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Your data matches 10 different statistics following compositions of up to 3 maps.
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Matching statistic: St000259
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Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => ([],2)
=> ([],1)
=> 0
[[2,2]]
=> [1,2] => ([],2)
=> ([],1)
=> 0
[[1],[2]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,3]]
=> [1,2] => ([],2)
=> ([],1)
=> 0
[[2,3]]
=> [1,2] => ([],2)
=> ([],1)
=> 0
[[3,3]]
=> [1,2] => ([],2)
=> ([],1)
=> 0
[[1],[3]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[2],[3]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,2]]
=> [1,2,3] => ([],3)
=> ([],1)
=> 0
[[1,2,2]]
=> [1,2,3] => ([],3)
=> ([],1)
=> 0
[[2,2,2]]
=> [1,2,3] => ([],3)
=> ([],1)
=> 0
[[1,1],[2]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
[[1,4]]
=> [1,2] => ([],2)
=> ([],1)
=> 0
[[2,4]]
=> [1,2] => ([],2)
=> ([],1)
=> 0
[[3,4]]
=> [1,2] => ([],2)
=> ([],1)
=> 0
[[4,4]]
=> [1,2] => ([],2)
=> ([],1)
=> 0
[[1],[4]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[2],[4]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[3],[4]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,3]]
=> [1,2,3] => ([],3)
=> ([],1)
=> 0
[[1,2,3]]
=> [1,2,3] => ([],3)
=> ([],1)
=> 0
[[1,3,3]]
=> [1,2,3] => ([],3)
=> ([],1)
=> 0
[[2,2,3]]
=> [1,2,3] => ([],3)
=> ([],1)
=> 0
[[2,3,3]]
=> [1,2,3] => ([],3)
=> ([],1)
=> 0
[[3,3,3]]
=> [1,2,3] => ([],3)
=> ([],1)
=> 0
[[1,1],[3]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
[[1,2],[3]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
[[2,2],[3]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
[[1],[2],[3]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[[1,1,1,2]]
=> [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[[1,1,2,2]]
=> [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[[1,2,2,2]]
=> [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[[2,2,2,2]]
=> [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[[1,1,1],[2]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[[1,1],[2,2]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 1
[[1,5]]
=> [1,2] => ([],2)
=> ([],1)
=> 0
[[2,5]]
=> [1,2] => ([],2)
=> ([],1)
=> 0
[[3,5]]
=> [1,2] => ([],2)
=> ([],1)
=> 0
[[4,5]]
=> [1,2] => ([],2)
=> ([],1)
=> 0
[[5,5]]
=> [1,2] => ([],2)
=> ([],1)
=> 0
[[1],[5]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[2],[5]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[3],[5]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[4],[5]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,4]]
=> [1,2,3] => ([],3)
=> ([],1)
=> 0
[[1,2,4]]
=> [1,2,3] => ([],3)
=> ([],1)
=> 0
[[1,3,4]]
=> [1,2,3] => ([],3)
=> ([],1)
=> 0
[[1,4,4]]
=> [1,2,3] => ([],3)
=> ([],1)
=> 0
[[2,2,4]]
=> [1,2,3] => ([],3)
=> ([],1)
=> 0
[[2,3,4]]
=> [1,2,3] => ([],3)
=> ([],1)
=> 0
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St001621
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001621: Lattices ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 50%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001621: Lattices ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 50%
Values
[[1,2]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[2,2]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[1],[2]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,3]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[2,3]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[3,3]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[1],[3]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[2],[3]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,1,2]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[1,2,2]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[2,2,2]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[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 = 1 + 1
[[1,4]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[2,4]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[3,4]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[4,4]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[1],[4]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[2],[4]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[3],[4]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,1,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[1,2,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[1,3,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[2,2,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[2,3,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[3,3,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[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 = 1 + 1
[[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 = 1 + 1
[[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 = 1 + 1
[[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 + 1
[[1,1,1,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[1,1,2,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[1,2,2,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[2,2,2,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[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)
=> ? = 1 + 1
[[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 + 1
[[1,5]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[2,5]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[3,5]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[4,5]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[5,5]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[1],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[2],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[3],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[4],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,1,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[1,2,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[1,3,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[1,4,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[2,2,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[2,3,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[2,4,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[3,3,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[3,4,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[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 + 1
[[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 + 1
[[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 + 1
[[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)
=> ? = 1 + 1
[[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)
=> ? = 1 + 1
[[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)
=> ? = 1 + 1
[[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)
=> ? = 1 + 1
[[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 + 1
[[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 + 1
[[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)
=> ? = 3 + 1
[[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 + 1
[[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 + 1
[[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 + 1
[[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)
=> ? = 2 + 1
[[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)
=> ? = 1 + 1
[[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 + 1
[[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 + 1
[[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 + 1
[[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 + 1
[[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 + 1
[[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 + 1
[[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 + 1
[[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)
=> ? = 1 + 1
[[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)
=> ? = 1 + 1
[[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)
=> ? = 1 + 1
[[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)
=> ? = 1 + 1
[[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)
=> ? = 1 + 1
[[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)
=> ? = 1 + 1
[[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)
=> ? = 1 + 1
[[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)
=> ? = 1 + 1
[[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)
=> ? = 1 + 1
[[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)
=> ? = 1 + 1
[[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 + 1
[[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 + 1
[[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 + 1
[[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)
=> ? = 3 + 1
[[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 + 1
[[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)
=> ? = 3 + 1
[[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 + 1
[[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)
=> ? = 3 + 1
[[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 + 1
[[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 + 1
[[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 + 1
[[2,3],[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)
=> ? = 3 + 1
[[2,3],[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 + 1
[[3,3],[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 + 1
[[1,1],[2],[4]]
=> [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 + 1
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: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 50%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001624: Lattices ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 50%
Values
[[1,2]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[2,2]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[1],[2]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,3]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[2,3]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[3,3]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[1],[3]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[2],[3]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,1,2]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[1,2,2]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[2,2,2]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[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 = 1 + 1
[[1,4]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[2,4]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[3,4]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[4,4]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[1],[4]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[2],[4]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[3],[4]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,1,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[1,2,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[1,3,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[2,2,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[2,3,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[3,3,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[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 = 1 + 1
[[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 = 1 + 1
[[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 = 1 + 1
[[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 + 1
[[1,1,1,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[1,1,2,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[1,2,2,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[2,2,2,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[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)
=> ? = 1 + 1
[[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 + 1
[[1,5]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[2,5]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[3,5]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[4,5]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[5,5]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[1],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[2],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[3],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[4],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,1,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[1,2,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[1,3,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[1,4,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[2,2,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[2,3,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[2,4,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[3,3,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[3,4,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[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 + 1
[[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 + 1
[[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 + 1
[[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)
=> ? = 1 + 1
[[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)
=> ? = 1 + 1
[[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)
=> ? = 1 + 1
[[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)
=> ? = 1 + 1
[[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 + 1
[[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 + 1
[[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)
=> ? = 3 + 1
[[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 + 1
[[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 + 1
[[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 + 1
[[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)
=> ? = 2 + 1
[[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)
=> ? = 1 + 1
[[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 + 1
[[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 + 1
[[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 + 1
[[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 + 1
[[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 + 1
[[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 + 1
[[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 + 1
[[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)
=> ? = 1 + 1
[[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)
=> ? = 1 + 1
[[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)
=> ? = 1 + 1
[[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)
=> ? = 1 + 1
[[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)
=> ? = 1 + 1
[[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)
=> ? = 1 + 1
[[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)
=> ? = 1 + 1
[[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)
=> ? = 1 + 1
[[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)
=> ? = 1 + 1
[[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)
=> ? = 1 + 1
[[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 + 1
[[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 + 1
[[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 + 1
[[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)
=> ? = 3 + 1
[[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 + 1
[[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)
=> ? = 3 + 1
[[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 + 1
[[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)
=> ? = 3 + 1
[[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 + 1
[[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 + 1
[[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 + 1
[[2,3],[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)
=> ? = 3 + 1
[[2,3],[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 + 1
[[3,3],[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 + 1
[[1,1],[2],[4]]
=> [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 + 1
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: St001878
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 50%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 50%
Values
[[1,2]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[2,2]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[1],[2]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,3]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[2,3]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[3,3]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[1],[3]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[2],[3]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,1,2]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[1,2,2]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[2,2,2]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[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 = 1 + 1
[[1,4]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[2,4]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[3,4]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[4,4]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[1],[4]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[2],[4]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[3],[4]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,1,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[1,2,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[1,3,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[2,2,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[2,3,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[3,3,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[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 = 1 + 1
[[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 = 1 + 1
[[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 = 1 + 1
[[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 + 1
[[1,1,1,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[1,1,2,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[1,2,2,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[2,2,2,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[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)
=> ? = 1 + 1
[[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 + 1
[[1,5]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[2,5]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[3,5]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[4,5]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[5,5]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[1],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[2],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[3],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[4],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,1,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[1,2,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[1,3,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[1,4,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[2,2,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[2,3,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[2,4,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[3,3,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[3,4,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[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 + 1
[[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 + 1
[[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 + 1
[[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)
=> ? = 1 + 1
[[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)
=> ? = 1 + 1
[[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)
=> ? = 1 + 1
[[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)
=> ? = 1 + 1
[[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 + 1
[[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 + 1
[[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)
=> ? = 3 + 1
[[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 + 1
[[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 + 1
[[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 + 1
[[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)
=> ? = 2 + 1
[[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)
=> ? = 1 + 1
[[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 + 1
[[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 + 1
[[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 + 1
[[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 + 1
[[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 + 1
[[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 + 1
[[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 + 1
[[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)
=> ? = 1 + 1
[[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)
=> ? = 1 + 1
[[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)
=> ? = 1 + 1
[[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)
=> ? = 1 + 1
[[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)
=> ? = 1 + 1
[[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)
=> ? = 1 + 1
[[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)
=> ? = 1 + 1
[[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)
=> ? = 1 + 1
[[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)
=> ? = 1 + 1
[[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)
=> ? = 1 + 1
[[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 + 1
[[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 + 1
[[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 + 1
[[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)
=> ? = 3 + 1
[[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 + 1
[[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)
=> ? = 3 + 1
[[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 + 1
[[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)
=> ? = 3 + 1
[[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 + 1
[[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 + 1
[[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 + 1
[[2,3],[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)
=> ? = 3 + 1
[[2,3],[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 + 1
[[3,3],[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 + 1
[[1,1],[2],[4]]
=> [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 + 1
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
Matching statistic: St001232
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 50%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 50%
Values
[[1,2]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[2,2]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[1],[2]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[[1,3]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[2,3]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[3,3]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[1],[3]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[[2],[3]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[[1,1,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[1,2,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[2,2,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[1,1],[2]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[[1,4]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[2,4]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[3,4]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[4,4]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[1],[4]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[[2],[4]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[[3],[4]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[[1,1,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[1,2,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[1,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[2,2,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[2,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[3,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[1,1],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[[1,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[[2,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[[1],[2],[3]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[[1,1,1,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[[1,1,2,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[[1,2,2,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[[2,2,2,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[[1,1,1],[2]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1
[[1,1],[2,2]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? = 1
[[1,5]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[2,5]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[3,5]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[4,5]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[5,5]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[1],[5]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[[2],[5]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[[3],[5]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[[4],[5]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[[1,1,4]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[1,2,4]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[1,3,4]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[1,4,4]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[2,2,4]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[2,3,4]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[2,4,4]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[3,3,4]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[3,4,4]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[4,4,4]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[1,1],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[[1,2],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[[1,3],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[[2,2],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[[2,3],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[[3,3],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[[1],[2],[4]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[[1],[3],[4]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[[2],[3],[4]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[[1,1,1,3]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[[1,1,2,3]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[[1,1,3,3]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[[1,1,1],[3]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 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]
=> ? = 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]
=> ? = 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]
=> ? = 1
[[1,1],[2,3]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? = 1
[[1,1],[3,3]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? = 1
[[1,2],[2,3]]
=> [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 3
[[1,2],[3,3]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? = 1
[[2,2],[3,3]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? = 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
[[1,2],[2],[3]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 2
[[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]
=> ? = 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 1
[[1,1],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[[1,2],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[[1,3],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[[1,4],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[[2,2],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[[2,3],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[[2,4],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[[3,3],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[[3,4],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[[4,4],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[[1],[2],[5]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[[1],[3],[5]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[[1],[4],[5]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[[2],[3],[5]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[[2],[4],[5]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[[3],[4],[5]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[[1,1,1],[4]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1
[[1,1,2],[4]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1
[[1,1,3],[4]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1
[[1,2,2],[4]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1
[[1,2,3],[4]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001491
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 25%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 25%
Values
[[1,2]]
=> [1,2] => [1,2] => 1 => 1 = 0 + 1
[[2,2]]
=> [1,2] => [1,2] => 1 => 1 = 0 + 1
[[1],[2]]
=> [2,1] => [2,1] => 0 => ? = 1 + 1
[[1,3]]
=> [1,2] => [1,2] => 1 => 1 = 0 + 1
[[2,3]]
=> [1,2] => [1,2] => 1 => 1 = 0 + 1
[[3,3]]
=> [1,2] => [1,2] => 1 => 1 = 0 + 1
[[1],[3]]
=> [2,1] => [2,1] => 0 => ? = 1 + 1
[[2],[3]]
=> [2,1] => [2,1] => 0 => ? = 1 + 1
[[1,1,2]]
=> [1,2,3] => [1,3,2] => 10 => 1 = 0 + 1
[[1,2,2]]
=> [1,2,3] => [1,3,2] => 10 => 1 = 0 + 1
[[2,2,2]]
=> [1,2,3] => [1,3,2] => 10 => 1 = 0 + 1
[[1,1],[2]]
=> [3,1,2] => [3,1,2] => 00 => ? = 1 + 1
[[1,4]]
=> [1,2] => [1,2] => 1 => 1 = 0 + 1
[[2,4]]
=> [1,2] => [1,2] => 1 => 1 = 0 + 1
[[3,4]]
=> [1,2] => [1,2] => 1 => 1 = 0 + 1
[[4,4]]
=> [1,2] => [1,2] => 1 => 1 = 0 + 1
[[1],[4]]
=> [2,1] => [2,1] => 0 => ? = 1 + 1
[[2],[4]]
=> [2,1] => [2,1] => 0 => ? = 1 + 1
[[3],[4]]
=> [2,1] => [2,1] => 0 => ? = 1 + 1
[[1,1,3]]
=> [1,2,3] => [1,3,2] => 10 => 1 = 0 + 1
[[1,2,3]]
=> [1,2,3] => [1,3,2] => 10 => 1 = 0 + 1
[[1,3,3]]
=> [1,2,3] => [1,3,2] => 10 => 1 = 0 + 1
[[2,2,3]]
=> [1,2,3] => [1,3,2] => 10 => 1 = 0 + 1
[[2,3,3]]
=> [1,2,3] => [1,3,2] => 10 => 1 = 0 + 1
[[3,3,3]]
=> [1,2,3] => [1,3,2] => 10 => 1 = 0 + 1
[[1,1],[3]]
=> [3,1,2] => [3,1,2] => 00 => ? = 1 + 1
[[1,2],[3]]
=> [3,1,2] => [3,1,2] => 00 => ? = 1 + 1
[[2,2],[3]]
=> [3,1,2] => [3,1,2] => 00 => ? = 1 + 1
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 00 => ? = 1 + 1
[[1,1,1,2]]
=> [1,2,3,4] => [1,4,3,2] => 100 => 1 = 0 + 1
[[1,1,2,2]]
=> [1,2,3,4] => [1,4,3,2] => 100 => 1 = 0 + 1
[[1,2,2,2]]
=> [1,2,3,4] => [1,4,3,2] => 100 => 1 = 0 + 1
[[2,2,2,2]]
=> [1,2,3,4] => [1,4,3,2] => 100 => 1 = 0 + 1
[[1,1,1],[2]]
=> [4,1,2,3] => [4,1,3,2] => 000 => ? = 1 + 1
[[1,1],[2,2]]
=> [3,4,1,2] => [3,4,1,2] => 000 => ? = 1 + 1
[[1,5]]
=> [1,2] => [1,2] => 1 => 1 = 0 + 1
[[2,5]]
=> [1,2] => [1,2] => 1 => 1 = 0 + 1
[[3,5]]
=> [1,2] => [1,2] => 1 => 1 = 0 + 1
[[4,5]]
=> [1,2] => [1,2] => 1 => 1 = 0 + 1
[[5,5]]
=> [1,2] => [1,2] => 1 => 1 = 0 + 1
[[1],[5]]
=> [2,1] => [2,1] => 0 => ? = 1 + 1
[[2],[5]]
=> [2,1] => [2,1] => 0 => ? = 1 + 1
[[3],[5]]
=> [2,1] => [2,1] => 0 => ? = 1 + 1
[[4],[5]]
=> [2,1] => [2,1] => 0 => ? = 1 + 1
[[1,1,4]]
=> [1,2,3] => [1,3,2] => 10 => 1 = 0 + 1
[[1,2,4]]
=> [1,2,3] => [1,3,2] => 10 => 1 = 0 + 1
[[1,3,4]]
=> [1,2,3] => [1,3,2] => 10 => 1 = 0 + 1
[[1,4,4]]
=> [1,2,3] => [1,3,2] => 10 => 1 = 0 + 1
[[2,2,4]]
=> [1,2,3] => [1,3,2] => 10 => 1 = 0 + 1
[[2,3,4]]
=> [1,2,3] => [1,3,2] => 10 => 1 = 0 + 1
[[2,4,4]]
=> [1,2,3] => [1,3,2] => 10 => 1 = 0 + 1
[[3,3,4]]
=> [1,2,3] => [1,3,2] => 10 => 1 = 0 + 1
[[3,4,4]]
=> [1,2,3] => [1,3,2] => 10 => 1 = 0 + 1
[[4,4,4]]
=> [1,2,3] => [1,3,2] => 10 => 1 = 0 + 1
[[1,1],[4]]
=> [3,1,2] => [3,1,2] => 00 => ? = 1 + 1
[[1,2],[4]]
=> [3,1,2] => [3,1,2] => 00 => ? = 1 + 1
[[1,3],[4]]
=> [3,1,2] => [3,1,2] => 00 => ? = 1 + 1
[[2,2],[4]]
=> [3,1,2] => [3,1,2] => 00 => ? = 1 + 1
[[2,3],[4]]
=> [3,1,2] => [3,1,2] => 00 => ? = 1 + 1
[[3,3],[4]]
=> [3,1,2] => [3,1,2] => 00 => ? = 1 + 1
[[1],[2],[4]]
=> [3,2,1] => [3,2,1] => 00 => ? = 1 + 1
[[1],[3],[4]]
=> [3,2,1] => [3,2,1] => 00 => ? = 1 + 1
[[2],[3],[4]]
=> [3,2,1] => [3,2,1] => 00 => ? = 1 + 1
[[1,1,1,3]]
=> [1,2,3,4] => [1,4,3,2] => 100 => 1 = 0 + 1
[[1,1,2,3]]
=> [1,2,3,4] => [1,4,3,2] => 100 => 1 = 0 + 1
[[1,1,3,3]]
=> [1,2,3,4] => [1,4,3,2] => 100 => 1 = 0 + 1
[[1,2,2,3]]
=> [1,2,3,4] => [1,4,3,2] => 100 => 1 = 0 + 1
[[1,2,3,3]]
=> [1,2,3,4] => [1,4,3,2] => 100 => 1 = 0 + 1
[[1,3,3,3]]
=> [1,2,3,4] => [1,4,3,2] => 100 => 1 = 0 + 1
[[2,2,2,3]]
=> [1,2,3,4] => [1,4,3,2] => 100 => 1 = 0 + 1
[[2,2,3,3]]
=> [1,2,3,4] => [1,4,3,2] => 100 => 1 = 0 + 1
[[2,3,3,3]]
=> [1,2,3,4] => [1,4,3,2] => 100 => 1 = 0 + 1
[[3,3,3,3]]
=> [1,2,3,4] => [1,4,3,2] => 100 => 1 = 0 + 1
[[1,1,1],[3]]
=> [4,1,2,3] => [4,1,3,2] => 000 => ? = 1 + 1
[[1,1,2],[3]]
=> [4,1,2,3] => [4,1,3,2] => 000 => ? = 1 + 1
[[1,2,2],[3]]
=> [4,1,2,3] => [4,1,3,2] => 000 => ? = 1 + 1
[[2,2,2],[3]]
=> [4,1,2,3] => [4,1,3,2] => 000 => ? = 1 + 1
[[1,1],[2,3]]
=> [3,4,1,2] => [3,4,1,2] => 000 => ? = 1 + 1
[[1,1],[3,3]]
=> [3,4,1,2] => [3,4,1,2] => 000 => ? = 1 + 1
[[1,2],[2,3]]
=> [2,4,1,3] => [2,4,1,3] => 000 => ? = 3 + 1
[[1,2],[3,3]]
=> [3,4,1,2] => [3,4,1,2] => 000 => ? = 1 + 1
[[2,2],[3,3]]
=> [3,4,1,2] => [3,4,1,2] => 000 => ? = 1 + 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [4,3,1,2] => 000 => ? = 1 + 1
[[1,2],[2],[3]]
=> [4,2,1,3] => [4,2,1,3] => 000 => ? = 2 + 1
[[1,1,1,1,2]]
=> [1,2,3,4,5] => [1,5,4,3,2] => 1000 => 1 = 0 + 1
[[1,1,1,2,2]]
=> [1,2,3,4,5] => [1,5,4,3,2] => 1000 => 1 = 0 + 1
[[1,1,2,2,2]]
=> [1,2,3,4,5] => [1,5,4,3,2] => 1000 => 1 = 0 + 1
[[1,1,1,1],[2]]
=> [5,1,2,3,4] => [5,1,4,3,2] => 0000 => ? = 1 + 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [4,5,1,3,2] => 0000 => ? = 1 + 1
[[1],[6]]
=> [2,1] => [2,1] => 0 => ? = 1 + 1
[[2],[6]]
=> [2,1] => [2,1] => 0 => ? = 1 + 1
[[3],[6]]
=> [2,1] => [2,1] => 0 => ? = 1 + 1
[[4],[6]]
=> [2,1] => [2,1] => 0 => ? = 1 + 1
[[5],[6]]
=> [2,1] => [2,1] => 0 => ? = 1 + 1
[[1,1],[5]]
=> [3,1,2] => [3,1,2] => 00 => ? = 1 + 1
[[1,2],[5]]
=> [3,1,2] => [3,1,2] => 00 => ? = 1 + 1
[[1,3],[5]]
=> [3,1,2] => [3,1,2] => 00 => ? = 1 + 1
[[1,4],[5]]
=> [3,1,2] => [3,1,2] => 00 => ? = 1 + 1
[[2,2],[5]]
=> [3,1,2] => [3,1,2] => 00 => ? = 1 + 1
[[2,3],[5]]
=> [3,1,2] => [3,1,2] => 00 => ? = 1 + 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: St000260
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 75%
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 75%
Values
[[1,2]]
=> [1,2] => [1,2] => ([],2)
=> ? = 0
[[2,2]]
=> [1,2] => [1,2] => ([],2)
=> ? = 0
[[1],[2]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[1,3]]
=> [1,2] => [1,2] => ([],2)
=> ? = 0
[[2,3]]
=> [1,2] => [1,2] => ([],2)
=> ? = 0
[[3,3]]
=> [1,2] => [1,2] => ([],2)
=> ? = 0
[[1],[3]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[2],[3]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[1,1,2]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? = 0
[[1,2,2]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? = 0
[[2,2,2]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? = 0
[[1,1],[2]]
=> [3,1,2] => [1,3,2] => ([(1,2)],3)
=> ? = 1
[[1,4]]
=> [1,2] => [1,2] => ([],2)
=> ? = 0
[[2,4]]
=> [1,2] => [1,2] => ([],2)
=> ? = 0
[[3,4]]
=> [1,2] => [1,2] => ([],2)
=> ? = 0
[[4,4]]
=> [1,2] => [1,2] => ([],2)
=> ? = 0
[[1],[4]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[2],[4]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[3],[4]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[1,1,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? = 0
[[1,2,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? = 0
[[1,3,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? = 0
[[2,2,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? = 0
[[2,3,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? = 0
[[3,3,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? = 0
[[1,1],[3]]
=> [3,1,2] => [1,3,2] => ([(1,2)],3)
=> ? = 1
[[1,2],[3]]
=> [3,1,2] => [1,3,2] => ([(1,2)],3)
=> ? = 1
[[2,2],[3]]
=> [3,1,2] => [1,3,2] => ([(1,2)],3)
=> ? = 1
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1,1,1,2]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? = 0
[[1,1,2,2]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? = 0
[[1,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? = 0
[[2,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? = 0
[[1,1,1],[2]]
=> [4,1,2,3] => [1,2,4,3] => ([(2,3)],4)
=> ? = 1
[[1,1],[2,2]]
=> [3,4,1,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? = 1
[[1,5]]
=> [1,2] => [1,2] => ([],2)
=> ? = 0
[[2,5]]
=> [1,2] => [1,2] => ([],2)
=> ? = 0
[[3,5]]
=> [1,2] => [1,2] => ([],2)
=> ? = 0
[[4,5]]
=> [1,2] => [1,2] => ([],2)
=> ? = 0
[[5,5]]
=> [1,2] => [1,2] => ([],2)
=> ? = 0
[[1],[5]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[2],[5]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[3],[5]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[4],[5]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[1,1,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? = 0
[[1,2,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? = 0
[[1,3,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? = 0
[[1,4,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? = 0
[[2,2,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? = 0
[[2,3,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? = 0
[[2,4,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? = 0
[[3,3,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? = 0
[[3,4,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? = 0
[[4,4,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? = 0
[[1,1],[4]]
=> [3,1,2] => [1,3,2] => ([(1,2)],3)
=> ? = 1
[[1,2],[4]]
=> [3,1,2] => [1,3,2] => ([(1,2)],3)
=> ? = 1
[[1,3],[4]]
=> [3,1,2] => [1,3,2] => ([(1,2)],3)
=> ? = 1
[[2,2],[4]]
=> [3,1,2] => [1,3,2] => ([(1,2)],3)
=> ? = 1
[[2,3],[4]]
=> [3,1,2] => [1,3,2] => ([(1,2)],3)
=> ? = 1
[[3,3],[4]]
=> [3,1,2] => [1,3,2] => ([(1,2)],3)
=> ? = 1
[[1],[2],[4]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1],[3],[4]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[2],[3],[4]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1,1,1,3]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? = 0
[[1,2],[2],[3]]
=> [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[1],[6]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[2],[6]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[3],[6]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[4],[6]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[5],[6]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[1],[2],[5]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1],[3],[5]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1],[4],[5]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[2],[3],[5]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[2],[4],[5]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[3],[4],[5]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1,2],[2],[4]]
=> [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,3],[2],[4]]
=> [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,3],[3],[4]]
=> [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[2,3],[3],[4]]
=> [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1,2],[2,3],[3]]
=> [4,2,5,1,3] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[[1],[7]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[2],[7]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[3],[7]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[4],[7]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[5],[7]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[6],[7]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[1],[2],[6]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1],[3],[6]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1],[4],[6]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1],[5],[6]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[2],[3],[6]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[2],[4],[6]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[2],[5],[6]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[3],[4],[6]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[3],[5],[6]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[4],[5],[6]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1,2],[2],[5]]
=> [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,3],[2],[5]]
=> [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St000777
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 50%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 50%
Values
[[1,2]]
=> [1,2] => [2] => ([],2)
=> ? = 0 + 1
[[2,2]]
=> [1,2] => [2] => ([],2)
=> ? = 0 + 1
[[1],[2]]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[1,3]]
=> [1,2] => [2] => ([],2)
=> ? = 0 + 1
[[2,3]]
=> [1,2] => [2] => ([],2)
=> ? = 0 + 1
[[3,3]]
=> [1,2] => [2] => ([],2)
=> ? = 0 + 1
[[1],[3]]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[2],[3]]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[1,1,2]]
=> [1,2,3] => [3] => ([],3)
=> ? = 0 + 1
[[1,2,2]]
=> [1,2,3] => [3] => ([],3)
=> ? = 0 + 1
[[2,2,2]]
=> [1,2,3] => [3] => ([],3)
=> ? = 0 + 1
[[1,1],[2]]
=> [3,1,2] => [1,2] => ([(1,2)],3)
=> ? = 1 + 1
[[1,4]]
=> [1,2] => [2] => ([],2)
=> ? = 0 + 1
[[2,4]]
=> [1,2] => [2] => ([],2)
=> ? = 0 + 1
[[3,4]]
=> [1,2] => [2] => ([],2)
=> ? = 0 + 1
[[4,4]]
=> [1,2] => [2] => ([],2)
=> ? = 0 + 1
[[1],[4]]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[2],[4]]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[3],[4]]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[1,1,3]]
=> [1,2,3] => [3] => ([],3)
=> ? = 0 + 1
[[1,2,3]]
=> [1,2,3] => [3] => ([],3)
=> ? = 0 + 1
[[1,3,3]]
=> [1,2,3] => [3] => ([],3)
=> ? = 0 + 1
[[2,2,3]]
=> [1,2,3] => [3] => ([],3)
=> ? = 0 + 1
[[2,3,3]]
=> [1,2,3] => [3] => ([],3)
=> ? = 0 + 1
[[3,3,3]]
=> [1,2,3] => [3] => ([],3)
=> ? = 0 + 1
[[1,1],[3]]
=> [3,1,2] => [1,2] => ([(1,2)],3)
=> ? = 1 + 1
[[1,2],[3]]
=> [3,1,2] => [1,2] => ([(1,2)],3)
=> ? = 1 + 1
[[2,2],[3]]
=> [3,1,2] => [1,2] => ([(1,2)],3)
=> ? = 1 + 1
[[1],[2],[3]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1,1,1,2]]
=> [1,2,3,4] => [4] => ([],4)
=> ? = 0 + 1
[[1,1,2,2]]
=> [1,2,3,4] => [4] => ([],4)
=> ? = 0 + 1
[[1,2,2,2]]
=> [1,2,3,4] => [4] => ([],4)
=> ? = 0 + 1
[[2,2,2,2]]
=> [1,2,3,4] => [4] => ([],4)
=> ? = 0 + 1
[[1,1,1],[2]]
=> [4,1,2,3] => [1,3] => ([(2,3)],4)
=> ? = 1 + 1
[[1,1],[2,2]]
=> [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[[1,5]]
=> [1,2] => [2] => ([],2)
=> ? = 0 + 1
[[2,5]]
=> [1,2] => [2] => ([],2)
=> ? = 0 + 1
[[3,5]]
=> [1,2] => [2] => ([],2)
=> ? = 0 + 1
[[4,5]]
=> [1,2] => [2] => ([],2)
=> ? = 0 + 1
[[5,5]]
=> [1,2] => [2] => ([],2)
=> ? = 0 + 1
[[1],[5]]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[2],[5]]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[3],[5]]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[4],[5]]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[1,1,4]]
=> [1,2,3] => [3] => ([],3)
=> ? = 0 + 1
[[1,2,4]]
=> [1,2,3] => [3] => ([],3)
=> ? = 0 + 1
[[1,3,4]]
=> [1,2,3] => [3] => ([],3)
=> ? = 0 + 1
[[1,4,4]]
=> [1,2,3] => [3] => ([],3)
=> ? = 0 + 1
[[2,2,4]]
=> [1,2,3] => [3] => ([],3)
=> ? = 0 + 1
[[2,3,4]]
=> [1,2,3] => [3] => ([],3)
=> ? = 0 + 1
[[2,4,4]]
=> [1,2,3] => [3] => ([],3)
=> ? = 0 + 1
[[3,3,4]]
=> [1,2,3] => [3] => ([],3)
=> ? = 0 + 1
[[3,4,4]]
=> [1,2,3] => [3] => ([],3)
=> ? = 0 + 1
[[4,4,4]]
=> [1,2,3] => [3] => ([],3)
=> ? = 0 + 1
[[1,1],[4]]
=> [3,1,2] => [1,2] => ([(1,2)],3)
=> ? = 1 + 1
[[1,2],[4]]
=> [3,1,2] => [1,2] => ([(1,2)],3)
=> ? = 1 + 1
[[1,3],[4]]
=> [3,1,2] => [1,2] => ([(1,2)],3)
=> ? = 1 + 1
[[2,2],[4]]
=> [3,1,2] => [1,2] => ([(1,2)],3)
=> ? = 1 + 1
[[2,3],[4]]
=> [3,1,2] => [1,2] => ([(1,2)],3)
=> ? = 1 + 1
[[3,3],[4]]
=> [3,1,2] => [1,2] => ([(1,2)],3)
=> ? = 1 + 1
[[1],[2],[4]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1],[3],[4]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[[2],[3],[4]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1,1,1,3]]
=> [1,2,3,4] => [4] => ([],4)
=> ? = 0 + 1
[[1],[6]]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[2],[6]]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[3],[6]]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[4],[6]]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[5],[6]]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[1],[2],[5]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1],[3],[5]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1],[4],[5]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[[2],[3],[5]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[[2],[4],[5]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[[3],[4],[5]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1],[7]]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[2],[7]]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[3],[7]]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[4],[7]]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[5],[7]]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[6],[7]]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[1],[2],[6]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1],[3],[6]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1],[4],[6]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1],[5],[6]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[[2],[3],[6]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[[2],[4],[6]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[[2],[5],[6]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[[3],[4],[6]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[[3],[5],[6]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[[4],[5],[6]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1],[2],[3],[5]]
=> [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1],[2],[4],[5]]
=> [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1],[3],[4],[5]]
=> [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[2],[3],[4],[5]]
=> [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1],[8]]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[2],[8]]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[3],[8]]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[4],[8]]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Matching statistic: St000112
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
St000112: Semistandard tableaux ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 50%
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
St000112: Semistandard tableaux ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 50%
Values
[[1,2]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[2,2]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[1],[2]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1
[[1,3]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[2,3]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[3,3]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[1],[3]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1
[[2],[3]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1
[[1,1,2]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0
[[1,2,2]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0
[[2,2,2]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0
[[1,1],[2]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 1
[[1,4]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[2,4]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[3,4]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[4,4]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[1],[4]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1
[[2],[4]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1
[[3],[4]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1
[[1,1,3]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0
[[1,2,3]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0
[[1,3,3]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0
[[2,2,3]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0
[[2,3,3]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0
[[3,3,3]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0
[[1,1],[3]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 1
[[1,2],[3]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 1
[[2,2],[3]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 1
[[1],[2],[3]]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> ? = 1
[[1,1,1,2]]
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 0
[[1,1,2,2]]
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 0
[[1,2,2,2]]
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 0
[[2,2,2,2]]
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 0
[[1,1,1],[2]]
=> [4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,2],[2,3,3],[3,4],[4]]
=> ? = 1
[[1,1],[2,2]]
=> [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,3],[2,3,4],[3,4],[4]]
=> ? = 1
[[1,5]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[2,5]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[3,5]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[4,5]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[5,5]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[1],[5]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1
[[2],[5]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1
[[3],[5]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1
[[4],[5]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1
[[1,1,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0
[[1,2,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0
[[1,3,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0
[[1,4,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0
[[2,2,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0
[[2,3,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0
[[2,4,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0
[[3,3,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0
[[3,4,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0
[[4,4,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0
[[1,1],[4]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 1
[[1,2],[4]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 1
[[1,3],[4]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 1
[[2,2],[4]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 1
[[2,3],[4]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 1
[[3,3],[4]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 1
[[1],[2],[4]]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> ? = 1
[[1],[3],[4]]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> ? = 1
[[2],[3],[4]]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> ? = 1
[[1,1,1,3]]
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 0
[[1,1,2,3]]
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 0
[[1,1,3,3]]
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 0
[[1,2,2,3]]
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 0
[[1,2,3,3]]
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 0
[[1,3,3,3]]
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 0
[[2,2,2,3]]
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 0
[[2,2,3,3]]
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 0
[[2,3,3,3]]
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 0
[[3,3,3,3]]
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 0
[[1,1,1],[3]]
=> [4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,2],[2,3,3],[3,4],[4]]
=> ? = 1
[[1,6]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[2,6]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[3,6]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[4,6]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[5,6]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[6,6]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[1],[6]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1
[[2],[6]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1
[[3],[6]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1
[[4],[6]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1
[[5],[6]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1
[[1,7]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[2,7]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[3,7]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[4,7]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[5,7]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[6,7]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[7,7]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[1],[7]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1
[[2],[7]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1
[[3],[7]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1
[[4],[7]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1
[[5],[7]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1
[[6],[7]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1
[[1,8]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[2,8]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
Description
The sum of the entries reduced by the index of their row in a semistandard tableau.
This is also the depth of a semistandard tableau $T$ in the crystal $B(\lambda)$ where $\lambda$ is the shape of $T$, independent of the Cartan rank.
Matching statistic: St000736
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
St000736: Semistandard tableaux ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 50%
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
St000736: Semistandard tableaux ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 50%
Values
[[1,2]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 0 + 1
[[2,2]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 0 + 1
[[1],[2]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2 = 1 + 1
[[1,3]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 0 + 1
[[2,3]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 0 + 1
[[3,3]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 0 + 1
[[1],[3]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2 = 1 + 1
[[2],[3]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2 = 1 + 1
[[1,1,2]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0 + 1
[[1,2,2]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0 + 1
[[2,2,2]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0 + 1
[[1,1],[2]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 1 + 1
[[1,4]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 0 + 1
[[2,4]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 0 + 1
[[3,4]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 0 + 1
[[4,4]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 0 + 1
[[1],[4]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2 = 1 + 1
[[2],[4]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2 = 1 + 1
[[3],[4]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2 = 1 + 1
[[1,1,3]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0 + 1
[[1,2,3]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0 + 1
[[1,3,3]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0 + 1
[[2,2,3]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0 + 1
[[2,3,3]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0 + 1
[[3,3,3]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0 + 1
[[1,1],[3]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 1 + 1
[[1,2],[3]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 1 + 1
[[2,2],[3]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 1 + 1
[[1],[2],[3]]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> ? = 1 + 1
[[1,1,1,2]]
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 0 + 1
[[1,1,2,2]]
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 0 + 1
[[1,2,2,2]]
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 0 + 1
[[2,2,2,2]]
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 0 + 1
[[1,1,1],[2]]
=> [4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,2],[2,3,3],[3,4],[4]]
=> ? = 1 + 1
[[1,1],[2,2]]
=> [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,3],[2,3,4],[3,4],[4]]
=> ? = 1 + 1
[[1,5]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 0 + 1
[[2,5]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 0 + 1
[[3,5]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 0 + 1
[[4,5]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 0 + 1
[[5,5]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 0 + 1
[[1],[5]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2 = 1 + 1
[[2],[5]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2 = 1 + 1
[[3],[5]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2 = 1 + 1
[[4],[5]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2 = 1 + 1
[[1,1,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0 + 1
[[1,2,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0 + 1
[[1,3,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0 + 1
[[1,4,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0 + 1
[[2,2,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0 + 1
[[2,3,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0 + 1
[[2,4,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0 + 1
[[3,3,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0 + 1
[[3,4,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0 + 1
[[4,4,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0 + 1
[[1,1],[4]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 1 + 1
[[1,2],[4]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 1 + 1
[[1,3],[4]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 1 + 1
[[2,2],[4]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 1 + 1
[[2,3],[4]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 1 + 1
[[3,3],[4]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 1 + 1
[[1],[2],[4]]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> ? = 1 + 1
[[1],[3],[4]]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> ? = 1 + 1
[[2],[3],[4]]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> ? = 1 + 1
[[1,1,1,3]]
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 0 + 1
[[1,1,2,3]]
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 0 + 1
[[1,1,3,3]]
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 0 + 1
[[1,2,2,3]]
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 0 + 1
[[1,2,3,3]]
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 0 + 1
[[1,3,3,3]]
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 0 + 1
[[2,2,2,3]]
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 0 + 1
[[2,2,3,3]]
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 0 + 1
[[2,3,3,3]]
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 0 + 1
[[3,3,3,3]]
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 0 + 1
[[1,1,1],[3]]
=> [4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,2],[2,3,3],[3,4],[4]]
=> ? = 1 + 1
[[1,6]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 0 + 1
[[2,6]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 0 + 1
[[3,6]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 0 + 1
[[4,6]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 0 + 1
[[5,6]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 0 + 1
[[6,6]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 0 + 1
[[1],[6]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2 = 1 + 1
[[2],[6]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2 = 1 + 1
[[3],[6]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2 = 1 + 1
[[4],[6]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2 = 1 + 1
[[5],[6]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2 = 1 + 1
[[1,7]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 0 + 1
[[2,7]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 0 + 1
[[3,7]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 0 + 1
[[4,7]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 0 + 1
[[5,7]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 0 + 1
[[6,7]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 0 + 1
[[7,7]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 0 + 1
[[1],[7]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2 = 1 + 1
[[2],[7]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2 = 1 + 1
[[3],[7]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2 = 1 + 1
[[4],[7]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2 = 1 + 1
[[5],[7]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2 = 1 + 1
[[6],[7]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2 = 1 + 1
[[1,8]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 0 + 1
[[2,8]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 0 + 1
Description
The last entry in the first row of a semistandard tableau.
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