Your data matches 55 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000228
(load all 36 compositions to match this statistic)
Mapping
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1]
 => 1
([],2)
 => [1,1]
 => 2
([(0,1)],2)
 => [2]
 => 2
([],3)
 => [1,1,1]
 => 3
([(1,2)],3)
 => [2,1]
 => 3
([(0,1),(0,2)],3)
 => [2,1]
 => 3
([(0,2),(2,1)],3)
 => [3]
 => 3
([(0,2),(1,2)],3)
 => [2,1]
 => 3
([],4)
 => [1,1,1,1]
 => 4
([(2,3)],4)
 => [2,1,1]
 => 4
([(1,2),(1,3)],4)
 => [2,1,1]
 => 4
([(0,1),(0,2),(0,3)],4)
 => [2,1,1]
 => 4
([(0,2),(0,3),(3,1)],4)
 => [3,1]
 => 4
([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => 4
([(1,2),(2,3)],4)
 => [3,1]
 => 4
([(0,3),(3,1),(3,2)],4)
 => [3,1]
 => 4
([(1,3),(2,3)],4)
 => [2,1,1]
 => 4
([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => 4
([(0,3),(1,3),(2,3)],4)
 => [2,1,1]
 => 4
([(0,3),(1,2)],4)
 => [2,2]
 => 4
([(0,3),(1,2),(1,3)],4)
 => [2,2]
 => 4
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => 4
([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => 4
([],5)
 => [1,1,1,1,1]
 => 5
([(3,4)],5)
 => [2,1,1,1]
 => 5
([(2,3),(2,4)],5)
 => [2,1,1,1]
 => 5
([(1,2),(1,3),(1,4)],5)
 => [2,1,1,1]
 => 5
([(0,1),(0,2),(0,3),(0,4)],5)
 => [2,1,1,1]
 => 5
([(0,2),(0,3),(0,4),(4,1)],5)
 => [3,1,1]
 => 5
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 5
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => 5
([(1,3),(1,4),(4,2)],5)
 => [3,1,1]
 => 5
([(0,3),(0,4),(4,1),(4,2)],5)
 => [3,1,1]
 => 5
([(1,2),(1,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 5
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [4,1]
 => 5
([(0,3),(0,4),(3,2),(4,1)],5)
 => [3,2]
 => 5
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2]
 => 5
([(2,3),(3,4)],5)
 => [3,1,1]
 => 5
([(1,4),(4,2),(4,3)],5)
 => [3,1,1]
 => 5
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,1,1]
 => 5
([(2,4),(3,4)],5)
 => [2,1,1,1]
 => 5
([(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => 5
([(0,4),(1,4),(4,2),(4,3)],5)
 => [3,2]
 => 5
([(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => 5
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => 5
([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => 5
([(0,4),(1,4),(2,3)],5)
 => [2,2,1]
 => 5
([(0,4),(1,3),(2,3),(2,4)],5)
 => [2,2,1]
 => 5
Description
The size of a partition. This statistic is the constant statistic of the level sets.
Matching statistic: St000293
(load all 8 compositions to match this statistic)
Mapping
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1]
 => 10 => 1
([],2)
 => [1,1]
 => 110 => 2
([(0,1)],2)
 => [2]
 => 100 => 2
([],3)
 => [1,1,1]
 => 1110 => 3
([(1,2)],3)
 => [2,1]
 => 1010 => 3
([(0,1),(0,2)],3)
 => [2,1]
 => 1010 => 3
([(0,2),(2,1)],3)
 => [3]
 => 1000 => 3
([(0,2),(1,2)],3)
 => [2,1]
 => 1010 => 3
([],4)
 => [1,1,1,1]
 => 11110 => 4
([(2,3)],4)
 => [2,1,1]
 => 10110 => 4
([(1,2),(1,3)],4)
 => [2,1,1]
 => 10110 => 4
([(0,1),(0,2),(0,3)],4)
 => [2,1,1]
 => 10110 => 4
([(0,2),(0,3),(3,1)],4)
 => [3,1]
 => 10010 => 4
([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => 10010 => 4
([(1,2),(2,3)],4)
 => [3,1]
 => 10010 => 4
([(0,3),(3,1),(3,2)],4)
 => [3,1]
 => 10010 => 4
([(1,3),(2,3)],4)
 => [2,1,1]
 => 10110 => 4
([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => 10010 => 4
([(0,3),(1,3),(2,3)],4)
 => [2,1,1]
 => 10110 => 4
([(0,3),(1,2)],4)
 => [2,2]
 => 1100 => 4
([(0,3),(1,2),(1,3)],4)
 => [2,2]
 => 1100 => 4
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => 1100 => 4
([(0,3),(2,1),(3,2)],4)
 => [4]
 => 10000 => 4
([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => 10010 => 4
([],5)
 => [1,1,1,1,1]
 => 111110 => 5
([(3,4)],5)
 => [2,1,1,1]
 => 101110 => 5
([(2,3),(2,4)],5)
 => [2,1,1,1]
 => 101110 => 5
([(1,2),(1,3),(1,4)],5)
 => [2,1,1,1]
 => 101110 => 5
([(0,1),(0,2),(0,3),(0,4)],5)
 => [2,1,1,1]
 => 101110 => 5
([(0,2),(0,3),(0,4),(4,1)],5)
 => [3,1,1]
 => 100110 => 5
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 100110 => 5
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => 100110 => 5
([(1,3),(1,4),(4,2)],5)
 => [3,1,1]
 => 100110 => 5
([(0,3),(0,4),(4,1),(4,2)],5)
 => [3,1,1]
 => 100110 => 5
([(1,2),(1,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 100110 => 5
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [4,1]
 => 100010 => 5
([(0,3),(0,4),(3,2),(4,1)],5)
 => [3,2]
 => 10100 => 5
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => 10100 => 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2]
 => 10100 => 5
([(2,3),(3,4)],5)
 => [3,1,1]
 => 100110 => 5
([(1,4),(4,2),(4,3)],5)
 => [3,1,1]
 => 100110 => 5
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,1,1]
 => 100110 => 5
([(2,4),(3,4)],5)
 => [2,1,1,1]
 => 101110 => 5
([(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => 100110 => 5
([(0,4),(1,4),(4,2),(4,3)],5)
 => [3,2]
 => 10100 => 5
([(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => 101110 => 5
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => 100110 => 5
([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => 101110 => 5
([(0,4),(1,4),(2,3)],5)
 => [2,2,1]
 => 11010 => 5
([(0,4),(1,3),(2,3),(2,4)],5)
 => [2,2,1]
 => 11010 => 5
Description
The number of inversions of a binary word.
Matching statistic: St001034
(load all 9 compositions to match this statistic)
Mapping
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001034: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1]
 => [1,0]
 => 1
([],2)
 => [1,1]
 => [1,1,0,0]
 => 2
([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 2
([],3)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 3
([(1,2)],3)
 => [2,1]
 => [1,0,1,1,0,0]
 => 3
([(0,1),(0,2)],3)
 => [2,1]
 => [1,0,1,1,0,0]
 => 3
([(0,2),(2,1)],3)
 => [3]
 => [1,0,1,0,1,0]
 => 3
([(0,2),(1,2)],3)
 => [2,1]
 => [1,0,1,1,0,0]
 => 3
([],4)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 4
([(2,3)],4)
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 4
([(1,2),(1,3)],4)
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 4
([(0,1),(0,2),(0,3)],4)
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 4
([(0,2),(0,3),(3,1)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 4
([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 4
([(1,2),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 4
([(0,3),(3,1),(3,2)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 4
([(1,3),(2,3)],4)
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 4
([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 4
([(0,3),(1,3),(2,3)],4)
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 4
([(0,3),(1,2)],4)
 => [2,2]
 => [1,1,1,0,0,0]
 => 4
([(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [1,1,1,0,0,0]
 => 4
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [1,1,1,0,0,0]
 => 4
([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 4
([],5)
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 5
([(3,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 5
([(2,3),(2,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 5
([(1,2),(1,3),(1,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 5
([(0,1),(0,2),(0,3),(0,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 5
([(0,2),(0,3),(0,4),(4,1)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 5
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 5
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 5
([(1,3),(1,4),(4,2)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 5
([(0,3),(0,4),(4,1),(4,2)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 5
([(1,2),(1,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 5
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 5
([(0,3),(0,4),(3,2),(4,1)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 5
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 5
([(2,3),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 5
([(1,4),(4,2),(4,3)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 5
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 5
([(2,4),(3,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 5
([(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 5
([(0,4),(1,4),(4,2),(4,3)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 5
([(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 5
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 5
([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 5
([(0,4),(1,4),(2,3)],5)
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 5
([(0,4),(1,3),(2,3),(2,4)],5)
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 5
Description
The area of the parallelogram polyomino associated with the Dyck path. The (bivariate) generating function is given in [1].
Matching statistic: St000018
Mapping
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000018: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1]
 => [1,0,1,0]
 => [2,1] => 1
([],2)
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 2
([(0,1)],2)
 => [2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 2
([],3)
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 3
([(1,2)],3)
 => [2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => 3
([(0,1),(0,2)],3)
 => [2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => 3
([(0,2),(2,1)],3)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 3
([(0,2),(1,2)],3)
 => [2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => 3
([],4)
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 4
([(2,3)],4)
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => 4
([(1,2),(1,3)],4)
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => 4
([(0,1),(0,2),(0,3)],4)
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => 4
([(0,2),(0,3),(3,1)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => 4
([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => 4
([(1,2),(2,3)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => 4
([(0,3),(3,1),(3,2)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => 4
([(1,3),(2,3)],4)
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => 4
([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => 4
([(0,3),(1,3),(2,3)],4)
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => 4
([(0,3),(1,2)],4)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 4
([(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 4
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 4
([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => 4
([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => 4
([],5)
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => 5
([(3,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => 5
([(2,3),(2,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => 5
([(1,2),(1,3),(1,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => 5
([(0,1),(0,2),(0,3),(0,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => 5
([(0,2),(0,3),(0,4),(4,1)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 5
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 5
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 5
([(1,3),(1,4),(4,2)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 5
([(0,3),(0,4),(4,1),(4,2)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 5
([(1,2),(1,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 5
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => 5
([(0,3),(0,4),(3,2),(4,1)],5)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => 5
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => 5
([(2,3),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 5
([(1,4),(4,2),(4,3)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 5
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 5
([(2,4),(3,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => 5
([(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 5
([(0,4),(1,4),(4,2),(4,3)],5)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => 5
([(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => 5
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 5
([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => 5
([(0,4),(1,4),(2,3)],5)
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => 5
([(0,4),(1,3),(2,3),(2,4)],5)
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => 5
Description
The number of inversions of a permutation. This equals the minimal number of simple transpositions $(i,i+1)$ needed to write $\pi$. Thus, it is also the Coxeter length of $\pi$.
Matching statistic: St000246
(load all 2 compositions to match this statistic)
Mapping
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000246: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1]
 => [1,0,1,0]
 => [1,2] => 1
([],2)
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 2
([(0,1)],2)
 => [2]
 => [1,1,0,0,1,0]
 => [2,1,3] => 2
([],3)
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 3
([(1,2)],3)
 => [2,1]
 => [1,0,1,0,1,0]
 => [1,2,3] => 3
([(0,1),(0,2)],3)
 => [2,1]
 => [1,0,1,0,1,0]
 => [1,2,3] => 3
([(0,2),(2,1)],3)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 3
([(0,2),(1,2)],3)
 => [2,1]
 => [1,0,1,0,1,0]
 => [1,2,3] => 3
([],4)
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => 4
([(2,3)],4)
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 4
([(1,2),(1,3)],4)
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 4
([(0,1),(0,2),(0,3)],4)
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 4
([(0,2),(0,3),(3,1)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 4
([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 4
([(1,2),(2,3)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 4
([(0,3),(3,1),(3,2)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 4
([(1,3),(2,3)],4)
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 4
([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 4
([(0,3),(1,3),(2,3)],4)
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 4
([(0,3),(1,2)],4)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 4
([(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 4
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 4
([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => 4
([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 4
([],5)
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,6,5,4,3,2] => 5
([(3,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => 5
([(2,3),(2,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => 5
([(1,2),(1,3),(1,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => 5
([(0,1),(0,2),(0,3),(0,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => 5
([(0,2),(0,3),(0,4),(4,1)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 5
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 5
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 5
([(1,3),(1,4),(4,2)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 5
([(0,3),(0,4),(4,1),(4,2)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 5
([(1,2),(1,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 5
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [3,4,2,1,5] => 5
([(0,3),(0,4),(3,2),(4,1)],5)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 5
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 5
([(2,3),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 5
([(1,4),(4,2),(4,3)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 5
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 5
([(2,4),(3,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => 5
([(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 5
([(0,4),(1,4),(4,2),(4,3)],5)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 5
([(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => 5
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 5
([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => 5
([(0,4),(1,4),(2,3)],5)
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 5
([(0,4),(1,3),(2,3),(2,4)],5)
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 5
Description
The number of non-inversions of a permutation. For a permutation of $\{1,\ldots,n\}$, this is given by $\operatorname{noninv}(\pi) = \binom{n}{2}-\operatorname{inv}(\pi)$.
Matching statistic: St000290
Mapping
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
St000290: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1]
 => 10 => 10 => 1
([],2)
 => [1,1]
 => 110 => 110 => 2
([(0,1)],2)
 => [2]
 => 100 => 010 => 2
([],3)
 => [1,1,1]
 => 1110 => 1110 => 3
([(1,2)],3)
 => [2,1]
 => 1010 => 0110 => 3
([(0,1),(0,2)],3)
 => [2,1]
 => 1010 => 0110 => 3
([(0,2),(2,1)],3)
 => [3]
 => 1000 => 0010 => 3
([(0,2),(1,2)],3)
 => [2,1]
 => 1010 => 0110 => 3
([],4)
 => [1,1,1,1]
 => 11110 => 11110 => 4
([(2,3)],4)
 => [2,1,1]
 => 10110 => 01110 => 4
([(1,2),(1,3)],4)
 => [2,1,1]
 => 10110 => 01110 => 4
([(0,1),(0,2),(0,3)],4)
 => [2,1,1]
 => 10110 => 01110 => 4
([(0,2),(0,3),(3,1)],4)
 => [3,1]
 => 10010 => 00110 => 4
([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => 10010 => 00110 => 4
([(1,2),(2,3)],4)
 => [3,1]
 => 10010 => 00110 => 4
([(0,3),(3,1),(3,2)],4)
 => [3,1]
 => 10010 => 00110 => 4
([(1,3),(2,3)],4)
 => [2,1,1]
 => 10110 => 01110 => 4
([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => 10010 => 00110 => 4
([(0,3),(1,3),(2,3)],4)
 => [2,1,1]
 => 10110 => 01110 => 4
([(0,3),(1,2)],4)
 => [2,2]
 => 1100 => 1010 => 4
([(0,3),(1,2),(1,3)],4)
 => [2,2]
 => 1100 => 1010 => 4
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => 1100 => 1010 => 4
([(0,3),(2,1),(3,2)],4)
 => [4]
 => 10000 => 00010 => 4
([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => 10010 => 00110 => 4
([],5)
 => [1,1,1,1,1]
 => 111110 => 111110 => 5
([(3,4)],5)
 => [2,1,1,1]
 => 101110 => 011110 => 5
([(2,3),(2,4)],5)
 => [2,1,1,1]
 => 101110 => 011110 => 5
([(1,2),(1,3),(1,4)],5)
 => [2,1,1,1]
 => 101110 => 011110 => 5
([(0,1),(0,2),(0,3),(0,4)],5)
 => [2,1,1,1]
 => 101110 => 011110 => 5
([(0,2),(0,3),(0,4),(4,1)],5)
 => [3,1,1]
 => 100110 => 001110 => 5
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 100110 => 001110 => 5
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => 100110 => 001110 => 5
([(1,3),(1,4),(4,2)],5)
 => [3,1,1]
 => 100110 => 001110 => 5
([(0,3),(0,4),(4,1),(4,2)],5)
 => [3,1,1]
 => 100110 => 001110 => 5
([(1,2),(1,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 100110 => 001110 => 5
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [4,1]
 => 100010 => 000110 => 5
([(0,3),(0,4),(3,2),(4,1)],5)
 => [3,2]
 => 10100 => 10010 => 5
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => 10100 => 10010 => 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2]
 => 10100 => 10010 => 5
([(2,3),(3,4)],5)
 => [3,1,1]
 => 100110 => 001110 => 5
([(1,4),(4,2),(4,3)],5)
 => [3,1,1]
 => 100110 => 001110 => 5
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,1,1]
 => 100110 => 001110 => 5
([(2,4),(3,4)],5)
 => [2,1,1,1]
 => 101110 => 011110 => 5
([(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => 100110 => 001110 => 5
([(0,4),(1,4),(4,2),(4,3)],5)
 => [3,2]
 => 10100 => 10010 => 5
([(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => 101110 => 011110 => 5
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => 100110 => 001110 => 5
([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => 101110 => 011110 => 5
([(0,4),(1,4),(2,3)],5)
 => [2,2,1]
 => 11010 => 10110 => 5
([(0,4),(1,3),(2,3),(2,4)],5)
 => [2,2,1]
 => 11010 => 10110 => 5
Description
The major index of a binary word. This is the sum of the positions of descents, i.e., a one followed by a zero. For words of length $n$ with $a$ zeros, the generating function for the major index is the $q$-binomial coefficient $\binom{n}{a}_q$.
Matching statistic: St000395
(load all 4 compositions to match this statistic)
Mapping
Mp00074: Posets —to graph⟶ Graphs
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000395: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],1)
 => [1] => [1,0]
 => 1
([],2)
 => ([],2)
 => [2] => [1,1,0,0]
 => 2
([(0,1)],2)
 => ([(0,1)],2)
 => [1,1] => [1,0,1,0]
 => 2
([],3)
 => ([],3)
 => [3] => [1,1,1,0,0,0]
 => 3
([(1,2)],3)
 => ([(1,2)],3)
 => [1,2] => [1,0,1,1,0,0]
 => 3
([(0,1),(0,2)],3)
 => ([(0,2),(1,2)],3)
 => [1,1,1] => [1,0,1,0,1,0]
 => 3
([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => [1,1,1] => [1,0,1,0,1,0]
 => 3
([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => [1,1,1] => [1,0,1,0,1,0]
 => 3
([],4)
 => ([],4)
 => [4] => [1,1,1,1,0,0,0,0]
 => 4
([(2,3)],4)
 => ([(2,3)],4)
 => [1,3] => [1,0,1,1,1,0,0,0]
 => 4
([(1,2),(1,3)],4)
 => ([(1,3),(2,3)],4)
 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 4
([(0,1),(0,2),(0,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 4
([(0,2),(0,3),(3,1)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 4
([(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 4
([(0,3),(3,1),(3,2)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 4
([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 4
([(0,3),(1,3),(3,2)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 4
([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 4
([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 4
([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 4
([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4
([],5)
 => ([],5)
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 5
([(3,4)],5)
 => ([(3,4)],5)
 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 5
([(2,3),(2,4)],5)
 => ([(2,4),(3,4)],5)
 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 5
([(1,2),(1,3),(1,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 5
([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 5
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 5
([(1,3),(1,4),(4,2)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 5
([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 5
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 5
([(2,3),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 5
([(1,4),(4,2),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 5
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 5
([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 5
([(1,4),(2,4),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 5
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 5
([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 5
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 5
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 5
([(0,4),(1,4),(2,3)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 5
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5
Description
The sum of the heights of the peaks of a Dyck path.
Matching statistic: St001020
(load all 2 compositions to match this statistic)
Mapping
Mp00074: Posets —to graph⟶ Graphs
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001020: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],1)
 => [1] => [1,0]
 => 1
([],2)
 => ([],2)
 => [2] => [1,1,0,0]
 => 2
([(0,1)],2)
 => ([(0,1)],2)
 => [1,1] => [1,0,1,0]
 => 2
([],3)
 => ([],3)
 => [3] => [1,1,1,0,0,0]
 => 3
([(1,2)],3)
 => ([(1,2)],3)
 => [1,2] => [1,0,1,1,0,0]
 => 3
([(0,1),(0,2)],3)
 => ([(0,2),(1,2)],3)
 => [1,1,1] => [1,0,1,0,1,0]
 => 3
([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => [1,1,1] => [1,0,1,0,1,0]
 => 3
([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => [1,1,1] => [1,0,1,0,1,0]
 => 3
([],4)
 => ([],4)
 => [4] => [1,1,1,1,0,0,0,0]
 => 4
([(2,3)],4)
 => ([(2,3)],4)
 => [1,3] => [1,0,1,1,1,0,0,0]
 => 4
([(1,2),(1,3)],4)
 => ([(1,3),(2,3)],4)
 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 4
([(0,1),(0,2),(0,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 4
([(0,2),(0,3),(3,1)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 4
([(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 4
([(0,3),(3,1),(3,2)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 4
([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 4
([(0,3),(1,3),(3,2)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 4
([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 4
([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 4
([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 4
([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4
([],5)
 => ([],5)
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 5
([(3,4)],5)
 => ([(3,4)],5)
 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 5
([(2,3),(2,4)],5)
 => ([(2,4),(3,4)],5)
 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 5
([(1,2),(1,3),(1,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 5
([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 5
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 5
([(1,3),(1,4),(4,2)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 5
([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 5
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 5
([(2,3),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 5
([(1,4),(4,2),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 5
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 5
([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 5
([(1,4),(2,4),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 5
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 5
([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 5
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 5
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 5
([(0,4),(1,4),(2,3)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 5
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5
Description
Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000998
(load all 2 compositions to match this statistic)
Mapping
Mp00074: Posets —to graph⟶ Graphs
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000998: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],1)
 => [1] => [1,0]
 => 2 = 1 + 1
([],2)
 => ([],2)
 => [2] => [1,1,0,0]
 => 3 = 2 + 1
([(0,1)],2)
 => ([(0,1)],2)
 => [1,1] => [1,0,1,0]
 => 3 = 2 + 1
([],3)
 => ([],3)
 => [3] => [1,1,1,0,0,0]
 => 4 = 3 + 1
([(1,2)],3)
 => ([(1,2)],3)
 => [1,2] => [1,0,1,1,0,0]
 => 4 = 3 + 1
([(0,1),(0,2)],3)
 => ([(0,2),(1,2)],3)
 => [1,1,1] => [1,0,1,0,1,0]
 => 4 = 3 + 1
([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => [1,1,1] => [1,0,1,0,1,0]
 => 4 = 3 + 1
([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => [1,1,1] => [1,0,1,0,1,0]
 => 4 = 3 + 1
([],4)
 => ([],4)
 => [4] => [1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
([(2,3)],4)
 => ([(2,3)],4)
 => [1,3] => [1,0,1,1,1,0,0,0]
 => 5 = 4 + 1
([(1,2),(1,3)],4)
 => ([(1,3),(2,3)],4)
 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 5 = 4 + 1
([(0,1),(0,2),(0,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 5 = 4 + 1
([(0,2),(0,3),(3,1)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 5 = 4 + 1
([(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 5 = 4 + 1
([(0,3),(3,1),(3,2)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 5 = 4 + 1
([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 5 = 4 + 1
([(0,3),(1,3),(3,2)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 5 = 4 + 1
([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 5 = 4 + 1
([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 5 = 4 + 1
([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 5 = 4 + 1
([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
([],5)
 => ([],5)
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
([(3,4)],5)
 => ([(3,4)],5)
 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 6 = 5 + 1
([(2,3),(2,4)],5)
 => ([(2,4),(3,4)],5)
 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 6 = 5 + 1
([(1,2),(1,3),(1,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 6 = 5 + 1
([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 6 = 5 + 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 6 = 5 + 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 6 = 5 + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 6 = 5 + 1
([(1,3),(1,4),(4,2)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 6 = 5 + 1
([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 6 = 5 + 1
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 6 = 5 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 6 = 5 + 1
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 6 = 5 + 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 6 = 5 + 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 6 = 5 + 1
([(2,3),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 6 = 5 + 1
([(1,4),(4,2),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 6 = 5 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 6 = 5 + 1
([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 6 = 5 + 1
([(1,4),(2,4),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 6 = 5 + 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 6 = 5 + 1
([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 6 = 5 + 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 6 = 5 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 6 = 5 + 1
([(0,4),(1,4),(2,3)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 6 = 5 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 6 = 5 + 1
Description
Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001641
Mapping
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St001641: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1]
 => [[1]]
 => {{1}}
 => 0 = 1 - 1
([],2)
 => [1,1]
 => [[1],[2]]
 => {{1},{2}}
 => 1 = 2 - 1
([(0,1)],2)
 => [2]
 => [[1,2]]
 => {{1,2}}
 => 1 = 2 - 1
([],3)
 => [1,1,1]
 => [[1],[2],[3]]
 => {{1},{2},{3}}
 => 2 = 3 - 1
([(1,2)],3)
 => [2,1]
 => [[1,2],[3]]
 => {{1,2},{3}}
 => 2 = 3 - 1
([(0,1),(0,2)],3)
 => [2,1]
 => [[1,2],[3]]
 => {{1,2},{3}}
 => 2 = 3 - 1
([(0,2),(2,1)],3)
 => [3]
 => [[1,2,3]]
 => {{1,2,3}}
 => 2 = 3 - 1
([(0,2),(1,2)],3)
 => [2,1]
 => [[1,2],[3]]
 => {{1,2},{3}}
 => 2 = 3 - 1
([],4)
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => {{1},{2},{3},{4}}
 => 3 = 4 - 1
([(2,3)],4)
 => [2,1,1]
 => [[1,2],[3],[4]]
 => {{1,2},{3},{4}}
 => 3 = 4 - 1
([(1,2),(1,3)],4)
 => [2,1,1]
 => [[1,2],[3],[4]]
 => {{1,2},{3},{4}}
 => 3 = 4 - 1
([(0,1),(0,2),(0,3)],4)
 => [2,1,1]
 => [[1,2],[3],[4]]
 => {{1,2},{3},{4}}
 => 3 = 4 - 1
([(0,2),(0,3),(3,1)],4)
 => [3,1]
 => [[1,2,3],[4]]
 => {{1,2,3},{4}}
 => 3 = 4 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => [[1,2,3],[4]]
 => {{1,2,3},{4}}
 => 3 = 4 - 1
([(1,2),(2,3)],4)
 => [3,1]
 => [[1,2,3],[4]]
 => {{1,2,3},{4}}
 => 3 = 4 - 1
([(0,3),(3,1),(3,2)],4)
 => [3,1]
 => [[1,2,3],[4]]
 => {{1,2,3},{4}}
 => 3 = 4 - 1
([(1,3),(2,3)],4)
 => [2,1,1]
 => [[1,2],[3],[4]]
 => {{1,2},{3},{4}}
 => 3 = 4 - 1
([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [[1,2,3],[4]]
 => {{1,2,3},{4}}
 => 3 = 4 - 1
([(0,3),(1,3),(2,3)],4)
 => [2,1,1]
 => [[1,2],[3],[4]]
 => {{1,2},{3},{4}}
 => 3 = 4 - 1
([(0,3),(1,2)],4)
 => [2,2]
 => [[1,2],[3,4]]
 => {{1,2},{3,4}}
 => 3 = 4 - 1
([(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [[1,2],[3,4]]
 => {{1,2},{3,4}}
 => 3 = 4 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [[1,2],[3,4]]
 => {{1,2},{3,4}}
 => 3 = 4 - 1
([(0,3),(2,1),(3,2)],4)
 => [4]
 => [[1,2,3,4]]
 => {{1,2,3,4}}
 => 3 = 4 - 1
([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [[1,2,3],[4]]
 => {{1,2,3},{4}}
 => 3 = 4 - 1
([],5)
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => {{1},{2},{3},{4},{5}}
 => 4 = 5 - 1
([(3,4)],5)
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => {{1,2},{3},{4},{5}}
 => 4 = 5 - 1
([(2,3),(2,4)],5)
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => {{1,2},{3},{4},{5}}
 => 4 = 5 - 1
([(1,2),(1,3),(1,4)],5)
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => {{1,2},{3},{4},{5}}
 => 4 = 5 - 1
([(0,1),(0,2),(0,3),(0,4)],5)
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => {{1,2},{3},{4},{5}}
 => 4 = 5 - 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 4 = 5 - 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 4 = 5 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 4 = 5 - 1
([(1,3),(1,4),(4,2)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 4 = 5 - 1
([(0,3),(0,4),(4,1),(4,2)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 4 = 5 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 4 = 5 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [4,1]
 => [[1,2,3,4],[5]]
 => {{1,2,3,4},{5}}
 => 4 = 5 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
 => [3,2]
 => [[1,2,3],[4,5]]
 => {{1,2,3},{4,5}}
 => 4 = 5 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => [[1,2,3],[4,5]]
 => {{1,2,3},{4,5}}
 => 4 = 5 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2]
 => [[1,2,3],[4,5]]
 => {{1,2,3},{4,5}}
 => 4 = 5 - 1
([(2,3),(3,4)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 4 = 5 - 1
([(1,4),(4,2),(4,3)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 4 = 5 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 4 = 5 - 1
([(2,4),(3,4)],5)
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => {{1,2},{3},{4},{5}}
 => 4 = 5 - 1
([(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 4 = 5 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => [3,2]
 => [[1,2,3],[4,5]]
 => {{1,2,3},{4,5}}
 => 4 = 5 - 1
([(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => {{1,2},{3},{4},{5}}
 => 4 = 5 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 4 = 5 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => {{1,2},{3},{4},{5}}
 => 4 = 5 - 1
([(0,4),(1,4),(2,3)],5)
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => {{1,2},{3,4},{5}}
 => 4 = 5 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => {{1,2},{3,4},{5}}
 => 4 = 5 - 1
Description
The number of ascent tops in the flattened set partition such that all smaller elements appear before. Let $P$ be a set partition. The flattened set partition is the permutation obtained by sorting the set of blocks of $P$ according to their minimal element and the elements in each block in increasing order. Given a set partition $P$, this statistic is the binary logarithm of the number of set partitions that flatten to the same permutation as $P$.
The following 45 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001746The coalition number of a graph. St001759The Rajchgot index of a permutation. St001645The pebbling number of a connected graph. St000719The number of alignments in a perfect matching. St000189The number of elements in the poset. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001342The number of vertices in the center of a graph. St000144The pyramid weight of the Dyck path. St000229Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St001723The differential of a graph. St001724The 2-packing differential of a graph. St000171The degree of the graph. St001120The length of a longest path in a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001622The number of join-irreducible elements of a lattice. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001725The harmonious chromatic number of a graph. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St000147The largest part of an integer partition. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000384The maximal part of the shifted composition of an integer partition. St000784The maximum of the length and the largest part of the integer partition. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St000093The cardinality of a maximal independent set of vertices of a graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000186The sum of the first row in a Gelfand-Tsetlin pattern. St000259The diameter of a connected graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St000273The domination number of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000916The packing number of a graph. St001286The annihilation number of a graph. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001829The common independence number of a graph.