4,5},{2},{3}}
 => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,8,9,10} + 1
{{1,4},{2,5},{3}}
 => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
{{1,4},{2},{3,5}}
 => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,8,9,10} + 1
{{1,4},{2},{3},{5}}
 => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,8,9,10} + 1
{{1,5},{2,4},{3}}
 => [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)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
{{1},{2,4,5},{3}}
 => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,8,9,10} + 1
{{1},{2,4},{3,5}}
 => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,8,9,10} + 1
{{1},{2,4},{3},{5}}
 => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,8,9,10} + 1
{{1,5},{2},{3,4}}
 => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,8,9,10} + 1
{{1},{2,5},{3,4}}
 => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,7,8,9,10} + 1
{{1,5},{2},{3},{4}}
 => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
{{1,2,3,4,5,6}}
 => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 1 + 1
{{1,2,3,5},{4,6}}
 => [2,3,5,6,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 8 = 7 + 1
{{1,2,4},{3,5,6}}
 => [2,4,5,1,6,3] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 16 = 15 + 1
{{1,3,5},{2,4,6}}
 => [3,4,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => 2 = 1 + 1
{{1,3},{2,4,5,6}}
 => [3,4,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 8 = 7 + 1
{{1,6},{2,3,4,5}}
 => [6,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
{{1,6},{2,3,5},{4}}
 => [6,3,5,4,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(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)
 => 6 = 5 + 1
{{1,4},{2,5},{3,6}}
 => [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,1)],2)
 => 2 = 1 + 1
{{1,4},{2,6},{3,5}}
 => [4,6,5,1,3,2] => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
{{1,5},{2,4},{3,6}}
 => [5,4,6,2,1,3] => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
{{1,6},{2,4,5},{3}}
 => [6,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(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)
 => 6 = 5 + 1
{{1,6},{2,4},{3,5}}
 => [6,4,5,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
{{1,5},{2,6},{3,4}}
 => [5,6,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
{{1,6},{2,5},{3,4}}
 => [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)
 => ([(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)
 => 6 = 5 + 1
{{1,5},{2,6},{3},{4}}
 => [5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
{{1,6},{2,5},{3},{4}}
 => [6,5,3,4,2,1] => ([(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)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
{{1,6},{2},{3},{4},{5}}
 => [6,2,3,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
{{1},{2},{3},{4},{5},{6}}
 => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
{{1,2,3,4,5,6,7}}
 => [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,1)],2)
 => 2 = 1 + 1
{{1,2,3,4,6},{5,7}}
 => [2,3,4,6,7,1,5] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,3),(1,2),(2,3)],4)
 => 8 = 7 + 1
{{1,2,3,5},{4,6,7}}
 => [2,3,5,6,1,7,4] => ([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 16 = 15 + 1
{{1,2,4,6},{3,5,7}}
 => [2,4,5,6,7,1,3] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,3),(1,2),(2,3)],4)
 => 8 = 7 + 1
{{1,2,4},{3,5,6,7}}
 => [2,4,5,1,6,7,3] => ([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 16 = 15 + 1
{{1,2,5},{3,6},{4,7}}
 => [2,5,6,7,1,3,4] => ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
 => ([(0,3),(1,2),(2,3)],4)
 => 8 = 7 + 1
{{1,3,5,7},{2,4,6}}
 => [3,4,5,6,7,2,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
{{1,3,5},{2,4,6,7}}
 => [3,4,5,6,1,7,2] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,3),(1,2),(2,3)],4)
 => 8 = 7 + 1
{{1,3},{2,4,5,6,7}}
 => [3,4,1,5,6,7,2] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,3),(1,2),(2,3)],4)
 => 8 = 7 + 1
{{1,3},{2,4,6},{5,7}}
 => [3,4,1,6,7,2,5] => ([(0,4),(0,5),(1,2),(1,3),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 16 = 15 + 1
{{1,7},{2,3,4,5,6}}
 => [7,3,4,5,6,2,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
{{1,7},{2,3,4,6},{5}}
 => [7,3,4,6,5,2,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(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)
 => 6 = 5 + 1
{{1,5},{2,3,7},{4,6}}
 => [5,3,7,6,1,4,2] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
 => 7 = 6 + 1
Description
The pebbling number of a connected graph.
Matching statistic: St001879
(load all 4 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 55%
Values
{{1,2}}
 => [2,1] => [2,1] => ([],2)
 => ? ∊ {0,1}
{{1},{2}}
 => [1,2] => [1,2] => ([(0,1)],2)
 => ? ∊ {0,1}
{{1,2,3}}
 => [2,3,1] => [3,1,2] => ([(1,2)],3)
 => ? ∊ {0,0,1,3}
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
 => ? ∊ {0,0,1,3}
{{1,3},{2}}
 => [3,2,1] => [2,3,1] => ([(1,2)],3)
 => ? ∊ {0,0,1,3}
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
 => ? ∊ {0,0,1,3}
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2
{{1,2,3,4}}
 => [2,3,4,1] => [4,1,2,3] => ([(1,2),(2,3)],4)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6}
{{1,2,3},{4}}
 => [2,3,1,4] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6}
{{1,2,4},{3}}
 => [2,4,3,1] => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6}
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6}
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6}
{{1,3,4},{2}}
 => [3,2,4,1] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6}
{{1,3},{2,4}}
 => [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6}
{{1,3},{2},{4}}
 => [3,2,1,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6}
{{1,4},{2,3}}
 => [4,3,2,1] => [3,2,4,1] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6}
{{1},{2,3,4}}
 => [1,3,4,2] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6}
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 4
{{1,4},{2},{3}}
 => [4,2,3,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6}
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6}
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6}
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,6,7,7,7,8,9,10}
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,6,7,7,7,8,9,10}
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,6,7,7,7,8,9,10}
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,6,7,7,7,8,9,10}
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,6,7,7,7,8,9,10}
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,6,7,7,7,8,9,10}
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,6,7,7,7,8,9,10}
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,6,7,7,7,8,9,10}
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,6,7,7,7,8,9,10}
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,6,7,7,7,8,9,10}
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,6,7,7,7,8,9,10}
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,6,7,7,7,8,9,10}
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,6,7,7,7,8,9,10}
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,6,7,7,7,8,9,10}
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,6,7,7,7,8,9,10}
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,6,7,7,7,8,9,10}
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [4,1,3,5,2] => ([(0,4),(1,2),(1,3),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,6,7,7,7,8,9,10}
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,6,7,7,7,8,9,10}
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,6,7,7,7,8,9,10}
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [3,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,6,7,7,7,8,9,10}
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [3,1,4,2,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,6,7,7,7,8,9,10}
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,6,7,7,7,8,9,10}
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [3,1,4,5,2] => ([(0,4),(1,2),(1,4),(4,3)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,6,7,7,7,8,9,10}
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,6,7,7,7,8,9,10}
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,6,7,7,7,8,9,10}
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [3,2,5,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,6,7,7,7,8,9,10}
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [4,1,5,2,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,6,7,7,7,8,9,10}
{{1,4},{2,3},{5}}
 => [4,3,2,1,5] => [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,6,7,7,7,8,9,10}
{{1,5},{2,3,4}}
 => [5,3,4,2,1] => [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,6,7,7,7,8,9,10}
{{1},{2,3,4,5}}
 => [1,3,4,5,2] => [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,6,7,7,7,8,9,10}
{{1},{2,3,4},{5}}
 => [1,3,4,2,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 6
{{1,5},{2,3},{4}}
 => [5,3,2,4,1] => [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,6,7,7,7,8,9,10}
{{1},{2,3},{4},{5}}
 => [1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 5
{{1},{2,4},{3},{5}}
 => [1,4,3,2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 6
{{1},{2},{3,4},{5}}
 => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 5
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
{{1},{2,3,4,5},{6}}
 => [1,3,4,5,2,6] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => 8
{{1},{2,3,4},{5},{6}}
 => [1,3,4,2,5,6] => [1,4,2,3,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => 7
{{1},{2,3,5},{4},{6}}
 => [1,3,5,4,2,6] => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => 8
{{1},{2,3},{4},{5},{6}}
 => [1,3,2,4,5,6] => [1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 6
{{1},{2,4,5},{3},{6}}
 => [1,4,3,5,2,6] => [1,3,5,2,4,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 7
{{1},{2,4},{3,5},{6}}
 => [1,4,5,2,3,6] => [1,4,2,5,3,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 7
{{1},{2,4},{3},{5},{6}}
 => [1,4,3,2,5,6] => [1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => 7
{{1},{2,5},{3,4},{6}}
 => [1,5,4,3,2,6] => [1,4,3,5,2,6] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 10
{{1},{2},{3,4,5},{6}}
 => [1,2,4,5,3,6] => [1,2,5,3,4,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
 => 7
{{1},{2},{3,4},{5},{6}}
 => [1,2,4,3,5,6] => [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 6
{{1},{2,5},{3},{4},{6}}
 => [1,5,3,4,2,6] => [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => 8
{{1},{2},{3,5},{4},{6}}
 => [1,2,5,4,3,6] => [1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
 => 7
{{1},{2},{3},{4,5},{6}}
 => [1,2,3,5,4,6] => [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 6
{{1},{2},{3},{4},{5},{6}}
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
{{1},{2,3,4,5,6},{7}}
 => [1,3,4,5,6,2,7] => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
 => 10
{{1},{2,3,4,5},{6},{7}}
 => [1,3,4,5,2,6,7] => [1,5,2,3,4,6,7] => ([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7)
 => 9
{{1},{2,3,4,6},{5},{7}}
 => [1,3,4,6,5,2,7] => [1,5,6,2,3,4,7] => ([(0,4),(0,5),(1,6),(2,6),(3,2),(4,3),(5,1)],7)
 => 10
{{1},{2,3,4},{5},{6},{7}}
 => [1,3,4,2,5,6,7] => [1,4,2,3,5,6,7] => ([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7)
 => 8
{{1},{2,3,5,6},{4},{7}}
 => [1,3,5,4,6,2,7] => [1,4,6,2,3,5,7] => ([(0,3),(0,4),(1,6),(2,5),(3,2),(4,1),(4,5),(5,6)],7)
 => 9
{{1},{2,3,5},{4,6},{7}}
 => [1,3,5,6,2,4,7] => [1,5,2,3,6,4,7] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,5),(4,3),(5,6)],7)
 => 9
{{1},{2,3,5},{4},{6},{7}}
 => [1,3,5,4,2,6,7] => [1,4,5,2,3,6,7] => ([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7)
 => 9
{{1},{2,3,6},{4,5},{7}}
 => [1,3,6,5,4,2,7] => [1,5,4,6,2,3,7] => ([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
 => 13
{{1},{2,3,6},{4},{5},{7}}
 => [1,3,6,4,5,2,7] => [1,4,5,6,2,3,7] => ([(0,4),(0,5),(1,6),(2,6),(3,2),(4,3),(5,1)],7)
 => 10
{{1},{2,3},{4},{5,6},{7}}
 => [1,3,2,4,6,5,7] => [1,3,2,4,6,5,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => 8
{{1},{2,3},{4},{5},{6},{7}}
 => [1,3,2,4,5,6,7] => [1,3,2,4,5,6,7] => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
 => 7
{{1},{2,4,5,6},{3},{7}}
 => [1,4,3,5,6,2,7] => [1,3,6,2,4,5,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7)
 => 9
{{1},{2,4,5},{3,6},{7}}
 => [1,4,6,5,2,3,7] => [1,5,2,4,6,3,7] => ([(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(5,6)],7)
 => 11
{{1},{2,4,5},{3},{6},{7}}
 => [1,4,3,5,2,6,7] => [1,3,5,2,4,6,7] => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => 8
{{1},{2,4,6},{3,5},{7}}
 => [1,4,5,6,3,2,7] => [1,5,3,6,2,4,7] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => 11
{{1},{2,4},{3,5},{6},{7}}
 => [1,4,5,2,3,6,7] => [1,4,2,5,3,6,7] => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => 8
{{1},{2,4,6},{3},{5},{7}}
 => [1,4,3,6,5,2,7] => [1,3,5,6,2,4,7] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7)
 => 9
{{1},{2,4},{3,6},{5},{7}}
 => [1,4,6,2,5,3,7] => [1,4,2,5,6,3,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7)
 => 9
{{1},{2,4},{3},{5},{6},{7}}
 => [1,4,3,2,5,6,7] => [1,3,4,2,5,6,7] => ([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7)
 => 8
{{1},{2,5},{3,4,6},{7}}
 => [1,5,4,6,2,3,7] => [1,5,2,6,3,4,7] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7)
 => 9
{{1},{2,5},{3,4},{6},{7}}
 => [1,5,4,3,2,6,7] => [1,4,3,5,2,6,7] => ([(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(5,6),(6,1)],7)
 => 11
{{1},{2,6},{3,4,5},{7}}
 => [1,6,4,5,3,2,7] => [1,5,3,4,6,2,7] => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,5),(4,1),(5,6)],7)
 => 13
{{1},{2},{3,4,5,6},{7}}
 => [1,2,4,5,6,3,7] => [1,2,6,3,4,5,7] => ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7)
 => 9
{{1},{2},{3,4,5},{6},{7}}
 => [1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
 => 8
{{1},{2,6},{3,4},{5},{7}}
 => [1,6,4,3,5,2,7] => [1,4,3,5,6,2,7] => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
 => 12
{{1},{2},{3,4,6},{5},{7}}
 => [1,2,4,6,5,3,7] => [1,2,5,6,3,4,7] => ([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
 => 9
{{1},{2},{3,4},{5},{6},{7}}
 => [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => 7
{{1},{2,5,6},{3},{4},{7}}
 => [1,5,3,4,6,2,7] => [1,3,4,6,2,5,7] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,5),(4,3),(5,6)],7)
 => 9
Description
The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.
Matching statistic: St001880
(load all 4 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 27%
Values
{{1,2}}
 => [2,1] => [1] => ([],1)
 => ? ∊ {0,1}
{{1},{2}}
 => [1,2] => [1] => ([],1)
 => ? ∊ {0,1}
{{1,2,3}}
 => [2,3,1] => [2,1] => ([],2)
 => ? ∊ {0,0,1,2,3}
{{1,2},{3}}
 => [2,1,3] => [2,1] => ([],2)
 => ? ∊ {0,0,1,2,3}
{{1,3},{2}}
 => [3,2,1] => [2,1] => ([],2)
 => ? ∊ {0,0,1,2,3}
{{1},{2,3}}
 => [1,3,2] => [1,2] => ([(0,1)],2)
 => ? ∊ {0,0,1,2,3}
{{1},{2},{3}}
 => [1,2,3] => [1,2] => ([(0,1)],2)
 => ? ∊ {0,0,1,2,3}
{{1,2,3,4}}
 => [2,3,4,1] => [2,3,1] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,4,5,6}
{{1,2,3},{4}}
 => [2,3,1,4] => [2,3,1] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,4,5,6}
{{1,2,4},{3}}
 => [2,4,3,1] => [2,3,1] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,4,5,6}
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,3] => ([(0,2),(1,2)],3)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,4,5,6}
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3] => ([(0,2),(1,2)],3)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,4,5,6}
{{1,3,4},{2}}
 => [3,2,4,1] => [3,2,1] => ([],3)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,4,5,6}
{{1,3},{2,4}}
 => [3,4,1,2] => [3,1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,4,5,6}
{{1,3},{2},{4}}
 => [3,2,1,4] => [3,2,1] => ([],3)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,4,5,6}
{{1,4},{2,3}}
 => [4,3,2,1] => [3,2,1] => ([],3)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,4,5,6}
{{1},{2,3,4}}
 => [1,3,4,2] => [1,3,2] => ([(0,1),(0,2)],3)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,4,5,6}
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2] => ([(0,1),(0,2)],3)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,4,5,6}
{{1,4},{2},{3}}
 => [4,2,3,1] => [2,3,1] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,4,5,6}
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,4,5,6}
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [2,4,3,1] => ([(1,2),(1,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [2,4,3,1] => ([(1,2),(1,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [2,4,3,1] => ([(1,2),(1,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [3,2,4,1] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [3,2,4,1] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [3,4,2,1] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [3,2,4,1] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [4,3,2,1] => ([],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [4,3,1,2] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,4},{2,3},{5}}
 => [4,3,2,1,5] => [4,3,2,1] => ([],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1,5},{2,3,4}}
 => [5,3,4,2,1] => [3,4,2,1] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1},{2,3,4,5}}
 => [1,3,4,5,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8,9,10}
{{1},{2,3},{4,5}}
 => [1,3,2,5,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 4
{{1},{2,3},{4},{5}}
 => [1,3,2,4,5] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 4
{{1},{2},{3},{4,5}}
 => [1,2,3,5,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 4
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 4
{{1},{2,3,4},{5,6}}
 => [1,3,4,2,6,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 4
{{1},{2,3,4},{5},{6}}
 => [1,3,4,2,5,6] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 4
{{1},{2,3},{4},{5,6}}
 => [1,3,2,4,6,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 5
{{1},{2,3},{4},{5},{6}}
 => [1,3,2,4,5,6] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 5
{{1},{2,4},{3},{5,6}}
 => [1,4,3,2,6,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 1
{{1},{2,4},{3},{5},{6}}
 => [1,4,3,2,5,6] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 1
{{1},{2},{3,4},{5,6}}
 => [1,2,4,3,6,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 5
{{1},{2},{3,4},{5},{6}}
 => [1,2,4,3,5,6] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 5
{{1},{2},{3},{4},{5,6}}
 => [1,2,3,4,6,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
{{1},{2},{3},{4},{5},{6}}
 => [1,2,3,4,5,6] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
{{1},{2,3,4,5},{6,7}}
 => [1,3,4,5,2,7,6] => [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => 4
{{1},{2,3,4,5},{6},{7}}
 => [1,3,4,5,2,6,7] => [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => 4
{{1},{2,3,4},{5},{6,7}}
 => [1,3,4,2,5,7,6] => [1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => 5
{{1},{2,3,4},{5},{6},{7}}
 => [1,3,4,2,5,6,7] => [1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => 5
{{1},{2,3,5},{4},{6,7}}
 => [1,3,5,4,2,7,6] => [1,3,5,4,2,6] => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
 => 2
{{1},{2,3,5},{4},{6},{7}}
 => [1,3,5,4,2,6,7] => [1,3,5,4,2,6] => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
 => 2
{{1},{2,3},{4},{5},{6,7}}
 => [1,3,2,4,5,7,6] => [1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 6
{{1},{2,3},{4},{5},{6},{7}}
 => [1,3,2,4,5,6,7] => [1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 6
{{1},{2,4,5},{3},{6,7}}
 => [1,4,3,5,2,7,6] => [1,4,3,5,2,6] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 2
{{1},{2,4,5},{3},{6},{7}}
 => [1,4,3,5,2,6,7] => [1,4,3,5,2,6] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 2
{{1},{2,4},{3,5},{6,7}}
 => [1,4,5,2,3,7,6] => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => 4
{{1},{2,4},{3,5},{6},{7}}
 => [1,4,5,2,3,6,7] => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => 4
{{1},{2,4},{3},{5},{6,7}}
 => [1,4,3,2,5,7,6] => [1,4,3,2,5,6] => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => 2
{{1},{2,4},{3},{5},{6},{7}}
 => [1,4,3,2,5,6,7] => [1,4,3,2,5,6] => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => 2
{{1},{2,5},{3,4},{6,7}}
 => [1,5,4,3,2,7,6] => [1,5,4,3,2,6] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
{{1},{2,5},{3,4},{6},{7}}
 => [1,5,4,3,2,6,7] => [1,5,4,3,2,6] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
{{1},{2},{3,4,5},{6,7}}
 => [1,2,4,5,3,7,6] => [1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
 => 5
{{1},{2},{3,4,5},{6},{7}}
 => [1,2,4,5,3,6,7] => [1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
 => 5
{{1},{2},{3,4},{5},{6,7}}
 => [1,2,4,3,5,7,6] => [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 6
{{1},{2},{3,4},{5},{6},{7}}
 => [1,2,4,3,5,6,7] => [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 6
{{1},{2,5},{3},{4},{6,7}}
 => [1,5,3,4,2,7,6] => [1,5,3,4,2,6] => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
 => 1
{{1},{2,5},{3},{4},{6},{7}}
 => [1,5,3,4,2,6,7] => [1,5,3,4,2,6] => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
 => 1
{{1},{2},{3,5},{4},{6,7}}
 => [1,2,5,4,3,7,6] => [1,2,5,4,3,6] => ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
 => 2
{{1},{2},{3,5},{4},{6},{7}}
 => [1,2,5,4,3,6,7] => [1,2,5,4,3,6] => ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
 => 2
{{1},{2},{3},{4,5},{6,7}}
 => [1,2,3,5,4,7,6] => [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 6
{{1},{2},{3},{4,5},{6},{7}}
 => [1,2,3,5,4,6,7] => [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 6
{{1},{2},{3},{4},{5},{6,7}}
 => [1,2,3,4,5,7,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
{{1},{2},{3},{4},{5},{6},{7}}
 => [1,2,3,4,5,6,7] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.