searching the database
Your data matches 96 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St001912
Mp00252: Permutations —restriction⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001912: 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
St001912: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [1,2] => [1,1]
=> [1]
=> 0
[1,3,2] => [1,2] => [1,1]
=> [1]
=> 0
[3,1,2] => [1,2] => [1,1]
=> [1]
=> 0
[1,2,3,4] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[1,2,4,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[1,3,2,4] => [1,3,2] => [2,1]
=> [1]
=> 0
[1,3,4,2] => [1,3,2] => [2,1]
=> [1]
=> 0
[1,4,2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[1,4,3,2] => [1,3,2] => [2,1]
=> [1]
=> 0
[2,1,3,4] => [2,1,3] => [2,1]
=> [1]
=> 0
[2,1,4,3] => [2,1,3] => [2,1]
=> [1]
=> 0
[2,4,1,3] => [2,1,3] => [2,1]
=> [1]
=> 0
[3,2,1,4] => [3,2,1] => [2,1]
=> [1]
=> 0
[3,2,4,1] => [3,2,1] => [2,1]
=> [1]
=> 0
[3,4,2,1] => [3,2,1] => [2,1]
=> [1]
=> 0
[4,1,2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[4,1,3,2] => [1,3,2] => [2,1]
=> [1]
=> 0
[4,2,1,3] => [2,1,3] => [2,1]
=> [1]
=> 0
[4,3,2,1] => [3,2,1] => [2,1]
=> [1]
=> 0
[1,2,3,4,5] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 2
[1,2,3,5,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 2
[1,2,4,3,5] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,2,4,5,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,2,5,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 2
[1,2,5,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,3,2,4,5] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[1,3,2,5,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[1,3,4,2,5] => [1,3,4,2] => [3,1]
=> [1]
=> 0
[1,3,4,5,2] => [1,3,4,2] => [3,1]
=> [1]
=> 0
[1,3,5,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[1,3,5,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> 0
[1,4,2,3,5] => [1,4,2,3] => [3,1]
=> [1]
=> 0
[1,4,2,5,3] => [1,4,2,3] => [3,1]
=> [1]
=> 0
[1,4,3,2,5] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[1,4,3,5,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[1,4,5,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> 0
[1,4,5,3,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[1,5,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 2
[1,5,2,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,5,3,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[1,5,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> 0
[1,5,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> 0
[1,5,4,3,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[2,1,3,4,5] => [2,1,3,4] => [2,1,1]
=> [1,1]
=> 0
[2,1,3,5,4] => [2,1,3,4] => [2,1,1]
=> [1,1]
=> 0
[2,1,4,3,5] => [2,1,4,3] => [2,2]
=> [2]
=> 0
[2,1,4,5,3] => [2,1,4,3] => [2,2]
=> [2]
=> 0
[2,1,5,3,4] => [2,1,3,4] => [2,1,1]
=> [1,1]
=> 0
[2,1,5,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 0
[2,3,1,4,5] => [2,3,1,4] => [3,1]
=> [1]
=> 0
Description
The length of the preperiod in Bulgarian solitaire corresponding to an integer partition.
Bulgarian solitaire is the dynamical system where a move consists of removing the first column of the Ferrers diagram and inserting it as a row.
Matching statistic: St000068
(load all 31 compositions to match this statistic)
(load all 31 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 14%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 14%
Values
[1,2,3] => [3,2,1] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,3,2] => [2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[3,1,2] => [2,1,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,2,3,4] => [4,3,2,1] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
[1,2,4,3] => [3,4,2,1] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
[1,3,2,4] => [4,2,3,1] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
[1,3,4,2] => [2,4,3,1] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,4,2,3] => [3,2,4,1] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
[1,4,3,2] => [2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,1,3,4] => [4,3,1,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
[2,1,4,3] => [3,4,1,2] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
[2,4,1,3] => [3,1,4,2] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
[3,2,1,4] => [4,1,2,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[3,2,4,1] => [1,4,2,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[3,4,2,1] => [1,2,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[4,1,2,3] => [3,2,1,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
[4,1,3,2] => [2,3,1,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[4,2,1,3] => [3,1,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 2 + 1
[1,2,3,5,4] => [4,5,3,2,1] => [1,4,2,5,3] => ([(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
[1,2,4,3,5] => [5,3,4,2,1] => [1,5,2,3,4] => ([(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)
=> ? = 0 + 1
[1,2,4,5,3] => [3,5,4,2,1] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 0 + 1
[1,2,5,3,4] => [4,3,5,2,1] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2 + 1
[1,2,5,4,3] => [3,4,5,2,1] => [1,3,5,2,4] => ([(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)
=> ? = 0 + 1
[1,3,2,4,5] => [5,4,2,3,1] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 0 + 1
[1,3,2,5,4] => [4,5,2,3,1] => [1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 + 1
[1,3,4,2,5] => [5,2,4,3,1] => [1,5,2,3,4] => ([(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)
=> ? = 0 + 1
[1,3,4,5,2] => [2,5,4,3,1] => [1,2,5,3,4] => ([(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)
=> ? = 0 + 1
[1,3,5,2,4] => [4,2,5,3,1] => [1,4,3,5,2] => ([(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)
=> ? = 0 + 1
[1,3,5,4,2] => [2,4,5,3,1] => [1,2,4,3,5] => ([(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)
=> ? = 0 + 1
[1,4,2,3,5] => [5,3,2,4,1] => [1,5,2,3,4] => ([(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)
=> ? = 0 + 1
[1,4,2,5,3] => [3,5,2,4,1] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 0 + 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,5,2,3,4] => ([(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)
=> ? = 0 + 1
[1,4,3,5,2] => [2,5,3,4,1] => [1,2,5,3,4] => ([(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)
=> ? = 0 + 1
[1,4,5,2,3] => [3,2,5,4,1] => [1,3,5,2,4] => ([(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)
=> ? = 0 + 1
[1,4,5,3,2] => [2,3,5,4,1] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
[1,5,2,3,4] => [4,3,2,5,1] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 2 + 1
[1,5,2,4,3] => [3,4,2,5,1] => [1,3,2,4,5] => ([(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)
=> ? = 0 + 1
[1,5,3,2,4] => [4,2,3,5,1] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 0 + 1
[1,5,3,4,2] => [2,4,3,5,1] => [1,2,4,5,3] => ([(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)
=> ? = 0 + 1
[1,5,4,2,3] => [3,2,4,5,1] => [1,3,4,5,2] => ([(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)
=> ? = 0 + 1
[1,5,4,3,2] => [2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,1,3,4,5] => [5,4,3,1,2] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 0 + 1
[2,1,3,5,4] => [4,5,3,1,2] => [1,4,2,5,3] => ([(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)
=> ? = 0 + 1
[2,1,4,3,5] => [5,3,4,1,2] => [1,5,2,3,4] => ([(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)
=> ? = 0 + 1
[2,1,4,5,3] => [3,5,4,1,2] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 0 + 1
[2,1,5,3,4] => [4,3,5,1,2] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 0 + 1
[2,1,5,4,3] => [3,4,5,1,2] => [1,3,5,2,4] => ([(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)
=> ? = 0 + 1
[2,3,1,4,5] => [5,4,1,3,2] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 0 + 1
[2,3,1,5,4] => [4,5,1,3,2] => [1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 + 1
[2,3,5,1,4] => [4,1,5,3,2] => [1,4,3,5,2] => ([(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)
=> ? = 0 + 1
[2,4,3,1,5] => [5,1,3,4,2] => [1,5,2,3,4] => ([(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)
=> ? = 0 + 1
[2,4,3,5,1] => [1,5,3,4,2] => [1,2,5,3,4] => ([(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)
=> ? = 0 + 1
[2,4,5,3,1] => [1,3,5,4,2] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
[2,5,1,3,4] => [4,3,1,5,2] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 0 + 1
[2,5,1,4,3] => [3,4,1,5,2] => [1,3,2,4,5] => ([(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)
=> ? = 0 + 1
[2,5,3,1,4] => [4,1,3,5,2] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 0 + 1
[2,5,4,3,1] => [1,3,4,5,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[3,1,2,4,5] => [5,4,2,1,3] => [1,5,3,2,4] => ([(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)
=> ? = 0 + 1
[3,1,2,5,4] => [4,5,2,1,3] => [1,4,2,5,3] => ([(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)
=> ? = 0 + 1
[3,1,5,2,4] => [4,2,5,1,3] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 0 + 1
[3,2,1,4,5] => [5,4,1,2,3] => [1,5,3,2,4] => ([(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)
=> ? = 0 + 1
[3,2,1,5,4] => [4,5,1,2,3] => [1,4,2,5,3] => ([(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)
=> ? = 0 + 1
[3,2,4,1,5] => [5,1,4,2,3] => [1,5,3,4,2] => ([(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)
=> ? = 0 + 1
[3,2,4,5,1] => [1,5,4,2,3] => [1,2,5,3,4] => ([(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)
=> ? = 0 + 1
[3,2,5,1,4] => [4,1,5,2,3] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 0 + 1
[3,2,5,4,1] => [1,4,5,2,3] => [1,2,4,3,5] => ([(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)
=> ? = 0 + 1
[3,4,1,2,5] => [5,2,1,4,3] => [1,5,3,2,4] => ([(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)
=> ? = 0 + 1
[3,4,1,5,2] => [2,5,1,4,3] => [1,2,5,3,4] => ([(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)
=> ? = 0 + 1
[3,4,5,1,2] => [2,1,5,4,3] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
[3,5,1,2,4] => [4,2,1,5,3] => [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 + 1
[3,5,2,1,4] => [4,1,2,5,3] => [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 + 1
[3,5,2,4,1] => [1,4,2,5,3] => [1,2,4,5,3] => ([(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)
=> ? = 0 + 1
[3,5,4,1,2] => [2,1,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,1,3,5,2] => [2,5,3,1,4] => [1,2,5,4,3] => ([(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 = 0 + 1
[4,1,5,3,2] => [2,3,5,1,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
[4,2,3,5,1] => [1,5,3,2,4] => [1,2,5,4,3] => ([(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 = 0 + 1
[4,2,5,3,1] => [1,3,5,2,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
[4,3,2,1,5] => [5,1,2,3,4] => [1,5,4,3,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
[4,3,2,5,1] => [1,5,2,3,4] => [1,2,5,4,3] => ([(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 = 0 + 1
[4,3,5,2,1] => [1,2,5,3,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
[4,5,1,3,2] => [2,3,1,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,5,2,3,1] => [1,3,2,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,5,3,2,1] => [1,2,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,1,4,3,2] => [2,3,4,1,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,2,4,3,1] => [1,3,4,2,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,3,4,1,2] => [2,1,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,4,1,3,2] => [2,3,1,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,4,2,3,1] => [1,3,2,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,5,6,4,3,2] => [2,3,4,6,5,1] => [1,2,3,4,6,5] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> 1 = 0 + 1
[1,6,5,4,3,2] => [2,3,4,5,6,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,5,6,4,3,1] => [1,3,4,6,5,2] => [1,2,3,4,6,5] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> 1 = 0 + 1
[2,6,5,4,3,1] => [1,3,4,5,6,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[3,5,6,4,1,2] => [2,1,4,6,5,3] => [1,2,3,4,6,5] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> 1 = 0 + 1
[3,6,5,4,1,2] => [2,1,4,5,6,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[4,5,6,1,3,2] => [2,3,1,6,5,4] => [1,2,3,4,6,5] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> 1 = 0 + 1
[4,5,6,3,1,2] => [2,1,3,6,5,4] => [1,2,3,4,6,5] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> 1 = 0 + 1
[4,5,6,3,2,1] => [1,2,3,6,5,4] => [1,2,3,4,6,5] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> 1 = 0 + 1
Description
The number of minimal elements in a poset.
Matching statistic: St001568
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 14%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 14%
Values
[1,2,3] => [3,2,1] => ([],3)
=> [2,2,2,2]
=> 1 = 0 + 1
[1,3,2] => [2,3,1] => ([(1,2)],3)
=> [6]
=> 1 = 0 + 1
[3,1,2] => [2,1,3] => ([(0,2),(1,2)],3)
=> [3,2]
=> 1 = 0 + 1
[1,2,3,4] => [4,3,2,1] => ([],4)
=> [2,2,2,2,2,2,2,2]
=> ? = 0 + 1
[1,2,4,3] => [3,4,2,1] => ([(2,3)],4)
=> [6,6]
=> ? = 0 + 1
[1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
=> [6,6]
=> ? = 0 + 1
[1,3,4,2] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> [6,2,2]
=> 1 = 0 + 1
[1,4,2,3] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> [6,2,2]
=> 1 = 0 + 1
[1,4,3,2] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> [4,4]
=> 1 = 0 + 1
[2,1,3,4] => [4,3,1,2] => ([(2,3)],4)
=> [6,6]
=> ? = 0 + 1
[2,1,4,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> [3,3,3]
=> 1 = 0 + 1
[2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> 1 = 0 + 1
[3,2,1,4] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> [4,4]
=> 1 = 0 + 1
[3,2,4,1] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> [7]
=> 1 = 0 + 1
[3,4,2,1] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> 1 = 0 + 1
[4,1,2,3] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> 1 = 0 + 1
[4,1,3,2] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> [7]
=> 1 = 0 + 1
[4,2,1,3] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> [7]
=> 1 = 0 + 1
[4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1 = 0 + 1
[1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> ? = 2 + 1
[1,2,3,5,4] => [4,5,3,2,1] => ([(3,4)],5)
=> [6,6,6,6]
=> ? = 2 + 1
[1,2,4,3,5] => [5,3,4,2,1] => ([(3,4)],5)
=> [6,6,6,6]
=> ? = 0 + 1
[1,2,4,5,3] => [3,5,4,2,1] => ([(2,3),(2,4)],5)
=> [6,6,2,2,2,2]
=> ? = 0 + 1
[1,2,5,3,4] => [4,3,5,2,1] => ([(2,4),(3,4)],5)
=> [6,6,2,2,2,2]
=> ? = 2 + 1
[1,2,5,4,3] => [3,4,5,2,1] => ([(2,3),(3,4)],5)
=> [4,4,4,4]
=> ? = 0 + 1
[1,3,2,4,5] => [5,4,2,3,1] => ([(3,4)],5)
=> [6,6,6,6]
=> ? = 0 + 1
[1,3,2,5,4] => [4,5,2,3,1] => ([(1,4),(2,3)],5)
=> [6,6,6]
=> ? = 0 + 1
[1,3,4,2,5] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
=> [6,6,2,2,2,2]
=> ? = 0 + 1
[1,3,4,5,2] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> [6,2,2,2,2,2,2]
=> ? = 0 + 1
[1,3,5,2,4] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
=> [10,6]
=> ? = 0 + 1
[1,3,5,4,2] => [2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
=> [14]
=> ? = 0 + 1
[1,4,2,3,5] => [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> [6,6,2,2,2,2]
=> ? = 0 + 1
[1,4,2,5,3] => [3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
=> [10,6]
=> ? = 0 + 1
[1,4,3,2,5] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> [4,4,4,4]
=> ? = 0 + 1
[1,4,3,5,2] => [2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
=> [14]
=> ? = 0 + 1
[1,4,5,2,3] => [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [6,2,2,2,2]
=> ? = 0 + 1
[1,4,5,3,2] => [2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> ? = 0 + 1
[1,5,2,3,4] => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> [6,2,2,2,2,2,2]
=> ? = 2 + 1
[1,5,2,4,3] => [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
=> [14]
=> ? = 0 + 1
[1,5,3,2,4] => [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
=> [14]
=> ? = 0 + 1
[1,5,3,4,2] => [2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> ? = 0 + 1
[1,5,4,2,3] => [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> ? = 0 + 1
[1,5,4,3,2] => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> [10]
=> 1 = 0 + 1
[2,1,3,4,5] => [5,4,3,1,2] => ([(3,4)],5)
=> [6,6,6,6]
=> ? = 0 + 1
[2,1,3,5,4] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> [6,6,6]
=> ? = 0 + 1
[2,1,4,3,5] => [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> [6,6,6]
=> ? = 0 + 1
[2,1,4,5,3] => [3,5,4,1,2] => ([(0,4),(1,2),(1,3)],5)
=> [6,3,3,3]
=> ? = 0 + 1
[2,1,5,3,4] => [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> [6,3,3,3]
=> ? = 0 + 1
[2,1,5,4,3] => [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> [12]
=> ? = 0 + 1
[2,3,1,4,5] => [5,4,1,3,2] => ([(2,3),(2,4)],5)
=> [6,6,2,2,2,2]
=> ? = 0 + 1
[2,3,1,5,4] => [4,5,1,3,2] => ([(0,4),(1,2),(1,3)],5)
=> [6,3,3,3]
=> ? = 0 + 1
[2,3,5,1,4] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> [6,5,3]
=> ? = 0 + 1
[2,4,3,1,5] => [5,1,3,4,2] => ([(1,3),(1,4),(4,2)],5)
=> [14]
=> ? = 0 + 1
[2,4,3,5,1] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> [7,6]
=> ? = 0 + 1
[2,4,5,3,1] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> [7,2,2]
=> ? = 0 + 1
[2,5,1,3,4] => [4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5)
=> [6,5,3]
=> ? = 0 + 1
[2,5,1,4,3] => [3,4,1,5,2] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> [8,3]
=> ? = 0 + 1
[2,5,3,1,4] => [4,1,3,5,2] => ([(0,4),(1,2),(1,3),(3,4)],5)
=> [10,2]
=> ? = 0 + 1
[2,5,4,3,1] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> 1 = 0 + 1
[3,1,2,4,5] => [5,4,2,1,3] => ([(2,4),(3,4)],5)
=> [6,6,2,2,2,2]
=> ? = 0 + 1
[3,1,2,5,4] => [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
=> [6,3,3,3]
=> ? = 0 + 1
[3,1,5,2,4] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> [8,3,2]
=> ? = 0 + 1
[3,2,1,4,5] => [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> [4,4,4,4]
=> ? = 0 + 1
[3,2,1,5,4] => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> [12]
=> ? = 0 + 1
[3,2,4,1,5] => [5,1,4,2,3] => ([(1,3),(1,4),(4,2)],5)
=> [14]
=> ? = 0 + 1
[3,2,4,5,1] => [1,5,4,2,3] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> [7,6]
=> ? = 0 + 1
[3,2,5,1,4] => [4,1,5,2,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> [8,3]
=> ? = 0 + 1
[3,2,5,4,1] => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> 1 = 0 + 1
[3,5,2,1,4] => [4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> [10]
=> 1 = 0 + 1
[3,5,2,4,1] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> 1 = 0 + 1
[3,5,4,1,2] => [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> 1 = 0 + 1
[4,1,5,3,2] => [2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> [10]
=> 1 = 0 + 1
[4,2,5,1,3] => [3,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [10]
=> 1 = 0 + 1
[4,2,5,3,1] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> 1 = 0 + 1
[4,3,2,1,5] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> [10]
=> 1 = 0 + 1
[4,3,2,5,1] => [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> 1 = 0 + 1
[4,3,5,2,1] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> [8]
=> 1 = 0 + 1
[4,5,1,3,2] => [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [7,2]
=> 1 = 0 + 1
[4,5,2,1,3] => [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [7,2]
=> 1 = 0 + 1
[4,5,2,3,1] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> 1 = 0 + 1
[4,5,3,2,1] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> [5,2]
=> 1 = 0 + 1
[5,1,4,3,2] => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 1 = 0 + 1
[5,2,1,4,3] => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> 1 = 0 + 1
[5,2,4,3,1] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [8]
=> 1 = 0 + 1
[5,3,2,1,4] => [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 1 = 0 + 1
[5,3,2,4,1] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [8]
=> 1 = 0 + 1
[5,3,4,1,2] => [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [4,2,2]
=> 1 = 0 + 1
[5,4,1,3,2] => [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> 1 = 0 + 1
[5,4,2,1,3] => [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> 1 = 0 + 1
[5,4,2,3,1] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> 1 = 0 + 1
[5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 1 = 0 + 1
[4,6,5,3,1,2] => [2,1,3,5,6,4] => ([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
=> [8,2]
=> 1 = 0 + 1
[4,6,5,3,2,1] => [1,2,3,5,6,4] => ([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
=> [9]
=> 1 = 0 + 1
[5,3,6,4,2,1] => [1,2,4,6,3,5] => ([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
=> [5,5]
=> 1 = 0 + 1
[5,4,3,6,2,1] => [1,2,6,3,4,5] => ([(0,5),(3,4),(4,2),(5,1),(5,3)],6)
=> [6,4]
=> 1 = 0 + 1
[5,4,6,2,3,1] => [1,3,2,6,4,5] => ([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> [8,2]
=> 1 = 0 + 1
[5,4,6,3,1,2] => [2,1,3,6,4,5] => ([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
=> [8,2]
=> 1 = 0 + 1
[5,4,6,3,2,1] => [1,2,3,6,4,5] => ([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
=> [9]
=> 1 = 0 + 1
[5,6,2,4,3,1] => [1,3,4,2,6,5] => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6)
=> [8,2]
=> 1 = 0 + 1
[5,6,3,2,4,1] => [1,4,2,3,6,5] => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6)
=> [8,2]
=> 1 = 0 + 1
Description
The smallest positive integer that does not appear twice in the partition.
Matching statistic: St001371
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001371: Binary words ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 14%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001371: Binary words ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 14%
Values
[1,2,3] => [3]
=> [1,0,1,0,1,0]
=> 101010 => 0
[1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 0
[3,1,2] => [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 0
[1,2,3,4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
[1,2,4,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
[1,3,2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
[1,3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
[1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
[2,1,3,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
[2,1,4,3] => [2,2]
=> [1,1,1,0,0,0]
=> 111000 => 0
[2,4,1,3] => [2,2]
=> [1,1,1,0,0,0]
=> 111000 => 0
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
[3,2,4,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
[3,4,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
[4,1,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
[4,1,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
[4,2,1,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
[4,3,2,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => 0
[1,2,3,4,5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? = 2
[1,2,3,5,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 2
[1,2,4,3,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 0
[1,2,4,5,3] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 0
[1,2,5,3,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 2
[1,2,5,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[1,3,2,4,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 0
[1,3,2,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
[1,3,4,2,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 0
[1,3,4,5,2] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 0
[1,3,5,2,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[1,4,2,3,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 0
[1,4,2,5,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
[1,4,3,2,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[1,4,5,2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
[1,4,5,3,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[1,5,2,3,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 2
[1,5,2,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[1,5,3,2,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[1,5,3,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[1,5,4,2,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[1,5,4,3,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 0
[2,1,3,4,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 0
[2,1,3,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
[2,1,4,3,5] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
[2,1,4,5,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
[2,1,5,3,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
[2,1,5,4,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
[2,3,1,4,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 0
[2,3,1,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
[2,3,5,1,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
[2,4,3,1,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[2,4,3,5,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[2,4,5,3,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[2,5,1,3,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
[2,5,1,4,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
[2,5,3,1,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[2,5,4,3,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 0
[3,1,2,4,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 0
[3,1,2,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
[3,1,5,2,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
[3,2,1,4,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[3,2,1,5,4] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
[3,2,4,1,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[3,2,4,5,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[3,2,5,1,4] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
[3,2,5,4,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
[3,4,1,2,5] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
[3,4,1,5,2] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
[3,4,5,1,2] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
[3,5,1,2,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
[3,5,2,1,4] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
[3,5,2,4,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
[3,5,4,1,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
[4,1,3,2,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[4,1,3,5,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[4,1,5,3,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
[4,2,1,3,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[4,2,1,5,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
[4,2,3,1,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[4,2,3,5,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[4,2,5,1,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
[4,2,5,3,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
[4,3,2,1,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 0
[4,3,2,5,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 0
[4,3,5,2,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 0
[4,5,1,3,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
[4,5,2,1,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
[4,5,3,2,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 0
[5,1,2,3,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 2
[5,1,2,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[5,1,3,2,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[5,1,3,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[5,1,4,2,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[5,1,4,3,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 0
[5,2,1,3,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[5,2,3,1,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[5,2,4,3,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 0
[5,3,1,2,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
Description
The length of the longest Yamanouchi prefix of a binary word.
This is the largest index $i$ such that in each of the prefixes $w_1$, $w_1w_2$, $w_1w_2\dots w_i$ the number of zeros is greater than or equal to the number of ones.
Matching statistic: St001730
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001730: Binary words ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 14%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001730: Binary words ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 14%
Values
[1,2,3] => [3]
=> [1,0,1,0,1,0]
=> 101010 => 0
[1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 0
[3,1,2] => [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 0
[1,2,3,4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
[1,2,4,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
[1,3,2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
[1,3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
[1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
[2,1,3,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
[2,1,4,3] => [2,2]
=> [1,1,1,0,0,0]
=> 111000 => 0
[2,4,1,3] => [2,2]
=> [1,1,1,0,0,0]
=> 111000 => 0
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
[3,2,4,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
[3,4,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
[4,1,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
[4,1,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
[4,2,1,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
[4,3,2,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => 0
[1,2,3,4,5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? = 2
[1,2,3,5,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 2
[1,2,4,3,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 0
[1,2,4,5,3] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 0
[1,2,5,3,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 2
[1,2,5,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[1,3,2,4,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 0
[1,3,2,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
[1,3,4,2,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 0
[1,3,4,5,2] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 0
[1,3,5,2,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[1,4,2,3,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 0
[1,4,2,5,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
[1,4,3,2,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[1,4,5,2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
[1,4,5,3,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[1,5,2,3,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 2
[1,5,2,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[1,5,3,2,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[1,5,3,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[1,5,4,2,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[1,5,4,3,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 0
[2,1,3,4,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 0
[2,1,3,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
[2,1,4,3,5] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
[2,1,4,5,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
[2,1,5,3,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
[2,1,5,4,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
[2,3,1,4,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 0
[2,3,1,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
[2,3,5,1,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
[2,4,3,1,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[2,4,3,5,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[2,4,5,3,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[2,5,1,3,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
[2,5,1,4,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
[2,5,3,1,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[2,5,4,3,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 0
[3,1,2,4,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 0
[3,1,2,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
[3,1,5,2,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
[3,2,1,4,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[3,2,1,5,4] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
[3,2,4,1,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[3,2,4,5,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[3,2,5,1,4] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
[3,2,5,4,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
[3,4,1,2,5] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
[3,4,1,5,2] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
[3,4,5,1,2] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
[3,5,1,2,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
[3,5,2,1,4] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
[3,5,2,4,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
[3,5,4,1,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
[4,1,3,2,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[4,1,3,5,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[4,1,5,3,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
[4,2,1,3,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[4,2,1,5,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
[4,2,3,1,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[4,2,3,5,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[4,2,5,1,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
[4,2,5,3,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
[4,3,2,1,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 0
[4,3,2,5,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 0
[4,3,5,2,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 0
[4,5,1,3,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
[4,5,2,1,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
[4,5,3,2,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 0
[5,1,2,3,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 2
[5,1,2,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[5,1,3,2,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[5,1,3,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[5,1,4,2,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[5,1,4,3,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 0
[5,2,1,3,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[5,2,3,1,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[5,2,4,3,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 0
[5,3,1,2,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
Description
The number of times the path corresponding to a binary word crosses the base line.
Interpret each $0$ as a step $(1,-1)$ and $1$ as a step $(1,1)$. Then this statistic counts the number of times the path crosses the $x$-axis.
Matching statistic: St001195
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 14%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 14%
Values
[1,2,3] => [3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[3,1,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,2,3,4] => [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,2,4,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[2,1,3,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[2,1,4,3] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[2,4,1,3] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[3,2,4,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[3,4,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[4,1,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[4,1,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[4,2,1,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[4,3,2,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,2,3,4,5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 + 1
[1,2,3,5,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 2 + 1
[1,2,4,3,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 0 + 1
[1,2,4,5,3] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 0 + 1
[1,2,5,3,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 2 + 1
[1,2,5,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 1
[1,3,2,4,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 0 + 1
[1,3,2,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,3,4,2,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 0 + 1
[1,3,4,5,2] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 0 + 1
[1,3,5,2,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 1
[1,4,2,3,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 0 + 1
[1,4,2,5,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,4,3,2,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 1
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 1
[1,4,5,2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,4,5,3,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 1
[1,5,2,3,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 2 + 1
[1,5,2,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 1
[1,5,3,2,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 1
[1,5,3,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 1
[1,5,4,2,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 1
[1,5,4,3,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 0 + 1
[2,1,3,4,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 0 + 1
[2,1,3,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[2,1,4,3,5] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[2,1,4,5,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[2,1,5,3,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[2,1,5,4,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[2,3,1,4,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 0 + 1
[2,3,1,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[2,3,5,1,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[2,4,3,1,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 1
[2,4,3,5,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 1
[2,4,5,3,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 1
[2,5,1,3,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[2,5,1,4,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[2,5,3,1,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 1
[2,5,4,3,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 0 + 1
[3,1,2,4,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 0 + 1
[3,1,2,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[3,1,5,2,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[3,2,1,4,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 1
[3,2,1,5,4] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[3,2,4,1,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 1
[3,2,4,5,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 1
[3,2,5,1,4] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[3,2,5,4,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[3,4,1,2,5] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[3,4,1,5,2] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[3,4,5,1,2] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[3,5,1,2,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[3,5,2,1,4] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[3,5,2,4,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[3,5,4,1,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[4,1,3,2,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 1
[4,1,3,5,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 1
[4,1,5,3,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[4,2,1,3,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 1
[4,2,1,5,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[4,2,3,1,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 1
[4,2,3,5,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 1
[4,2,5,1,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[4,2,5,3,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[4,3,2,1,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 0 + 1
[4,3,2,5,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 0 + 1
[4,3,5,2,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 0 + 1
[4,5,1,3,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[4,5,2,1,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[4,5,3,2,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 0 + 1
[5,1,2,3,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 2 + 1
[5,1,2,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 1
[5,1,3,2,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 1
[5,1,3,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 1
[5,1,4,2,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 1
[5,1,4,3,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 0 + 1
[5,2,1,3,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 1
[5,2,3,1,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 1
[5,2,4,3,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 0 + 1
[5,3,1,2,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 1
Description
The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$.
Matching statistic: St001208
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001208: Permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 14%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001208: Permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 14%
Values
[1,2,3] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1 = 0 + 1
[1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 1 = 0 + 1
[3,1,2] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 1 = 0 + 1
[1,2,3,4] => [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1 = 0 + 1
[1,2,4,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1 = 0 + 1
[1,3,2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1 = 0 + 1
[1,3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1 = 0 + 1
[1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1 = 0 + 1
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 0 + 1
[2,1,3,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1 = 0 + 1
[2,1,4,3] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 1 = 0 + 1
[2,4,1,3] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 1 = 0 + 1
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 0 + 1
[3,2,4,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 0 + 1
[3,4,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 0 + 1
[4,1,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1 = 0 + 1
[4,1,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 0 + 1
[4,2,1,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 0 + 1
[4,3,2,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 1 = 0 + 1
[1,2,3,4,5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => ? = 2 + 1
[1,2,3,5,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ? = 2 + 1
[1,2,4,3,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ? = 0 + 1
[1,2,4,5,3] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ? = 0 + 1
[1,2,5,3,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ? = 2 + 1
[1,2,5,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 0 + 1
[1,3,2,4,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ? = 0 + 1
[1,3,2,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1 = 0 + 1
[1,3,4,2,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ? = 0 + 1
[1,3,4,5,2] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ? = 0 + 1
[1,3,5,2,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1 = 0 + 1
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 0 + 1
[1,4,2,3,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ? = 0 + 1
[1,4,2,5,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1 = 0 + 1
[1,4,3,2,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 0 + 1
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 0 + 1
[1,4,5,2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1 = 0 + 1
[1,4,5,3,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 0 + 1
[1,5,2,3,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ? = 2 + 1
[1,5,2,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 0 + 1
[1,5,3,2,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 0 + 1
[1,5,3,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 0 + 1
[1,5,4,2,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 0 + 1
[1,5,4,3,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? = 0 + 1
[2,1,3,4,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ? = 0 + 1
[2,1,3,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1 = 0 + 1
[2,1,4,3,5] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1 = 0 + 1
[2,1,4,5,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1 = 0 + 1
[2,1,5,3,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1 = 0 + 1
[2,1,5,4,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 1 = 0 + 1
[2,3,1,4,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ? = 0 + 1
[2,3,1,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1 = 0 + 1
[2,3,5,1,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1 = 0 + 1
[2,4,3,1,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 0 + 1
[2,4,3,5,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 0 + 1
[2,4,5,3,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 0 + 1
[2,5,1,3,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1 = 0 + 1
[2,5,1,4,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 1 = 0 + 1
[2,5,3,1,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 0 + 1
[2,5,4,3,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? = 0 + 1
[3,1,2,4,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ? = 0 + 1
[3,1,2,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1 = 0 + 1
[3,1,5,2,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1 = 0 + 1
[3,2,1,4,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 0 + 1
[3,2,1,5,4] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 1 = 0 + 1
[3,2,4,1,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 0 + 1
[3,2,4,5,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 0 + 1
[3,2,5,1,4] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 1 = 0 + 1
[3,2,5,4,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 1 = 0 + 1
[3,4,1,2,5] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1 = 0 + 1
[3,4,1,5,2] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1 = 0 + 1
[3,4,5,1,2] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1 = 0 + 1
[3,5,1,2,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1 = 0 + 1
[3,5,2,1,4] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 1 = 0 + 1
[3,5,2,4,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 1 = 0 + 1
[3,5,4,1,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 1 = 0 + 1
[4,1,3,2,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 0 + 1
[4,1,3,5,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 0 + 1
[4,1,5,3,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 1 = 0 + 1
[4,2,1,3,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 0 + 1
[4,2,1,5,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 1 = 0 + 1
[4,2,3,1,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 0 + 1
[4,2,3,5,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 0 + 1
[4,2,5,1,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 1 = 0 + 1
[4,2,5,3,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 1 = 0 + 1
[4,3,2,1,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? = 0 + 1
[4,3,2,5,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? = 0 + 1
[4,3,5,2,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? = 0 + 1
[4,5,1,3,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 1 = 0 + 1
[4,5,2,1,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 1 = 0 + 1
[4,5,3,2,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? = 0 + 1
[5,1,2,3,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ? = 2 + 1
[5,1,2,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 0 + 1
[5,1,3,2,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 0 + 1
[5,1,3,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 0 + 1
[5,1,4,2,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 0 + 1
[5,1,4,3,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? = 0 + 1
[5,2,1,3,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 0 + 1
[5,2,3,1,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 0 + 1
[5,2,4,3,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? = 0 + 1
[5,3,1,2,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 0 + 1
Description
The 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 $K[x]/(x^n)$.
Matching statistic: St001236
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St001236: Integer compositions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 14%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St001236: Integer compositions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 14%
Values
[1,2,3] => [3]
=> 1000 => [1,4] => 1 = 0 + 1
[1,3,2] => [2,1]
=> 1010 => [1,2,2] => 1 = 0 + 1
[3,1,2] => [2,1]
=> 1010 => [1,2,2] => 1 = 0 + 1
[1,2,3,4] => [4]
=> 10000 => [1,5] => 1 = 0 + 1
[1,2,4,3] => [3,1]
=> 10010 => [1,3,2] => 1 = 0 + 1
[1,3,2,4] => [3,1]
=> 10010 => [1,3,2] => 1 = 0 + 1
[1,3,4,2] => [3,1]
=> 10010 => [1,3,2] => 1 = 0 + 1
[1,4,2,3] => [3,1]
=> 10010 => [1,3,2] => 1 = 0 + 1
[1,4,3,2] => [2,1,1]
=> 10110 => [1,2,1,2] => 1 = 0 + 1
[2,1,3,4] => [3,1]
=> 10010 => [1,3,2] => 1 = 0 + 1
[2,1,4,3] => [2,2]
=> 1100 => [1,1,3] => 1 = 0 + 1
[2,4,1,3] => [2,2]
=> 1100 => [1,1,3] => 1 = 0 + 1
[3,2,1,4] => [2,1,1]
=> 10110 => [1,2,1,2] => 1 = 0 + 1
[3,2,4,1] => [2,1,1]
=> 10110 => [1,2,1,2] => 1 = 0 + 1
[3,4,2,1] => [2,1,1]
=> 10110 => [1,2,1,2] => 1 = 0 + 1
[4,1,2,3] => [3,1]
=> 10010 => [1,3,2] => 1 = 0 + 1
[4,1,3,2] => [2,1,1]
=> 10110 => [1,2,1,2] => 1 = 0 + 1
[4,2,1,3] => [2,1,1]
=> 10110 => [1,2,1,2] => 1 = 0 + 1
[4,3,2,1] => [1,1,1,1]
=> 11110 => [1,1,1,1,2] => 1 = 0 + 1
[1,2,3,4,5] => [5]
=> 100000 => [1,6] => ? = 2 + 1
[1,2,3,5,4] => [4,1]
=> 100010 => [1,4,2] => ? = 2 + 1
[1,2,4,3,5] => [4,1]
=> 100010 => [1,4,2] => ? = 0 + 1
[1,2,4,5,3] => [4,1]
=> 100010 => [1,4,2] => ? = 0 + 1
[1,2,5,3,4] => [4,1]
=> 100010 => [1,4,2] => ? = 2 + 1
[1,2,5,4,3] => [3,1,1]
=> 100110 => [1,3,1,2] => ? = 0 + 1
[1,3,2,4,5] => [4,1]
=> 100010 => [1,4,2] => ? = 0 + 1
[1,3,2,5,4] => [3,2]
=> 10100 => [1,2,3] => 1 = 0 + 1
[1,3,4,2,5] => [4,1]
=> 100010 => [1,4,2] => ? = 0 + 1
[1,3,4,5,2] => [4,1]
=> 100010 => [1,4,2] => ? = 0 + 1
[1,3,5,2,4] => [3,2]
=> 10100 => [1,2,3] => 1 = 0 + 1
[1,3,5,4,2] => [3,1,1]
=> 100110 => [1,3,1,2] => ? = 0 + 1
[1,4,2,3,5] => [4,1]
=> 100010 => [1,4,2] => ? = 0 + 1
[1,4,2,5,3] => [3,2]
=> 10100 => [1,2,3] => 1 = 0 + 1
[1,4,3,2,5] => [3,1,1]
=> 100110 => [1,3,1,2] => ? = 0 + 1
[1,4,3,5,2] => [3,1,1]
=> 100110 => [1,3,1,2] => ? = 0 + 1
[1,4,5,2,3] => [3,2]
=> 10100 => [1,2,3] => 1 = 0 + 1
[1,4,5,3,2] => [3,1,1]
=> 100110 => [1,3,1,2] => ? = 0 + 1
[1,5,2,3,4] => [4,1]
=> 100010 => [1,4,2] => ? = 2 + 1
[1,5,2,4,3] => [3,1,1]
=> 100110 => [1,3,1,2] => ? = 0 + 1
[1,5,3,2,4] => [3,1,1]
=> 100110 => [1,3,1,2] => ? = 0 + 1
[1,5,3,4,2] => [3,1,1]
=> 100110 => [1,3,1,2] => ? = 0 + 1
[1,5,4,2,3] => [3,1,1]
=> 100110 => [1,3,1,2] => ? = 0 + 1
[1,5,4,3,2] => [2,1,1,1]
=> 101110 => [1,2,1,1,2] => ? = 0 + 1
[2,1,3,4,5] => [4,1]
=> 100010 => [1,4,2] => ? = 0 + 1
[2,1,3,5,4] => [3,2]
=> 10100 => [1,2,3] => 1 = 0 + 1
[2,1,4,3,5] => [3,2]
=> 10100 => [1,2,3] => 1 = 0 + 1
[2,1,4,5,3] => [3,2]
=> 10100 => [1,2,3] => 1 = 0 + 1
[2,1,5,3,4] => [3,2]
=> 10100 => [1,2,3] => 1 = 0 + 1
[2,1,5,4,3] => [2,2,1]
=> 11010 => [1,1,2,2] => 1 = 0 + 1
[2,3,1,4,5] => [4,1]
=> 100010 => [1,4,2] => ? = 0 + 1
[2,3,1,5,4] => [3,2]
=> 10100 => [1,2,3] => 1 = 0 + 1
[2,3,5,1,4] => [3,2]
=> 10100 => [1,2,3] => 1 = 0 + 1
[2,4,3,1,5] => [3,1,1]
=> 100110 => [1,3,1,2] => ? = 0 + 1
[2,4,3,5,1] => [3,1,1]
=> 100110 => [1,3,1,2] => ? = 0 + 1
[2,4,5,3,1] => [3,1,1]
=> 100110 => [1,3,1,2] => ? = 0 + 1
[2,5,1,3,4] => [3,2]
=> 10100 => [1,2,3] => 1 = 0 + 1
[2,5,1,4,3] => [2,2,1]
=> 11010 => [1,1,2,2] => 1 = 0 + 1
[2,5,3,1,4] => [3,1,1]
=> 100110 => [1,3,1,2] => ? = 0 + 1
[2,5,4,3,1] => [2,1,1,1]
=> 101110 => [1,2,1,1,2] => ? = 0 + 1
[3,1,2,4,5] => [4,1]
=> 100010 => [1,4,2] => ? = 0 + 1
[3,1,2,5,4] => [3,2]
=> 10100 => [1,2,3] => 1 = 0 + 1
[3,1,5,2,4] => [3,2]
=> 10100 => [1,2,3] => 1 = 0 + 1
[3,2,1,4,5] => [3,1,1]
=> 100110 => [1,3,1,2] => ? = 0 + 1
[3,2,1,5,4] => [2,2,1]
=> 11010 => [1,1,2,2] => 1 = 0 + 1
[3,2,4,1,5] => [3,1,1]
=> 100110 => [1,3,1,2] => ? = 0 + 1
[3,2,4,5,1] => [3,1,1]
=> 100110 => [1,3,1,2] => ? = 0 + 1
[3,2,5,1,4] => [2,2,1]
=> 11010 => [1,1,2,2] => 1 = 0 + 1
[3,2,5,4,1] => [2,2,1]
=> 11010 => [1,1,2,2] => 1 = 0 + 1
[3,4,1,2,5] => [3,2]
=> 10100 => [1,2,3] => 1 = 0 + 1
[3,4,1,5,2] => [3,2]
=> 10100 => [1,2,3] => 1 = 0 + 1
[3,4,5,1,2] => [3,2]
=> 10100 => [1,2,3] => 1 = 0 + 1
[3,5,1,2,4] => [3,2]
=> 10100 => [1,2,3] => 1 = 0 + 1
[3,5,2,1,4] => [2,2,1]
=> 11010 => [1,1,2,2] => 1 = 0 + 1
[3,5,2,4,1] => [2,2,1]
=> 11010 => [1,1,2,2] => 1 = 0 + 1
[3,5,4,1,2] => [2,2,1]
=> 11010 => [1,1,2,2] => 1 = 0 + 1
[4,1,3,2,5] => [3,1,1]
=> 100110 => [1,3,1,2] => ? = 0 + 1
[4,1,3,5,2] => [3,1,1]
=> 100110 => [1,3,1,2] => ? = 0 + 1
[4,1,5,3,2] => [2,2,1]
=> 11010 => [1,1,2,2] => 1 = 0 + 1
[4,2,1,3,5] => [3,1,1]
=> 100110 => [1,3,1,2] => ? = 0 + 1
[4,2,1,5,3] => [2,2,1]
=> 11010 => [1,1,2,2] => 1 = 0 + 1
[4,2,3,1,5] => [3,1,1]
=> 100110 => [1,3,1,2] => ? = 0 + 1
[4,2,3,5,1] => [3,1,1]
=> 100110 => [1,3,1,2] => ? = 0 + 1
[4,2,5,1,3] => [2,2,1]
=> 11010 => [1,1,2,2] => 1 = 0 + 1
[4,2,5,3,1] => [2,2,1]
=> 11010 => [1,1,2,2] => 1 = 0 + 1
[4,3,2,1,5] => [2,1,1,1]
=> 101110 => [1,2,1,1,2] => ? = 0 + 1
[4,3,2,5,1] => [2,1,1,1]
=> 101110 => [1,2,1,1,2] => ? = 0 + 1
[4,3,5,2,1] => [2,1,1,1]
=> 101110 => [1,2,1,1,2] => ? = 0 + 1
[4,5,1,3,2] => [2,2,1]
=> 11010 => [1,1,2,2] => 1 = 0 + 1
[4,5,2,1,3] => [2,2,1]
=> 11010 => [1,1,2,2] => 1 = 0 + 1
[4,5,3,2,1] => [2,1,1,1]
=> 101110 => [1,2,1,1,2] => ? = 0 + 1
[5,1,2,3,4] => [4,1]
=> 100010 => [1,4,2] => ? = 2 + 1
[5,1,2,4,3] => [3,1,1]
=> 100110 => [1,3,1,2] => ? = 0 + 1
[5,1,3,2,4] => [3,1,1]
=> 100110 => [1,3,1,2] => ? = 0 + 1
[5,1,3,4,2] => [3,1,1]
=> 100110 => [1,3,1,2] => ? = 0 + 1
[5,1,4,2,3] => [3,1,1]
=> 100110 => [1,3,1,2] => ? = 0 + 1
[5,1,4,3,2] => [2,1,1,1]
=> 101110 => [1,2,1,1,2] => ? = 0 + 1
[5,2,1,3,4] => [3,1,1]
=> 100110 => [1,3,1,2] => ? = 0 + 1
[5,2,3,1,4] => [3,1,1]
=> 100110 => [1,3,1,2] => ? = 0 + 1
[5,2,4,3,1] => [2,1,1,1]
=> 101110 => [1,2,1,1,2] => ? = 0 + 1
[5,3,1,2,4] => [3,1,1]
=> 100110 => [1,3,1,2] => ? = 0 + 1
Description
The dominant dimension of the corresponding Comp-Nakayama algebra.
Matching statistic: St001301
(load all 57 compositions to match this statistic)
(load all 57 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001301: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 14%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001301: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 14%
Values
[1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> 0
[3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> 0
[1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0
[1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0
[1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0
[1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 0
[1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 0
[1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0
[2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 0
[3,2,1,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0
[3,2,4,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 0
[3,4,2,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 0
[4,1,3,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[4,2,1,3] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 0
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ([(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
[1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2
[1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ([(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
[1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 2
[1,2,5,4,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ([(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
[1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[1,3,4,2,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ([(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
[1,3,4,5,2] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 0
[1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[1,3,5,4,2] => [2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 0
[1,4,2,3,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ([(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
[1,4,2,5,3] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ([(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
[1,4,3,5,2] => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 0
[1,4,5,2,3] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 0
[1,4,5,3,2] => [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 0
[1,5,2,3,4] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 2
[1,5,2,4,3] => [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 0
[1,5,3,2,4] => [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 0
[1,5,3,4,2] => [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 0
[1,5,4,2,3] => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 0
[1,5,4,3,2] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,1,3,4,5] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ([(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
[2,1,3,5,4] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[2,1,4,3,5] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ([(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
[2,1,4,5,3] => [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[2,1,5,3,4] => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[2,1,5,4,3] => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[2,3,1,4,5] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> ([(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
[2,3,1,5,4] => [4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[2,3,5,1,4] => [4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[2,4,3,1,5] => [5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> ([(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
[2,4,3,5,1] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 0
[2,4,5,3,1] => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 0
[2,5,1,3,4] => [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 0
[2,5,1,4,3] => [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 0
[2,5,3,1,4] => [4,1,3,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 0
[2,5,4,3,1] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[3,1,2,4,5] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> ([(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
[3,1,2,5,4] => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[3,1,5,2,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[3,2,1,4,5] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ([(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
[3,2,1,5,4] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[3,2,4,1,5] => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ([(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
[3,2,4,5,1] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 0
[3,2,5,1,4] => [4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[3,2,5,4,1] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 0
[3,4,1,2,5] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> ([(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
[3,4,1,5,2] => [2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 0
[3,4,5,1,2] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 0
[3,5,1,2,4] => [4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 0
[3,5,2,1,4] => [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 0
[3,5,2,4,1] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 0
[3,5,4,1,2] => [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[4,1,3,2,5] => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> ([(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
[4,1,3,5,2] => [2,5,3,1,4] => [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 0
[4,1,5,3,2] => [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 0
[4,2,1,3,5] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> ([(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
[4,2,1,5,3] => [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[4,2,5,1,3] => [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 0
[4,2,5,3,1] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 0
[4,3,5,2,1] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 0
[4,5,1,3,2] => [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[4,5,2,1,3] => [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 0
[4,5,2,3,1] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[4,5,3,2,1] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[5,1,3,4,2] => [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 0
[5,1,4,2,3] => [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 0
[5,1,4,3,2] => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[5,2,4,3,1] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[5,3,2,4,1] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 0
[5,3,4,1,2] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[5,4,1,3,2] => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[5,4,2,1,3] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 0
[5,4,2,3,1] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[5,4,3,2,1] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,6,4,5,3,2] => [2,3,5,4,6,1] => [1,1,0,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 0
[1,6,5,4,3,2] => [2,3,4,5,6,1] => [1,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[2,6,5,4,3,1] => [1,3,4,5,6,2] => [1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
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.
Matching statistic: St001396
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001396: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 14%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001396: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 14%
Values
[1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> 0
[3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> 0
[1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0
[1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0
[1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0
[1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 0
[1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 0
[1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0
[2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 0
[3,2,1,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0
[3,2,4,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 0
[3,4,2,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 0
[4,1,3,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[4,2,1,3] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 0
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ([(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
[1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2
[1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ([(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
[1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 2
[1,2,5,4,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ([(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
[1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[1,3,4,2,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ([(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
[1,3,4,5,2] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 0
[1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[1,3,5,4,2] => [2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 0
[1,4,2,3,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ([(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
[1,4,2,5,3] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ([(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
[1,4,3,5,2] => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 0
[1,4,5,2,3] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 0
[1,4,5,3,2] => [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 0
[1,5,2,3,4] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 2
[1,5,2,4,3] => [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 0
[1,5,3,2,4] => [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 0
[1,5,3,4,2] => [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 0
[1,5,4,2,3] => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 0
[1,5,4,3,2] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,1,3,4,5] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ([(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
[2,1,3,5,4] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[2,1,4,3,5] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ([(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
[2,1,4,5,3] => [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[2,1,5,3,4] => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[2,1,5,4,3] => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[2,3,1,4,5] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> ([(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
[2,3,1,5,4] => [4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[2,3,5,1,4] => [4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[2,4,3,1,5] => [5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> ([(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
[2,4,3,5,1] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 0
[2,4,5,3,1] => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 0
[2,5,1,3,4] => [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 0
[2,5,1,4,3] => [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 0
[2,5,3,1,4] => [4,1,3,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 0
[2,5,4,3,1] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[3,1,2,4,5] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> ([(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
[3,1,2,5,4] => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[3,1,5,2,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[3,2,1,4,5] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ([(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
[3,2,1,5,4] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[3,2,4,1,5] => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ([(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
[3,2,4,5,1] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 0
[3,2,5,1,4] => [4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[3,2,5,4,1] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 0
[3,4,1,2,5] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> ([(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
[3,4,1,5,2] => [2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 0
[3,4,5,1,2] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 0
[3,5,1,2,4] => [4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 0
[3,5,2,1,4] => [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 0
[3,5,2,4,1] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 0
[3,5,4,1,2] => [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[4,1,3,2,5] => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> ([(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
[4,1,3,5,2] => [2,5,3,1,4] => [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 0
[4,1,5,3,2] => [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 0
[4,2,1,3,5] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> ([(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
[4,2,1,5,3] => [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[4,2,5,1,3] => [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 0
[4,2,5,3,1] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 0
[4,3,5,2,1] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 0
[4,5,1,3,2] => [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[4,5,2,1,3] => [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 0
[4,5,2,3,1] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[4,5,3,2,1] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[5,1,3,4,2] => [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 0
[5,1,4,2,3] => [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 0
[5,1,4,3,2] => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[5,2,4,3,1] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[5,3,2,4,1] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 0
[5,3,4,1,2] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[5,4,1,3,2] => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[5,4,2,1,3] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 0
[5,4,2,3,1] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[5,4,3,2,1] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,6,4,5,3,2] => [2,3,5,4,6,1] => [1,1,0,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 0
[1,6,5,4,3,2] => [2,3,4,5,6,1] => [1,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[2,6,5,4,3,1] => [1,3,4,5,6,2] => [1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
Description
Number of triples of incomparable elements in a finite poset.
For a finite poset this is the number of 3-element sets $S \in \binom{P}{3}$ that are pairwise incomparable.
The following 86 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001555The order of a signed permutation. St001895The oddness of a signed permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St000256The number of parts from which one can substract 2 and still get an integer partition. St001490The number of connected components of a skew partition. 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. St001624The breadth of a lattice. St001845The number of join irreducibles minus the rank of a lattice. St001964The interval resolution global dimension of a poset. St001625The Möbius invariant of a lattice. St001621The number of atoms of a lattice. St001429The number of negative entries in a signed permutation. 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. St000022The number of fixed points of a permutation. St000731The number of double exceedences of a permutation. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001889The size of the connectivity set of a signed permutation. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001866The nesting alignments of a signed permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001851The number of Hecke atoms of a signed permutation. St001877Number of indecomposable injective modules with projective dimension 2. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000323The minimal crossing number of a graph. St000351The determinant of the adjacency matrix of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000370The genus of a graph. St000379The number of Hamiltonian cycles in a graph. St000403The Szeged index minus the Wiener index of a graph. St000671The maximin edge-connectivity for choosing a subgraph. St000699The toughness times the least common multiple of 1,. St000948The chromatic discriminant of a graph. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001119The length of a shortest maximal path in a graph. St001271The competition number of a graph. St001281The normalized isoperimetric number of a graph. St001305The number of induced cycles on four vertices in a graph. St001307The number of induced stars on four vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001323The independence gap of a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001357The maximal degree of a regular spanning subgraph of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001395The number of strictly unfriendly partitions of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001793The difference between the clique number and the chromatic number of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001797The number of overfull subgraphs of a graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St001316The domatic number of a graph. St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001496The number of graphs with the same Laplacian spectrum as the given graph. St000636The hull number of a graph. St001029The size of the core of a graph. St001109The number of proper colourings of a graph with as few colours as possible. St001111The weak 2-dynamic chromatic number of a graph. St001654The monophonic hull number of a graph. St001716The 1-improper chromatic number of a graph. St001116The game chromatic number of a graph.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!