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Your data matches 42 different statistics following compositions of up to 3 maps.
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Matching statistic: St001630
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001630: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001630: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,3,2,5,4,6] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[1,3,2,5,6,4] => [1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,3,2,6,4,5] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> ([(0,3),(2,1),(3,2)],4)
=> 2
[1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> ([(0,3),(2,1),(3,2)],4)
=> 2
[1,3,4,2,6,5] => [1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,3,5,2,4,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,3,5,2,6,4] => [1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,3,5,4,2,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,3,5,4,6,2] => [1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,4,2,3,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,4,2,5,3,6] => [1,0,1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2,1],[2]]
=> ([(0,2),(2,1)],3)
=> 1
[1,4,2,6,3,5] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,4,2,6,5,3] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,4,3,5,2,6] => [1,0,1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2,1],[2]]
=> ([(0,2),(2,1)],3)
=> 1
[1,4,3,6,2,5] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,4,3,6,5,2] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,1,3,5,4,6] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,4,3,5,6] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,1,4,3,6,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[2,1,4,5,3,6] => [1,1,0,0,1,1,0,1,0,0,1,0]
=> [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,4,6,3,5] => [1,1,0,0,1,1,0,1,1,0,0,0]
=> [[4,4,2],[2,1]]
=> ([(0,3),(2,1),(3,2)],4)
=> 2
[2,1,4,6,5,3] => [1,1,0,0,1,1,0,1,1,0,0,0]
=> [[4,4,2],[2,1]]
=> ([(0,3),(2,1),(3,2)],4)
=> 2
[2,1,5,3,4,6] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [[3,3,3,2],[2,1,1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,1,5,3,6,4] => [1,1,0,0,1,1,1,0,0,1,0,0]
=> [[4,3,2],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,1,5,4,3,6] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [[3,3,3,2],[2,1,1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,1,5,4,6,3] => [1,1,0,0,1,1,1,0,0,1,0,0]
=> [[4,3,2],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,3,1,5,4,6] => [1,1,0,1,0,0,1,1,0,0,1,0]
=> [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
[2,3,1,5,6,4] => [1,1,0,1,0,0,1,1,0,1,0,0]
=> [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
[2,4,1,3,6,5] => [1,1,0,1,1,0,0,0,1,1,0,0]
=> [[4,3,3],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,4,1,5,3,6] => [1,1,0,1,1,0,0,1,0,0,1,0]
=> [[4,4,3],[3,1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,4,1,6,3,5] => [1,1,0,1,1,0,0,1,1,0,0,0]
=> [[4,4,3],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,4,1,6,5,3] => [1,1,0,1,1,0,0,1,1,0,0,0]
=> [[4,4,3],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,4,3,1,6,5] => [1,1,0,1,1,0,0,0,1,1,0,0]
=> [[4,3,3],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,4,3,5,1,6] => [1,1,0,1,1,0,0,1,0,0,1,0]
=> [[4,4,3],[3,1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,4,3,6,1,5] => [1,1,0,1,1,0,0,1,1,0,0,0]
=> [[4,4,3],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,4,3,6,5,1] => [1,1,0,1,1,0,0,1,1,0,0,0]
=> [[4,4,3],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[3,1,2,5,4,6] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [[3,3,2,2],[2,1]]
=> ([(0,3),(2,1),(3,2)],4)
=> 2
[3,1,2,6,4,5] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [[3,3,2,2],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
[3,1,2,6,5,4] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [[3,3,2,2],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
[3,1,4,2,6,5] => [1,1,1,0,0,1,0,0,1,1,0,0]
=> [[4,3,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 2
[3,1,4,6,2,5] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [[4,4,2],[2]]
=> ([(0,2),(2,1)],3)
=> 1
[3,1,4,6,5,2] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [[4,4,2],[2]]
=> ([(0,2),(2,1)],3)
=> 1
[3,1,5,2,4,6] => [1,1,1,0,0,1,1,0,0,0,1,0]
=> [[3,3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[3,1,5,2,6,4] => [1,1,1,0,0,1,1,0,0,1,0,0]
=> [[4,3,2],[1]]
=> ([(0,2),(2,1)],3)
=> 1
Description
The global dimension of the incidence algebra of the lattice over the rational numbers.
Matching statistic: St001645
(load all 35 compositions to match this statistic)
(load all 35 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 50%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 50%
Values
[1,3,2,5,4] => [1,3,2,5,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 6
[2,1,4,3,5] => [2,1,4,3,5] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 6
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 6
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 6
[1,3,2,5,6,4] => [1,3,2,6,5,4] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
[1,3,2,6,4,5] => [1,3,2,6,5,4] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 6
[1,3,2,6,5,4] => [1,3,2,6,5,4] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 6
[1,3,4,2,6,5] => [1,4,3,2,6,5] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 6
[1,3,5,2,4,6] => [1,4,5,2,3,6] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
[1,3,5,2,6,4] => [1,4,6,2,5,3] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
[1,3,5,4,2,6] => [1,5,4,3,2,6] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
[1,3,5,4,6,2] => [1,6,4,3,5,2] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
[1,4,2,3,6,5] => [1,4,3,2,6,5] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
[1,4,2,5,3,6] => [1,5,3,4,2,6] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
[1,4,2,6,3,5] => [1,5,3,6,2,4] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
[1,4,2,6,5,3] => [1,6,3,5,4,2] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
[1,4,3,2,6,5] => [1,4,3,2,6,5] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
[1,4,3,5,2,6] => [1,5,3,4,2,6] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
[1,4,3,6,2,5] => [1,5,3,6,2,4] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
[1,4,3,6,5,2] => [1,6,3,5,4,2] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
[2,1,3,5,4,6] => [2,1,3,5,4,6] => [2,1,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 6
[2,1,4,3,5,6] => [2,1,4,3,5,6] => [2,1,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
[2,1,4,3,6,5] => [2,1,4,3,6,5] => [2,1,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 6
[2,1,4,5,3,6] => [2,1,5,4,3,6] => [2,1,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 6
[2,1,4,6,3,5] => [2,1,5,6,3,4] => [2,1,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 6
[2,1,4,6,5,3] => [2,1,6,5,4,3] => [2,1,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 6
[2,1,5,3,4,6] => [2,1,5,4,3,6] => [2,1,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
[2,1,5,3,6,4] => [2,1,6,4,5,3] => [2,1,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
[2,1,5,4,3,6] => [2,1,5,4,3,6] => [2,1,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
[2,1,5,4,6,3] => [2,1,6,4,5,3] => [2,1,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
[2,3,1,5,4,6] => [3,2,1,5,4,6] => [3,2,1,6,5,4] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ? = 1 + 6
[2,3,1,5,6,4] => [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)
=> ? = 1 + 6
[2,4,1,3,6,5] => [3,4,1,2,6,5] => [3,6,1,5,4,2] => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
[2,4,1,5,3,6] => [3,5,1,4,2,6] => [3,6,1,5,4,2] => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
[2,4,1,6,3,5] => [3,5,1,6,2,4] => [3,6,1,5,4,2] => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
[2,4,1,6,5,3] => [3,6,1,5,4,2] => [3,6,1,5,4,2] => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
[2,4,3,1,6,5] => [4,3,2,1,6,5] => [4,3,2,1,6,5] => ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
[2,4,3,5,1,6] => [5,3,2,4,1,6] => [5,3,2,6,1,4] => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
[2,4,3,6,1,5] => [5,3,2,6,1,4] => [5,3,2,6,1,4] => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
[2,4,3,6,5,1] => [6,3,2,5,4,1] => [6,3,2,5,4,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
[3,1,2,5,4,6] => [3,2,1,5,4,6] => [3,2,1,6,5,4] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ? = 2 + 6
[3,1,2,6,4,5] => [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)
=> ? = 1 + 6
[3,1,2,6,5,4] => [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)
=> ? = 1 + 6
[3,1,4,2,6,5] => [4,2,3,1,6,5] => [4,2,6,1,5,3] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
=> ? = 2 + 6
[3,1,4,6,2,5] => [5,2,3,6,1,4] => [5,2,6,4,1,3] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
[3,1,4,6,5,2] => [6,2,3,5,4,1] => [6,2,5,4,3,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
[3,1,5,2,4,6] => [4,2,5,1,3,6] => [4,2,6,1,5,3] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
=> ? = 1 + 6
[3,1,5,2,6,4] => [4,2,6,1,5,3] => [4,2,6,1,5,3] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
=> ? = 1 + 6
[2,4,6,1,3,7,5] => [4,5,7,1,2,6,3] => [4,7,6,1,5,3,2] => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[2,4,6,1,5,7,3] => [4,7,5,1,3,6,2] => [4,7,6,1,5,3,2] => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[2,4,6,1,7,3,5] => [4,6,7,1,5,2,3] => [4,7,6,1,5,3,2] => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[2,4,6,1,7,5,3] => [4,7,6,1,5,3,2] => [4,7,6,1,5,3,2] => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[2,4,6,3,5,7,1] => [7,4,5,2,3,6,1] => [7,4,6,2,5,3,1] => ([(0,1),(0,4),(0,5),(0,6),(1,3),(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)
=> 7 = 1 + 6
[2,4,6,3,7,1,5] => [6,4,7,2,5,1,3] => [6,4,7,2,5,1,3] => ([(0,1),(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[2,4,6,3,7,5,1] => [7,4,6,2,5,3,1] => [7,4,6,2,5,3,1] => ([(0,1),(0,4),(0,5),(0,6),(1,3),(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)
=> 7 = 1 + 6
[2,4,6,5,7,1,3] => [6,7,4,3,5,1,2] => [6,7,4,3,5,1,2] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 7 = 1 + 6
[2,4,6,5,7,3,1] => [7,6,4,3,5,2,1] => [7,6,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[2,5,3,6,1,7,4] => [5,7,3,4,1,6,2] => [5,7,3,6,1,4,2] => ([(0,1),(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[2,5,3,6,4,7,1] => [7,5,3,4,2,6,1] => [7,5,3,6,2,4,1] => ([(0,1),(0,4),(0,5),(0,6),(1,3),(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)
=> 7 = 1 + 6
[2,5,3,7,1,4,6] => [5,6,3,7,1,2,4] => [5,7,3,6,1,4,2] => ([(0,1),(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[2,5,3,7,1,6,4] => [5,7,3,6,1,4,2] => [5,7,3,6,1,4,2] => ([(0,1),(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[2,5,3,7,4,1,6] => [6,5,3,7,2,1,4] => [6,5,3,7,2,1,4] => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[2,5,3,7,4,6,1] => [7,5,3,6,2,4,1] => [7,5,3,6,2,4,1] => ([(0,1),(0,4),(0,5),(0,6),(1,3),(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)
=> 7 = 1 + 6
[2,5,3,7,6,1,4] => [6,7,3,5,4,1,2] => [6,7,3,5,4,1,2] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 7 = 1 + 6
[2,5,3,7,6,4,1] => [7,6,3,5,4,2,1] => [7,6,3,5,4,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[2,5,4,6,1,7,3] => [5,7,3,4,1,6,2] => [5,7,3,6,1,4,2] => ([(0,1),(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[2,5,4,6,3,7,1] => [7,5,3,4,2,6,1] => [7,5,3,6,2,4,1] => ([(0,1),(0,4),(0,5),(0,6),(1,3),(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)
=> 7 = 1 + 6
[2,5,4,7,1,3,6] => [5,6,3,7,1,2,4] => [5,7,3,6,1,4,2] => ([(0,1),(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[2,5,4,7,1,6,3] => [5,7,3,6,1,4,2] => [5,7,3,6,1,4,2] => ([(0,1),(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[2,5,4,7,3,1,6] => [6,5,3,7,2,1,4] => [6,5,3,7,2,1,4] => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[2,5,4,7,3,6,1] => [7,5,3,6,2,4,1] => [7,5,3,6,2,4,1] => ([(0,1),(0,4),(0,5),(0,6),(1,3),(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)
=> 7 = 1 + 6
[2,5,4,7,6,1,3] => [6,7,3,5,4,1,2] => [6,7,3,5,4,1,2] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 7 = 1 + 6
[2,5,4,7,6,3,1] => [7,6,3,5,4,2,1] => [7,6,3,5,4,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[3,5,1,6,2,7,4] => [5,7,3,4,1,6,2] => [5,7,3,6,1,4,2] => ([(0,1),(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[3,5,1,6,4,7,2] => [7,5,3,4,2,6,1] => [7,5,3,6,2,4,1] => ([(0,1),(0,4),(0,5),(0,6),(1,3),(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)
=> 7 = 1 + 6
[3,5,1,7,2,4,6] => [5,6,3,7,1,2,4] => [5,7,3,6,1,4,2] => ([(0,1),(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[3,5,1,7,2,6,4] => [5,7,3,6,1,4,2] => [5,7,3,6,1,4,2] => ([(0,1),(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[3,5,1,7,4,2,6] => [6,5,3,7,2,1,4] => [6,5,3,7,2,1,4] => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[3,5,1,7,4,6,2] => [7,5,3,6,2,4,1] => [7,5,3,6,2,4,1] => ([(0,1),(0,4),(0,5),(0,6),(1,3),(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)
=> 7 = 1 + 6
[3,5,1,7,6,2,4] => [6,7,3,5,4,1,2] => [6,7,3,5,4,1,2] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 7 = 1 + 6
[3,5,1,7,6,4,2] => [7,6,3,5,4,2,1] => [7,6,3,5,4,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[3,5,2,6,1,7,4] => [5,7,3,4,1,6,2] => [5,7,3,6,1,4,2] => ([(0,1),(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[3,5,2,6,4,7,1] => [7,5,3,4,2,6,1] => [7,5,3,6,2,4,1] => ([(0,1),(0,4),(0,5),(0,6),(1,3),(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)
=> 7 = 1 + 6
[3,5,2,7,1,4,6] => [5,6,3,7,1,2,4] => [5,7,3,6,1,4,2] => ([(0,1),(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[3,5,2,7,1,6,4] => [5,7,3,6,1,4,2] => [5,7,3,6,1,4,2] => ([(0,1),(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[3,5,2,7,4,1,6] => [6,5,3,7,2,1,4] => [6,5,3,7,2,1,4] => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[3,5,2,7,4,6,1] => [7,5,3,6,2,4,1] => [7,5,3,6,2,4,1] => ([(0,1),(0,4),(0,5),(0,6),(1,3),(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)
=> 7 = 1 + 6
[3,5,2,7,6,1,4] => [6,7,3,5,4,1,2] => [6,7,3,5,4,1,2] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 7 = 1 + 6
[3,5,2,7,6,4,1] => [7,6,3,5,4,2,1] => [7,6,3,5,4,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[3,5,4,6,1,7,2] => [7,5,3,4,2,6,1] => [7,5,3,6,2,4,1] => ([(0,1),(0,4),(0,5),(0,6),(1,3),(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)
=> 7 = 1 + 6
[3,5,4,6,2,7,1] => [7,5,3,4,2,6,1] => [7,5,3,6,2,4,1] => ([(0,1),(0,4),(0,5),(0,6),(1,3),(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)
=> 7 = 1 + 6
[3,5,4,7,1,2,6] => [6,5,3,7,2,1,4] => [6,5,3,7,2,1,4] => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[3,5,4,7,1,6,2] => [7,5,3,6,2,4,1] => [7,5,3,6,2,4,1] => ([(0,1),(0,4),(0,5),(0,6),(1,3),(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)
=> 7 = 1 + 6
[3,5,4,7,2,1,6] => [6,5,3,7,2,1,4] => [6,5,3,7,2,1,4] => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[3,5,4,7,2,6,1] => [7,5,3,6,2,4,1] => [7,5,3,6,2,4,1] => ([(0,1),(0,4),(0,5),(0,6),(1,3),(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)
=> 7 = 1 + 6
[3,5,4,7,6,1,2] => [7,6,3,5,4,2,1] => [7,6,3,5,4,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[3,5,4,7,6,2,1] => [7,6,3,5,4,2,1] => [7,6,3,5,4,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[3,5,6,1,7,2,4] => [6,7,4,3,5,1,2] => [6,7,4,3,5,1,2] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 7 = 1 + 6
Description
The pebbling number of a connected graph.
Matching statistic: St000455
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 50%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 50%
Values
[1,3,2,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[2,1,4,3,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[1,2,4,3,5,6] => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 1
[1,2,4,3,6,5] => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 1
[1,3,2,4,6,5] => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 2
[1,3,2,5,4,6] => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 2
[1,3,2,5,6,4] => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 1
[1,3,2,6,4,5] => [1,2,3,4,6,5] => [.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 2
[1,3,2,6,5,4] => [1,2,3,4,6,5] => [.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 2
[1,3,4,2,6,5] => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 2
[1,3,5,2,4,6] => [1,2,3,5,4,6] => [.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 1
[1,3,5,2,6,4] => [1,2,3,5,6,4] => [.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 1
[1,3,5,4,2,6] => [1,2,3,5,4,6] => [.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 1
[1,3,5,4,6,2] => [1,2,3,5,6,4] => [.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 1
[1,4,2,3,6,5] => [1,2,4,3,5,6] => [.,[.,[[.,.],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1
[1,4,2,5,3,6] => [1,2,4,5,3,6] => [.,[.,[[.,.],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1
[1,4,2,6,3,5] => [1,2,4,6,5,3] => [.,[.,[[.,.],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1
[1,4,2,6,5,3] => [1,2,4,6,3,5] => [.,[.,[[.,.],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1
[1,4,3,2,6,5] => [1,2,4,3,5,6] => [.,[.,[[.,.],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1
[1,4,3,5,2,6] => [1,2,4,5,3,6] => [.,[.,[[.,.],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1
[1,4,3,6,2,5] => [1,2,4,6,5,3] => [.,[.,[[.,.],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1
[1,4,3,6,5,2] => [1,2,4,6,3,5] => [.,[.,[[.,.],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1
[2,1,3,5,4,6] => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 2
[2,1,4,3,5,6] => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 1
[2,1,4,3,6,5] => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 2
[2,1,4,5,3,6] => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 2
[2,1,4,6,3,5] => [1,2,3,4,6,5] => [.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 2
[2,1,4,6,5,3] => [1,2,3,4,6,5] => [.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 2
[2,1,5,3,4,6] => [1,2,3,5,4,6] => [.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 1
[2,1,5,3,6,4] => [1,2,3,5,6,4] => [.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 1
[2,1,5,4,3,6] => [1,2,3,5,4,6] => [.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 1
[2,1,5,4,6,3] => [1,2,3,5,6,4] => [.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 1
[2,3,1,5,4,6] => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 1
[2,3,1,5,6,4] => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 1
[2,4,1,3,6,5] => [1,2,4,3,5,6] => [.,[.,[[.,.],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1
[2,4,1,5,3,6] => [1,2,4,5,3,6] => [.,[.,[[.,.],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1
[2,4,1,6,3,5] => [1,2,4,6,5,3] => [.,[.,[[.,.],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1
[2,4,1,6,5,3] => [1,2,4,6,3,5] => [.,[.,[[.,.],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1
[2,4,3,1,6,5] => [1,2,4,3,5,6] => [.,[.,[[.,.],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1
[2,4,3,5,1,6] => [1,2,4,5,3,6] => [.,[.,[[.,.],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1
[2,4,3,6,1,5] => [1,2,4,6,5,3] => [.,[.,[[.,.],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1
[2,4,3,6,5,1] => [1,2,4,6,3,5] => [.,[.,[[.,.],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1
[3,1,2,5,4,6] => [1,3,2,4,5,6] => [.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 2
[3,1,2,6,4,5] => [1,3,2,4,6,5] => [.,[[.,.],[.,[[.,.],.]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 1
[3,1,2,6,5,4] => [1,3,2,4,6,5] => [.,[[.,.],[.,[[.,.],.]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 1
[3,1,4,2,6,5] => [1,3,4,2,5,6] => [.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 2
[3,1,4,6,2,5] => [1,3,4,6,5,2] => [.,[[.,.],[.,[[.,.],.]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 1
[3,1,4,6,5,2] => [1,3,4,6,2,5] => [.,[[.,.],[.,[[.,.],.]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 1
[3,1,5,2,4,6] => [1,3,5,4,2,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 1
[3,1,5,2,6,4] => [1,3,5,6,4,2] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 1
[3,1,5,4,2,6] => [1,3,5,2,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 1
[3,1,5,4,6,2] => [1,3,5,6,2,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 1
[3,2,1,5,4,6] => [1,3,2,4,5,6] => [.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 2
[3,2,1,6,4,5] => [1,3,2,4,6,5] => [.,[[.,.],[.,[[.,.],.]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 1
[3,2,1,6,5,4] => [1,3,2,4,6,5] => [.,[[.,.],[.,[[.,.],.]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 1
[3,2,4,1,6,5] => [1,3,4,2,5,6] => [.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 2
[3,2,4,6,1,5] => [1,3,4,6,5,2] => [.,[[.,.],[.,[[.,.],.]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 1
[3,2,4,6,5,1] => [1,3,4,6,2,5] => [.,[[.,.],[.,[[.,.],.]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 1
[3,2,5,1,4,6] => [1,3,5,4,2,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 1
[3,2,5,1,6,4] => [1,3,5,6,4,2] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 1
[3,2,5,4,1,6] => [1,3,5,2,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 1
[3,2,5,4,6,1] => [1,3,5,6,2,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 1
[1,2,3,5,4,6,7] => [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ? = 1
[1,2,3,5,4,7,6] => [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ? = 1
[1,2,4,3,5,6,7] => [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ? = 1
[1,2,4,3,5,7,6] => [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ? = 2
[1,2,4,3,6,5,7] => [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ? = 2
[1,2,4,3,6,7,5] => [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ? = 1
[1,2,4,5,3,6,7] => [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ? = 1
[1,2,4,5,3,7,6] => [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ? = 2
[1,3,2,4,5,7,6] => [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ? = 2
[1,3,2,4,6,5,7] => [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ? = 2
[1,3,2,4,6,7,5] => [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ? = 2
[1,3,2,5,4,6,7] => [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ? = 2
[1,3,2,5,4,7,6] => [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ? = 2
[1,3,2,5,6,4,7] => [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ? = 2
[1,5,2,7,3,4,6] => [1,2,5,3,4,7,6] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 1
[1,5,2,7,3,6,4] => [1,2,5,3,4,7,6] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 1
[1,5,2,7,4,3,6] => [1,2,5,4,7,6,3] => [.,[.,[[[.,.],.],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 1
[1,5,2,7,4,6,3] => [1,2,5,4,7,3,6] => [.,[.,[[[.,.],.],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 1
[1,5,2,7,6,3,4] => [1,2,5,6,3,4,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 1
[1,5,2,7,6,4,3] => [1,2,5,6,4,7,3] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 1
[1,5,3,7,2,4,6] => [1,2,5,3,4,7,6] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 1
[1,5,3,7,2,6,4] => [1,2,5,3,4,7,6] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 1
[1,5,3,7,4,2,6] => [1,2,5,4,7,6,3] => [.,[.,[[[.,.],.],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 1
[1,5,3,7,4,6,2] => [1,2,5,4,7,3,6] => [.,[.,[[[.,.],.],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 1
[1,5,3,7,6,2,4] => [1,2,5,6,3,4,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 1
[1,5,3,7,6,4,2] => [1,2,5,6,4,7,3] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 1
[1,5,4,7,2,3,6] => [1,2,5,3,4,7,6] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 1
[1,5,4,7,2,6,3] => [1,2,5,3,4,7,6] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 1
[1,5,4,7,3,2,6] => [1,2,5,3,4,7,6] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 1
[1,5,4,7,3,6,2] => [1,2,5,3,4,7,6] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 1
[1,5,4,7,6,2,3] => [1,2,5,6,3,4,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 1
[1,5,4,7,6,3,2] => [1,2,5,6,3,4,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 1
[2,5,1,3,7,4,6] => [1,2,5,7,6,4,3] => [.,[.,[[[.,.],.],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 1
[2,5,1,3,7,6,4] => [1,2,5,7,4,3,6] => [.,[.,[[[.,.],.],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 1
[2,5,1,4,7,3,6] => [1,2,5,7,6,3,4] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 1
[2,5,1,4,7,6,3] => [1,2,5,7,3,4,6] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 1
[2,5,1,6,3,7,4] => [1,2,5,3,4,6,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 1
[2,5,1,6,4,7,3] => [1,2,5,4,6,7,3] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 1
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St001553
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001553: Dyck paths ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 50%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001553: Dyck paths ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 50%
Values
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 1 + 1
[1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 2 + 1
[1,3,2,5,4,6] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2 + 1
[1,3,2,5,6,4] => [1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 1 + 1
[1,3,2,6,4,5] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 2 + 1
[1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 2 + 1
[1,3,4,2,6,5] => [1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 2 + 1
[1,3,5,2,4,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 1 + 1
[1,3,5,2,6,4] => [1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> ? = 1 + 1
[1,3,5,4,2,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 1 + 1
[1,3,5,4,6,2] => [1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> ? = 1 + 1
[1,4,2,3,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 1
[1,4,2,5,3,6] => [1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 1
[1,4,2,6,3,5] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> ? = 1 + 1
[1,4,2,6,5,3] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> ? = 1 + 1
[1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 1
[1,4,3,5,2,6] => [1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 1
[1,4,3,6,2,5] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> ? = 1 + 1
[1,4,3,6,5,2] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> ? = 1 + 1
[2,1,3,5,4,6] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 2 + 1
[2,1,4,3,5,6] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[2,1,4,3,6,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 2 + 1
[2,1,4,5,3,6] => [1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 2 + 1
[2,1,4,6,3,5] => [1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> ? = 2 + 1
[2,1,4,6,5,3] => [1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> ? = 2 + 1
[2,1,5,3,4,6] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 1 + 1
[2,1,5,3,6,4] => [1,1,0,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> ? = 1 + 1
[2,1,5,4,3,6] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 1 + 1
[2,1,5,4,6,3] => [1,1,0,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> ? = 1 + 1
[2,3,1,5,4,6] => [1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 1 + 1
[2,3,1,5,6,4] => [1,1,0,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 1 + 1
[2,4,1,3,6,5] => [1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 1
[2,4,1,5,3,6] => [1,1,0,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 1
[2,4,1,6,3,5] => [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[2,4,1,6,5,3] => [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[2,4,3,1,6,5] => [1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 1
[2,4,3,5,1,6] => [1,1,0,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 1
[2,4,3,6,1,5] => [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[2,4,3,6,5,1] => [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,1,2,5,4,6] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 2 + 1
[3,1,2,6,4,5] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 1 + 1
[3,1,2,6,5,4] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 1 + 1
[3,1,4,2,6,5] => [1,1,1,0,0,1,0,0,1,1,0,0]
=> [4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> ? = 2 + 1
[3,1,4,6,2,5] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[3,1,4,6,5,2] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[3,1,5,2,4,6] => [1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> ? = 1 + 1
[3,1,5,2,6,4] => [1,1,1,0,0,1,1,0,0,1,0,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[3,1,5,4,2,6] => [1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> ? = 1 + 1
[3,1,5,4,6,2] => [1,1,1,0,0,1,1,0,0,1,0,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[3,2,1,5,4,6] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 2 + 1
[3,2,1,6,4,5] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 1 + 1
[3,2,1,6,5,4] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 1 + 1
[3,2,4,1,6,5] => [1,1,1,0,0,1,0,0,1,1,0,0]
=> [4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> ? = 2 + 1
[3,2,4,6,1,5] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[3,2,4,6,5,1] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[3,2,5,1,4,6] => [1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> ? = 1 + 1
[3,2,5,1,6,4] => [1,1,1,0,0,1,1,0,0,1,0,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[3,2,5,4,1,6] => [1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> ? = 1 + 1
[3,2,5,4,6,1] => [1,1,1,0,0,1,1,0,0,1,0,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,2,3,5,4,6,7] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[1,2,3,5,4,7,6] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 1 + 1
[3,5,1,7,2,4,6] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,5,1,7,2,6,4] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,5,1,7,4,2,6] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,5,1,7,4,6,2] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,5,1,7,6,2,4] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,5,1,7,6,4,2] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,5,2,7,1,4,6] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,5,2,7,1,6,4] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,5,2,7,4,1,6] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,5,2,7,4,6,1] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,5,2,7,6,1,4] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,5,2,7,6,4,1] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,5,4,7,1,2,6] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,5,4,7,1,6,2] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,5,4,7,2,1,6] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,5,4,7,2,6,1] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,5,4,7,6,1,2] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,5,4,7,6,2,1] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,5,6,1,7,2,4] => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[3,5,6,1,7,4,2] => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[3,5,6,2,7,1,4] => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[3,5,6,2,7,4,1] => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[3,5,6,4,7,1,2] => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[3,5,6,4,7,2,1] => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[4,1,5,7,2,3,6] => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[4,1,5,7,2,6,3] => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[4,1,5,7,3,2,6] => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[4,1,5,7,3,6,2] => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[4,1,5,7,6,2,3] => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[4,1,5,7,6,3,2] => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[4,1,6,2,7,3,5] => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[4,1,6,2,7,5,3] => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[4,1,6,3,7,2,5] => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[4,1,6,3,7,5,2] => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[4,1,6,5,7,2,3] => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[4,1,6,5,7,3,2] => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
Description
The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path.
The statistic returns zero in case that bimodule is the zero module.
Matching statistic: St001695
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001695: Standard tableaux ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 50%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001695: Standard tableaux ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 50%
Values
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> 0 = 1 - 1
[1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13],[14]]
=> ? = 1 - 1
[1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12],[13]]
=> ? = 1 - 1
[1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12],[13]]
=> ? = 2 - 1
[1,3,2,5,4,6] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12],[13]]
=> ? = 2 - 1
[1,3,2,5,6,4] => [1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11],[12]]
=> ? = 1 - 1
[1,3,2,6,4,5] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10],[11]]
=> ? = 2 - 1
[1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10],[11]]
=> ? = 2 - 1
[1,3,4,2,6,5] => [1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11],[12]]
=> ? = 2 - 1
[1,3,5,2,4,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10],[11]]
=> ? = 1 - 1
[1,3,5,2,6,4] => [1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10]]
=> ? = 1 - 1
[1,3,5,4,2,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10],[11]]
=> ? = 1 - 1
[1,3,5,4,6,2] => [1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10]]
=> ? = 1 - 1
[1,4,2,3,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11]]
=> ? = 1 - 1
[1,4,2,5,3,6] => [1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10],[11]]
=> ? = 1 - 1
[1,4,2,6,3,5] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> ? = 1 - 1
[1,4,2,6,5,3] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> ? = 1 - 1
[1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11]]
=> ? = 1 - 1
[1,4,3,5,2,6] => [1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10],[11]]
=> ? = 1 - 1
[1,4,3,6,2,5] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> ? = 1 - 1
[1,4,3,6,5,2] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> ? = 1 - 1
[2,1,3,5,4,6] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13]]
=> ? = 2 - 1
[2,1,4,3,5,6] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13]]
=> ? = 1 - 1
[2,1,4,3,6,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12]]
=> ? = 2 - 1
[2,1,4,5,3,6] => [1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
=> ? = 2 - 1
[2,1,4,6,3,5] => [1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10]]
=> ? = 2 - 1
[2,1,4,6,5,3] => [1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10]]
=> ? = 2 - 1
[2,1,5,3,4,6] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> [[1,2,3,4,5],[6,7],[8,9],[10,11]]
=> ? = 1 - 1
[2,1,5,3,6,4] => [1,1,0,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10]]
=> ? = 1 - 1
[2,1,5,4,3,6] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> [[1,2,3,4,5],[6,7],[8,9],[10,11]]
=> ? = 1 - 1
[2,1,5,4,6,3] => [1,1,0,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10]]
=> ? = 1 - 1
[2,3,1,5,4,6] => [1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12]]
=> ? = 1 - 1
[2,3,1,5,6,4] => [1,1,0,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11]]
=> ? = 1 - 1
[2,4,1,3,6,5] => [1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10]]
=> ? = 1 - 1
[2,4,1,5,3,6] => [1,1,0,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10]]
=> ? = 1 - 1
[2,4,1,6,3,5] => [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[2,4,1,6,5,3] => [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[2,4,3,1,6,5] => [1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10]]
=> ? = 1 - 1
[2,4,3,5,1,6] => [1,1,0,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10]]
=> ? = 1 - 1
[2,4,3,6,1,5] => [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[2,4,3,6,5,1] => [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,1,2,5,4,6] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> [[1,2,3,4,5],[6,7,8],[9,10,11]]
=> ? = 2 - 1
[3,1,2,6,4,5] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> ? = 1 - 1
[3,1,2,6,5,4] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> ? = 1 - 1
[3,1,4,2,6,5] => [1,1,1,0,0,1,0,0,1,1,0,0]
=> [4,4,2]
=> [[1,2,3,4],[5,6,7,8],[9,10]]
=> ? = 2 - 1
[3,1,4,6,2,5] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> 0 = 1 - 1
[3,1,4,6,5,2] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> 0 = 1 - 1
[3,1,5,2,4,6] => [1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,2,2]
=> [[1,2,3,4,5],[6,7],[8,9]]
=> ? = 1 - 1
[3,1,5,2,6,4] => [1,1,1,0,0,1,1,0,0,1,0,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> 0 = 1 - 1
[3,1,5,4,2,6] => [1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,2,2]
=> [[1,2,3,4,5],[6,7],[8,9]]
=> ? = 1 - 1
[3,1,5,4,6,2] => [1,1,1,0,0,1,1,0,0,1,0,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> 0 = 1 - 1
[3,2,1,5,4,6] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> [[1,2,3,4,5],[6,7,8],[9,10,11]]
=> ? = 2 - 1
[3,2,1,6,4,5] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> ? = 1 - 1
[3,2,1,6,5,4] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> ? = 1 - 1
[3,2,4,1,6,5] => [1,1,1,0,0,1,0,0,1,1,0,0]
=> [4,4,2]
=> [[1,2,3,4],[5,6,7,8],[9,10]]
=> ? = 2 - 1
[3,2,4,6,1,5] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> 0 = 1 - 1
[3,2,4,6,5,1] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> 0 = 1 - 1
[3,2,5,1,4,6] => [1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,2,2]
=> [[1,2,3,4,5],[6,7],[8,9]]
=> ? = 1 - 1
[3,2,5,1,6,4] => [1,1,1,0,0,1,1,0,0,1,0,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> 0 = 1 - 1
[3,2,5,4,1,6] => [1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,2,2]
=> [[1,2,3,4,5],[6,7],[8,9]]
=> ? = 1 - 1
[3,2,5,4,6,1] => [1,1,1,0,0,1,1,0,0,1,0,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> 0 = 1 - 1
[1,2,3,5,4,6,7] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2,1]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13,14],[15,16,17],[18,19],[20]]
=> ? = 1 - 1
[1,2,3,5,4,7,6] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13],[14,15,16],[17,18],[19]]
=> ? = 1 - 1
[3,5,1,7,2,4,6] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,1,7,2,6,4] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,1,7,4,2,6] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,1,7,4,6,2] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,1,7,6,2,4] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,1,7,6,4,2] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,2,7,1,4,6] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,2,7,1,6,4] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,2,7,4,1,6] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,2,7,4,6,1] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,2,7,6,1,4] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,2,7,6,4,1] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,4,7,1,2,6] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,4,7,1,6,2] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,4,7,2,1,6] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,4,7,2,6,1] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,4,7,6,1,2] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,4,7,6,2,1] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,6,1,7,2,4] => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,6,1,7,4,2] => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,6,2,7,1,4] => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,6,2,7,4,1] => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,6,4,7,1,2] => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,6,4,7,2,1] => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> 0 = 1 - 1
[4,1,5,7,2,3,6] => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> 0 = 1 - 1
[4,1,5,7,2,6,3] => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> 0 = 1 - 1
[4,1,5,7,3,2,6] => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> 0 = 1 - 1
[4,1,5,7,3,6,2] => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> 0 = 1 - 1
[4,1,5,7,6,2,3] => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> 0 = 1 - 1
[4,1,5,7,6,3,2] => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> 0 = 1 - 1
[4,1,6,2,7,3,5] => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> 0 = 1 - 1
[4,1,6,2,7,5,3] => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> 0 = 1 - 1
[4,1,6,3,7,2,5] => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> 0 = 1 - 1
[4,1,6,3,7,5,2] => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> 0 = 1 - 1
[4,1,6,5,7,2,3] => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> 0 = 1 - 1
[4,1,6,5,7,3,2] => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> 0 = 1 - 1
Description
The natural comajor index of a standard Young tableau.
A natural descent of a standard tableau $T$ is an entry $i$ such that $i+1$ appears in a higher row than $i$ in English notation.
The natural comajor index of a tableau of size $n$ with natural descent set $D$ is then $\sum_{d\in D} n-d$.
Matching statistic: St001698
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001698: Standard tableaux ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 50%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001698: Standard tableaux ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 50%
Values
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> 0 = 1 - 1
[1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13],[14]]
=> ? = 1 - 1
[1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12],[13]]
=> ? = 1 - 1
[1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12],[13]]
=> ? = 2 - 1
[1,3,2,5,4,6] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12],[13]]
=> ? = 2 - 1
[1,3,2,5,6,4] => [1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11],[12]]
=> ? = 1 - 1
[1,3,2,6,4,5] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10],[11]]
=> ? = 2 - 1
[1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10],[11]]
=> ? = 2 - 1
[1,3,4,2,6,5] => [1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11],[12]]
=> ? = 2 - 1
[1,3,5,2,4,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10],[11]]
=> ? = 1 - 1
[1,3,5,2,6,4] => [1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10]]
=> ? = 1 - 1
[1,3,5,4,2,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10],[11]]
=> ? = 1 - 1
[1,3,5,4,6,2] => [1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10]]
=> ? = 1 - 1
[1,4,2,3,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11]]
=> ? = 1 - 1
[1,4,2,5,3,6] => [1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10],[11]]
=> ? = 1 - 1
[1,4,2,6,3,5] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> ? = 1 - 1
[1,4,2,6,5,3] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> ? = 1 - 1
[1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11]]
=> ? = 1 - 1
[1,4,3,5,2,6] => [1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10],[11]]
=> ? = 1 - 1
[1,4,3,6,2,5] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> ? = 1 - 1
[1,4,3,6,5,2] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> ? = 1 - 1
[2,1,3,5,4,6] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13]]
=> ? = 2 - 1
[2,1,4,3,5,6] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13]]
=> ? = 1 - 1
[2,1,4,3,6,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12]]
=> ? = 2 - 1
[2,1,4,5,3,6] => [1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
=> ? = 2 - 1
[2,1,4,6,3,5] => [1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10]]
=> ? = 2 - 1
[2,1,4,6,5,3] => [1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10]]
=> ? = 2 - 1
[2,1,5,3,4,6] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> [[1,2,3,4,5],[6,7],[8,9],[10,11]]
=> ? = 1 - 1
[2,1,5,3,6,4] => [1,1,0,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10]]
=> ? = 1 - 1
[2,1,5,4,3,6] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> [[1,2,3,4,5],[6,7],[8,9],[10,11]]
=> ? = 1 - 1
[2,1,5,4,6,3] => [1,1,0,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10]]
=> ? = 1 - 1
[2,3,1,5,4,6] => [1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12]]
=> ? = 1 - 1
[2,3,1,5,6,4] => [1,1,0,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11]]
=> ? = 1 - 1
[2,4,1,3,6,5] => [1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10]]
=> ? = 1 - 1
[2,4,1,5,3,6] => [1,1,0,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10]]
=> ? = 1 - 1
[2,4,1,6,3,5] => [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[2,4,1,6,5,3] => [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[2,4,3,1,6,5] => [1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10]]
=> ? = 1 - 1
[2,4,3,5,1,6] => [1,1,0,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10]]
=> ? = 1 - 1
[2,4,3,6,1,5] => [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[2,4,3,6,5,1] => [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,1,2,5,4,6] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> [[1,2,3,4,5],[6,7,8],[9,10,11]]
=> ? = 2 - 1
[3,1,2,6,4,5] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> ? = 1 - 1
[3,1,2,6,5,4] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> ? = 1 - 1
[3,1,4,2,6,5] => [1,1,1,0,0,1,0,0,1,1,0,0]
=> [4,4,2]
=> [[1,2,3,4],[5,6,7,8],[9,10]]
=> ? = 2 - 1
[3,1,4,6,2,5] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> 0 = 1 - 1
[3,1,4,6,5,2] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> 0 = 1 - 1
[3,1,5,2,4,6] => [1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,2,2]
=> [[1,2,3,4,5],[6,7],[8,9]]
=> ? = 1 - 1
[3,1,5,2,6,4] => [1,1,1,0,0,1,1,0,0,1,0,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> 0 = 1 - 1
[3,1,5,4,2,6] => [1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,2,2]
=> [[1,2,3,4,5],[6,7],[8,9]]
=> ? = 1 - 1
[3,1,5,4,6,2] => [1,1,1,0,0,1,1,0,0,1,0,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> 0 = 1 - 1
[3,2,1,5,4,6] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> [[1,2,3,4,5],[6,7,8],[9,10,11]]
=> ? = 2 - 1
[3,2,1,6,4,5] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> ? = 1 - 1
[3,2,1,6,5,4] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> ? = 1 - 1
[3,2,4,1,6,5] => [1,1,1,0,0,1,0,0,1,1,0,0]
=> [4,4,2]
=> [[1,2,3,4],[5,6,7,8],[9,10]]
=> ? = 2 - 1
[3,2,4,6,1,5] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> 0 = 1 - 1
[3,2,4,6,5,1] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> 0 = 1 - 1
[3,2,5,1,4,6] => [1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,2,2]
=> [[1,2,3,4,5],[6,7],[8,9]]
=> ? = 1 - 1
[3,2,5,1,6,4] => [1,1,1,0,0,1,1,0,0,1,0,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> 0 = 1 - 1
[3,2,5,4,1,6] => [1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,2,2]
=> [[1,2,3,4,5],[6,7],[8,9]]
=> ? = 1 - 1
[3,2,5,4,6,1] => [1,1,1,0,0,1,1,0,0,1,0,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> 0 = 1 - 1
[1,2,3,5,4,6,7] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2,1]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13,14],[15,16,17],[18,19],[20]]
=> ? = 1 - 1
[1,2,3,5,4,7,6] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13],[14,15,16],[17,18],[19]]
=> ? = 1 - 1
[3,5,1,7,2,4,6] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,1,7,2,6,4] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,1,7,4,2,6] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,1,7,4,6,2] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,1,7,6,2,4] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,1,7,6,4,2] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,2,7,1,4,6] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,2,7,1,6,4] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,2,7,4,1,6] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,2,7,4,6,1] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,2,7,6,1,4] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,2,7,6,4,1] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,4,7,1,2,6] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,4,7,1,6,2] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,4,7,2,1,6] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,4,7,2,6,1] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,4,7,6,1,2] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,4,7,6,2,1] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,6,1,7,2,4] => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,6,1,7,4,2] => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,6,2,7,1,4] => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,6,2,7,4,1] => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,6,4,7,1,2] => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,6,4,7,2,1] => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> 0 = 1 - 1
[4,1,5,7,2,3,6] => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> 0 = 1 - 1
[4,1,5,7,2,6,3] => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> 0 = 1 - 1
[4,1,5,7,3,2,6] => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> 0 = 1 - 1
[4,1,5,7,3,6,2] => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> 0 = 1 - 1
[4,1,5,7,6,2,3] => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> 0 = 1 - 1
[4,1,5,7,6,3,2] => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> 0 = 1 - 1
[4,1,6,2,7,3,5] => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> 0 = 1 - 1
[4,1,6,2,7,5,3] => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> 0 = 1 - 1
[4,1,6,3,7,2,5] => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> 0 = 1 - 1
[4,1,6,3,7,5,2] => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> 0 = 1 - 1
[4,1,6,5,7,2,3] => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> 0 = 1 - 1
[4,1,6,5,7,3,2] => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> 0 = 1 - 1
Description
The comajor index of a standard tableau minus the weighted size of its shape.
Matching statistic: St001699
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St001699: Standard tableaux ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 50%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St001699: Standard tableaux ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 50%
Values
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 0 = 1 - 1
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> 0 = 1 - 1
[1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [[1,3,8,9,14],[2,5,12,13],[4,7],[6,11],[10]]
=> ? = 1 - 1
[1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [[1,3,8,9],[2,5,12,13],[4,7],[6,11],[10]]
=> ? = 1 - 1
[1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [[1,4,5,9],[2,7,8,13],[3,11,12],[6],[10]]
=> ? = 2 - 1
[1,3,2,5,4,6] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [[1,4,5,12,13],[2,7,8],[3,10,11],[6],[9]]
=> ? = 2 - 1
[1,3,2,5,6,4] => [1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [[1,4,5,12],[2,7,8],[3,10,11],[6],[9]]
=> ? = 1 - 1
[1,3,2,6,4,5] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [[1,4,5],[2,7,8],[3,10,11],[6],[9]]
=> ? = 2 - 1
[1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [[1,4,5],[2,7,8],[3,10,11],[6],[9]]
=> ? = 2 - 1
[1,3,4,2,6,5] => [1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [[1,4,7,8],[2,6,11,12],[3,10],[5],[9]]
=> ? = 2 - 1
[1,3,5,2,4,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [[1,4,9,10,11],[2,6],[3,8],[5],[7]]
=> ? = 1 - 1
[1,3,5,2,6,4] => [1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [[1,4,9,10],[2,6],[3,8],[5],[7]]
=> ? = 1 - 1
[1,3,5,4,2,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [[1,4,9,10,11],[2,6],[3,8],[5],[7]]
=> ? = 1 - 1
[1,3,5,4,6,2] => [1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [[1,4,9,10],[2,6],[3,8],[5],[7]]
=> ? = 1 - 1
[1,4,2,3,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [[1,5,6,7],[2,9,10,11],[3],[4],[8]]
=> ? = 1 - 1
[1,4,2,5,3,6] => [1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [[1,5,6,10,11],[2,8,9],[3],[4],[7]]
=> ? = 1 - 1
[1,4,2,6,3,5] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3],[4],[7]]
=> ? = 1 - 1
[1,4,2,6,5,3] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3],[4],[7]]
=> ? = 1 - 1
[1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [[1,5,6,7],[2,9,10,11],[3],[4],[8]]
=> ? = 1 - 1
[1,4,3,5,2,6] => [1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [[1,5,6,10,11],[2,8,9],[3],[4],[7]]
=> ? = 1 - 1
[1,4,3,6,2,5] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3],[4],[7]]
=> ? = 1 - 1
[1,4,3,6,5,2] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3],[4],[7]]
=> ? = 1 - 1
[2,1,3,5,4,6] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [[1,2,5,12,13],[3,4,8],[6,7,11],[9,10]]
=> ? = 2 - 1
[2,1,4,3,5,6] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> [[1,2,7,8,13],[3,4,11,12],[5,6],[9,10]]
=> ? = 1 - 1
[2,1,4,3,6,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
=> ? = 2 - 1
[2,1,4,5,3,6] => [1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> [[1,2,7,11,12],[3,4,10],[5,6],[8,9]]
=> ? = 2 - 1
[2,1,4,6,3,5] => [1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [[1,2,7],[3,4,10],[5,6],[8,9]]
=> ? = 2 - 1
[2,1,4,6,5,3] => [1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [[1,2,7],[3,4,10],[5,6],[8,9]]
=> ? = 2 - 1
[2,1,5,3,4,6] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> [[1,2,9,10,11],[3,4],[5,6],[7,8]]
=> ? = 1 - 1
[2,1,5,3,6,4] => [1,1,0,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2]
=> [[1,2,9,10],[3,4],[5,6],[7,8]]
=> ? = 1 - 1
[2,1,5,4,3,6] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> [[1,2,9,10,11],[3,4],[5,6],[7,8]]
=> ? = 1 - 1
[2,1,5,4,6,3] => [1,1,0,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2]
=> [[1,2,9,10],[3,4],[5,6],[7,8]]
=> ? = 1 - 1
[2,3,1,5,4,6] => [1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1]
=> [[1,3,4,11,12],[2,6,7],[5,9,10],[8]]
=> ? = 1 - 1
[2,3,1,5,6,4] => [1,1,0,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1]
=> [[1,3,4,11],[2,6,7],[5,9,10],[8]]
=> ? = 1 - 1
[2,4,1,3,6,5] => [1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1]
=> [[1,4,5,6],[2,8,9,10],[3],[7]]
=> ? = 1 - 1
[2,4,1,5,3,6] => [1,1,0,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1]
=> [[1,4,5,9,10],[2,7,8],[3],[6]]
=> ? = 1 - 1
[2,4,1,6,3,5] => [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 0 = 1 - 1
[2,4,1,6,5,3] => [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 0 = 1 - 1
[2,4,3,1,6,5] => [1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1]
=> [[1,4,5,6],[2,8,9,10],[3],[7]]
=> ? = 1 - 1
[2,4,3,5,1,6] => [1,1,0,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1]
=> [[1,4,5,9,10],[2,7,8],[3],[6]]
=> ? = 1 - 1
[2,4,3,6,1,5] => [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 0 = 1 - 1
[2,4,3,6,5,1] => [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 0 = 1 - 1
[3,1,2,5,4,6] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> [[1,2,3,10,11],[4,5,6],[7,8,9]]
=> ? = 2 - 1
[3,1,2,6,4,5] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> ? = 1 - 1
[3,1,2,6,5,4] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> ? = 1 - 1
[3,1,4,2,6,5] => [1,1,1,0,0,1,0,0,1,1,0,0]
=> [4,4,2]
=> [[1,2,5,6],[3,4,9,10],[7,8]]
=> ? = 2 - 1
[3,1,4,6,2,5] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> 0 = 1 - 1
[3,1,4,6,5,2] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> 0 = 1 - 1
[3,1,5,2,4,6] => [1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> ? = 1 - 1
[3,1,5,2,6,4] => [1,1,1,0,0,1,1,0,0,1,0,0]
=> [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> 0 = 1 - 1
[3,1,5,4,2,6] => [1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> ? = 1 - 1
[3,1,5,4,6,2] => [1,1,1,0,0,1,1,0,0,1,0,0]
=> [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> 0 = 1 - 1
[3,2,1,5,4,6] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> [[1,2,3,10,11],[4,5,6],[7,8,9]]
=> ? = 2 - 1
[3,2,1,6,4,5] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> ? = 1 - 1
[3,2,1,6,5,4] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> ? = 1 - 1
[3,2,4,1,6,5] => [1,1,1,0,0,1,0,0,1,1,0,0]
=> [4,4,2]
=> [[1,2,5,6],[3,4,9,10],[7,8]]
=> ? = 2 - 1
[3,2,4,6,1,5] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> 0 = 1 - 1
[3,2,4,6,5,1] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> 0 = 1 - 1
[3,2,5,1,4,6] => [1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> ? = 1 - 1
[3,2,5,1,6,4] => [1,1,1,0,0,1,1,0,0,1,0,0]
=> [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> 0 = 1 - 1
[3,2,5,4,1,6] => [1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> ? = 1 - 1
[3,2,5,4,6,1] => [1,1,1,0,0,1,1,0,0,1,0,0]
=> [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> 0 = 1 - 1
[1,2,3,5,4,6,7] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2,1]
=> [[1,3,6,13,14,20],[2,5,9,18,19],[4,8,12],[7,11,17],[10,16],[15]]
=> ? = 1 - 1
[1,2,3,5,4,7,6] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2,1]
=> [[1,3,6,13,14],[2,5,9,18,19],[4,8,12],[7,11,17],[10,16],[15]]
=> ? = 1 - 1
[3,5,1,7,2,4,6] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 0 = 1 - 1
[3,5,1,7,2,6,4] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 0 = 1 - 1
[3,5,1,7,4,2,6] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 0 = 1 - 1
[3,5,1,7,4,6,2] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 0 = 1 - 1
[3,5,1,7,6,2,4] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 0 = 1 - 1
[3,5,1,7,6,4,2] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 0 = 1 - 1
[3,5,2,7,1,4,6] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 0 = 1 - 1
[3,5,2,7,1,6,4] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 0 = 1 - 1
[3,5,2,7,4,1,6] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 0 = 1 - 1
[3,5,2,7,4,6,1] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 0 = 1 - 1
[3,5,2,7,6,1,4] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 0 = 1 - 1
[3,5,2,7,6,4,1] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 0 = 1 - 1
[3,5,4,7,1,2,6] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 0 = 1 - 1
[3,5,4,7,1,6,2] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 0 = 1 - 1
[3,5,4,7,2,1,6] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 0 = 1 - 1
[3,5,4,7,2,6,1] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 0 = 1 - 1
[3,5,4,7,6,1,2] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 0 = 1 - 1
[3,5,4,7,6,2,1] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 0 = 1 - 1
[3,5,6,1,7,2,4] => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> 0 = 1 - 1
[3,5,6,1,7,4,2] => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> 0 = 1 - 1
[3,5,6,2,7,1,4] => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> 0 = 1 - 1
[3,5,6,2,7,4,1] => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> 0 = 1 - 1
[3,5,6,4,7,1,2] => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> 0 = 1 - 1
[3,5,6,4,7,2,1] => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> 0 = 1 - 1
[4,1,5,7,2,3,6] => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> 0 = 1 - 1
[4,1,5,7,2,6,3] => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> 0 = 1 - 1
[4,1,5,7,3,2,6] => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> 0 = 1 - 1
[4,1,5,7,3,6,2] => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> 0 = 1 - 1
[4,1,5,7,6,2,3] => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> 0 = 1 - 1
[4,1,5,7,6,3,2] => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> 0 = 1 - 1
[4,1,6,2,7,3,5] => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> 0 = 1 - 1
[4,1,6,2,7,5,3] => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> 0 = 1 - 1
[4,1,6,3,7,2,5] => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> 0 = 1 - 1
[4,1,6,3,7,5,2] => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> 0 = 1 - 1
[4,1,6,5,7,2,3] => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> 0 = 1 - 1
[4,1,6,5,7,3,2] => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> 0 = 1 - 1
Description
The major index of a standard tableau minus the weighted size of its shape.
Matching statistic: St001712
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001712: Standard tableaux ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 50%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001712: Standard tableaux ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 50%
Values
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> 0 = 1 - 1
[1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13],[14]]
=> ? = 1 - 1
[1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12],[13]]
=> ? = 1 - 1
[1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12],[13]]
=> ? = 2 - 1
[1,3,2,5,4,6] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12],[13]]
=> ? = 2 - 1
[1,3,2,5,6,4] => [1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11],[12]]
=> ? = 1 - 1
[1,3,2,6,4,5] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10],[11]]
=> ? = 2 - 1
[1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10],[11]]
=> ? = 2 - 1
[1,3,4,2,6,5] => [1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11],[12]]
=> ? = 2 - 1
[1,3,5,2,4,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10],[11]]
=> ? = 1 - 1
[1,3,5,2,6,4] => [1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10]]
=> ? = 1 - 1
[1,3,5,4,2,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10],[11]]
=> ? = 1 - 1
[1,3,5,4,6,2] => [1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10]]
=> ? = 1 - 1
[1,4,2,3,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11]]
=> ? = 1 - 1
[1,4,2,5,3,6] => [1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10],[11]]
=> ? = 1 - 1
[1,4,2,6,3,5] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> ? = 1 - 1
[1,4,2,6,5,3] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> ? = 1 - 1
[1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11]]
=> ? = 1 - 1
[1,4,3,5,2,6] => [1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10],[11]]
=> ? = 1 - 1
[1,4,3,6,2,5] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> ? = 1 - 1
[1,4,3,6,5,2] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> ? = 1 - 1
[2,1,3,5,4,6] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13]]
=> ? = 2 - 1
[2,1,4,3,5,6] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13]]
=> ? = 1 - 1
[2,1,4,3,6,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12]]
=> ? = 2 - 1
[2,1,4,5,3,6] => [1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
=> ? = 2 - 1
[2,1,4,6,3,5] => [1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10]]
=> ? = 2 - 1
[2,1,4,6,5,3] => [1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10]]
=> ? = 2 - 1
[2,1,5,3,4,6] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> [[1,2,3,4,5],[6,7],[8,9],[10,11]]
=> ? = 1 - 1
[2,1,5,3,6,4] => [1,1,0,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10]]
=> ? = 1 - 1
[2,1,5,4,3,6] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> [[1,2,3,4,5],[6,7],[8,9],[10,11]]
=> ? = 1 - 1
[2,1,5,4,6,3] => [1,1,0,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10]]
=> ? = 1 - 1
[2,3,1,5,4,6] => [1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12]]
=> ? = 1 - 1
[2,3,1,5,6,4] => [1,1,0,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11]]
=> ? = 1 - 1
[2,4,1,3,6,5] => [1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10]]
=> ? = 1 - 1
[2,4,1,5,3,6] => [1,1,0,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10]]
=> ? = 1 - 1
[2,4,1,6,3,5] => [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[2,4,1,6,5,3] => [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[2,4,3,1,6,5] => [1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10]]
=> ? = 1 - 1
[2,4,3,5,1,6] => [1,1,0,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10]]
=> ? = 1 - 1
[2,4,3,6,1,5] => [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[2,4,3,6,5,1] => [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,1,2,5,4,6] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> [[1,2,3,4,5],[6,7,8],[9,10,11]]
=> ? = 2 - 1
[3,1,2,6,4,5] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> ? = 1 - 1
[3,1,2,6,5,4] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> ? = 1 - 1
[3,1,4,2,6,5] => [1,1,1,0,0,1,0,0,1,1,0,0]
=> [4,4,2]
=> [[1,2,3,4],[5,6,7,8],[9,10]]
=> ? = 2 - 1
[3,1,4,6,2,5] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> 0 = 1 - 1
[3,1,4,6,5,2] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> 0 = 1 - 1
[3,1,5,2,4,6] => [1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,2,2]
=> [[1,2,3,4,5],[6,7],[8,9]]
=> ? = 1 - 1
[3,1,5,2,6,4] => [1,1,1,0,0,1,1,0,0,1,0,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> 0 = 1 - 1
[3,1,5,4,2,6] => [1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,2,2]
=> [[1,2,3,4,5],[6,7],[8,9]]
=> ? = 1 - 1
[3,1,5,4,6,2] => [1,1,1,0,0,1,1,0,0,1,0,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> 0 = 1 - 1
[3,2,1,5,4,6] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> [[1,2,3,4,5],[6,7,8],[9,10,11]]
=> ? = 2 - 1
[3,2,1,6,4,5] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> ? = 1 - 1
[3,2,1,6,5,4] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> ? = 1 - 1
[3,2,4,1,6,5] => [1,1,1,0,0,1,0,0,1,1,0,0]
=> [4,4,2]
=> [[1,2,3,4],[5,6,7,8],[9,10]]
=> ? = 2 - 1
[3,2,4,6,1,5] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> 0 = 1 - 1
[3,2,4,6,5,1] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> 0 = 1 - 1
[3,2,5,1,4,6] => [1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,2,2]
=> [[1,2,3,4,5],[6,7],[8,9]]
=> ? = 1 - 1
[3,2,5,1,6,4] => [1,1,1,0,0,1,1,0,0,1,0,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> 0 = 1 - 1
[3,2,5,4,1,6] => [1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,2,2]
=> [[1,2,3,4,5],[6,7],[8,9]]
=> ? = 1 - 1
[3,2,5,4,6,1] => [1,1,1,0,0,1,1,0,0,1,0,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> 0 = 1 - 1
[1,2,3,5,4,6,7] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2,1]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13,14],[15,16,17],[18,19],[20]]
=> ? = 1 - 1
[1,2,3,5,4,7,6] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13],[14,15,16],[17,18],[19]]
=> ? = 1 - 1
[3,5,1,7,2,4,6] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,1,7,2,6,4] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,1,7,4,2,6] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,1,7,4,6,2] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,1,7,6,2,4] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,1,7,6,4,2] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,2,7,1,4,6] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,2,7,1,6,4] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,2,7,4,1,6] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,2,7,4,6,1] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,2,7,6,1,4] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,2,7,6,4,1] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,4,7,1,2,6] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,4,7,1,6,2] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,4,7,2,1,6] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,4,7,2,6,1] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,4,7,6,1,2] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,4,7,6,2,1] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,6,1,7,2,4] => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,6,1,7,4,2] => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,6,2,7,1,4] => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,6,2,7,4,1] => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,6,4,7,1,2] => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> 0 = 1 - 1
[3,5,6,4,7,2,1] => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> 0 = 1 - 1
[4,1,5,7,2,3,6] => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> 0 = 1 - 1
[4,1,5,7,2,6,3] => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> 0 = 1 - 1
[4,1,5,7,3,2,6] => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> 0 = 1 - 1
[4,1,5,7,3,6,2] => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> 0 = 1 - 1
[4,1,5,7,6,2,3] => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> 0 = 1 - 1
[4,1,5,7,6,3,2] => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> 0 = 1 - 1
[4,1,6,2,7,3,5] => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> 0 = 1 - 1
[4,1,6,2,7,5,3] => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> 0 = 1 - 1
[4,1,6,3,7,2,5] => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> 0 = 1 - 1
[4,1,6,3,7,5,2] => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> 0 = 1 - 1
[4,1,6,5,7,2,3] => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> 0 = 1 - 1
[4,1,6,5,7,3,2] => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> 0 = 1 - 1
Description
The number of natural descents of a standard Young tableau.
A natural descent of a standard tableau $T$ is an entry $i$ such that $i+1$ appears in a higher row than $i$ in English notation.
Matching statistic: St000744
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000744: Standard tableaux ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 50%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000744: Standard tableaux ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 50%
Values
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 3 = 1 + 2
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> 3 = 1 + 2
[1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [[1,3,8,9,14],[2,5,12,13],[4,7],[6,11],[10]]
=> ? = 1 + 2
[1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [[1,3,8,9],[2,5,12,13],[4,7],[6,11],[10]]
=> ? = 1 + 2
[1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [[1,4,5,9],[2,7,8,13],[3,11,12],[6],[10]]
=> ? = 2 + 2
[1,3,2,5,4,6] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [[1,4,5,12,13],[2,7,8],[3,10,11],[6],[9]]
=> ? = 2 + 2
[1,3,2,5,6,4] => [1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [[1,4,5,12],[2,7,8],[3,10,11],[6],[9]]
=> ? = 1 + 2
[1,3,2,6,4,5] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [[1,4,5],[2,7,8],[3,10,11],[6],[9]]
=> ? = 2 + 2
[1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [[1,4,5],[2,7,8],[3,10,11],[6],[9]]
=> ? = 2 + 2
[1,3,4,2,6,5] => [1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [[1,4,7,8],[2,6,11,12],[3,10],[5],[9]]
=> ? = 2 + 2
[1,3,5,2,4,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [[1,4,9,10,11],[2,6],[3,8],[5],[7]]
=> ? = 1 + 2
[1,3,5,2,6,4] => [1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [[1,4,9,10],[2,6],[3,8],[5],[7]]
=> ? = 1 + 2
[1,3,5,4,2,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [[1,4,9,10,11],[2,6],[3,8],[5],[7]]
=> ? = 1 + 2
[1,3,5,4,6,2] => [1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [[1,4,9,10],[2,6],[3,8],[5],[7]]
=> ? = 1 + 2
[1,4,2,3,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [[1,5,6,7],[2,9,10,11],[3],[4],[8]]
=> ? = 1 + 2
[1,4,2,5,3,6] => [1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [[1,5,6,10,11],[2,8,9],[3],[4],[7]]
=> ? = 1 + 2
[1,4,2,6,3,5] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3],[4],[7]]
=> ? = 1 + 2
[1,4,2,6,5,3] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3],[4],[7]]
=> ? = 1 + 2
[1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [[1,5,6,7],[2,9,10,11],[3],[4],[8]]
=> ? = 1 + 2
[1,4,3,5,2,6] => [1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [[1,5,6,10,11],[2,8,9],[3],[4],[7]]
=> ? = 1 + 2
[1,4,3,6,2,5] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3],[4],[7]]
=> ? = 1 + 2
[1,4,3,6,5,2] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3],[4],[7]]
=> ? = 1 + 2
[2,1,3,5,4,6] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [[1,2,5,12,13],[3,4,8],[6,7,11],[9,10]]
=> ? = 2 + 2
[2,1,4,3,5,6] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> [[1,2,7,8,13],[3,4,11,12],[5,6],[9,10]]
=> ? = 1 + 2
[2,1,4,3,6,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
=> ? = 2 + 2
[2,1,4,5,3,6] => [1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> [[1,2,7,11,12],[3,4,10],[5,6],[8,9]]
=> ? = 2 + 2
[2,1,4,6,3,5] => [1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [[1,2,7],[3,4,10],[5,6],[8,9]]
=> ? = 2 + 2
[2,1,4,6,5,3] => [1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [[1,2,7],[3,4,10],[5,6],[8,9]]
=> ? = 2 + 2
[2,1,5,3,4,6] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> [[1,2,9,10,11],[3,4],[5,6],[7,8]]
=> ? = 1 + 2
[2,1,5,3,6,4] => [1,1,0,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2]
=> [[1,2,9,10],[3,4],[5,6],[7,8]]
=> ? = 1 + 2
[2,1,5,4,3,6] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> [[1,2,9,10,11],[3,4],[5,6],[7,8]]
=> ? = 1 + 2
[2,1,5,4,6,3] => [1,1,0,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2]
=> [[1,2,9,10],[3,4],[5,6],[7,8]]
=> ? = 1 + 2
[2,3,1,5,4,6] => [1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1]
=> [[1,3,4,11,12],[2,6,7],[5,9,10],[8]]
=> ? = 1 + 2
[2,3,1,5,6,4] => [1,1,0,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1]
=> [[1,3,4,11],[2,6,7],[5,9,10],[8]]
=> ? = 1 + 2
[2,4,1,3,6,5] => [1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1]
=> [[1,4,5,6],[2,8,9,10],[3],[7]]
=> ? = 1 + 2
[2,4,1,5,3,6] => [1,1,0,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1]
=> [[1,4,5,9,10],[2,7,8],[3],[6]]
=> ? = 1 + 2
[2,4,1,6,3,5] => [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 3 = 1 + 2
[2,4,1,6,5,3] => [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 3 = 1 + 2
[2,4,3,1,6,5] => [1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1]
=> [[1,4,5,6],[2,8,9,10],[3],[7]]
=> ? = 1 + 2
[2,4,3,5,1,6] => [1,1,0,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1]
=> [[1,4,5,9,10],[2,7,8],[3],[6]]
=> ? = 1 + 2
[2,4,3,6,1,5] => [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 3 = 1 + 2
[2,4,3,6,5,1] => [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 3 = 1 + 2
[3,1,2,5,4,6] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> [[1,2,3,10,11],[4,5,6],[7,8,9]]
=> ? = 2 + 2
[3,1,2,6,4,5] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> ? = 1 + 2
[3,1,2,6,5,4] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> ? = 1 + 2
[3,1,4,2,6,5] => [1,1,1,0,0,1,0,0,1,1,0,0]
=> [4,4,2]
=> [[1,2,5,6],[3,4,9,10],[7,8]]
=> ? = 2 + 2
[3,1,4,6,2,5] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> 3 = 1 + 2
[3,1,4,6,5,2] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> 3 = 1 + 2
[3,1,5,2,4,6] => [1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> ? = 1 + 2
[3,1,5,2,6,4] => [1,1,1,0,0,1,1,0,0,1,0,0]
=> [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> 3 = 1 + 2
[3,1,5,4,2,6] => [1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> ? = 1 + 2
[3,1,5,4,6,2] => [1,1,1,0,0,1,1,0,0,1,0,0]
=> [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> 3 = 1 + 2
[3,2,1,5,4,6] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> [[1,2,3,10,11],[4,5,6],[7,8,9]]
=> ? = 2 + 2
[3,2,1,6,4,5] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> ? = 1 + 2
[3,2,1,6,5,4] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> ? = 1 + 2
[3,2,4,1,6,5] => [1,1,1,0,0,1,0,0,1,1,0,0]
=> [4,4,2]
=> [[1,2,5,6],[3,4,9,10],[7,8]]
=> ? = 2 + 2
[3,2,4,6,1,5] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> 3 = 1 + 2
[3,2,4,6,5,1] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> 3 = 1 + 2
[3,2,5,1,4,6] => [1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> ? = 1 + 2
[3,2,5,1,6,4] => [1,1,1,0,0,1,1,0,0,1,0,0]
=> [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> 3 = 1 + 2
[3,2,5,4,1,6] => [1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> ? = 1 + 2
[3,2,5,4,6,1] => [1,1,1,0,0,1,1,0,0,1,0,0]
=> [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> 3 = 1 + 2
[1,2,3,5,4,6,7] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2,1]
=> [[1,3,6,13,14,20],[2,5,9,18,19],[4,8,12],[7,11,17],[10,16],[15]]
=> ? = 1 + 2
[1,2,3,5,4,7,6] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2,1]
=> [[1,3,6,13,14],[2,5,9,18,19],[4,8,12],[7,11,17],[10,16],[15]]
=> ? = 1 + 2
[3,5,1,7,2,4,6] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 3 = 1 + 2
[3,5,1,7,2,6,4] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 3 = 1 + 2
[3,5,1,7,4,2,6] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 3 = 1 + 2
[3,5,1,7,4,6,2] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 3 = 1 + 2
[3,5,1,7,6,2,4] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 3 = 1 + 2
[3,5,1,7,6,4,2] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 3 = 1 + 2
[3,5,2,7,1,4,6] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 3 = 1 + 2
[3,5,2,7,1,6,4] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 3 = 1 + 2
[3,5,2,7,4,1,6] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 3 = 1 + 2
[3,5,2,7,4,6,1] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 3 = 1 + 2
[3,5,2,7,6,1,4] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 3 = 1 + 2
[3,5,2,7,6,4,1] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 3 = 1 + 2
[3,5,4,7,1,2,6] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 3 = 1 + 2
[3,5,4,7,1,6,2] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 3 = 1 + 2
[3,5,4,7,2,1,6] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 3 = 1 + 2
[3,5,4,7,2,6,1] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 3 = 1 + 2
[3,5,4,7,6,1,2] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 3 = 1 + 2
[3,5,4,7,6,2,1] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 3 = 1 + 2
[3,5,6,1,7,2,4] => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> 3 = 1 + 2
[3,5,6,1,7,4,2] => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> 3 = 1 + 2
[3,5,6,2,7,1,4] => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> 3 = 1 + 2
[3,5,6,2,7,4,1] => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> 3 = 1 + 2
[3,5,6,4,7,1,2] => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> 3 = 1 + 2
[3,5,6,4,7,2,1] => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> 3 = 1 + 2
[4,1,5,7,2,3,6] => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> 3 = 1 + 2
[4,1,5,7,2,6,3] => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> 3 = 1 + 2
[4,1,5,7,3,2,6] => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> 3 = 1 + 2
[4,1,5,7,3,6,2] => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> 3 = 1 + 2
[4,1,5,7,6,2,3] => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> 3 = 1 + 2
[4,1,5,7,6,3,2] => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> 3 = 1 + 2
[4,1,6,2,7,3,5] => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> 3 = 1 + 2
[4,1,6,2,7,5,3] => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> 3 = 1 + 2
[4,1,6,3,7,2,5] => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> 3 = 1 + 2
[4,1,6,3,7,5,2] => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> 3 = 1 + 2
[4,1,6,5,7,2,3] => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> 3 = 1 + 2
[4,1,6,5,7,3,2] => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> 3 = 1 + 2
Description
The length of the path to the largest entry in a standard Young tableau.
Matching statistic: St001200
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 50%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 50%
Values
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 1 + 2
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 1 + 2
[1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 1 + 2
[1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 1 + 2
[1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 2 + 2
[1,3,2,5,4,6] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2 + 2
[1,3,2,5,6,4] => [1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 1 + 2
[1,3,2,6,4,5] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 2 + 2
[1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 2 + 2
[1,3,4,2,6,5] => [1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 2 + 2
[1,3,5,2,4,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 1 + 2
[1,3,5,2,6,4] => [1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> ? = 1 + 2
[1,3,5,4,2,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 1 + 2
[1,3,5,4,6,2] => [1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> ? = 1 + 2
[1,4,2,3,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 2
[1,4,2,5,3,6] => [1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 2
[1,4,2,6,3,5] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> ? = 1 + 2
[1,4,2,6,5,3] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> ? = 1 + 2
[1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 2
[1,4,3,5,2,6] => [1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 2
[1,4,3,6,2,5] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> ? = 1 + 2
[1,4,3,6,5,2] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> ? = 1 + 2
[2,1,3,5,4,6] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 2 + 2
[2,1,4,3,5,6] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 1 + 2
[2,1,4,3,6,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 2 + 2
[2,1,4,5,3,6] => [1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 2 + 2
[2,1,4,6,3,5] => [1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> ? = 2 + 2
[2,1,4,6,5,3] => [1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> ? = 2 + 2
[2,1,5,3,4,6] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 1 + 2
[2,1,5,3,6,4] => [1,1,0,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> ? = 1 + 2
[2,1,5,4,3,6] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 1 + 2
[2,1,5,4,6,3] => [1,1,0,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> ? = 1 + 2
[2,3,1,5,4,6] => [1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 1 + 2
[2,3,1,5,6,4] => [1,1,0,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 1 + 2
[2,4,1,3,6,5] => [1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 2
[2,4,1,5,3,6] => [1,1,0,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 2
[2,4,1,6,3,5] => [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 1 + 2
[2,4,1,6,5,3] => [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 1 + 2
[2,4,3,1,6,5] => [1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 2
[2,4,3,5,1,6] => [1,1,0,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 2
[2,4,3,6,1,5] => [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 1 + 2
[2,4,3,6,5,1] => [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 1 + 2
[3,1,2,5,4,6] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 2 + 2
[3,1,2,6,4,5] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 1 + 2
[3,1,2,6,5,4] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 1 + 2
[3,1,4,2,6,5] => [1,1,1,0,0,1,0,0,1,1,0,0]
=> [4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> ? = 2 + 2
[3,1,4,6,2,5] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 1 + 2
[3,1,4,6,5,2] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 1 + 2
[3,1,5,2,4,6] => [1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> ? = 1 + 2
[3,1,5,2,6,4] => [1,1,1,0,0,1,1,0,0,1,0,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 1 + 2
[3,1,5,4,2,6] => [1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> ? = 1 + 2
[3,1,5,4,6,2] => [1,1,1,0,0,1,1,0,0,1,0,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 1 + 2
[3,2,1,5,4,6] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 2 + 2
[3,2,1,6,4,5] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 1 + 2
[3,2,1,6,5,4] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 1 + 2
[3,2,4,1,6,5] => [1,1,1,0,0,1,0,0,1,1,0,0]
=> [4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> ? = 2 + 2
[3,2,4,6,1,5] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 1 + 2
[3,2,4,6,5,1] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 1 + 2
[3,2,5,1,4,6] => [1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> ? = 1 + 2
[3,2,5,1,6,4] => [1,1,1,0,0,1,1,0,0,1,0,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 1 + 2
[3,2,5,4,1,6] => [1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> ? = 1 + 2
[3,2,5,4,6,1] => [1,1,1,0,0,1,1,0,0,1,0,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 1 + 2
[1,2,3,5,4,6,7] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 1 + 2
[1,2,3,5,4,7,6] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 1 + 2
[3,5,1,7,2,4,6] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 1 + 2
[3,5,1,7,2,6,4] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 1 + 2
[3,5,1,7,4,2,6] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 1 + 2
[3,5,1,7,4,6,2] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 1 + 2
[3,5,1,7,6,2,4] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 1 + 2
[3,5,1,7,6,4,2] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 1 + 2
[3,5,2,7,1,4,6] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 1 + 2
[3,5,2,7,1,6,4] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 1 + 2
[3,5,2,7,4,1,6] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 1 + 2
[3,5,2,7,4,6,1] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 1 + 2
[3,5,2,7,6,1,4] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 1 + 2
[3,5,2,7,6,4,1] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 1 + 2
[3,5,4,7,1,2,6] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 1 + 2
[3,5,4,7,1,6,2] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 1 + 2
[3,5,4,7,2,1,6] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 1 + 2
[3,5,4,7,2,6,1] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 1 + 2
[3,5,4,7,6,1,2] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 1 + 2
[3,5,4,7,6,2,1] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 1 + 2
[3,5,6,1,7,2,4] => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3 = 1 + 2
[3,5,6,1,7,4,2] => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3 = 1 + 2
[3,5,6,2,7,1,4] => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3 = 1 + 2
[3,5,6,2,7,4,1] => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3 = 1 + 2
[3,5,6,4,7,1,2] => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3 = 1 + 2
[3,5,6,4,7,2,1] => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3 = 1 + 2
[4,1,5,7,2,3,6] => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 1 + 2
[4,1,5,7,2,6,3] => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 1 + 2
[4,1,5,7,3,2,6] => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 1 + 2
[4,1,5,7,3,6,2] => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 1 + 2
[4,1,5,7,6,2,3] => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 1 + 2
[4,1,5,7,6,3,2] => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 1 + 2
[4,1,6,2,7,3,5] => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 1 + 2
[4,1,6,2,7,5,3] => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 1 + 2
[4,1,6,3,7,2,5] => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 1 + 2
[4,1,6,3,7,5,2] => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 1 + 2
[4,1,6,5,7,2,3] => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 1 + 2
[4,1,6,5,7,3,2] => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 1 + 2
Description
The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
The following 32 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000454The largest eigenvalue of a graph if it is integral. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000124The cardinality of the preimage of the Simion-Schmidt map. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001564The value of the forgotten symmetric functions when all variables set to 1. St001563The value of the power-sum symmetric function evaluated at 1. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St000422The energy of a graph, if it is integral. St000022The number of fixed points of a permutation. St000153The number of adjacent cycles of a permutation. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000359The number of occurrences of the pattern 23-1. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St000647The number of big descents of a permutation. St000662The staircase size of the code of a permutation. St001330The hat guessing number of a graph. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000929The constant term of the character polynomial of an integer partition. St000352The Elizalde-Pak rank of a permutation. St000546The number of global descents of a permutation. St000648The number of 2-excedences of a permutation. St000834The number of right outer peaks of a permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000007The number of saliances of the permutation.
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