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Your data matches 88 different statistics following compositions of up to 3 maps.
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Matching statistic: St001128
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001128: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001128: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,1,2]]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[[1,2,2]]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[[2,2,2]]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[[1,1,3]]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[[1,2,3]]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[[1,3,3]]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[[2,2,3]]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[[2,3,3]]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[[3,3,3]]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[[1,1,1,2]]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[[1,1,2,2]]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[[1,2,2,2]]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[[2,2,2,2]]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[[1,2,2],[2]]
=> [2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[[1,1],[2,2]]
=> [3,4,1,2] => [2,2]
=> [2]
=> 1
[[1,1,4]]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[[1,2,4]]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[[1,3,4]]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[[1,4,4]]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[[2,2,4]]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[[2,3,4]]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[[2,4,4]]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[[3,3,4]]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[[3,4,4]]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[[4,4,4]]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[[1,1,1,3]]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[[1,1,2,3]]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[[1,1,3,3]]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[[1,2,2,3]]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[[1,2,3,3]]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[[1,3,3,3]]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[[2,2,2,3]]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[[2,2,3,3]]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[[2,3,3,3]]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[[3,3,3,3]]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[[1,2,3],[2]]
=> [2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[[1,3,3],[2]]
=> [2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[[1,3,3],[3]]
=> [2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[[2,3,3],[3]]
=> [2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[[1,1],[2,3]]
=> [3,4,1,2] => [2,2]
=> [2]
=> 1
[[1,1],[3,3]]
=> [3,4,1,2] => [2,2]
=> [2]
=> 1
[[1,2],[3,3]]
=> [3,4,1,2] => [2,2]
=> [2]
=> 1
[[2,2],[3,3]]
=> [3,4,1,2] => [2,2]
=> [2]
=> 1
[[1,3],[2],[3]]
=> [3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[[1,1,1,1,2]]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[[1,1,1,2,2]]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[[1,1,2,2,2]]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[[1,2,2,2,2]]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[[2,2,2,2,2]]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[[1,1,2,2],[2]]
=> [3,1,2,4,5] => [3,1,1]
=> [1,1]
=> 1
Description
The exponens consonantiae of a partition.
This is the quotient of the least common multiple and the greatest common divior of the parts of the partiton. See [1, Caput sextum, §19-§22].
Matching statistic: St001845
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Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001845: Lattices ⟶ ℤResult quality: 33% ●values known / values provided: 56%●distinct values known / distinct values provided: 33%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001845: Lattices ⟶ ℤResult quality: 33% ●values known / values provided: 56%●distinct values known / distinct values provided: 33%
Values
[[1,1,2]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[1,2,2]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[2,2,2]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[1,1,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[1,2,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[1,3,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[2,2,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[2,3,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[3,3,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 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)
=> 0 = 1 - 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)
=> 0 = 1 - 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)
=> 0 = 1 - 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)
=> 0 = 1 - 1
[[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)
=> 0 = 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)
=> 0 = 1 - 1
[[1,1,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[1,2,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[1,3,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[1,4,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[2,2,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[2,3,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[2,4,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[3,3,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[3,4,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[4,4,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[1,1,1,3]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[[1,1,2,3]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[[1,1,3,3]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[[1,2,2,3]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[[1,2,3,3]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[[1,3,3,3]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[[2,2,2,3]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[[2,2,3,3]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[[2,3,3,3]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[[3,3,3,3]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[[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)
=> 0 = 1 - 1
[[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)
=> 0 = 1 - 1
[[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)
=> 0 = 1 - 1
[[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)
=> 0 = 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)
=> 0 = 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)
=> 0 = 1 - 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)
=> 0 = 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)
=> 0 = 1 - 1
[[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)
=> 0 = 1 - 1
[[1,1,1,1,2]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[[1,1,1,2,2]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[[1,1,2,2,2]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[[1,2,2,2,2]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[[2,2,2,2,2]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[[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 = 1 - 1
[[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)
=> ? = 2 - 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 1 - 1
[[1,1,2],[2,3]]
=> [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ? = 1 - 1
[[1,1,3],[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)
=> ? = 2 - 1
[[1,1,3],[2,3]]
=> [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)
=> ? = 2 - 1
[[1,1,3],[3,3]]
=> [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)
=> ? = 2 - 1
[[1,2,3],[3,3]]
=> [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)
=> ? = 2 - 1
[[2,2,3],[3,3]]
=> [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)
=> ? = 2 - 1
[[1,1,1],[2],[3]]
=> [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,6),(1,7),(2,1),(2,9),(2,10),(3,8),(3,12),(4,8),(4,11),(5,2),(5,11),(5,12),(6,14),(7,14),(8,13),(9,6),(9,15),(10,7),(10,15),(11,9),(11,13),(12,10),(12,13),(13,15),(15,14)],16)
=> ? = 1 - 1
[[1,2,3],[2],[3]]
=> [4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 1 - 1
[[1,3,3],[2],[3]]
=> [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,2),(0,3),(0,4),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,1),(6,9),(7,9),(8,9),(9,5)],10)
=> ? = 1 - 1
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(0,5),(1,10),(2,7),(2,8),(3,9),(3,12),(4,9),(4,11),(5,2),(5,11),(5,12),(7,14),(8,14),(9,1),(9,13),(10,6),(11,7),(11,13),(12,8),(12,13),(13,10),(13,14),(14,6)],15)
=> ? = 1 - 1
[[1,2],[2,3],[3]]
=> [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,11),(2,10),(3,8),(3,9),(4,7),(4,8),(5,7),(5,9),(7,12),(8,2),(8,12),(9,1),(9,12),(10,6),(11,6),(12,10),(12,11)],13)
=> ? = 2 - 1
[[1,1,1,2,2],[2]]
=> [4,1,2,3,5,6] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 1 - 1
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => ([(0,5),(1,3),(4,2),(5,4)],6)
=> ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> ? = 1 - 1
[[1,1,2,2],[2,2]]
=> [3,4,1,2,5,6] => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
=> ? = 2 - 1
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => ([(0,5),(1,4),(4,2),(5,3)],6)
=> ([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> ? = 1 - 1
[[1],[2],[3],[5]]
=> [4,3,2,1] => ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 1 - 1
[[1],[2],[4],[5]]
=> [4,3,2,1] => ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 1 - 1
[[1],[3],[4],[5]]
=> [4,3,2,1] => ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 1 - 1
[[2],[3],[4],[5]]
=> [4,3,2,1] => ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 1 - 1
[[1,1,2],[2,4]]
=> [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ? = 1 - 1
[[1,1,4],[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)
=> ? = 2 - 1
[[1,1,3],[2,4]]
=> [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ? = 1 - 1
[[1,1,4],[2,3]]
=> [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)
=> ? = 2 - 1
[[1,1,4],[2,4]]
=> [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)
=> ? = 2 - 1
[[1,1,3],[3,4]]
=> [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ? = 1 - 1
[[1,1,4],[3,3]]
=> [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)
=> ? = 2 - 1
[[1,1,4],[3,4]]
=> [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)
=> ? = 2 - 1
[[1,1,4],[4,4]]
=> [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)
=> ? = 2 - 1
[[1,2,3],[3,4]]
=> [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ? = 1 - 1
[[1,2,4],[3,3]]
=> [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)
=> ? = 2 - 1
[[1,2,4],[3,4]]
=> [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)
=> ? = 2 - 1
[[1,2,4],[4,4]]
=> [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)
=> ? = 2 - 1
[[1,3,4],[4,4]]
=> [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)
=> ? = 2 - 1
[[2,2,3],[3,4]]
=> [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ? = 1 - 1
[[2,2,4],[3,3]]
=> [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)
=> ? = 2 - 1
[[2,2,4],[3,4]]
=> [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)
=> ? = 2 - 1
[[2,2,4],[4,4]]
=> [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)
=> ? = 2 - 1
[[2,3,4],[4,4]]
=> [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)
=> ? = 2 - 1
[[3,3,4],[4,4]]
=> [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)
=> ? = 2 - 1
[[1,1,1],[2],[4]]
=> [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,6),(1,7),(2,1),(2,9),(2,10),(3,8),(3,12),(4,8),(4,11),(5,2),(5,11),(5,12),(6,14),(7,14),(8,13),(9,6),(9,15),(10,7),(10,15),(11,9),(11,13),(12,10),(12,13),(13,15),(15,14)],16)
=> ? = 1 - 1
[[1,1,1],[3],[4]]
=> [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,6),(1,7),(2,1),(2,9),(2,10),(3,8),(3,12),(4,8),(4,11),(5,2),(5,11),(5,12),(6,14),(7,14),(8,13),(9,6),(9,15),(10,7),(10,15),(11,9),(11,13),(12,10),(12,13),(13,15),(15,14)],16)
=> ? = 1 - 1
[[1,1,2],[3],[4]]
=> [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,6),(1,7),(2,1),(2,9),(2,10),(3,8),(3,12),(4,8),(4,11),(5,2),(5,11),(5,12),(6,14),(7,14),(8,13),(9,6),(9,15),(10,7),(10,15),(11,9),(11,13),(12,10),(12,13),(13,15),(15,14)],16)
=> ? = 1 - 1
[[1,2,2],[3],[4]]
=> [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,6),(1,7),(2,1),(2,9),(2,10),(3,8),(3,12),(4,8),(4,11),(5,2),(5,11),(5,12),(6,14),(7,14),(8,13),(9,6),(9,15),(10,7),(10,15),(11,9),(11,13),(12,10),(12,13),(13,15),(15,14)],16)
=> ? = 1 - 1
[[1,2,4],[2],[3]]
=> [4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 1 - 1
[[1,2,4],[2],[4]]
=> [4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 1 - 1
[[1,3,4],[2],[3]]
=> [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,2),(0,3),(0,4),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,1),(6,9),(7,9),(8,9),(9,5)],10)
=> ? = 1 - 1
[[1,3,4],[2],[4]]
=> [4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 1 - 1
[[1,4,4],[2],[3]]
=> [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,2),(0,3),(0,4),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,1),(6,9),(7,9),(8,9),(9,5)],10)
=> ? = 1 - 1
Description
The number of join irreducibles minus the rank of a lattice.
A lattice is join-extremal, if this statistic is 0.
Matching statistic: St000068
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 33% ●values known / values provided: 54%●distinct values known / distinct values provided: 33%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 33% ●values known / values provided: 54%●distinct values known / distinct values provided: 33%
Values
[[1,1,2]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[1,2,2]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[2,2,2]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[1,1,3]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[1,2,3]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[1,3,3]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[2,2,3]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[2,3,3]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[3,3,3]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[1,1,1,2]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,1,2,2]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[2,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,2,2],[2]]
=> [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[[1,1],[2,2]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,1,4]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[1,2,4]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[1,3,4]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[1,4,4]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[2,2,4]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[2,3,4]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[2,4,4]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[3,3,4]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[3,4,4]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[4,4,4]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[1,1,1,3]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,1,2,3]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,1,3,3]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,2,2,3]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,2,3,3]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,3,3,3]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[2,2,2,3]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[2,2,3,3]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[2,3,3,3]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[3,3,3,3]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,2,3],[2]]
=> [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[[1,3,3],[2]]
=> [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[[1,3,3],[3]]
=> [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[[2,3,3],[3]]
=> [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[[1,1],[2,3]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,1],[3,3]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,2],[3,3]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[2,2],[3,3]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,3],[2],[3]]
=> [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1
[[1,1,1,1,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[1,1,1,2,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[1,1,2,2,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[1,2,2,2,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[2,2,2,2,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[1,1,2,2],[2]]
=> [3,1,2,4,5] => [3,2,1,4,5] => ([(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
[[1,2,2,2],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(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],[2,2]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2
[[1,1,5]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[1,2,5]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[1,3,5]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[1,4,5]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[1,1],[2,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,1],[3,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,1],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,2],[3,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,2],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,3],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[2,2],[3,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[2,2],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[2,3],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[3,3],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,1,2],[2,3]]
=> [3,5,1,2,4] => [3,1,5,4,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 1
[[1,1,3],[2,2]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2
[[1,1,3],[2,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2
[[1,1,3],[3,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2
[[1,2,3],[3,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2
[[2,2,3],[3,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2
[[1,1,1],[2],[3]]
=> [5,4,1,2,3] => [4,2,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1
[[1,2,3],[2],[3]]
=> [4,2,1,3,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 1
[[1,3,3],[2],[3]]
=> [3,2,1,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [4,1,5,2,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 1
[[1,2],[2,3],[3]]
=> [4,2,5,1,3] => [2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ? = 2
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [5,3,1,6,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(1,16),(1,17),(1,24),(2,11),(2,15),(2,17),(2,24),(3,9),(3,13),(3,15),(3,24),(4,10),(4,14),(4,16),(4,24),(5,7),(5,9),(5,11),(5,14),(5,24),(6,7),(6,10),(6,12),(6,13),(6,24),(7,21),(7,22),(7,25),(9,21),(9,25),(10,22),(10,25),(11,19),(11,21),(11,25),(12,20),(12,22),(12,25),(13,19),(13,22),(13,25),(14,20),(14,21),(14,25),(15,19),(15,25),(16,20),(16,25),(17,19),(17,20),(18,8),(19,18),(19,23),(20,18),(20,23),(21,18),(21,23),(22,18),(22,23),(23,8),(24,19),(24,20),(24,21),(24,22),(25,23)],26)
=> ? = 1
[[1,1,2,2],[2,2]]
=> [3,4,1,2,5,6] => [3,1,4,2,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,6),(1,14),(1,18),(2,13),(2,14),(2,18),(3,12),(3,14),(3,18),(4,11),(4,14),(4,18),(5,8),(5,9),(5,10),(5,16),(6,5),(6,11),(6,12),(6,13),(6,18),(8,15),(8,19),(9,15),(9,19),(10,15),(10,19),(11,8),(11,16),(11,17),(12,9),(12,16),(12,17),(13,10),(13,16),(13,17),(14,17),(15,7),(16,15),(16,19),(17,19),(18,16),(18,17),(19,7)],20)
=> ? = 2
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [4,1,5,2,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(1,16),(1,17),(1,24),(2,11),(2,15),(2,17),(2,24),(3,9),(3,13),(3,15),(3,24),(4,10),(4,14),(4,16),(4,24),(5,7),(5,9),(5,11),(5,14),(5,24),(6,7),(6,10),(6,12),(6,13),(6,24),(7,21),(7,22),(7,25),(9,21),(9,25),(10,22),(10,25),(11,19),(11,21),(11,25),(12,20),(12,22),(12,25),(13,19),(13,22),(13,25),(14,20),(14,21),(14,25),(15,19),(15,25),(16,20),(16,25),(17,19),(17,20),(18,8),(19,18),(19,23),(20,18),(20,23),(21,18),(21,23),(22,18),(22,23),(23,8),(24,19),(24,20),(24,21),(24,22),(25,23)],26)
=> ? = 1
[[1,1],[2,5]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,1],[3,5]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,1],[4,5]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,1],[5,5]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,2],[3,5]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,2],[4,5]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,2],[5,5]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,3],[4,5]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,3],[5,5]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,4],[5,5]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[2,2],[3,5]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[2,2],[4,5]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[2,2],[5,5]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[2,3],[4,5]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[2,3],[5,5]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[2,4],[5,5]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[3,3],[4,5]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[3,3],[5,5]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[3,4],[5,5]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[4,4],[5,5]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
Description
The number of minimal elements in a poset.
Matching statistic: St001771
(load all 21 compositions to match this statistic)
(load all 21 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001771: Signed permutations ⟶ ℤResult quality: 33% ●values known / values provided: 43%●distinct values known / distinct values provided: 33%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001771: Signed permutations ⟶ ℤResult quality: 33% ●values known / values provided: 43%●distinct values known / distinct values provided: 33%
Values
[[1,1,2]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,2,2]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[2,2,2]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,1,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,2,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,3,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[2,2,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[2,3,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[3,3,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,1,1,2]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,1,2,2]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[2,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,2,2],[2]]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0 = 1 - 1
[[1,1],[2,2]]
=> [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 0 = 1 - 1
[[1,1,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,2,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,3,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,4,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[2,2,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[2,3,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[2,4,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[3,3,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[3,4,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[4,4,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,1,1,3]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,1,2,3]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,1,3,3]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,2,2,3]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,2,3,3]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,3,3,3]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[2,2,2,3]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[2,2,3,3]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[2,3,3,3]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[3,3,3,3]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,2,3],[2]]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0 = 1 - 1
[[1,3,3],[2]]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0 = 1 - 1
[[1,3,3],[3]]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0 = 1 - 1
[[2,3,3],[3]]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0 = 1 - 1
[[1,1],[2,3]]
=> [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 0 = 1 - 1
[[1,1],[3,3]]
=> [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 0 = 1 - 1
[[1,2],[3,3]]
=> [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 0 = 1 - 1
[[2,2],[3,3]]
=> [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 0 = 1 - 1
[[1,3],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 0 = 1 - 1
[[1,1,1,1,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[[1,1,1,2,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[[1,1,2,2,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[[1,2,2,2,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[[2,2,2,2,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[[1,1,2,2],[2]]
=> [3,1,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0 = 1 - 1
[[1,2,2,2],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 2 - 1
[[1,2,2,3],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[1,2,3,3],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[1,3,3,3],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[1,3,3,3],[3]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[2,3,3,3],[3]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[1,1,3],[2,2]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 2 - 1
[[1,1,3],[2,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 2 - 1
[[1,1,3],[3,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 2 - 1
[[1,2,3],[3,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 2 - 1
[[2,2,3],[3,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 2 - 1
[[1,2,3],[2],[3]]
=> [4,2,1,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 1 - 1
[[1,3,3],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 1 - 1
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [4,3,1,5,2] => [4,3,1,5,2] => ? = 1 - 1
[[1,2],[2,3],[3]]
=> [4,2,5,1,3] => [2,4,1,5,3] => [2,4,1,5,3] => ? = 2 - 1
[[1,1,1,1,1,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[1,1,1,1,2,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[1,1,1,2,2,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[1,1,2,2,2,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[1,2,2,2,2,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[2,2,2,2,2,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[1,1,1,2,2],[2]]
=> [4,1,2,3,5,6] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => ? = 1 - 1
[[1,1,2,2,2],[2]]
=> [3,1,2,4,5,6] => [1,3,2,4,5,6] => [1,3,2,4,5,6] => ? = 1 - 1
[[1,2,2,2,2],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => [2,1,3,4,5,6] => ? = 1 - 1
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [1,2,5,3,6,4] => [1,2,5,3,6,4] => ? = 1 - 1
[[1,1,2,2],[2,2]]
=> [3,4,1,2,5,6] => [3,1,4,2,5,6] => [3,1,4,2,5,6] => ? = 2 - 1
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [4,1,5,2,6,3] => [4,1,5,2,6,3] => ? = 1 - 1
[[1,2,2,4],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[1,2,3,4],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[1,2,4,4],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[1,3,3,4],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[1,3,4,4],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[1,4,4,4],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[1,3,3,4],[3]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[1,3,4,4],[3]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[1,4,4,4],[3]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[1,4,4,4],[4]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[2,3,3,4],[3]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[2,3,4,4],[3]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[2,4,4,4],[3]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[2,4,4,4],[4]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[3,4,4,4],[4]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[1,1,4],[2,2]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 2 - 1
[[1,1,4],[2,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 2 - 1
[[1,1,4],[2,4]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 2 - 1
[[1,1,4],[3,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 2 - 1
[[1,1,4],[3,4]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 2 - 1
[[1,1,4],[4,4]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 2 - 1
[[1,2,4],[3,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 2 - 1
Description
The number of occurrences of the signed pattern 1-2 in a signed permutation.
This is the number of pairs 1≤i<j≤n such that 0<π(i)<−π(j).
Matching statistic: St001868
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001868: Signed permutations ⟶ ℤResult quality: 33% ●values known / values provided: 43%●distinct values known / distinct values provided: 33%
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001868: Signed permutations ⟶ ℤResult quality: 33% ●values known / values provided: 43%●distinct values known / distinct values provided: 33%
Values
[[1,1,2]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,2,2]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[2,2,2]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,1,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,2,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,3,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[2,2,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[2,3,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[3,3,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,1,1,2]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,1,2,2]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[2,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,2,2],[2]]
=> [2,1,3,4] => [2,3,4,1] => [2,3,4,1] => 0 = 1 - 1
[[1,1],[2,2]]
=> [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 0 = 1 - 1
[[1,1,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,2,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,3,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,4,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[2,2,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[2,3,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[2,4,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[3,3,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[3,4,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[4,4,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,1,1,3]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,1,2,3]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,1,3,3]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,2,2,3]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,2,3,3]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,3,3,3]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[2,2,2,3]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[2,2,3,3]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[2,3,3,3]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[3,3,3,3]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,2,3],[2]]
=> [2,1,3,4] => [2,3,4,1] => [2,3,4,1] => 0 = 1 - 1
[[1,3,3],[2]]
=> [2,1,3,4] => [2,3,4,1] => [2,3,4,1] => 0 = 1 - 1
[[1,3,3],[3]]
=> [2,1,3,4] => [2,3,4,1] => [2,3,4,1] => 0 = 1 - 1
[[2,3,3],[3]]
=> [2,1,3,4] => [2,3,4,1] => [2,3,4,1] => 0 = 1 - 1
[[1,1],[2,3]]
=> [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 0 = 1 - 1
[[1,1],[3,3]]
=> [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 0 = 1 - 1
[[1,2],[3,3]]
=> [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 0 = 1 - 1
[[2,2],[3,3]]
=> [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 0 = 1 - 1
[[1,3],[2],[3]]
=> [3,2,1,4] => [3,4,2,1] => [3,4,2,1] => 0 = 1 - 1
[[1,1,1,1,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[[1,1,1,2,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[[1,1,2,2,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[[1,2,2,2,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[[2,2,2,2,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[[1,1,2,2],[2]]
=> [3,1,2,4,5] => [1,3,4,5,2] => [1,3,4,5,2] => 0 = 1 - 1
[[1,2,2,2],[2]]
=> [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 1 - 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 2 - 1
[[1,2,2,3],[2]]
=> [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 1 - 1
[[1,2,3,3],[2]]
=> [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 1 - 1
[[1,3,3,3],[2]]
=> [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 1 - 1
[[1,3,3,3],[3]]
=> [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 1 - 1
[[2,3,3,3],[3]]
=> [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 1 - 1
[[1,1,3],[2,2]]
=> [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 2 - 1
[[1,1,3],[2,3]]
=> [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 2 - 1
[[1,1,3],[3,3]]
=> [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 2 - 1
[[1,2,3],[3,3]]
=> [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 2 - 1
[[2,2,3],[3,3]]
=> [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 2 - 1
[[1,2,3],[2],[3]]
=> [4,2,1,3,5] => [2,4,5,3,1] => [2,4,5,3,1] => ? = 1 - 1
[[1,3,3],[2],[3]]
=> [3,2,1,4,5] => [3,4,5,2,1] => [3,4,5,2,1] => ? = 1 - 1
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [4,5,1,3,2] => [4,5,1,3,2] => ? = 1 - 1
[[1,2],[2,3],[3]]
=> [4,2,5,1,3] => [4,5,2,3,1] => [4,5,2,3,1] => ? = 2 - 1
[[1,1,1,1,1,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[1,1,1,1,2,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[1,1,1,2,2,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[1,1,2,2,2,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[1,2,2,2,2,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[2,2,2,2,2,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[1,1,1,2,2],[2]]
=> [4,1,2,3,5,6] => [1,2,4,5,6,3] => [1,2,4,5,6,3] => ? = 1 - 1
[[1,1,2,2,2],[2]]
=> [3,1,2,4,5,6] => [1,3,4,5,6,2] => [1,3,4,5,6,2] => ? = 1 - 1
[[1,2,2,2,2],[2]]
=> [2,1,3,4,5,6] => [2,3,4,5,6,1] => [2,3,4,5,6,1] => ? = 1 - 1
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [1,2,5,6,3,4] => [1,2,5,6,3,4] => ? = 1 - 1
[[1,1,2,2],[2,2]]
=> [3,4,1,2,5,6] => [3,4,5,6,1,2] => [3,4,5,6,1,2] => ? = 2 - 1
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [4,5,6,1,2,3] => [4,5,6,1,2,3] => ? = 1 - 1
[[1,2,2,4],[2]]
=> [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 1 - 1
[[1,2,3,4],[2]]
=> [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 1 - 1
[[1,2,4,4],[2]]
=> [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 1 - 1
[[1,3,3,4],[2]]
=> [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 1 - 1
[[1,3,4,4],[2]]
=> [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 1 - 1
[[1,4,4,4],[2]]
=> [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 1 - 1
[[1,3,3,4],[3]]
=> [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 1 - 1
[[1,3,4,4],[3]]
=> [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 1 - 1
[[1,4,4,4],[3]]
=> [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 1 - 1
[[1,4,4,4],[4]]
=> [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 1 - 1
[[2,3,3,4],[3]]
=> [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 1 - 1
[[2,3,4,4],[3]]
=> [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 1 - 1
[[2,4,4,4],[3]]
=> [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 1 - 1
[[2,4,4,4],[4]]
=> [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 1 - 1
[[3,4,4,4],[4]]
=> [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 1 - 1
[[1,1,4],[2,2]]
=> [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 2 - 1
[[1,1,4],[2,3]]
=> [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 2 - 1
[[1,1,4],[2,4]]
=> [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 2 - 1
[[1,1,4],[3,3]]
=> [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 2 - 1
[[1,1,4],[3,4]]
=> [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 2 - 1
[[1,1,4],[4,4]]
=> [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 2 - 1
[[1,2,4],[3,3]]
=> [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 2 - 1
Description
The number of alignments of type NE of a signed permutation.
An alignment of type NE of a signed permutation π∈Hn is a pair 1≤i,j≤n such that π(i)<i<j≤π(j).
Matching statistic: St001870
(load all 21 compositions to match this statistic)
(load all 21 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001870: Signed permutations ⟶ ℤResult quality: 33% ●values known / values provided: 43%●distinct values known / distinct values provided: 33%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001870: Signed permutations ⟶ ℤResult quality: 33% ●values known / values provided: 43%●distinct values known / distinct values provided: 33%
Values
[[1,1,2]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,2,2]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[2,2,2]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,1,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,2,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,3,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[2,2,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[2,3,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[3,3,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,1,1,2]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,1,2,2]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[2,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,2,2],[2]]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0 = 1 - 1
[[1,1],[2,2]]
=> [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 0 = 1 - 1
[[1,1,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,2,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,3,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,4,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[2,2,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[2,3,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[2,4,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[3,3,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[3,4,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[4,4,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,1,1,3]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,1,2,3]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,1,3,3]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,2,2,3]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,2,3,3]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,3,3,3]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[2,2,2,3]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[2,2,3,3]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[2,3,3,3]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[3,3,3,3]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,2,3],[2]]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0 = 1 - 1
[[1,3,3],[2]]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0 = 1 - 1
[[1,3,3],[3]]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0 = 1 - 1
[[2,3,3],[3]]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0 = 1 - 1
[[1,1],[2,3]]
=> [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 0 = 1 - 1
[[1,1],[3,3]]
=> [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 0 = 1 - 1
[[1,2],[3,3]]
=> [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 0 = 1 - 1
[[2,2],[3,3]]
=> [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 0 = 1 - 1
[[1,3],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 0 = 1 - 1
[[1,1,1,1,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[[1,1,1,2,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[[1,1,2,2,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[[1,2,2,2,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[[2,2,2,2,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[[1,1,2,2],[2]]
=> [3,1,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0 = 1 - 1
[[1,2,2,2],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 2 - 1
[[1,2,2,3],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[1,2,3,3],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[1,3,3,3],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[1,3,3,3],[3]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[2,3,3,3],[3]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[1,1,3],[2,2]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 2 - 1
[[1,1,3],[2,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 2 - 1
[[1,1,3],[3,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 2 - 1
[[1,2,3],[3,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 2 - 1
[[2,2,3],[3,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 2 - 1
[[1,2,3],[2],[3]]
=> [4,2,1,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 1 - 1
[[1,3,3],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 1 - 1
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [4,3,1,5,2] => [4,3,1,5,2] => ? = 1 - 1
[[1,2],[2,3],[3]]
=> [4,2,5,1,3] => [2,4,1,5,3] => [2,4,1,5,3] => ? = 2 - 1
[[1,1,1,1,1,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[1,1,1,1,2,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[1,1,1,2,2,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[1,1,2,2,2,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[1,2,2,2,2,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[2,2,2,2,2,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[1,1,1,2,2],[2]]
=> [4,1,2,3,5,6] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => ? = 1 - 1
[[1,1,2,2,2],[2]]
=> [3,1,2,4,5,6] => [1,3,2,4,5,6] => [1,3,2,4,5,6] => ? = 1 - 1
[[1,2,2,2,2],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => [2,1,3,4,5,6] => ? = 1 - 1
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [1,2,5,3,6,4] => [1,2,5,3,6,4] => ? = 1 - 1
[[1,1,2,2],[2,2]]
=> [3,4,1,2,5,6] => [3,1,4,2,5,6] => [3,1,4,2,5,6] => ? = 2 - 1
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [4,1,5,2,6,3] => [4,1,5,2,6,3] => ? = 1 - 1
[[1,2,2,4],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[1,2,3,4],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[1,2,4,4],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[1,3,3,4],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[1,3,4,4],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[1,4,4,4],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[1,3,3,4],[3]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[1,3,4,4],[3]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[1,4,4,4],[3]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[1,4,4,4],[4]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[2,3,3,4],[3]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[2,3,4,4],[3]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[2,4,4,4],[3]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[2,4,4,4],[4]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[3,4,4,4],[4]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[1,1,4],[2,2]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 2 - 1
[[1,1,4],[2,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 2 - 1
[[1,1,4],[2,4]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 2 - 1
[[1,1,4],[3,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 2 - 1
[[1,1,4],[3,4]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 2 - 1
[[1,1,4],[4,4]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 2 - 1
[[1,2,4],[3,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 2 - 1
Description
The number of positive entries followed by a negative entry in a signed permutation.
For a signed permutation π∈Hn, this is the number of positive entries followed by a negative entry in π(−n),…,π(−1),π(1),…,π(n).
Matching statistic: St001895
(load all 20 compositions to match this statistic)
(load all 20 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001895: Signed permutations ⟶ ℤResult quality: 33% ●values known / values provided: 43%●distinct values known / distinct values provided: 33%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001895: Signed permutations ⟶ ℤResult quality: 33% ●values known / values provided: 43%●distinct values known / distinct values provided: 33%
Values
[[1,1,2]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,2,2]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[2,2,2]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,1,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,2,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,3,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[2,2,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[2,3,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[3,3,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,1,1,2]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,1,2,2]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[2,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,2,2],[2]]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0 = 1 - 1
[[1,1],[2,2]]
=> [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 0 = 1 - 1
[[1,1,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,2,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,3,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,4,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[2,2,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[2,3,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[2,4,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[3,3,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[3,4,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[4,4,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,1,1,3]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,1,2,3]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,1,3,3]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,2,2,3]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,2,3,3]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,3,3,3]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[2,2,2,3]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[2,2,3,3]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[2,3,3,3]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[3,3,3,3]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,2,3],[2]]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0 = 1 - 1
[[1,3,3],[2]]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0 = 1 - 1
[[1,3,3],[3]]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0 = 1 - 1
[[2,3,3],[3]]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0 = 1 - 1
[[1,1],[2,3]]
=> [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 0 = 1 - 1
[[1,1],[3,3]]
=> [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 0 = 1 - 1
[[1,2],[3,3]]
=> [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 0 = 1 - 1
[[2,2],[3,3]]
=> [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 0 = 1 - 1
[[1,3],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 0 = 1 - 1
[[1,1,1,1,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[[1,1,1,2,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[[1,1,2,2,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[[1,2,2,2,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[[2,2,2,2,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[[1,1,2,2],[2]]
=> [3,1,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0 = 1 - 1
[[1,2,2,2],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 2 - 1
[[1,2,2,3],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[1,2,3,3],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[1,3,3,3],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[1,3,3,3],[3]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[2,3,3,3],[3]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[1,1,3],[2,2]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 2 - 1
[[1,1,3],[2,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 2 - 1
[[1,1,3],[3,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 2 - 1
[[1,2,3],[3,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 2 - 1
[[2,2,3],[3,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 2 - 1
[[1,2,3],[2],[3]]
=> [4,2,1,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 1 - 1
[[1,3,3],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 1 - 1
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [4,3,1,5,2] => [4,3,1,5,2] => ? = 1 - 1
[[1,2],[2,3],[3]]
=> [4,2,5,1,3] => [2,4,1,5,3] => [2,4,1,5,3] => ? = 2 - 1
[[1,1,1,1,1,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[1,1,1,1,2,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[1,1,1,2,2,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[1,1,2,2,2,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[1,2,2,2,2,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[2,2,2,2,2,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[1,1,1,2,2],[2]]
=> [4,1,2,3,5,6] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => ? = 1 - 1
[[1,1,2,2,2],[2]]
=> [3,1,2,4,5,6] => [1,3,2,4,5,6] => [1,3,2,4,5,6] => ? = 1 - 1
[[1,2,2,2,2],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => [2,1,3,4,5,6] => ? = 1 - 1
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [1,2,5,3,6,4] => [1,2,5,3,6,4] => ? = 1 - 1
[[1,1,2,2],[2,2]]
=> [3,4,1,2,5,6] => [3,1,4,2,5,6] => [3,1,4,2,5,6] => ? = 2 - 1
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [4,1,5,2,6,3] => [4,1,5,2,6,3] => ? = 1 - 1
[[1,2,2,4],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[1,2,3,4],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[1,2,4,4],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[1,3,3,4],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[1,3,4,4],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[1,4,4,4],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[1,3,3,4],[3]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[1,3,4,4],[3]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[1,4,4,4],[3]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[1,4,4,4],[4]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[2,3,3,4],[3]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[2,3,4,4],[3]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[2,4,4,4],[3]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[2,4,4,4],[4]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[3,4,4,4],[4]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[1,1,4],[2,2]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 2 - 1
[[1,1,4],[2,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 2 - 1
[[1,1,4],[2,4]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 2 - 1
[[1,1,4],[3,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 2 - 1
[[1,1,4],[3,4]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 2 - 1
[[1,1,4],[4,4]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 2 - 1
[[1,2,4],[3,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 2 - 1
Description
The oddness of a signed permutation.
The direct sum of two signed permutations σ∈Hk and τ∈Hm is the signed permutation in Hk+m obtained by concatenating σ with the result of increasing the absolute value of every entry in τ by k.
This statistic records the number of blocks with an odd number of signs in the direct sum decomposition of a signed permutation.
Matching statistic: St000908
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000908: Posets ⟶ ℤResult quality: 33% ●values known / values provided: 43%●distinct values known / distinct values provided: 33%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000908: Posets ⟶ ℤResult quality: 33% ●values known / values provided: 43%●distinct values known / distinct values provided: 33%
Values
[[1,1,2]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[1,2,2]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[2,2,2]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[1,1,3]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[1,2,3]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[1,3,3]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[2,2,3]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[2,3,3]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[3,3,3]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[1,1,1,2]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,1,2,2]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[2,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,2,2],[2]]
=> [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[[1,1],[2,2]]
=> [3,4,1,2] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1
[[1,1,4]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[1,2,4]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[1,3,4]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[1,4,4]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[2,2,4]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[2,3,4]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[2,4,4]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[3,3,4]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[3,4,4]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[4,4,4]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[1,1,1,3]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,1,2,3]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,1,3,3]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,2,2,3]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,2,3,3]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,3,3,3]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[2,2,2,3]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[2,2,3,3]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[2,3,3,3]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[3,3,3,3]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,2,3],[2]]
=> [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[[1,3,3],[2]]
=> [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[[1,3,3],[3]]
=> [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[[2,3,3],[3]]
=> [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[[1,1],[2,3]]
=> [3,4,1,2] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1
[[1,1],[3,3]]
=> [3,4,1,2] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1
[[1,2],[3,3]]
=> [3,4,1,2] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1
[[2,2],[3,3]]
=> [3,4,1,2] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1
[[1,3],[2],[3]]
=> [3,2,1,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1
[[1,1,1,1,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[1,1,1,2,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[1,1,2,2,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[1,2,2,2,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[2,2,2,2,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[1,1,2,2],[2]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[[1,2,2,2],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(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],[2,2]]
=> [3,4,1,2,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 2
[[1,1,5]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[1,1,2,3],[2]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[[1,1,3,3],[2]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[[1,1,3,3],[3]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[[1,2,2,3],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(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],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(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],[3]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[[1,3,3,3],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(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],[3]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[[2,2,3,3],[3]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[[2,3,3,3],[3]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(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]]
=> [3,5,1,2,4] => [2,4,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 1
[[1,1,3],[2,2]]
=> [3,4,1,2,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 2
[[1,1,3],[2,3]]
=> [3,4,1,2,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 2
[[1,1,3],[3,3]]
=> [3,4,1,2,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 2
[[1,2,3],[3,3]]
=> [3,4,1,2,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 2
[[2,2,3],[3,3]]
=> [3,4,1,2,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 2
[[1,1,1],[2],[3]]
=> [5,4,1,2,3] => [2,3,5,1,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ? = 1
[[1,2,3],[2],[3]]
=> [4,2,1,3,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1
[[1,3,3],[2],[3]]
=> [3,2,1,4,5] => [3,1,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [2,5,4,3,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[[1,2],[2,3],[3]]
=> [4,2,5,1,3] => [3,4,5,2,1] => ([(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)
=> ? = 2
[[1,1,1,2,2],[2]]
=> [4,1,2,3,5,6] => [2,3,4,1,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? = 1
[[1,1,2,2,2],[2]]
=> [3,1,2,4,5,6] => [2,3,1,4,5,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 1
[[1,2,2,2,2],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => ([(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]]
=> [5,6,1,2,3,4] => [2,3,4,6,5,1] => ([(0,3),(0,4),(0,5),(1,14),(2,1),(2,6),(2,7),(3,9),(3,11),(4,9),(4,10),(5,2),(5,10),(5,11),(6,13),(6,14),(7,13),(7,14),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8),(14,8)],15)
=> ? = 1
[[1,1,2,2],[2,2]]
=> [3,4,1,2,5,6] => [2,4,3,1,5,6] => ([(0,1),(0,2),(0,3),(0,5),(1,11),(1,14),(2,10),(2,13),(2,14),(3,10),(3,12),(3,14),(4,7),(4,8),(4,9),(5,4),(5,11),(5,12),(5,13),(7,17),(8,17),(8,18),(9,17),(9,18),(10,15),(11,7),(11,16),(12,8),(12,15),(12,16),(13,9),(13,15),(13,16),(14,15),(14,16),(15,18),(16,17),(16,18),(17,6),(18,6)],19)
=> ? = 2
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [2,3,6,4,5,1] => ([(0,1),(0,3),(0,4),(0,5),(1,6),(1,15),(2,7),(2,8),(2,13),(3,10),(3,12),(3,15),(4,2),(4,11),(4,12),(4,15),(5,6),(5,10),(5,11),(6,16),(7,17),(8,17),(8,18),(10,14),(10,16),(11,8),(11,14),(11,16),(12,7),(12,13),(12,14),(13,17),(13,18),(14,17),(14,18),(15,13),(15,16),(16,18),(17,9),(18,9)],19)
=> ? = 1
[[1,1,2,4],[2]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[[1,1,3,4],[2]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[[1,1,4,4],[2]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[[1,1,3,4],[3]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[[1,1,4,4],[3]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[[1,1,4,4],[4]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[[1,2,2,4],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[[1,2,3,4],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[[1,2,4,4],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[[1,2,3,4],[3]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[[1,3,3,4],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[[1,2,4,4],[3]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[[1,3,4,4],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[[1,2,4,4],[4]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[[1,4,4,4],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[[1,3,3,4],[3]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[[1,3,4,4],[3]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[[1,3,4,4],[4]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[[1,4,4,4],[3]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[[1,4,4,4],[4]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
Description
The length of the shortest maximal antichain in a poset.
Matching statistic: St000914
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000914: Posets ⟶ ℤResult quality: 33% ●values known / values provided: 43%●distinct values known / distinct values provided: 33%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000914: Posets ⟶ ℤResult quality: 33% ●values known / values provided: 43%●distinct values known / distinct values provided: 33%
Values
[[1,1,2]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[1,2,2]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[2,2,2]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[1,1,3]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[1,2,3]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[1,3,3]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[2,2,3]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[2,3,3]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[3,3,3]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[1,1,1,2]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,1,2,2]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[2,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,2,2],[2]]
=> [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[[1,1],[2,2]]
=> [3,4,1,2] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1
[[1,1,4]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[1,2,4]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[1,3,4]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[1,4,4]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[2,2,4]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[2,3,4]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[2,4,4]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[3,3,4]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[3,4,4]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[4,4,4]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[1,1,1,3]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,1,2,3]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,1,3,3]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,2,2,3]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,2,3,3]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,3,3,3]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[2,2,2,3]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[2,2,3,3]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[2,3,3,3]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[3,3,3,3]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,2,3],[2]]
=> [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[[1,3,3],[2]]
=> [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[[1,3,3],[3]]
=> [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[[2,3,3],[3]]
=> [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[[1,1],[2,3]]
=> [3,4,1,2] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1
[[1,1],[3,3]]
=> [3,4,1,2] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1
[[1,2],[3,3]]
=> [3,4,1,2] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1
[[2,2],[3,3]]
=> [3,4,1,2] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1
[[1,3],[2],[3]]
=> [3,2,1,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1
[[1,1,1,1,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[1,1,1,2,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[1,1,2,2,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[1,2,2,2,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[2,2,2,2,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[1,1,2,2],[2]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[[1,2,2,2],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(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],[2,2]]
=> [3,4,1,2,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 2
[[1,1,5]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[1,1,2,3],[2]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[[1,1,3,3],[2]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[[1,1,3,3],[3]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[[1,2,2,3],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(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],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(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],[3]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[[1,3,3,3],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(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],[3]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[[2,2,3,3],[3]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[[2,3,3,3],[3]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(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]]
=> [3,5,1,2,4] => [2,4,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 1
[[1,1,3],[2,2]]
=> [3,4,1,2,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 2
[[1,1,3],[2,3]]
=> [3,4,1,2,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 2
[[1,1,3],[3,3]]
=> [3,4,1,2,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 2
[[1,2,3],[3,3]]
=> [3,4,1,2,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 2
[[2,2,3],[3,3]]
=> [3,4,1,2,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 2
[[1,1,1],[2],[3]]
=> [5,4,1,2,3] => [2,3,5,1,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ? = 1
[[1,2,3],[2],[3]]
=> [4,2,1,3,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1
[[1,3,3],[2],[3]]
=> [3,2,1,4,5] => [3,1,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [2,5,4,3,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[[1,2],[2,3],[3]]
=> [4,2,5,1,3] => [3,4,5,2,1] => ([(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)
=> ? = 2
[[1,1,1,2,2],[2]]
=> [4,1,2,3,5,6] => [2,3,4,1,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? = 1
[[1,1,2,2,2],[2]]
=> [3,1,2,4,5,6] => [2,3,1,4,5,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 1
[[1,2,2,2,2],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => ([(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]]
=> [5,6,1,2,3,4] => [2,3,4,6,5,1] => ([(0,3),(0,4),(0,5),(1,14),(2,1),(2,6),(2,7),(3,9),(3,11),(4,9),(4,10),(5,2),(5,10),(5,11),(6,13),(6,14),(7,13),(7,14),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8),(14,8)],15)
=> ? = 1
[[1,1,2,2],[2,2]]
=> [3,4,1,2,5,6] => [2,4,3,1,5,6] => ([(0,1),(0,2),(0,3),(0,5),(1,11),(1,14),(2,10),(2,13),(2,14),(3,10),(3,12),(3,14),(4,7),(4,8),(4,9),(5,4),(5,11),(5,12),(5,13),(7,17),(8,17),(8,18),(9,17),(9,18),(10,15),(11,7),(11,16),(12,8),(12,15),(12,16),(13,9),(13,15),(13,16),(14,15),(14,16),(15,18),(16,17),(16,18),(17,6),(18,6)],19)
=> ? = 2
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [2,3,6,4,5,1] => ([(0,1),(0,3),(0,4),(0,5),(1,6),(1,15),(2,7),(2,8),(2,13),(3,10),(3,12),(3,15),(4,2),(4,11),(4,12),(4,15),(5,6),(5,10),(5,11),(6,16),(7,17),(8,17),(8,18),(10,14),(10,16),(11,8),(11,14),(11,16),(12,7),(12,13),(12,14),(13,17),(13,18),(14,17),(14,18),(15,13),(15,16),(16,18),(17,9),(18,9)],19)
=> ? = 1
[[1,1,2,4],[2]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[[1,1,3,4],[2]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[[1,1,4,4],[2]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[[1,1,3,4],[3]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[[1,1,4,4],[3]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[[1,1,4,4],[4]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[[1,2,2,4],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[[1,2,3,4],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[[1,2,4,4],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[[1,2,3,4],[3]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[[1,3,3,4],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[[1,2,4,4],[3]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[[1,3,4,4],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[[1,2,4,4],[4]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[[1,4,4,4],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[[1,3,3,4],[3]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[[1,3,4,4],[3]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[[1,3,4,4],[4]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[[1,4,4,4],[3]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[[1,4,4,4],[4]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
Description
The sum of the values of the Möbius function of a poset.
The Möbius function μ of a finite poset is defined as
μ(x,y)={1if x=y−∑z:x≤z<yμ(x,z)for x<y0otherwise.
Since μ(x,y)=0 whenever x≰, this statistic is
\sum_{x\leq y} \mu(x,y).
If the poset has a minimal or a maximal element, then the definition implies immediately that the statistic equals 1. Moreover, the statistic equals the sum of the statistics of the connected components.
This statistic is also called the magnitude of a poset.
Matching statistic: St001301
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001301: Posets ⟶ ℤResult quality: 33% ●values known / values provided: 43%●distinct values known / distinct values provided: 33%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001301: Posets ⟶ ℤResult quality: 33% ●values known / values provided: 43%●distinct values known / distinct values provided: 33%
Values
[[1,1,2]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[1,2,2]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[2,2,2]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[1,1,3]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[1,2,3]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[1,3,3]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[2,2,3]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[2,3,3]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[3,3,3]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[1,1,1,2]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[1,1,2,2]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[1,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[2,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[1,2,2],[2]]
=> [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0 = 1 - 1
[[1,1],[2,2]]
=> [3,4,1,2] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 0 = 1 - 1
[[1,1,4]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[1,2,4]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[1,3,4]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[1,4,4]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[2,2,4]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[2,3,4]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[2,4,4]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[3,3,4]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[3,4,4]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[4,4,4]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[1,1,1,3]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[1,1,2,3]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[1,1,3,3]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[1,2,2,3]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[1,2,3,3]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[1,3,3,3]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[2,2,2,3]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[2,2,3,3]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[2,3,3,3]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[3,3,3,3]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[1,2,3],[2]]
=> [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0 = 1 - 1
[[1,3,3],[2]]
=> [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0 = 1 - 1
[[1,3,3],[3]]
=> [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0 = 1 - 1
[[2,3,3],[3]]
=> [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0 = 1 - 1
[[1,1],[2,3]]
=> [3,4,1,2] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 0 = 1 - 1
[[1,1],[3,3]]
=> [3,4,1,2] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 0 = 1 - 1
[[1,2],[3,3]]
=> [3,4,1,2] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 0 = 1 - 1
[[2,2],[3,3]]
=> [3,4,1,2] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 0 = 1 - 1
[[1,3],[2],[3]]
=> [3,2,1,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 0 = 1 - 1
[[1,1,1,1,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[[1,1,1,2,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[[1,1,2,2,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[[1,2,2,2,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[[2,2,2,2,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[[1,1,2,2],[2]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 - 1
[[1,2,2,2],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(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],[2,2]]
=> [3,4,1,2,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 2 - 1
[[1,1,5]]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[1,1,2,3],[2]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 - 1
[[1,1,3,3],[2]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 - 1
[[1,1,3,3],[3]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 - 1
[[1,2,2,3],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 - 1
[[1,2,3,3],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 - 1
[[1,2,3,3],[3]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 - 1
[[1,3,3,3],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 - 1
[[1,3,3,3],[3]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 - 1
[[2,2,3,3],[3]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 - 1
[[2,3,3,3],[3]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(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]]
=> [3,5,1,2,4] => [2,4,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 1 - 1
[[1,1,3],[2,2]]
=> [3,4,1,2,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 2 - 1
[[1,1,3],[2,3]]
=> [3,4,1,2,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 2 - 1
[[1,1,3],[3,3]]
=> [3,4,1,2,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 2 - 1
[[1,2,3],[3,3]]
=> [3,4,1,2,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 2 - 1
[[2,2,3],[3,3]]
=> [3,4,1,2,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 2 - 1
[[1,1,1],[2],[3]]
=> [5,4,1,2,3] => [2,3,5,1,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ? = 1 - 1
[[1,2,3],[2],[3]]
=> [4,2,1,3,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 - 1
[[1,3,3],[2],[3]]
=> [3,2,1,4,5] => [3,1,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 - 1
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [2,5,4,3,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 - 1
[[1,2],[2,3],[3]]
=> [4,2,5,1,3] => [3,4,5,2,1] => ([(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)
=> ? = 2 - 1
[[1,1,1,2,2],[2]]
=> [4,1,2,3,5,6] => [2,3,4,1,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? = 1 - 1
[[1,1,2,2,2],[2]]
=> [3,1,2,4,5,6] => [2,3,1,4,5,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 1 - 1
[[1,2,2,2,2],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => ([(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
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [2,3,4,6,5,1] => ([(0,3),(0,4),(0,5),(1,14),(2,1),(2,6),(2,7),(3,9),(3,11),(4,9),(4,10),(5,2),(5,10),(5,11),(6,13),(6,14),(7,13),(7,14),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8),(14,8)],15)
=> ? = 1 - 1
[[1,1,2,2],[2,2]]
=> [3,4,1,2,5,6] => [2,4,3,1,5,6] => ([(0,1),(0,2),(0,3),(0,5),(1,11),(1,14),(2,10),(2,13),(2,14),(3,10),(3,12),(3,14),(4,7),(4,8),(4,9),(5,4),(5,11),(5,12),(5,13),(7,17),(8,17),(8,18),(9,17),(9,18),(10,15),(11,7),(11,16),(12,8),(12,15),(12,16),(13,9),(13,15),(13,16),(14,15),(14,16),(15,18),(16,17),(16,18),(17,6),(18,6)],19)
=> ? = 2 - 1
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [2,3,6,4,5,1] => ([(0,1),(0,3),(0,4),(0,5),(1,6),(1,15),(2,7),(2,8),(2,13),(3,10),(3,12),(3,15),(4,2),(4,11),(4,12),(4,15),(5,6),(5,10),(5,11),(6,16),(7,17),(8,17),(8,18),(10,14),(10,16),(11,8),(11,14),(11,16),(12,7),(12,13),(12,14),(13,17),(13,18),(14,17),(14,18),(15,13),(15,16),(16,18),(17,9),(18,9)],19)
=> ? = 1 - 1
[[1,1,2,4],[2]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 - 1
[[1,1,3,4],[2]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 - 1
[[1,1,4,4],[2]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 - 1
[[1,1,3,4],[3]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 - 1
[[1,1,4,4],[3]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 - 1
[[1,1,4,4],[4]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 - 1
[[1,2,2,4],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(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],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 - 1
[[1,2,4,4],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(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],[3]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 - 1
[[1,3,3,4],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 - 1
[[1,2,4,4],[3]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 - 1
[[1,3,4,4],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 - 1
[[1,2,4,4],[4]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 - 1
[[1,4,4,4],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(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],[3]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 - 1
[[1,3,4,4],[3]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 - 1
[[1,3,4,4],[4]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 - 1
[[1,4,4,4],[3]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 - 1
[[1,4,4,4],[4]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 - 1
Description
The first Betti number of the order complex associated with the poset.
The order complex of a poset is the simplicial complex whose faces are the chains of the poset. This statistic is the rank of the first homology group of the order complex.
The following 78 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001625The Möbius invariant of a lattice. St001867The number of alignments of type EN of a signed permutation. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001862The number of crossings of a signed permutation. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001889The size of the connectivity set of a signed permutation. St001490The number of connected components of a skew partition. St001964The interval resolution global dimension of a poset. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000943The number of spots the most unlucky car had to go further in a parking function. St001195The global dimension of the algebra A/AfA of the corresponding Nakayama algebra A with minimal left faithful projective-injective module Af. St001208The number of connected components of the quiver of A/T when T is the 1-tilting module corresponding to the permutation in the Auslander algebra A of K[x]/(x^n). St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001768The number of reduced words of a signed permutation. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000298The order dimension or Dushnik-Miller dimension of a poset. St000640The rank of the largest boolean interval in a poset. St000907The number of maximal antichains of minimal length in a poset. St001171The vector space dimension of Ext_A^1(I_o,A) when I_o is the tilting module corresponding to the permutation o in the Auslander algebra A of K[x]/(x^n). St001207The 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). St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St001927Sparre Andersen's number of positives of a signed permutation. St000181The number of connected components of the Hasse diagram for the poset. St000635The number of strictly order preserving maps of a poset into itself. St001890The maximum magnitude of the Möbius function of a poset. St001632The number of indecomposable injective modules I with dim Ext^1(I,A)=1 for the incidence algebra A of a poset. St001613The binary logarithm of the size of the center of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000417The size of the automorphism group of the ordered tree. St000679The pruning number of an ordered tree. St001058The breadth of the ordered tree. St001626The number of maximal proper sublattices of a lattice. St001875The number of simple modules with projective dimension at most 1. 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. St001168The vector space dimension of the tilting module corresponding to the permutation in the Auslander algebra of K[x]/(x^n). St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000910The number of maximal chains of minimal length in a poset. St001857The number of edges in the reduced word graph of a signed permutation. St001926Sparre Andersen's position of the maximum of a signed permutation. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St000075The orbit size of a standard tableau under promotion. St000084The number of subtrees. St000168The number of internal nodes of an ordered tree. St000173The segment statistic of a semistandard tableau. St000174The flush statistic of a semistandard tableau. St000328The maximum number of child nodes in a tree. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001623The number of doubly irreducible elements of a lattice. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000166The depth minus 1 of an ordered tree. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001621The number of atoms of a lattice. St001713The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern. St001754The number of tolerances of a finite lattice. St000094The depth of an ordered tree. St000116The major index of a semistandard tableau obtained by standardizing. St000327The number of cover relations in a poset. St001877Number of indecomposable injective modules with projective dimension 2. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000529The number of permutations whose descent word is the given binary word. St000043The number of crossings plus two-nestings of a perfect matching.
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