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Matching statistic: St001767
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Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001767: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001767: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[2]]
=> [1,1]
=> [1]
=> 0
[[1],[3]]
=> [1,1]
=> [1]
=> 0
[[2],[3]]
=> [1,1]
=> [1]
=> 0
[[1,1],[2]]
=> [2,1]
=> [1]
=> 0
[[1,2],[2]]
=> [2,1]
=> [1]
=> 0
[[1],[4]]
=> [1,1]
=> [1]
=> 0
[[2],[4]]
=> [1,1]
=> [1]
=> 0
[[3],[4]]
=> [1,1]
=> [1]
=> 0
[[1,1],[3]]
=> [2,1]
=> [1]
=> 0
[[1,2],[3]]
=> [2,1]
=> [1]
=> 0
[[1,3],[2]]
=> [2,1]
=> [1]
=> 0
[[1,3],[3]]
=> [2,1]
=> [1]
=> 0
[[2,2],[3]]
=> [2,1]
=> [1]
=> 0
[[2,3],[3]]
=> [2,1]
=> [1]
=> 0
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,1],[2]]
=> [3,1]
=> [1]
=> 0
[[1,1,2],[2]]
=> [3,1]
=> [1]
=> 0
[[1,2,2],[2]]
=> [3,1]
=> [1]
=> 0
[[1,1],[2,2]]
=> [2,2]
=> [2]
=> 1
[[1],[5]]
=> [1,1]
=> [1]
=> 0
[[2],[5]]
=> [1,1]
=> [1]
=> 0
[[3],[5]]
=> [1,1]
=> [1]
=> 0
[[4],[5]]
=> [1,1]
=> [1]
=> 0
[[1,1],[4]]
=> [2,1]
=> [1]
=> 0
[[1,2],[4]]
=> [2,1]
=> [1]
=> 0
[[1,4],[2]]
=> [2,1]
=> [1]
=> 0
[[1,3],[4]]
=> [2,1]
=> [1]
=> 0
[[1,4],[3]]
=> [2,1]
=> [1]
=> 0
[[1,4],[4]]
=> [2,1]
=> [1]
=> 0
[[2,2],[4]]
=> [2,1]
=> [1]
=> 0
[[2,3],[4]]
=> [2,1]
=> [1]
=> 0
[[2,4],[3]]
=> [2,1]
=> [1]
=> 0
[[2,4],[4]]
=> [2,1]
=> [1]
=> 0
[[3,3],[4]]
=> [2,1]
=> [1]
=> 0
[[3,4],[4]]
=> [2,1]
=> [1]
=> 0
[[1],[2],[4]]
=> [1,1,1]
=> [1,1]
=> 1
[[1],[3],[4]]
=> [1,1,1]
=> [1,1]
=> 1
[[2],[3],[4]]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,1],[3]]
=> [3,1]
=> [1]
=> 0
[[1,1,2],[3]]
=> [3,1]
=> [1]
=> 0
[[1,1,3],[2]]
=> [3,1]
=> [1]
=> 0
[[1,1,3],[3]]
=> [3,1]
=> [1]
=> 0
[[1,2,2],[3]]
=> [3,1]
=> [1]
=> 0
[[1,2,3],[2]]
=> [3,1]
=> [1]
=> 0
[[1,2,3],[3]]
=> [3,1]
=> [1]
=> 0
[[1,3,3],[2]]
=> [3,1]
=> [1]
=> 0
[[1,3,3],[3]]
=> [3,1]
=> [1]
=> 0
[[2,2,2],[3]]
=> [3,1]
=> [1]
=> 0
[[2,2,3],[3]]
=> [3,1]
=> [1]
=> 0
[[2,3,3],[3]]
=> [3,1]
=> [1]
=> 0
Description
The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment.
Assign to each cell of the Ferrers diagram an arrow pointing north, east, south or west. Then compute for each cell the number of arrows pointing towards it, and take the minimum of those. This statistic is the maximal minimum that can be obtained by assigning arrows in any way.
Matching statistic: St001621
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001621: Lattices ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 33%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001621: Lattices ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 33%
Values
[[1],[2]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1],[3]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[2],[3]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,1],[2]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,2],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1],[4]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[2],[4]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[3],[4]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,1],[3]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,2],[3]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,3],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1,3],[3]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[2,2],[3]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[2,3],[3]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1],[2],[3]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[1,1,1],[2]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,1,2],[2]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 0 + 2
[[1,2,2],[2]]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 0 + 2
[[1,1],[2,2]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[2],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[3],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[4],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,1],[4]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,2],[4]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,4],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1,3],[4]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,4],[3]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1,4],[4]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[2,2],[4]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[2,3],[4]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[2,4],[3]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[2,4],[4]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[3,3],[4]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[3,4],[4]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1],[2],[4]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[1],[3],[4]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[2],[3],[4]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[1,1,1],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,1,2],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,1,3],[2]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 0 + 2
[[1,1,3],[3]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 0 + 2
[[1,2,2],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,2,3],[2]]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 0 + 2
[[1,2,3],[3]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 0 + 2
[[1,3,3],[2]]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 0 + 2
[[1,3,3],[3]]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 0 + 2
[[2,2,2],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[2,2,3],[3]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 0 + 2
[[2,3,3],[3]]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 0 + 2
[[1,1],[2,3]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,1],[3,3]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,2],[2,3]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 2
[[1,2],[3,3]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[2,2],[3,3]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,1],[2],[3]]
=> [4,3,1,2] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? = 1 + 2
[[1,2],[2],[3]]
=> [4,2,1,3] => ([(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> ? = 1 + 2
[[1,3],[2],[3]]
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> ? = 1 + 2
[[1,1,1,1],[2]]
=> [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 0 + 2
[[1,1,1,2],[2]]
=> [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 0 + 2
[[1,1,2,2],[2]]
=> [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 0 + 2
[[1,2,2,2],[2]]
=> [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 2 = 0 + 2
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 1 + 2
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 1 + 2
[[1],[6]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[2],[6]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[3],[6]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[4],[6]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[5],[6]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,1],[5]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,2],[5]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,5],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1,3],[5]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1],[2],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[1],[3],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[1],[4],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[2],[3],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[2],[4],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[3],[4],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[1,1,1],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,1,2],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,1,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,2,2],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,2,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,3,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[2,2,2],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[2,2,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[2,3,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[3,3,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,1],[2,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,1],[3,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,1],[4,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,2],[2,4]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 2
[[1,2],[3,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,3],[2,4]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 2
[[1,2],[4,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,3],[3,4]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 2
[[1,3],[4,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[2,2],[3,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[2,2],[4,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
Description
The number of atoms of a lattice.
An element of a lattice is an '''atom''' if it covers the least element.
Matching statistic: St001624
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001624: Lattices ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 33%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001624: Lattices ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 33%
Values
[[1],[2]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1],[3]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[2],[3]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,1],[2]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,2],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1],[4]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[2],[4]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[3],[4]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,1],[3]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,2],[3]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,3],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1,3],[3]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[2,2],[3]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[2,3],[3]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1],[2],[3]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[1,1,1],[2]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,1,2],[2]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 0 + 2
[[1,2,2],[2]]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 0 + 2
[[1,1],[2,2]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[2],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[3],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[4],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,1],[4]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,2],[4]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,4],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1,3],[4]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,4],[3]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1,4],[4]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[2,2],[4]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[2,3],[4]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[2,4],[3]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[2,4],[4]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[3,3],[4]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[3,4],[4]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1],[2],[4]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[1],[3],[4]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[2],[3],[4]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[1,1,1],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,1,2],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,1,3],[2]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 0 + 2
[[1,1,3],[3]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 0 + 2
[[1,2,2],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,2,3],[2]]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 0 + 2
[[1,2,3],[3]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 0 + 2
[[1,3,3],[2]]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 0 + 2
[[1,3,3],[3]]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 0 + 2
[[2,2,2],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[2,2,3],[3]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 0 + 2
[[2,3,3],[3]]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 0 + 2
[[1,1],[2,3]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,1],[3,3]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,2],[2,3]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 2
[[1,2],[3,3]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[2,2],[3,3]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,1],[2],[3]]
=> [4,3,1,2] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? = 1 + 2
[[1,2],[2],[3]]
=> [4,2,1,3] => ([(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> ? = 1 + 2
[[1,3],[2],[3]]
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> ? = 1 + 2
[[1,1,1,1],[2]]
=> [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 0 + 2
[[1,1,1,2],[2]]
=> [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 0 + 2
[[1,1,2,2],[2]]
=> [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 0 + 2
[[1,2,2,2],[2]]
=> [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 2 = 0 + 2
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 1 + 2
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 1 + 2
[[1],[6]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[2],[6]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[3],[6]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[4],[6]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[5],[6]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,1],[5]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,2],[5]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,5],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1,3],[5]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1],[2],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[1],[3],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[1],[4],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[2],[3],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[2],[4],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[3],[4],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[1,1,1],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,1,2],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,1,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,2,2],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,2,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,3,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[2,2,2],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[2,2,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[2,3,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[3,3,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,1],[2,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,1],[3,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,1],[4,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,2],[2,4]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 2
[[1,2],[3,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,3],[2,4]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 2
[[1,2],[4,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,3],[3,4]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 2
[[1,3],[4,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[2,2],[3,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[2,2],[4,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
Description
The breadth of a lattice.
The '''breadth''' of a lattice is the least integer $b$ such that any join $x_1\vee x_2\vee\cdots\vee x_n$, with $n > b$, can be expressed as a join over a proper subset of $\{x_1,x_2,\ldots,x_n\}$.
Matching statistic: St001630
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001630: Lattices ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 33%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001630: Lattices ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 33%
Values
[[1],[2]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1],[3]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[2],[3]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,1],[2]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,2],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1],[4]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[2],[4]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[3],[4]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,1],[3]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,2],[3]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,3],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1,3],[3]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[2,2],[3]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[2,3],[3]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1],[2],[3]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[1,1,1],[2]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,1,2],[2]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 0 + 2
[[1,2,2],[2]]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 0 + 2
[[1,1],[2,2]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[2],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[3],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[4],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,1],[4]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,2],[4]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,4],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1,3],[4]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,4],[3]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1,4],[4]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[2,2],[4]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[2,3],[4]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[2,4],[3]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[2,4],[4]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[3,3],[4]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[3,4],[4]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1],[2],[4]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[1],[3],[4]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[2],[3],[4]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[1,1,1],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,1,2],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,1,3],[2]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 0 + 2
[[1,1,3],[3]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 0 + 2
[[1,2,2],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,2,3],[2]]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 0 + 2
[[1,2,3],[3]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 0 + 2
[[1,3,3],[2]]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 0 + 2
[[1,3,3],[3]]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 0 + 2
[[2,2,2],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[2,2,3],[3]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 0 + 2
[[2,3,3],[3]]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 0 + 2
[[1,1],[2,3]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,1],[3,3]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,2],[2,3]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 2
[[1,2],[3,3]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[2,2],[3,3]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,1],[2],[3]]
=> [4,3,1,2] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? = 1 + 2
[[1,2],[2],[3]]
=> [4,2,1,3] => ([(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> ? = 1 + 2
[[1,3],[2],[3]]
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> ? = 1 + 2
[[1,1,1,1],[2]]
=> [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 0 + 2
[[1,1,1,2],[2]]
=> [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 0 + 2
[[1,1,2,2],[2]]
=> [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 0 + 2
[[1,2,2,2],[2]]
=> [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 2 = 0 + 2
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 1 + 2
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 1 + 2
[[1],[6]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[2],[6]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[3],[6]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[4],[6]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[5],[6]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,1],[5]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,2],[5]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,5],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1,3],[5]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1],[2],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[1],[3],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[1],[4],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[2],[3],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[2],[4],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[3],[4],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[1,1,1],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,1,2],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,1,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,2,2],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,2,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,3,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[2,2,2],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[2,2,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[2,3,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[3,3,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,1],[2,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,1],[3,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,1],[4,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,2],[2,4]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 2
[[1,2],[3,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,3],[2,4]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 2
[[1,2],[4,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,3],[3,4]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 2
[[1,3],[4,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[2,2],[3,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[2,2],[4,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
Description
The global dimension of the incidence algebra of the lattice over the rational numbers.
Matching statistic: St001878
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 33%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 33%
Values
[[1],[2]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1],[3]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[2],[3]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,1],[2]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,2],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1],[4]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[2],[4]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[3],[4]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,1],[3]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,2],[3]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,3],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1,3],[3]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[2,2],[3]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[2,3],[3]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1],[2],[3]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[1,1,1],[2]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,1,2],[2]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 0 + 2
[[1,2,2],[2]]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 0 + 2
[[1,1],[2,2]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[2],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[3],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[4],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,1],[4]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,2],[4]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,4],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1,3],[4]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,4],[3]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1,4],[4]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[2,2],[4]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[2,3],[4]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[2,4],[3]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[2,4],[4]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[3,3],[4]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[3,4],[4]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1],[2],[4]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[1],[3],[4]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[2],[3],[4]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[1,1,1],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,1,2],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,1,3],[2]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 0 + 2
[[1,1,3],[3]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 0 + 2
[[1,2,2],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,2,3],[2]]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 0 + 2
[[1,2,3],[3]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 0 + 2
[[1,3,3],[2]]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 0 + 2
[[1,3,3],[3]]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 0 + 2
[[2,2,2],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[2,2,3],[3]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 0 + 2
[[2,3,3],[3]]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 0 + 2
[[1,1],[2,3]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,1],[3,3]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,2],[2,3]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 2
[[1,2],[3,3]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[2,2],[3,3]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,1],[2],[3]]
=> [4,3,1,2] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? = 1 + 2
[[1,2],[2],[3]]
=> [4,2,1,3] => ([(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> ? = 1 + 2
[[1,3],[2],[3]]
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> ? = 1 + 2
[[1,1,1,1],[2]]
=> [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 0 + 2
[[1,1,1,2],[2]]
=> [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 0 + 2
[[1,1,2,2],[2]]
=> [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 0 + 2
[[1,2,2,2],[2]]
=> [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 2 = 0 + 2
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 1 + 2
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 1 + 2
[[1],[6]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[2],[6]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[3],[6]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[4],[6]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[5],[6]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,1],[5]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,2],[5]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,5],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1,3],[5]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1],[2],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[1],[3],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[1],[4],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[2],[3],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[2],[4],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[3],[4],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[1,1,1],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,1,2],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,1,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,2,2],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,2,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,3,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[2,2,2],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[2,2,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[2,3,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[3,3,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,1],[2,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,1],[3,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,1],[4,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,2],[2,4]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 2
[[1,2],[3,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,3],[2,4]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 2
[[1,2],[4,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,3],[3,4]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 2
[[1,3],[4,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[2,2],[3,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[2,2],[4,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
Matching statistic: St001491
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 67%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 67%
Values
[[1],[2]]
=> [1,1]
=> 110 => 110 => 1 = 0 + 1
[[1],[3]]
=> [1,1]
=> 110 => 110 => 1 = 0 + 1
[[2],[3]]
=> [1,1]
=> 110 => 110 => 1 = 0 + 1
[[1,1],[2]]
=> [2,1]
=> 1010 => 1100 => 1 = 0 + 1
[[1,2],[2]]
=> [2,1]
=> 1010 => 1100 => 1 = 0 + 1
[[1],[4]]
=> [1,1]
=> 110 => 110 => 1 = 0 + 1
[[2],[4]]
=> [1,1]
=> 110 => 110 => 1 = 0 + 1
[[3],[4]]
=> [1,1]
=> 110 => 110 => 1 = 0 + 1
[[1,1],[3]]
=> [2,1]
=> 1010 => 1100 => 1 = 0 + 1
[[1,2],[3]]
=> [2,1]
=> 1010 => 1100 => 1 = 0 + 1
[[1,3],[2]]
=> [2,1]
=> 1010 => 1100 => 1 = 0 + 1
[[1,3],[3]]
=> [2,1]
=> 1010 => 1100 => 1 = 0 + 1
[[2,2],[3]]
=> [2,1]
=> 1010 => 1100 => 1 = 0 + 1
[[2,3],[3]]
=> [2,1]
=> 1010 => 1100 => 1 = 0 + 1
[[1],[2],[3]]
=> [1,1,1]
=> 1110 => 1110 => 2 = 1 + 1
[[1,1,1],[2]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,1,2],[2]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,2,2],[2]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,1],[2,2]]
=> [2,2]
=> 1100 => 0110 => 2 = 1 + 1
[[1],[5]]
=> [1,1]
=> 110 => 110 => 1 = 0 + 1
[[2],[5]]
=> [1,1]
=> 110 => 110 => 1 = 0 + 1
[[3],[5]]
=> [1,1]
=> 110 => 110 => 1 = 0 + 1
[[4],[5]]
=> [1,1]
=> 110 => 110 => 1 = 0 + 1
[[1,1],[4]]
=> [2,1]
=> 1010 => 1100 => 1 = 0 + 1
[[1,2],[4]]
=> [2,1]
=> 1010 => 1100 => 1 = 0 + 1
[[1,4],[2]]
=> [2,1]
=> 1010 => 1100 => 1 = 0 + 1
[[1,3],[4]]
=> [2,1]
=> 1010 => 1100 => 1 = 0 + 1
[[1,4],[3]]
=> [2,1]
=> 1010 => 1100 => 1 = 0 + 1
[[1,4],[4]]
=> [2,1]
=> 1010 => 1100 => 1 = 0 + 1
[[2,2],[4]]
=> [2,1]
=> 1010 => 1100 => 1 = 0 + 1
[[2,3],[4]]
=> [2,1]
=> 1010 => 1100 => 1 = 0 + 1
[[2,4],[3]]
=> [2,1]
=> 1010 => 1100 => 1 = 0 + 1
[[2,4],[4]]
=> [2,1]
=> 1010 => 1100 => 1 = 0 + 1
[[3,3],[4]]
=> [2,1]
=> 1010 => 1100 => 1 = 0 + 1
[[3,4],[4]]
=> [2,1]
=> 1010 => 1100 => 1 = 0 + 1
[[1],[2],[4]]
=> [1,1,1]
=> 1110 => 1110 => 2 = 1 + 1
[[1],[3],[4]]
=> [1,1,1]
=> 1110 => 1110 => 2 = 1 + 1
[[2],[3],[4]]
=> [1,1,1]
=> 1110 => 1110 => 2 = 1 + 1
[[1,1,1],[3]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,1,2],[3]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,1,3],[2]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,1,3],[3]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,2,2],[3]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,2,3],[2]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,2,3],[3]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,3,3],[2]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,3,3],[3]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[2,2,2],[3]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[2,2,3],[3]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[2,3,3],[3]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,1],[2,3]]
=> [2,2]
=> 1100 => 0110 => 2 = 1 + 1
[[1,1],[3,3]]
=> [2,2]
=> 1100 => 0110 => 2 = 1 + 1
[[1,2],[2,3]]
=> [2,2]
=> 1100 => 0110 => 2 = 1 + 1
[[1,2],[3,3]]
=> [2,2]
=> 1100 => 0110 => 2 = 1 + 1
[[2,2],[3,3]]
=> [2,2]
=> 1100 => 0110 => 2 = 1 + 1
[[1,1],[2],[3]]
=> [2,1,1]
=> 10110 => 11010 => ? = 1 + 1
[[1,2],[2],[3]]
=> [2,1,1]
=> 10110 => 11010 => ? = 1 + 1
[[1,3],[2],[3]]
=> [2,1,1]
=> 10110 => 11010 => ? = 1 + 1
[[1,1,1,1],[2]]
=> [4,1]
=> 100010 => 100100 => ? = 0 + 1
[[1,1,1,2],[2]]
=> [4,1]
=> 100010 => 100100 => ? = 0 + 1
[[1,1,2,2],[2]]
=> [4,1]
=> 100010 => 100100 => ? = 0 + 1
[[1,2,2,2],[2]]
=> [4,1]
=> 100010 => 100100 => ? = 0 + 1
[[1,1,1],[2,2]]
=> [3,2]
=> 10100 => 01100 => ? = 1 + 1
[[1,1,2],[2,2]]
=> [3,2]
=> 10100 => 01100 => ? = 1 + 1
[[1],[6]]
=> [1,1]
=> 110 => 110 => 1 = 0 + 1
[[2],[6]]
=> [1,1]
=> 110 => 110 => 1 = 0 + 1
[[3],[6]]
=> [1,1]
=> 110 => 110 => 1 = 0 + 1
[[4],[6]]
=> [1,1]
=> 110 => 110 => 1 = 0 + 1
[[5],[6]]
=> [1,1]
=> 110 => 110 => 1 = 0 + 1
[[1,1],[5]]
=> [2,1]
=> 1010 => 1100 => 1 = 0 + 1
[[1,2],[5]]
=> [2,1]
=> 1010 => 1100 => 1 = 0 + 1
[[1,5],[2]]
=> [2,1]
=> 1010 => 1100 => 1 = 0 + 1
[[1,3],[5]]
=> [2,1]
=> 1010 => 1100 => 1 = 0 + 1
[[1,5],[3]]
=> [2,1]
=> 1010 => 1100 => 1 = 0 + 1
[[1,1,1],[4]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,1,2],[4]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,1,4],[2]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,1,3],[4]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,1,4],[3]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,1,4],[4]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,2,2],[4]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,2,4],[2]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,2,3],[4]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,2,4],[3]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,3,4],[2]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,2,4],[4]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,4,4],[2]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,3,3],[4]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,3,4],[3]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,3,4],[4]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,4,4],[3]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,4,4],[4]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[2,2,2],[4]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[2,2,3],[4]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[2,2,4],[3]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[2,2,4],[4]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[2,3,3],[4]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[2,3,4],[3]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[2,3,4],[4]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[2,4,4],[3]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
Let $A_n=K[x]/(x^n)$.
We associate to a nonempty subset S of an (n-1)-set the module $M_S$, which is the direct sum of $A_n$-modules with indecomposable non-projective direct summands of dimension $i$ when $i$ is in $S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of $M_S$. We decode the subset as a binary word so that for example the subset $S=\{1,3 \} $ of $\{1,2,3 \}$ is decoded as 101.
Matching statistic: St001207
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 67%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 67%
Values
[[1],[2]]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 0 + 2
[[1],[3]]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 0 + 2
[[2],[3]]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 0 + 2
[[1,1],[2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[[1,2],[2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[[1],[4]]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 0 + 2
[[2],[4]]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 0 + 2
[[3],[4]]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 0 + 2
[[1,1],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[[1,2],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[[1,3],[2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[[1,3],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[[2,2],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[[2,3],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[[1],[2],[3]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 3 = 1 + 2
[[1,1,1],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,1,2],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,2,2],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 1 + 2
[[1],[5]]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 0 + 2
[[2],[5]]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 0 + 2
[[3],[5]]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 0 + 2
[[4],[5]]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 0 + 2
[[1,1],[4]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[[1,2],[4]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[[1,4],[2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[[1,3],[4]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[[1,4],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[[1,4],[4]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[[2,2],[4]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[[2,3],[4]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[[2,4],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[[2,4],[4]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[[3,3],[4]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[[3,4],[4]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[[1],[2],[4]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 3 = 1 + 2
[[1],[3],[4]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 3 = 1 + 2
[[2],[3],[4]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 3 = 1 + 2
[[1,1,1],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,1,2],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,1,3],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,1,3],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,2,2],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,2,3],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,2,3],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,3,3],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,3,3],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[2,2,2],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[2,2,3],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[2,3,3],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,1],[2,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 1 + 2
[[1,1],[3,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 1 + 2
[[1,2],[2,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 1 + 2
[[1,2],[3,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 1 + 2
[[2,2],[3,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 1 + 2
[[1,1],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 1 + 2
[[1,2],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 1 + 2
[[1,3],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 1 + 2
[[1,1,1,1],[2]]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ? = 0 + 2
[[1,1,1,2],[2]]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ? = 0 + 2
[[1,1,2,2],[2]]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ? = 0 + 2
[[1,2,2,2],[2]]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ? = 0 + 2
[[1,1,1],[2,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? = 1 + 2
[[1,1,2],[2,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? = 1 + 2
[[1],[6]]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 0 + 2
[[2],[6]]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 0 + 2
[[3],[6]]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 0 + 2
[[4],[6]]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 0 + 2
[[5],[6]]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 0 + 2
[[1,1],[5]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[[1,2],[5]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[[1,5],[2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[[1,3],[5]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[[1,5],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[[1,1,1],[4]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,1,2],[4]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,1,4],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,1,3],[4]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,1,4],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,1,4],[4]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,2,2],[4]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,2,4],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,2,3],[4]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,2,4],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,3,4],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,2,4],[4]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,4,4],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,3,3],[4]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,3,4],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,3,4],[4]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,4,4],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,4,4],[4]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[2,2,2],[4]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[2,2,3],[4]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[2,2,4],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[2,2,4],[4]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[2,3,3],[4]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[2,3,4],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[2,3,4],[4]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[2,4,4],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
Description
The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
Matching statistic: St001232
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 33%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 33%
Values
[[1],[2]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[[1],[3]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[[2],[3]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[[1,1],[2]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 0 + 1
[[1,2],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[[1],[4]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[[2],[4]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[[3],[4]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[[1,1],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 0 + 1
[[1,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 0 + 1
[[1,3],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[[1,3],[3]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[[2,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 0 + 1
[[2,3],[3]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[[1],[2],[3]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[1,1,1],[2]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 0 + 1
[[1,1,2],[2]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 1
[[1,2,2],[2]]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[[1,1],[2,2]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> ? = 1 + 1
[[1],[5]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[[2],[5]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[[3],[5]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[[4],[5]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[[1,1],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 0 + 1
[[1,2],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 0 + 1
[[1,4],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[[1,3],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 0 + 1
[[1,4],[3]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[[1,4],[4]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[[2,2],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 0 + 1
[[2,3],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 0 + 1
[[2,4],[3]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[[2,4],[4]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[[3,3],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 0 + 1
[[3,4],[4]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[[1],[2],[4]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[1],[3],[4]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[2],[3],[4]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[1,1,1],[3]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 0 + 1
[[1,1,2],[3]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 0 + 1
[[1,1,3],[2]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 1
[[1,1,3],[3]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 1
[[1,2,2],[3]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 0 + 1
[[1,2,3],[2]]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[[1,2,3],[3]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 1
[[1,3,3],[2]]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[[1,3,3],[3]]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[[2,2,2],[3]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 0 + 1
[[2,2,3],[3]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 1
[[2,3,3],[3]]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[[1,1],[2,3]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> ? = 1 + 1
[[1,1],[3,3]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> ? = 1 + 1
[[1,2],[2,3]]
=> [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[[1,2],[3,3]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> ? = 1 + 1
[[2,2],[3,3]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> ? = 1 + 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1 + 1
[[1,2],[2],[3]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1 + 1
[[1,3],[2],[3]]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[[1,1,1,1],[2]]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 0 + 1
[[1,1,1,2],[2]]
=> [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 0 + 1
[[1,1,2,2],[2]]
=> [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 0 + 1
[[1,2,2,2],[2]]
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> ? = 1 + 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ? = 1 + 1
[[1],[6]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[[2],[6]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[[3],[6]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[[4],[6]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[[5],[6]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[[1,1],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 0 + 1
[[1,2],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 0 + 1
[[1,5],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[[1,3],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 0 + 1
[[1,5],[3]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[[1,4],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 0 + 1
[[1,5],[4]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[[1,5],[5]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[[2,2],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 0 + 1
[[2,3],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 0 + 1
[[2,5],[3]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[[2,4],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 0 + 1
[[2,5],[4]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[[2,5],[5]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[[3,3],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 0 + 1
[[3,4],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 0 + 1
[[3,5],[4]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[[3,5],[5]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[[4,4],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 0 + 1
[[4,5],[5]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[[1],[2],[5]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[1],[3],[5]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[1,2,4],[2]]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[[1,3,4],[2]]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[[1,4,4],[2]]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[[1,3,4],[3]]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[[1,4,4],[3]]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[[1,4,4],[4]]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[[2,3,4],[3]]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[[2,4,4],[3]]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[[2,4,4],[4]]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001857
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001857: Signed permutations ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 67%
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001857: Signed permutations ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 67%
Values
[[1],[2]]
=> [2,1] => [2,1] => 0
[[1],[3]]
=> [2,1] => [2,1] => 0
[[2],[3]]
=> [2,1] => [2,1] => 0
[[1,1],[2]]
=> [3,1,2] => [3,1,2] => 0
[[1,2],[2]]
=> [2,1,3] => [2,1,3] => 0
[[1],[4]]
=> [2,1] => [2,1] => 0
[[2],[4]]
=> [2,1] => [2,1] => 0
[[3],[4]]
=> [2,1] => [2,1] => 0
[[1,1],[3]]
=> [3,1,2] => [3,1,2] => 0
[[1,2],[3]]
=> [3,1,2] => [3,1,2] => 0
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => 0
[[1,3],[3]]
=> [2,1,3] => [2,1,3] => 0
[[2,2],[3]]
=> [3,1,2] => [3,1,2] => 0
[[2,3],[3]]
=> [2,1,3] => [2,1,3] => 0
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 1
[[1,1,1],[2]]
=> [4,1,2,3] => [4,1,2,3] => ? = 0
[[1,1,2],[2]]
=> [3,1,2,4] => [3,1,2,4] => ? = 0
[[1,2,2],[2]]
=> [2,1,3,4] => [2,1,3,4] => ? = 0
[[1,1],[2,2]]
=> [3,4,1,2] => [3,4,1,2] => ? = 1
[[1],[5]]
=> [2,1] => [2,1] => 0
[[2],[5]]
=> [2,1] => [2,1] => 0
[[3],[5]]
=> [2,1] => [2,1] => 0
[[4],[5]]
=> [2,1] => [2,1] => 0
[[1,1],[4]]
=> [3,1,2] => [3,1,2] => 0
[[1,2],[4]]
=> [3,1,2] => [3,1,2] => 0
[[1,4],[2]]
=> [2,1,3] => [2,1,3] => 0
[[1,3],[4]]
=> [3,1,2] => [3,1,2] => 0
[[1,4],[3]]
=> [2,1,3] => [2,1,3] => 0
[[1,4],[4]]
=> [2,1,3] => [2,1,3] => 0
[[2,2],[4]]
=> [3,1,2] => [3,1,2] => 0
[[2,3],[4]]
=> [3,1,2] => [3,1,2] => 0
[[2,4],[3]]
=> [2,1,3] => [2,1,3] => 0
[[2,4],[4]]
=> [2,1,3] => [2,1,3] => 0
[[3,3],[4]]
=> [3,1,2] => [3,1,2] => 0
[[3,4],[4]]
=> [2,1,3] => [2,1,3] => 0
[[1],[2],[4]]
=> [3,2,1] => [3,2,1] => 1
[[1],[3],[4]]
=> [3,2,1] => [3,2,1] => 1
[[2],[3],[4]]
=> [3,2,1] => [3,2,1] => 1
[[1,1,1],[3]]
=> [4,1,2,3] => [4,1,2,3] => ? = 0
[[1,1,2],[3]]
=> [4,1,2,3] => [4,1,2,3] => ? = 0
[[1,1,3],[2]]
=> [3,1,2,4] => [3,1,2,4] => ? = 0
[[1,1,3],[3]]
=> [3,1,2,4] => [3,1,2,4] => ? = 0
[[1,2,2],[3]]
=> [4,1,2,3] => [4,1,2,3] => ? = 0
[[1,2,3],[2]]
=> [2,1,3,4] => [2,1,3,4] => ? = 0
[[1,2,3],[3]]
=> [3,1,2,4] => [3,1,2,4] => ? = 0
[[1,3,3],[2]]
=> [2,1,3,4] => [2,1,3,4] => ? = 0
[[1,3,3],[3]]
=> [2,1,3,4] => [2,1,3,4] => ? = 0
[[2,2,2],[3]]
=> [4,1,2,3] => [4,1,2,3] => ? = 0
[[2,2,3],[3]]
=> [3,1,2,4] => [3,1,2,4] => ? = 0
[[2,3,3],[3]]
=> [2,1,3,4] => [2,1,3,4] => ? = 0
[[1,1],[2,3]]
=> [3,4,1,2] => [3,4,1,2] => ? = 1
[[1,1],[3,3]]
=> [3,4,1,2] => [3,4,1,2] => ? = 1
[[1,2],[2,3]]
=> [2,4,1,3] => [2,4,1,3] => ? = 1
[[1,2],[3,3]]
=> [3,4,1,2] => [3,4,1,2] => ? = 1
[[2,2],[3,3]]
=> [3,4,1,2] => [3,4,1,2] => ? = 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [4,3,1,2] => ? = 1
[[1,2],[2],[3]]
=> [4,2,1,3] => [4,2,1,3] => ? = 1
[[1,3],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => ? = 1
[[1,1,1,1],[2]]
=> [5,1,2,3,4] => [5,1,2,3,4] => ? = 0
[[1,1,1,2],[2]]
=> [4,1,2,3,5] => [4,1,2,3,5] => ? = 0
[[1,1,2,2],[2]]
=> [3,1,2,4,5] => [3,1,2,4,5] => ? = 0
[[1,2,2,2],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ? = 0
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 1
[[1],[6]]
=> [2,1] => [2,1] => 0
[[2],[6]]
=> [2,1] => [2,1] => 0
[[3],[6]]
=> [2,1] => [2,1] => 0
[[4],[6]]
=> [2,1] => [2,1] => 0
[[5],[6]]
=> [2,1] => [2,1] => 0
[[1,1],[5]]
=> [3,1,2] => [3,1,2] => 0
[[1,2],[5]]
=> [3,1,2] => [3,1,2] => 0
[[1,5],[2]]
=> [2,1,3] => [2,1,3] => 0
[[1,3],[5]]
=> [3,1,2] => [3,1,2] => 0
[[1,5],[3]]
=> [2,1,3] => [2,1,3] => 0
[[1,4],[5]]
=> [3,1,2] => [3,1,2] => 0
[[1,5],[4]]
=> [2,1,3] => [2,1,3] => 0
[[1,5],[5]]
=> [2,1,3] => [2,1,3] => 0
[[2,2],[5]]
=> [3,1,2] => [3,1,2] => 0
[[2,3],[5]]
=> [3,1,2] => [3,1,2] => 0
[[2,5],[3]]
=> [2,1,3] => [2,1,3] => 0
[[1,1,1],[4]]
=> [4,1,2,3] => [4,1,2,3] => ? = 0
[[1,1,2],[4]]
=> [4,1,2,3] => [4,1,2,3] => ? = 0
[[1,1,4],[2]]
=> [3,1,2,4] => [3,1,2,4] => ? = 0
[[1,1,3],[4]]
=> [4,1,2,3] => [4,1,2,3] => ? = 0
[[1,1,4],[3]]
=> [3,1,2,4] => [3,1,2,4] => ? = 0
[[1,1,4],[4]]
=> [3,1,2,4] => [3,1,2,4] => ? = 0
[[1,2,2],[4]]
=> [4,1,2,3] => [4,1,2,3] => ? = 0
[[1,2,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => ? = 0
[[1,2,3],[4]]
=> [4,1,2,3] => [4,1,2,3] => ? = 0
[[1,2,4],[3]]
=> [3,1,2,4] => [3,1,2,4] => ? = 0
[[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => ? = 0
[[1,2,4],[4]]
=> [3,1,2,4] => [3,1,2,4] => ? = 0
[[1,4,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => ? = 0
[[1,3,3],[4]]
=> [4,1,2,3] => [4,1,2,3] => ? = 0
[[1,3,4],[3]]
=> [2,1,3,4] => [2,1,3,4] => ? = 0
[[1,3,4],[4]]
=> [3,1,2,4] => [3,1,2,4] => ? = 0
[[1,4,4],[3]]
=> [2,1,3,4] => [2,1,3,4] => ? = 0
[[1,4,4],[4]]
=> [2,1,3,4] => [2,1,3,4] => ? = 0
[[2,2,2],[4]]
=> [4,1,2,3] => [4,1,2,3] => ? = 0
[[2,2,3],[4]]
=> [4,1,2,3] => [4,1,2,3] => ? = 0
Description
The number of edges in the reduced word graph of a signed permutation.
The reduced word graph of a signed permutation $\pi$ has the reduced words of $\pi$ as vertices and an edge between two reduced words if they differ by exactly one braid move.
Matching statistic: St000166
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00008: Binary trees —to complete tree⟶ Ordered trees
St000166: Ordered trees ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 67%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00008: Binary trees —to complete tree⟶ Ordered trees
St000166: Ordered trees ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 67%
Values
[[1],[2]]
=> [2,1] => [[.,.],.]
=> [[[],[]],[]]
=> 2 = 0 + 2
[[1],[3]]
=> [2,1] => [[.,.],.]
=> [[[],[]],[]]
=> 2 = 0 + 2
[[2],[3]]
=> [2,1] => [[.,.],.]
=> [[[],[]],[]]
=> 2 = 0 + 2
[[1,1],[2]]
=> [3,1,2] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 2 = 0 + 2
[[1,2],[2]]
=> [2,1,3] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 2 = 0 + 2
[[1],[4]]
=> [2,1] => [[.,.],.]
=> [[[],[]],[]]
=> 2 = 0 + 2
[[2],[4]]
=> [2,1] => [[.,.],.]
=> [[[],[]],[]]
=> 2 = 0 + 2
[[3],[4]]
=> [2,1] => [[.,.],.]
=> [[[],[]],[]]
=> 2 = 0 + 2
[[1,1],[3]]
=> [3,1,2] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 2 = 0 + 2
[[1,2],[3]]
=> [3,1,2] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 2 = 0 + 2
[[1,3],[2]]
=> [2,1,3] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 2 = 0 + 2
[[1,3],[3]]
=> [2,1,3] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 2 = 0 + 2
[[2,2],[3]]
=> [3,1,2] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 2 = 0 + 2
[[2,3],[3]]
=> [2,1,3] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 2 = 0 + 2
[[1],[2],[3]]
=> [3,2,1] => [[[.,.],.],.]
=> [[[[],[]],[]],[]]
=> 3 = 1 + 2
[[1,1,1],[2]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? = 0 + 2
[[1,1,2],[2]]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? = 0 + 2
[[1,2,2],[2]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? = 0 + 2
[[1,1],[2,2]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[[],[[],[]]],[[],[]]]
=> ? = 1 + 2
[[1],[5]]
=> [2,1] => [[.,.],.]
=> [[[],[]],[]]
=> 2 = 0 + 2
[[2],[5]]
=> [2,1] => [[.,.],.]
=> [[[],[]],[]]
=> 2 = 0 + 2
[[3],[5]]
=> [2,1] => [[.,.],.]
=> [[[],[]],[]]
=> 2 = 0 + 2
[[4],[5]]
=> [2,1] => [[.,.],.]
=> [[[],[]],[]]
=> 2 = 0 + 2
[[1,1],[4]]
=> [3,1,2] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 2 = 0 + 2
[[1,2],[4]]
=> [3,1,2] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 2 = 0 + 2
[[1,4],[2]]
=> [2,1,3] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 2 = 0 + 2
[[1,3],[4]]
=> [3,1,2] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 2 = 0 + 2
[[1,4],[3]]
=> [2,1,3] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 2 = 0 + 2
[[1,4],[4]]
=> [2,1,3] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 2 = 0 + 2
[[2,2],[4]]
=> [3,1,2] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 2 = 0 + 2
[[2,3],[4]]
=> [3,1,2] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 2 = 0 + 2
[[2,4],[3]]
=> [2,1,3] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 2 = 0 + 2
[[2,4],[4]]
=> [2,1,3] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 2 = 0 + 2
[[3,3],[4]]
=> [3,1,2] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 2 = 0 + 2
[[3,4],[4]]
=> [2,1,3] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 2 = 0 + 2
[[1],[2],[4]]
=> [3,2,1] => [[[.,.],.],.]
=> [[[[],[]],[]],[]]
=> 3 = 1 + 2
[[1],[3],[4]]
=> [3,2,1] => [[[.,.],.],.]
=> [[[[],[]],[]],[]]
=> 3 = 1 + 2
[[2],[3],[4]]
=> [3,2,1] => [[[.,.],.],.]
=> [[[[],[]],[]],[]]
=> 3 = 1 + 2
[[1,1,1],[3]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? = 0 + 2
[[1,1,2],[3]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? = 0 + 2
[[1,1,3],[2]]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? = 0 + 2
[[1,1,3],[3]]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? = 0 + 2
[[1,2,2],[3]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? = 0 + 2
[[1,2,3],[2]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? = 0 + 2
[[1,2,3],[3]]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? = 0 + 2
[[1,3,3],[2]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? = 0 + 2
[[1,3,3],[3]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? = 0 + 2
[[2,2,2],[3]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? = 0 + 2
[[2,2,3],[3]]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? = 0 + 2
[[2,3,3],[3]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? = 0 + 2
[[1,1],[2,3]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[[],[[],[]]],[[],[]]]
=> ? = 1 + 2
[[1,1],[3,3]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[[],[[],[]]],[[],[]]]
=> ? = 1 + 2
[[1,2],[2,3]]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [[[],[[],[]]],[[],[]]]
=> ? = 1 + 2
[[1,2],[3,3]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[[],[[],[]]],[[],[]]]
=> ? = 1 + 2
[[2,2],[3,3]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[[],[[],[]]],[[],[]]]
=> ? = 1 + 2
[[1,1],[2],[3]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> [[[[],[]],[]],[[],[]]]
=> ? = 1 + 2
[[1,2],[2],[3]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> [[[[],[]],[]],[[],[]]]
=> ? = 1 + 2
[[1,3],[2],[3]]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [[[[],[]],[]],[[],[]]]
=> ? = 1 + 2
[[1,1,1,1],[2]]
=> [5,1,2,3,4] => [[.,.],[.,[.,[.,.]]]]
=> [[[],[]],[[],[[],[[],[]]]]]
=> ? = 0 + 2
[[1,1,1,2],[2]]
=> [4,1,2,3,5] => [[.,.],[.,[.,[.,.]]]]
=> [[[],[]],[[],[[],[[],[]]]]]
=> ? = 0 + 2
[[1,1,2,2],[2]]
=> [3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [[[],[]],[[],[[],[[],[]]]]]
=> ? = 0 + 2
[[1,2,2,2],[2]]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [[[],[]],[[],[[],[[],[]]]]]
=> ? = 0 + 2
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
=> [[[],[[],[]]],[[],[[],[]]]]
=> ? = 1 + 2
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [[[],[[],[]]],[[],[[],[]]]]
=> ? = 1 + 2
[[1],[6]]
=> [2,1] => [[.,.],.]
=> [[[],[]],[]]
=> 2 = 0 + 2
[[2],[6]]
=> [2,1] => [[.,.],.]
=> [[[],[]],[]]
=> 2 = 0 + 2
[[3],[6]]
=> [2,1] => [[.,.],.]
=> [[[],[]],[]]
=> 2 = 0 + 2
[[4],[6]]
=> [2,1] => [[.,.],.]
=> [[[],[]],[]]
=> 2 = 0 + 2
[[5],[6]]
=> [2,1] => [[.,.],.]
=> [[[],[]],[]]
=> 2 = 0 + 2
[[1,1],[5]]
=> [3,1,2] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 2 = 0 + 2
[[1,2],[5]]
=> [3,1,2] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 2 = 0 + 2
[[1,5],[2]]
=> [2,1,3] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 2 = 0 + 2
[[1,3],[5]]
=> [3,1,2] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 2 = 0 + 2
[[1,5],[3]]
=> [2,1,3] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 2 = 0 + 2
[[1,4],[5]]
=> [3,1,2] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 2 = 0 + 2
[[1,5],[4]]
=> [2,1,3] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 2 = 0 + 2
[[1,5],[5]]
=> [2,1,3] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 2 = 0 + 2
[[2,2],[5]]
=> [3,1,2] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 2 = 0 + 2
[[2,3],[5]]
=> [3,1,2] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 2 = 0 + 2
[[2,5],[3]]
=> [2,1,3] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 2 = 0 + 2
[[1,1,1],[4]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? = 0 + 2
[[1,1,2],[4]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? = 0 + 2
[[1,1,4],[2]]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? = 0 + 2
[[1,1,3],[4]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? = 0 + 2
[[1,1,4],[3]]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? = 0 + 2
[[1,1,4],[4]]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? = 0 + 2
[[1,2,2],[4]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? = 0 + 2
[[1,2,4],[2]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? = 0 + 2
[[1,2,3],[4]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? = 0 + 2
[[1,2,4],[3]]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? = 0 + 2
[[1,3,4],[2]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? = 0 + 2
[[1,2,4],[4]]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? = 0 + 2
[[1,4,4],[2]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? = 0 + 2
[[1,3,3],[4]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? = 0 + 2
[[1,3,4],[3]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? = 0 + 2
[[1,3,4],[4]]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? = 0 + 2
[[1,4,4],[3]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? = 0 + 2
[[1,4,4],[4]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? = 0 + 2
[[2,2,2],[4]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? = 0 + 2
[[2,2,3],[4]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? = 0 + 2
Description
The depth minus 1 of an ordered tree.
The ordered trees of size $n$ are bijection with the Dyck paths of size $n-1$, and this statistic then corresponds to [[St000013]].
The following 30 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000522The number of 1-protected nodes of a rooted tree. St000094The depth of an ordered tree. St000521The number of distinct subtrees of an ordered tree. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000782The indicator function of whether a given perfect matching is an L & P matching. St001722The number of minimal chains with small intervals between a binary word and the top element. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001645The pebbling number of a connected graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000102The charge of a semistandard tableau. St001964The interval resolution global dimension of a poset. St000101The cocharge of a semistandard tableau. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000181The number of connected components of the Hasse diagram for the poset. St000454The largest eigenvalue of a graph if it is integral. St001408The number of maximal entries in a semistandard tableau. St001410The minimal entry of a semistandard tableau. St001890The maximum magnitude of the Möbius function of a poset. St000422The energy of a graph, if it is integral. St000736The last entry in the first row of a semistandard tableau. St000739The first entry in the last row of a semistandard tableau. St001401The number of distinct entries in a semistandard tableau. St001407The number of minimal entries in a semistandard tableau. St001409The maximal entry of a semistandard tableau. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St000327The number of cover relations in a poset. St000635The number of strictly order preserving maps of a poset into itself. St000103The sum of the entries of a semistandard tableau.
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