Your data matches 89 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000071
Mapping
Mp00195: Posets —order ideals⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St000071: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
([],2)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
([],3)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 6
([(1,2)],3)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 3
([(0,1),(0,2)],3)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 2
([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2
([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => 24
([(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
 => ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
 => 12
([(1,2),(1,3)],4)
 => ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
 => ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
 => 8
([(0,1),(0,2),(0,3)],4)
 => ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
 => ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
 => 6
([(0,2),(0,3),(3,1)],4)
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => 3
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 2
([(1,2),(2,3)],4)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => 4
([(0,3),(3,1),(3,2)],4)
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 2
([(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
 => ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
 => 8
([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 2
([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 6
([(0,3),(1,2)],4)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => 6
([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => 5
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => 4
([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => 3
Description
The number of maximal chains in a poset.
Matching statistic: St001034
(load all 3 compositions to match this statistic)
Mapping
Mp00307: Posets —promotion cycle type⟶ 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)
 => [2]
 => [1,0,1,0]
 => 2
([(0,1)],2)
 => [1]
 => [1,0]
 => 1
([],3)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 6
([(1,2)],3)
 => [3]
 => [1,0,1,0,1,0]
 => 3
([(0,1),(0,2)],3)
 => [2]
 => [1,0,1,0]
 => 2
([(0,2),(2,1)],3)
 => [1]
 => [1,0]
 => 1
([(0,2),(1,2)],3)
 => [2]
 => [1,0,1,0]
 => 2
([],4)
 => [4,4,4,4,4,4]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => 24
([(2,3)],4)
 => [4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 12
([(1,2),(1,3)],4)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 8
([(0,1),(0,2),(0,3)],4)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 6
([(0,2),(0,3),(3,1)],4)
 => [3]
 => [1,0,1,0,1,0]
 => 3
([(0,1),(0,2),(1,3),(2,3)],4)
 => [2]
 => [1,0,1,0]
 => 2
([(1,2),(2,3)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
([(0,3),(3,1),(3,2)],4)
 => [2]
 => [1,0,1,0]
 => 2
([(1,3),(2,3)],4)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 8
([(0,3),(1,3),(3,2)],4)
 => [2]
 => [1,0,1,0]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 6
([(0,3),(1,2)],4)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 6
([(0,3),(1,2),(1,3)],4)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 5
([(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)
 => [1]
 => [1,0]
 => 1
([(0,3),(1,2),(2,3)],4)
 => [3]
 => [1,0,1,0,1,0]
 => 3
Description
The area of the parallelogram polyomino associated with the Dyck path. The (bivariate) generating function is given in [1].
Matching statistic: St000100
(load all 2 compositions to match this statistic)
Mapping
St000100: Posets ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ? = 1
([],2)
 => 2
([(0,1)],2)
 => 1
([],3)
 => 6
([(1,2)],3)
 => 3
([(0,1),(0,2)],3)
 => 2
([(0,2),(2,1)],3)
 => 1
([(0,2),(1,2)],3)
 => 2
([],4)
 => 24
([(2,3)],4)
 => 12
([(1,2),(1,3)],4)
 => 8
([(0,1),(0,2),(0,3)],4)
 => 6
([(0,2),(0,3),(3,1)],4)
 => 3
([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
([(1,2),(2,3)],4)
 => 4
([(0,3),(3,1),(3,2)],4)
 => 2
([(1,3),(2,3)],4)
 => 8
([(0,3),(1,3),(3,2)],4)
 => 2
([(0,3),(1,3),(2,3)],4)
 => 6
([(0,3),(1,2)],4)
 => 6
([(0,3),(1,2),(1,3)],4)
 => 5
([(0,2),(0,3),(1,2),(1,3)],4)
 => 4
([(0,3),(2,1),(3,2)],4)
 => 1
([(0,3),(1,2),(2,3)],4)
 => 3
Description
The number of linear extensions of a poset.
Matching statistic: St000293
(load all 2 compositions to match this statistic)
Mapping
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 89% ●values known / values provided: 96%●distinct values known / distinct values provided: 89%
Values
([],1)
 => [1]
 => 10 => 1
([],2)
 => [2]
 => 100 => 2
([(0,1)],2)
 => [1]
 => 10 => 1
([],3)
 => [3,3]
 => 11000 => 6
([(1,2)],3)
 => [3]
 => 1000 => 3
([(0,1),(0,2)],3)
 => [2]
 => 100 => 2
([(0,2),(2,1)],3)
 => [1]
 => 10 => 1
([(0,2),(1,2)],3)
 => [2]
 => 100 => 2
([],4)
 => [4,4,4,4,4,4]
 => 1111110000 => ? = 24
([(2,3)],4)
 => [4,4,4]
 => 1110000 => 12
([(1,2),(1,3)],4)
 => [8]
 => 100000000 => 8
([(0,1),(0,2),(0,3)],4)
 => [3,3]
 => 11000 => 6
([(0,2),(0,3),(3,1)],4)
 => [3]
 => 1000 => 3
([(0,1),(0,2),(1,3),(2,3)],4)
 => [2]
 => 100 => 2
([(1,2),(2,3)],4)
 => [4]
 => 10000 => 4
([(0,3),(3,1),(3,2)],4)
 => [2]
 => 100 => 2
([(1,3),(2,3)],4)
 => [8]
 => 100000000 => 8
([(0,3),(1,3),(3,2)],4)
 => [2]
 => 100 => 2
([(0,3),(1,3),(2,3)],4)
 => [3,3]
 => 11000 => 6
([(0,3),(1,2)],4)
 => [4,2]
 => 100100 => 6
([(0,3),(1,2),(1,3)],4)
 => [3,2]
 => 10100 => 5
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => 1100 => 4
([(0,3),(2,1),(3,2)],4)
 => [1]
 => 10 => 1
([(0,3),(1,2),(2,3)],4)
 => [3]
 => 1000 => 3
Description
The number of inversions of a binary word.
Matching statistic: St000459
Mapping
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St000459: Integer partitions ⟶ ℤResult quality: 89% ●values known / values provided: 96%●distinct values known / distinct values provided: 89%
Values
([],1)
 => [1]
 => [1]
 => 1
([],2)
 => [2]
 => [1,1]
 => 2
([(0,1)],2)
 => [1]
 => [1]
 => 1
([],3)
 => [3,3]
 => [6]
 => 6
([(1,2)],3)
 => [3]
 => [3]
 => 3
([(0,1),(0,2)],3)
 => [2]
 => [1,1]
 => 2
([(0,2),(2,1)],3)
 => [1]
 => [1]
 => 1
([(0,2),(1,2)],3)
 => [2]
 => [1,1]
 => 2
([],4)
 => [4,4,4,4,4,4]
 => ?
 => ? = 24
([(2,3)],4)
 => [4,4,4]
 => [8,1,1,1,1]
 => 12
([(1,2),(1,3)],4)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => 8
([(0,1),(0,2),(0,3)],4)
 => [3,3]
 => [6]
 => 6
([(0,2),(0,3),(3,1)],4)
 => [3]
 => [3]
 => 3
([(0,1),(0,2),(1,3),(2,3)],4)
 => [2]
 => [1,1]
 => 2
([(1,2),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 4
([(0,3),(3,1),(3,2)],4)
 => [2]
 => [1,1]
 => 2
([(1,3),(2,3)],4)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => 8
([(0,3),(1,3),(3,2)],4)
 => [2]
 => [1,1]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [3,3]
 => [6]
 => 6
([(0,3),(1,2)],4)
 => [4,2]
 => [1,1,1,1,1,1]
 => 6
([(0,3),(1,2),(1,3)],4)
 => [3,2]
 => [3,1,1]
 => 5
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [4]
 => 4
([(0,3),(2,1),(3,2)],4)
 => [1]
 => [1]
 => 1
([(0,3),(1,2),(2,3)],4)
 => [3]
 => [3]
 => 3
Description
The hook length of the base cell of a partition. This is also known as the perimeter of a partition. In particular, the perimeter of the empty partition is zero.
Matching statistic: St000460
Mapping
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St000460: Integer partitions ⟶ ℤResult quality: 89% ●values known / values provided: 96%●distinct values known / distinct values provided: 89%
Values
([],1)
 => [1]
 => [1]
 => 1
([],2)
 => [2]
 => [1,1]
 => 2
([(0,1)],2)
 => [1]
 => [1]
 => 1
([],3)
 => [3,3]
 => [6]
 => 6
([(1,2)],3)
 => [3]
 => [3]
 => 3
([(0,1),(0,2)],3)
 => [2]
 => [1,1]
 => 2
([(0,2),(2,1)],3)
 => [1]
 => [1]
 => 1
([(0,2),(1,2)],3)
 => [2]
 => [1,1]
 => 2
([],4)
 => [4,4,4,4,4,4]
 => ?
 => ? = 24
([(2,3)],4)
 => [4,4,4]
 => [8,1,1,1,1]
 => 12
([(1,2),(1,3)],4)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => 8
([(0,1),(0,2),(0,3)],4)
 => [3,3]
 => [6]
 => 6
([(0,2),(0,3),(3,1)],4)
 => [3]
 => [3]
 => 3
([(0,1),(0,2),(1,3),(2,3)],4)
 => [2]
 => [1,1]
 => 2
([(1,2),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 4
([(0,3),(3,1),(3,2)],4)
 => [2]
 => [1,1]
 => 2
([(1,3),(2,3)],4)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => 8
([(0,3),(1,3),(3,2)],4)
 => [2]
 => [1,1]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [3,3]
 => [6]
 => 6
([(0,3),(1,2)],4)
 => [4,2]
 => [1,1,1,1,1,1]
 => 6
([(0,3),(1,2),(1,3)],4)
 => [3,2]
 => [3,1,1]
 => 5
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [4]
 => 4
([(0,3),(2,1),(3,2)],4)
 => [1]
 => [1]
 => 1
([(0,3),(1,2),(2,3)],4)
 => [3]
 => [3]
 => 3
Description
The hook length of the last cell along the main diagonal of an integer partition.
Matching statistic: St000870
Mapping
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St000870: Integer partitions ⟶ ℤResult quality: 89% ●values known / values provided: 96%●distinct values known / distinct values provided: 89%
Values
([],1)
 => [1]
 => [1]
 => 1
([],2)
 => [2]
 => [1,1]
 => 2
([(0,1)],2)
 => [1]
 => [1]
 => 1
([],3)
 => [3,3]
 => [6]
 => 6
([(1,2)],3)
 => [3]
 => [3]
 => 3
([(0,1),(0,2)],3)
 => [2]
 => [1,1]
 => 2
([(0,2),(2,1)],3)
 => [1]
 => [1]
 => 1
([(0,2),(1,2)],3)
 => [2]
 => [1,1]
 => 2
([],4)
 => [4,4,4,4,4,4]
 => ?
 => ? = 24
([(2,3)],4)
 => [4,4,4]
 => [8,1,1,1,1]
 => 12
([(1,2),(1,3)],4)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => 8
([(0,1),(0,2),(0,3)],4)
 => [3,3]
 => [6]
 => 6
([(0,2),(0,3),(3,1)],4)
 => [3]
 => [3]
 => 3
([(0,1),(0,2),(1,3),(2,3)],4)
 => [2]
 => [1,1]
 => 2
([(1,2),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 4
([(0,3),(3,1),(3,2)],4)
 => [2]
 => [1,1]
 => 2
([(1,3),(2,3)],4)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => 8
([(0,3),(1,3),(3,2)],4)
 => [2]
 => [1,1]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [3,3]
 => [6]
 => 6
([(0,3),(1,2)],4)
 => [4,2]
 => [1,1,1,1,1,1]
 => 6
([(0,3),(1,2),(1,3)],4)
 => [3,2]
 => [3,1,1]
 => 5
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [4]
 => 4
([(0,3),(2,1),(3,2)],4)
 => [1]
 => [1]
 => 1
([(0,3),(1,2),(2,3)],4)
 => [3]
 => [3]
 => 3
Description
The product of the hook lengths of the diagonal cells in an integer partition. For a cell in the Ferrers diagram of a partition, the hook length is given by the number of boxes to its right plus the number of boxes below + 1. This statistic is the product of the hook lengths of the diagonal cells $(i,i)$ of a partition.
Matching statistic: St001382
Mapping
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St001382: Integer partitions ⟶ ℤResult quality: 89% ●values known / values provided: 96%●distinct values known / distinct values provided: 89%
Values
([],1)
 => [1]
 => [1]
 => 0 = 1 - 1
([],2)
 => [2]
 => [1,1]
 => 1 = 2 - 1
([(0,1)],2)
 => [1]
 => [1]
 => 0 = 1 - 1
([],3)
 => [3,3]
 => [6]
 => 5 = 6 - 1
([(1,2)],3)
 => [3]
 => [3]
 => 2 = 3 - 1
([(0,1),(0,2)],3)
 => [2]
 => [1,1]
 => 1 = 2 - 1
([(0,2),(2,1)],3)
 => [1]
 => [1]
 => 0 = 1 - 1
([(0,2),(1,2)],3)
 => [2]
 => [1,1]
 => 1 = 2 - 1
([],4)
 => [4,4,4,4,4,4]
 => ?
 => ? = 24 - 1
([(2,3)],4)
 => [4,4,4]
 => [8,1,1,1,1]
 => 11 = 12 - 1
([(1,2),(1,3)],4)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => 7 = 8 - 1
([(0,1),(0,2),(0,3)],4)
 => [3,3]
 => [6]
 => 5 = 6 - 1
([(0,2),(0,3),(3,1)],4)
 => [3]
 => [3]
 => 2 = 3 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => [2]
 => [1,1]
 => 1 = 2 - 1
([(1,2),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 3 = 4 - 1
([(0,3),(3,1),(3,2)],4)
 => [2]
 => [1,1]
 => 1 = 2 - 1
([(1,3),(2,3)],4)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => 7 = 8 - 1
([(0,3),(1,3),(3,2)],4)
 => [2]
 => [1,1]
 => 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
 => [3,3]
 => [6]
 => 5 = 6 - 1
([(0,3),(1,2)],4)
 => [4,2]
 => [1,1,1,1,1,1]
 => 5 = 6 - 1
([(0,3),(1,2),(1,3)],4)
 => [3,2]
 => [3,1,1]
 => 4 = 5 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [4]
 => 3 = 4 - 1
([(0,3),(2,1),(3,2)],4)
 => [1]
 => [1]
 => 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
 => [3]
 => [3]
 => 2 = 3 - 1
Description
The number of boxes in the diagram of a partition that do not lie in its Durfee square.
Matching statistic: St000290
Mapping
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
St000290: Binary words ⟶ ℤResult quality: 89% ●values known / values provided: 96%●distinct values known / distinct values provided: 89%
Values
([],1)
 => [1]
 => 10 => 10 => 1
([],2)
 => [2]
 => 100 => 010 => 2
([(0,1)],2)
 => [1]
 => 10 => 10 => 1
([],3)
 => [3,3]
 => 11000 => 01010 => 6
([(1,2)],3)
 => [3]
 => 1000 => 0010 => 3
([(0,1),(0,2)],3)
 => [2]
 => 100 => 010 => 2
([(0,2),(2,1)],3)
 => [1]
 => 10 => 10 => 1
([(0,2),(1,2)],3)
 => [2]
 => 100 => 010 => 2
([],4)
 => [4,4,4,4,4,4]
 => 1111110000 => ? => ? = 24
([(2,3)],4)
 => [4,4,4]
 => 1110000 => 0101010 => 12
([(1,2),(1,3)],4)
 => [8]
 => 100000000 => 000000010 => 8
([(0,1),(0,2),(0,3)],4)
 => [3,3]
 => 11000 => 01010 => 6
([(0,2),(0,3),(3,1)],4)
 => [3]
 => 1000 => 0010 => 3
([(0,1),(0,2),(1,3),(2,3)],4)
 => [2]
 => 100 => 010 => 2
([(1,2),(2,3)],4)
 => [4]
 => 10000 => 00010 => 4
([(0,3),(3,1),(3,2)],4)
 => [2]
 => 100 => 010 => 2
([(1,3),(2,3)],4)
 => [8]
 => 100000000 => 000000010 => 8
([(0,3),(1,3),(3,2)],4)
 => [2]
 => 100 => 010 => 2
([(0,3),(1,3),(2,3)],4)
 => [3,3]
 => 11000 => 01010 => 6
([(0,3),(1,2)],4)
 => [4,2]
 => 100100 => 100010 => 6
([(0,3),(1,2),(1,3)],4)
 => [3,2]
 => 10100 => 10010 => 5
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => 1100 => 1010 => 4
([(0,3),(2,1),(3,2)],4)
 => [1]
 => 10 => 10 => 1
([(0,3),(1,2),(2,3)],4)
 => [3]
 => 1000 => 0010 => 3
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: St000391
Mapping
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
St000391: Binary words ⟶ ℤResult quality: 89% ●values known / values provided: 96%●distinct values known / distinct values provided: 89%
Values
([],1)
 => [1]
 => 10 => 10 => 1
([],2)
 => [2]
 => 100 => 010 => 2
([(0,1)],2)
 => [1]
 => 10 => 10 => 1
([],3)
 => [3,3]
 => 11000 => 01010 => 6
([(1,2)],3)
 => [3]
 => 1000 => 0010 => 3
([(0,1),(0,2)],3)
 => [2]
 => 100 => 010 => 2
([(0,2),(2,1)],3)
 => [1]
 => 10 => 10 => 1
([(0,2),(1,2)],3)
 => [2]
 => 100 => 010 => 2
([],4)
 => [4,4,4,4,4,4]
 => 1111110000 => ? => ? = 24
([(2,3)],4)
 => [4,4,4]
 => 1110000 => 0101010 => 12
([(1,2),(1,3)],4)
 => [8]
 => 100000000 => 000000010 => 8
([(0,1),(0,2),(0,3)],4)
 => [3,3]
 => 11000 => 01010 => 6
([(0,2),(0,3),(3,1)],4)
 => [3]
 => 1000 => 0010 => 3
([(0,1),(0,2),(1,3),(2,3)],4)
 => [2]
 => 100 => 010 => 2
([(1,2),(2,3)],4)
 => [4]
 => 10000 => 00010 => 4
([(0,3),(3,1),(3,2)],4)
 => [2]
 => 100 => 010 => 2
([(1,3),(2,3)],4)
 => [8]
 => 100000000 => 000000010 => 8
([(0,3),(1,3),(3,2)],4)
 => [2]
 => 100 => 010 => 2
([(0,3),(1,3),(2,3)],4)
 => [3,3]
 => 11000 => 01010 => 6
([(0,3),(1,2)],4)
 => [4,2]
 => 100100 => 100010 => 6
([(0,3),(1,2),(1,3)],4)
 => [3,2]
 => 10100 => 10010 => 5
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => 1100 => 1010 => 4
([(0,3),(2,1),(3,2)],4)
 => [1]
 => 10 => 10 => 1
([(0,3),(1,2),(2,3)],4)
 => [3]
 => 1000 => 0010 => 3
Description
The sum of the positions of the ones in a binary word.
The following 79 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000228The size of a partition. St000395The sum of the heights of the peaks of a Dyck path. St000909The number of maximal chains of maximal size in a poset. St001259The vector space dimension of the double dual of D(A) in the corresponding Nakayama algebra. St000438The position of the last up step in a Dyck path. St000363The number of minimal vertex covers of a graph. St001304The number of maximally independent sets of vertices of a graph. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001645The pebbling number of a connected graph. St000454The largest eigenvalue of a graph if it is integral. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000450The number of edges minus the number of vertices plus 2 of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001117The game chromatic index of a graph. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001742The difference of the maximal and the minimal degree in a graph. St001812The biclique partition number of a graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001545The second Elser number of a connected graph. St000456The monochromatic index of a connected graph. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000455The second largest eigenvalue of a graph if it is integral. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000474Dyson's crank of a partition. St000667The greatest common divisor of the parts of the partition. St000668The least common multiple of the parts of the partition. St000770The major index of an integer partition when read from bottom to top. St001279The sum of the parts of an integer partition that are at least two. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001330The hat guessing number of a graph. St000706The product of the factorials of the multiplicities of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001624The breadth of a lattice. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001877Number of indecomposable injective modules with projective dimension 2. St000939The number of characters of the symmetric group whose value on the partition is positive. St001060The distinguishing index of a graph. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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 $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000639The number of relations in a poset. St000641The number of non-empty boolean intervals in a poset. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000643The size of the largest orbit of antichains under Panyushev complementation. St000656The number of cuts of a poset. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000264The girth of a graph, which is not a tree. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000741The Colin de Verdière graph invariant. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000937The number of positive values of the symmetric group character corresponding to the partition. St001568The smallest positive integer that does not appear twice in the partition. St001875The number of simple modules with projective dimension at most 1. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition.