Your data matches 34 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000228
(load all 20 compositions to match this statistic)
Mapping
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1] => [1]
 => 1
[1,-2] => [1]
 => 1
[-1,2] => [1]
 => 1
[-1,-2] => [1,1]
 => 2
[2,-1] => [2]
 => 2
[-2,1] => [2]
 => 2
[1,2,-3] => [1]
 => 1
[1,-2,3] => [1]
 => 1
[1,-2,-3] => [1,1]
 => 2
[-1,2,3] => [1]
 => 1
[-1,2,-3] => [1,1]
 => 2
[-1,-2,3] => [1,1]
 => 2
[-1,-2,-3] => [1,1,1]
 => 3
[1,3,-2] => [2]
 => 2
[1,-3,2] => [2]
 => 2
[-1,3,2] => [1]
 => 1
[-1,3,-2] => [2,1]
 => 3
[-1,-3,2] => [2,1]
 => 3
[-1,-3,-2] => [1]
 => 1
[2,1,-3] => [1]
 => 1
[2,-1,3] => [2]
 => 2
[2,-1,-3] => [2,1]
 => 3
[-2,1,3] => [2]
 => 2
[-2,1,-3] => [2,1]
 => 3
[-2,-1,-3] => [1]
 => 1
[2,3,-1] => [3]
 => 3
[2,-3,1] => [3]
 => 3
[-2,3,1] => [3]
 => 3
[-2,-3,-1] => [3]
 => 3
[3,1,-2] => [3]
 => 3
[3,-1,2] => [3]
 => 3
[-3,1,2] => [3]
 => 3
[-3,-1,-2] => [3]
 => 3
[3,2,-1] => [2]
 => 2
[3,-2,1] => [1]
 => 1
[3,-2,-1] => [2,1]
 => 3
[-3,2,1] => [2]
 => 2
[-3,-2,1] => [2,1]
 => 3
[-3,-2,-1] => [1]
 => 1
[1,2,3,-4] => [1]
 => 1
[1,2,-3,4] => [1]
 => 1
[1,2,-3,-4] => [1,1]
 => 2
[1,-2,3,4] => [1]
 => 1
[1,-2,3,-4] => [1,1]
 => 2
[1,-2,-3,4] => [1,1]
 => 2
[1,-2,-3,-4] => [1,1,1]
 => 3
[-1,2,3,4] => [1]
 => 1
[-1,2,3,-4] => [1,1]
 => 2
[-1,2,-3,4] => [1,1]
 => 2
[-1,2,-3,-4] => [1,1,1]
 => 3
Description
The size of a partition. This statistic is the constant statistic of the level sets.
Matching statistic: St000293
(load all 6 compositions to match this statistic)
Mapping
Mp00169: Signed permutations —odd cycle type⟶ 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
[1,-2] => [1]
 => 10 => 1
[-1,2] => [1]
 => 10 => 1
[-1,-2] => [1,1]
 => 110 => 2
[2,-1] => [2]
 => 100 => 2
[-2,1] => [2]
 => 100 => 2
[1,2,-3] => [1]
 => 10 => 1
[1,-2,3] => [1]
 => 10 => 1
[1,-2,-3] => [1,1]
 => 110 => 2
[-1,2,3] => [1]
 => 10 => 1
[-1,2,-3] => [1,1]
 => 110 => 2
[-1,-2,3] => [1,1]
 => 110 => 2
[-1,-2,-3] => [1,1,1]
 => 1110 => 3
[1,3,-2] => [2]
 => 100 => 2
[1,-3,2] => [2]
 => 100 => 2
[-1,3,2] => [1]
 => 10 => 1
[-1,3,-2] => [2,1]
 => 1010 => 3
[-1,-3,2] => [2,1]
 => 1010 => 3
[-1,-3,-2] => [1]
 => 10 => 1
[2,1,-3] => [1]
 => 10 => 1
[2,-1,3] => [2]
 => 100 => 2
[2,-1,-3] => [2,1]
 => 1010 => 3
[-2,1,3] => [2]
 => 100 => 2
[-2,1,-3] => [2,1]
 => 1010 => 3
[-2,-1,-3] => [1]
 => 10 => 1
[2,3,-1] => [3]
 => 1000 => 3
[2,-3,1] => [3]
 => 1000 => 3
[-2,3,1] => [3]
 => 1000 => 3
[-2,-3,-1] => [3]
 => 1000 => 3
[3,1,-2] => [3]
 => 1000 => 3
[3,-1,2] => [3]
 => 1000 => 3
[-3,1,2] => [3]
 => 1000 => 3
[-3,-1,-2] => [3]
 => 1000 => 3
[3,2,-1] => [2]
 => 100 => 2
[3,-2,1] => [1]
 => 10 => 1
[3,-2,-1] => [2,1]
 => 1010 => 3
[-3,2,1] => [2]
 => 100 => 2
[-3,-2,1] => [2,1]
 => 1010 => 3
[-3,-2,-1] => [1]
 => 10 => 1
[1,2,3,-4] => [1]
 => 10 => 1
[1,2,-3,4] => [1]
 => 10 => 1
[1,2,-3,-4] => [1,1]
 => 110 => 2
[1,-2,3,4] => [1]
 => 10 => 1
[1,-2,3,-4] => [1,1]
 => 110 => 2
[1,-2,-3,4] => [1,1]
 => 110 => 2
[1,-2,-3,-4] => [1,1,1]
 => 1110 => 3
[-1,2,3,4] => [1]
 => 10 => 1
[-1,2,3,-4] => [1,1]
 => 110 => 2
[-1,2,-3,4] => [1,1]
 => 110 => 2
[-1,2,-3,-4] => [1,1,1]
 => 1110 => 3
Description
The number of inversions of a binary word.
Matching statistic: St001034
(load all 9 compositions to match this statistic)
Mapping
Mp00169: Signed permutations —odd 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
[1,-2] => [1]
 => [1,0]
 => 1
[-1,2] => [1]
 => [1,0]
 => 1
[-1,-2] => [1,1]
 => [1,1,0,0]
 => 2
[2,-1] => [2]
 => [1,0,1,0]
 => 2
[-2,1] => [2]
 => [1,0,1,0]
 => 2
[1,2,-3] => [1]
 => [1,0]
 => 1
[1,-2,3] => [1]
 => [1,0]
 => 1
[1,-2,-3] => [1,1]
 => [1,1,0,0]
 => 2
[-1,2,3] => [1]
 => [1,0]
 => 1
[-1,2,-3] => [1,1]
 => [1,1,0,0]
 => 2
[-1,-2,3] => [1,1]
 => [1,1,0,0]
 => 2
[-1,-2,-3] => [1,1,1]
 => [1,1,0,1,0,0]
 => 3
[1,3,-2] => [2]
 => [1,0,1,0]
 => 2
[1,-3,2] => [2]
 => [1,0,1,0]
 => 2
[-1,3,2] => [1]
 => [1,0]
 => 1
[-1,3,-2] => [2,1]
 => [1,0,1,1,0,0]
 => 3
[-1,-3,2] => [2,1]
 => [1,0,1,1,0,0]
 => 3
[-1,-3,-2] => [1]
 => [1,0]
 => 1
[2,1,-3] => [1]
 => [1,0]
 => 1
[2,-1,3] => [2]
 => [1,0,1,0]
 => 2
[2,-1,-3] => [2,1]
 => [1,0,1,1,0,0]
 => 3
[-2,1,3] => [2]
 => [1,0,1,0]
 => 2
[-2,1,-3] => [2,1]
 => [1,0,1,1,0,0]
 => 3
[-2,-1,-3] => [1]
 => [1,0]
 => 1
[2,3,-1] => [3]
 => [1,0,1,0,1,0]
 => 3
[2,-3,1] => [3]
 => [1,0,1,0,1,0]
 => 3
[-2,3,1] => [3]
 => [1,0,1,0,1,0]
 => 3
[-2,-3,-1] => [3]
 => [1,0,1,0,1,0]
 => 3
[3,1,-2] => [3]
 => [1,0,1,0,1,0]
 => 3
[3,-1,2] => [3]
 => [1,0,1,0,1,0]
 => 3
[-3,1,2] => [3]
 => [1,0,1,0,1,0]
 => 3
[-3,-1,-2] => [3]
 => [1,0,1,0,1,0]
 => 3
[3,2,-1] => [2]
 => [1,0,1,0]
 => 2
[3,-2,1] => [1]
 => [1,0]
 => 1
[3,-2,-1] => [2,1]
 => [1,0,1,1,0,0]
 => 3
[-3,2,1] => [2]
 => [1,0,1,0]
 => 2
[-3,-2,1] => [2,1]
 => [1,0,1,1,0,0]
 => 3
[-3,-2,-1] => [1]
 => [1,0]
 => 1
[1,2,3,-4] => [1]
 => [1,0]
 => 1
[1,2,-3,4] => [1]
 => [1,0]
 => 1
[1,2,-3,-4] => [1,1]
 => [1,1,0,0]
 => 2
[1,-2,3,4] => [1]
 => [1,0]
 => 1
[1,-2,3,-4] => [1,1]
 => [1,1,0,0]
 => 2
[1,-2,-3,4] => [1,1]
 => [1,1,0,0]
 => 2
[1,-2,-3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => 3
[-1,2,3,4] => [1]
 => [1,0]
 => 1
[-1,2,3,-4] => [1,1]
 => [1,1,0,0]
 => 2
[-1,2,-3,4] => [1,1]
 => [1,1,0,0]
 => 2
[-1,2,-3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => 3
Description
The area of the parallelogram polyomino associated with the Dyck path. The (bivariate) generating function is given in [1].
Matching statistic: St001643
(load all 6 compositions to match this statistic)
Mapping
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001643: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1] => [1]
 => [1,0,1,0]
 => 3 = 1 + 2
[1,-2] => [1]
 => [1,0,1,0]
 => 3 = 1 + 2
[-1,2] => [1]
 => [1,0,1,0]
 => 3 = 1 + 2
[-1,-2] => [1,1]
 => [1,0,1,1,0,0]
 => 4 = 2 + 2
[2,-1] => [2]
 => [1,1,0,0,1,0]
 => 4 = 2 + 2
[-2,1] => [2]
 => [1,1,0,0,1,0]
 => 4 = 2 + 2
[1,2,-3] => [1]
 => [1,0,1,0]
 => 3 = 1 + 2
[1,-2,3] => [1]
 => [1,0,1,0]
 => 3 = 1 + 2
[1,-2,-3] => [1,1]
 => [1,0,1,1,0,0]
 => 4 = 2 + 2
[-1,2,3] => [1]
 => [1,0,1,0]
 => 3 = 1 + 2
[-1,2,-3] => [1,1]
 => [1,0,1,1,0,0]
 => 4 = 2 + 2
[-1,-2,3] => [1,1]
 => [1,0,1,1,0,0]
 => 4 = 2 + 2
[-1,-2,-3] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 5 = 3 + 2
[1,3,-2] => [2]
 => [1,1,0,0,1,0]
 => 4 = 2 + 2
[1,-3,2] => [2]
 => [1,1,0,0,1,0]
 => 4 = 2 + 2
[-1,3,2] => [1]
 => [1,0,1,0]
 => 3 = 1 + 2
[-1,3,-2] => [2,1]
 => [1,0,1,0,1,0]
 => 5 = 3 + 2
[-1,-3,2] => [2,1]
 => [1,0,1,0,1,0]
 => 5 = 3 + 2
[-1,-3,-2] => [1]
 => [1,0,1,0]
 => 3 = 1 + 2
[2,1,-3] => [1]
 => [1,0,1,0]
 => 3 = 1 + 2
[2,-1,3] => [2]
 => [1,1,0,0,1,0]
 => 4 = 2 + 2
[2,-1,-3] => [2,1]
 => [1,0,1,0,1,0]
 => 5 = 3 + 2
[-2,1,3] => [2]
 => [1,1,0,0,1,0]
 => 4 = 2 + 2
[-2,1,-3] => [2,1]
 => [1,0,1,0,1,0]
 => 5 = 3 + 2
[-2,-1,-3] => [1]
 => [1,0,1,0]
 => 3 = 1 + 2
[2,3,-1] => [3]
 => [1,1,1,0,0,0,1,0]
 => 5 = 3 + 2
[2,-3,1] => [3]
 => [1,1,1,0,0,0,1,0]
 => 5 = 3 + 2
[-2,3,1] => [3]
 => [1,1,1,0,0,0,1,0]
 => 5 = 3 + 2
[-2,-3,-1] => [3]
 => [1,1,1,0,0,0,1,0]
 => 5 = 3 + 2
[3,1,-2] => [3]
 => [1,1,1,0,0,0,1,0]
 => 5 = 3 + 2
[3,-1,2] => [3]
 => [1,1,1,0,0,0,1,0]
 => 5 = 3 + 2
[-3,1,2] => [3]
 => [1,1,1,0,0,0,1,0]
 => 5 = 3 + 2
[-3,-1,-2] => [3]
 => [1,1,1,0,0,0,1,0]
 => 5 = 3 + 2
[3,2,-1] => [2]
 => [1,1,0,0,1,0]
 => 4 = 2 + 2
[3,-2,1] => [1]
 => [1,0,1,0]
 => 3 = 1 + 2
[3,-2,-1] => [2,1]
 => [1,0,1,0,1,0]
 => 5 = 3 + 2
[-3,2,1] => [2]
 => [1,1,0,0,1,0]
 => 4 = 2 + 2
[-3,-2,1] => [2,1]
 => [1,0,1,0,1,0]
 => 5 = 3 + 2
[-3,-2,-1] => [1]
 => [1,0,1,0]
 => 3 = 1 + 2
[1,2,3,-4] => [1]
 => [1,0,1,0]
 => 3 = 1 + 2
[1,2,-3,4] => [1]
 => [1,0,1,0]
 => 3 = 1 + 2
[1,2,-3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => 4 = 2 + 2
[1,-2,3,4] => [1]
 => [1,0,1,0]
 => 3 = 1 + 2
[1,-2,3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => 4 = 2 + 2
[1,-2,-3,4] => [1,1]
 => [1,0,1,1,0,0]
 => 4 = 2 + 2
[1,-2,-3,-4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 5 = 3 + 2
[-1,2,3,4] => [1]
 => [1,0,1,0]
 => 3 = 1 + 2
[-1,2,3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => 4 = 2 + 2
[-1,2,-3,4] => [1,1]
 => [1,0,1,1,0,0]
 => 4 = 2 + 2
[-1,2,-3,-4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 5 = 3 + 2
Description
The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000018
Mapping
Mp00169: Signed permutations —odd cycle type⟶ 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
[1,-2] => [1]
 => [1,0,1,0]
 => [2,1] => 1
[-1,2] => [1]
 => [1,0,1,0]
 => [2,1] => 1
[-1,-2] => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 2
[2,-1] => [2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 2
[-2,1] => [2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 2
[1,2,-3] => [1]
 => [1,0,1,0]
 => [2,1] => 1
[1,-2,3] => [1]
 => [1,0,1,0]
 => [2,1] => 1
[1,-2,-3] => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 2
[-1,2,3] => [1]
 => [1,0,1,0]
 => [2,1] => 1
[-1,2,-3] => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 2
[-1,-2,3] => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 2
[-1,-2,-3] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 3
[1,3,-2] => [2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 2
[1,-3,2] => [2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 2
[-1,3,2] => [1]
 => [1,0,1,0]
 => [2,1] => 1
[-1,3,-2] => [2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => 3
[-1,-3,2] => [2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => 3
[-1,-3,-2] => [1]
 => [1,0,1,0]
 => [2,1] => 1
[2,1,-3] => [1]
 => [1,0,1,0]
 => [2,1] => 1
[2,-1,3] => [2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 2
[2,-1,-3] => [2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => 3
[-2,1,3] => [2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 2
[-2,1,-3] => [2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => 3
[-2,-1,-3] => [1]
 => [1,0,1,0]
 => [2,1] => 1
[2,3,-1] => [3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 3
[2,-3,1] => [3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 3
[-2,3,1] => [3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 3
[-2,-3,-1] => [3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 3
[3,1,-2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 3
[3,-1,2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 3
[-3,1,2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 3
[-3,-1,-2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 3
[3,2,-1] => [2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 2
[3,-2,1] => [1]
 => [1,0,1,0]
 => [2,1] => 1
[3,-2,-1] => [2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => 3
[-3,2,1] => [2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 2
[-3,-2,1] => [2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => 3
[-3,-2,-1] => [1]
 => [1,0,1,0]
 => [2,1] => 1
[1,2,3,-4] => [1]
 => [1,0,1,0]
 => [2,1] => 1
[1,2,-3,4] => [1]
 => [1,0,1,0]
 => [2,1] => 1
[1,2,-3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 2
[1,-2,3,4] => [1]
 => [1,0,1,0]
 => [2,1] => 1
[1,-2,3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 2
[1,-2,-3,4] => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 2
[1,-2,-3,-4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 3
[-1,2,3,4] => [1]
 => [1,0,1,0]
 => [2,1] => 1
[-1,2,3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 2
[-1,2,-3,4] => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 2
[-1,2,-3,-4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 3
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: St000186
(load all 2 compositions to match this statistic)
Mapping
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St000186: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1] => [1]
 => [[1]]
 => [[1]]
 => 1
[1,-2] => [1]
 => [[1]]
 => [[1]]
 => 1
[-1,2] => [1]
 => [[1]]
 => [[1]]
 => 1
[-1,-2] => [1,1]
 => [[1],[2]]
 => [[1,1],[1]]
 => 2
[2,-1] => [2]
 => [[1,2]]
 => [[2,0],[1]]
 => 2
[-2,1] => [2]
 => [[1,2]]
 => [[2,0],[1]]
 => 2
[1,2,-3] => [1]
 => [[1]]
 => [[1]]
 => 1
[1,-2,3] => [1]
 => [[1]]
 => [[1]]
 => 1
[1,-2,-3] => [1,1]
 => [[1],[2]]
 => [[1,1],[1]]
 => 2
[-1,2,3] => [1]
 => [[1]]
 => [[1]]
 => 1
[-1,2,-3] => [1,1]
 => [[1],[2]]
 => [[1,1],[1]]
 => 2
[-1,-2,3] => [1,1]
 => [[1],[2]]
 => [[1,1],[1]]
 => 2
[-1,-2,-3] => [1,1,1]
 => [[1],[2],[3]]
 => [[1,1,1],[1,1],[1]]
 => 3
[1,3,-2] => [2]
 => [[1,2]]
 => [[2,0],[1]]
 => 2
[1,-3,2] => [2]
 => [[1,2]]
 => [[2,0],[1]]
 => 2
[-1,3,2] => [1]
 => [[1]]
 => [[1]]
 => 1
[-1,3,-2] => [2,1]
 => [[1,3],[2]]
 => [[2,1,0],[1,1],[1]]
 => 3
[-1,-3,2] => [2,1]
 => [[1,3],[2]]
 => [[2,1,0],[1,1],[1]]
 => 3
[-1,-3,-2] => [1]
 => [[1]]
 => [[1]]
 => 1
[2,1,-3] => [1]
 => [[1]]
 => [[1]]
 => 1
[2,-1,3] => [2]
 => [[1,2]]
 => [[2,0],[1]]
 => 2
[2,-1,-3] => [2,1]
 => [[1,3],[2]]
 => [[2,1,0],[1,1],[1]]
 => 3
[-2,1,3] => [2]
 => [[1,2]]
 => [[2,0],[1]]
 => 2
[-2,1,-3] => [2,1]
 => [[1,3],[2]]
 => [[2,1,0],[1,1],[1]]
 => 3
[-2,-1,-3] => [1]
 => [[1]]
 => [[1]]
 => 1
[2,3,-1] => [3]
 => [[1,2,3]]
 => [[3,0,0],[2,0],[1]]
 => 3
[2,-3,1] => [3]
 => [[1,2,3]]
 => [[3,0,0],[2,0],[1]]
 => 3
[-2,3,1] => [3]
 => [[1,2,3]]
 => [[3,0,0],[2,0],[1]]
 => 3
[-2,-3,-1] => [3]
 => [[1,2,3]]
 => [[3,0,0],[2,0],[1]]
 => 3
[3,1,-2] => [3]
 => [[1,2,3]]
 => [[3,0,0],[2,0],[1]]
 => 3
[3,-1,2] => [3]
 => [[1,2,3]]
 => [[3,0,0],[2,0],[1]]
 => 3
[-3,1,2] => [3]
 => [[1,2,3]]
 => [[3,0,0],[2,0],[1]]
 => 3
[-3,-1,-2] => [3]
 => [[1,2,3]]
 => [[3,0,0],[2,0],[1]]
 => 3
[3,2,-1] => [2]
 => [[1,2]]
 => [[2,0],[1]]
 => 2
[3,-2,1] => [1]
 => [[1]]
 => [[1]]
 => 1
[3,-2,-1] => [2,1]
 => [[1,3],[2]]
 => [[2,1,0],[1,1],[1]]
 => 3
[-3,2,1] => [2]
 => [[1,2]]
 => [[2,0],[1]]
 => 2
[-3,-2,1] => [2,1]
 => [[1,3],[2]]
 => [[2,1,0],[1,1],[1]]
 => 3
[-3,-2,-1] => [1]
 => [[1]]
 => [[1]]
 => 1
[1,2,3,-4] => [1]
 => [[1]]
 => [[1]]
 => 1
[1,2,-3,4] => [1]
 => [[1]]
 => [[1]]
 => 1
[1,2,-3,-4] => [1,1]
 => [[1],[2]]
 => [[1,1],[1]]
 => 2
[1,-2,3,4] => [1]
 => [[1]]
 => [[1]]
 => 1
[1,-2,3,-4] => [1,1]
 => [[1],[2]]
 => [[1,1],[1]]
 => 2
[1,-2,-3,4] => [1,1]
 => [[1],[2]]
 => [[1,1],[1]]
 => 2
[1,-2,-3,-4] => [1,1,1]
 => [[1],[2],[3]]
 => [[1,1,1],[1,1],[1]]
 => 3
[-1,2,3,4] => [1]
 => [[1]]
 => [[1]]
 => 1
[-1,2,3,-4] => [1,1]
 => [[1],[2]]
 => [[1,1],[1]]
 => 2
[-1,2,-3,4] => [1,1]
 => [[1],[2]]
 => [[1,1],[1]]
 => 2
[-1,2,-3,-4] => [1,1,1]
 => [[1],[2],[3]]
 => [[1,1,1],[1,1],[1]]
 => 3
Description
The sum of the first row in a Gelfand-Tsetlin pattern.
Matching statistic: St000189
Mapping
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000189: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1] => [1]
 => [[1],[]]
 => ([],1)
 => 1
[1,-2] => [1]
 => [[1],[]]
 => ([],1)
 => 1
[-1,2] => [1]
 => [[1],[]]
 => ([],1)
 => 1
[-1,-2] => [1,1]
 => [[1,1],[]]
 => ([(0,1)],2)
 => 2
[2,-1] => [2]
 => [[2],[]]
 => ([(0,1)],2)
 => 2
[-2,1] => [2]
 => [[2],[]]
 => ([(0,1)],2)
 => 2
[1,2,-3] => [1]
 => [[1],[]]
 => ([],1)
 => 1
[1,-2,3] => [1]
 => [[1],[]]
 => ([],1)
 => 1
[1,-2,-3] => [1,1]
 => [[1,1],[]]
 => ([(0,1)],2)
 => 2
[-1,2,3] => [1]
 => [[1],[]]
 => ([],1)
 => 1
[-1,2,-3] => [1,1]
 => [[1,1],[]]
 => ([(0,1)],2)
 => 2
[-1,-2,3] => [1,1]
 => [[1,1],[]]
 => ([(0,1)],2)
 => 2
[-1,-2,-3] => [1,1,1]
 => [[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => 3
[1,3,-2] => [2]
 => [[2],[]]
 => ([(0,1)],2)
 => 2
[1,-3,2] => [2]
 => [[2],[]]
 => ([(0,1)],2)
 => 2
[-1,3,2] => [1]
 => [[1],[]]
 => ([],1)
 => 1
[-1,3,-2] => [2,1]
 => [[2,1],[]]
 => ([(0,1),(0,2)],3)
 => 3
[-1,-3,2] => [2,1]
 => [[2,1],[]]
 => ([(0,1),(0,2)],3)
 => 3
[-1,-3,-2] => [1]
 => [[1],[]]
 => ([],1)
 => 1
[2,1,-3] => [1]
 => [[1],[]]
 => ([],1)
 => 1
[2,-1,3] => [2]
 => [[2],[]]
 => ([(0,1)],2)
 => 2
[2,-1,-3] => [2,1]
 => [[2,1],[]]
 => ([(0,1),(0,2)],3)
 => 3
[-2,1,3] => [2]
 => [[2],[]]
 => ([(0,1)],2)
 => 2
[-2,1,-3] => [2,1]
 => [[2,1],[]]
 => ([(0,1),(0,2)],3)
 => 3
[-2,-1,-3] => [1]
 => [[1],[]]
 => ([],1)
 => 1
[2,3,-1] => [3]
 => [[3],[]]
 => ([(0,2),(2,1)],3)
 => 3
[2,-3,1] => [3]
 => [[3],[]]
 => ([(0,2),(2,1)],3)
 => 3
[-2,3,1] => [3]
 => [[3],[]]
 => ([(0,2),(2,1)],3)
 => 3
[-2,-3,-1] => [3]
 => [[3],[]]
 => ([(0,2),(2,1)],3)
 => 3
[3,1,-2] => [3]
 => [[3],[]]
 => ([(0,2),(2,1)],3)
 => 3
[3,-1,2] => [3]
 => [[3],[]]
 => ([(0,2),(2,1)],3)
 => 3
[-3,1,2] => [3]
 => [[3],[]]
 => ([(0,2),(2,1)],3)
 => 3
[-3,-1,-2] => [3]
 => [[3],[]]
 => ([(0,2),(2,1)],3)
 => 3
[3,2,-1] => [2]
 => [[2],[]]
 => ([(0,1)],2)
 => 2
[3,-2,1] => [1]
 => [[1],[]]
 => ([],1)
 => 1
[3,-2,-1] => [2,1]
 => [[2,1],[]]
 => ([(0,1),(0,2)],3)
 => 3
[-3,2,1] => [2]
 => [[2],[]]
 => ([(0,1)],2)
 => 2
[-3,-2,1] => [2,1]
 => [[2,1],[]]
 => ([(0,1),(0,2)],3)
 => 3
[-3,-2,-1] => [1]
 => [[1],[]]
 => ([],1)
 => 1
[1,2,3,-4] => [1]
 => [[1],[]]
 => ([],1)
 => 1
[1,2,-3,4] => [1]
 => [[1],[]]
 => ([],1)
 => 1
[1,2,-3,-4] => [1,1]
 => [[1,1],[]]
 => ([(0,1)],2)
 => 2
[1,-2,3,4] => [1]
 => [[1],[]]
 => ([],1)
 => 1
[1,-2,3,-4] => [1,1]
 => [[1,1],[]]
 => ([(0,1)],2)
 => 2
[1,-2,-3,4] => [1,1]
 => [[1,1],[]]
 => ([(0,1)],2)
 => 2
[1,-2,-3,-4] => [1,1,1]
 => [[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => 3
[-1,2,3,4] => [1]
 => [[1],[]]
 => ([],1)
 => 1
[-1,2,3,-4] => [1,1]
 => [[1,1],[]]
 => ([(0,1)],2)
 => 2
[-1,2,-3,4] => [1,1]
 => [[1,1],[]]
 => ([(0,1)],2)
 => 2
[-1,2,-3,-4] => [1,1,1]
 => [[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => 3
Description
The number of elements in the poset.
Matching statistic: St000229
Mapping
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000229: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1] => [1]
 => [[1]]
 => {{1}}
 => 1
[1,-2] => [1]
 => [[1]]
 => {{1}}
 => 1
[-1,2] => [1]
 => [[1]]
 => {{1}}
 => 1
[-1,-2] => [1,1]
 => [[1],[2]]
 => {{1},{2}}
 => 2
[2,-1] => [2]
 => [[1,2]]
 => {{1,2}}
 => 2
[-2,1] => [2]
 => [[1,2]]
 => {{1,2}}
 => 2
[1,2,-3] => [1]
 => [[1]]
 => {{1}}
 => 1
[1,-2,3] => [1]
 => [[1]]
 => {{1}}
 => 1
[1,-2,-3] => [1,1]
 => [[1],[2]]
 => {{1},{2}}
 => 2
[-1,2,3] => [1]
 => [[1]]
 => {{1}}
 => 1
[-1,2,-3] => [1,1]
 => [[1],[2]]
 => {{1},{2}}
 => 2
[-1,-2,3] => [1,1]
 => [[1],[2]]
 => {{1},{2}}
 => 2
[-1,-2,-3] => [1,1,1]
 => [[1],[2],[3]]
 => {{1},{2},{3}}
 => 3
[1,3,-2] => [2]
 => [[1,2]]
 => {{1,2}}
 => 2
[1,-3,2] => [2]
 => [[1,2]]
 => {{1,2}}
 => 2
[-1,3,2] => [1]
 => [[1]]
 => {{1}}
 => 1
[-1,3,-2] => [2,1]
 => [[1,2],[3]]
 => {{1,2},{3}}
 => 3
[-1,-3,2] => [2,1]
 => [[1,2],[3]]
 => {{1,2},{3}}
 => 3
[-1,-3,-2] => [1]
 => [[1]]
 => {{1}}
 => 1
[2,1,-3] => [1]
 => [[1]]
 => {{1}}
 => 1
[2,-1,3] => [2]
 => [[1,2]]
 => {{1,2}}
 => 2
[2,-1,-3] => [2,1]
 => [[1,2],[3]]
 => {{1,2},{3}}
 => 3
[-2,1,3] => [2]
 => [[1,2]]
 => {{1,2}}
 => 2
[-2,1,-3] => [2,1]
 => [[1,2],[3]]
 => {{1,2},{3}}
 => 3
[-2,-1,-3] => [1]
 => [[1]]
 => {{1}}
 => 1
[2,3,-1] => [3]
 => [[1,2,3]]
 => {{1,2,3}}
 => 3
[2,-3,1] => [3]
 => [[1,2,3]]
 => {{1,2,3}}
 => 3
[-2,3,1] => [3]
 => [[1,2,3]]
 => {{1,2,3}}
 => 3
[-2,-3,-1] => [3]
 => [[1,2,3]]
 => {{1,2,3}}
 => 3
[3,1,-2] => [3]
 => [[1,2,3]]
 => {{1,2,3}}
 => 3
[3,-1,2] => [3]
 => [[1,2,3]]
 => {{1,2,3}}
 => 3
[-3,1,2] => [3]
 => [[1,2,3]]
 => {{1,2,3}}
 => 3
[-3,-1,-2] => [3]
 => [[1,2,3]]
 => {{1,2,3}}
 => 3
[3,2,-1] => [2]
 => [[1,2]]
 => {{1,2}}
 => 2
[3,-2,1] => [1]
 => [[1]]
 => {{1}}
 => 1
[3,-2,-1] => [2,1]
 => [[1,2],[3]]
 => {{1,2},{3}}
 => 3
[-3,2,1] => [2]
 => [[1,2]]
 => {{1,2}}
 => 2
[-3,-2,1] => [2,1]
 => [[1,2],[3]]
 => {{1,2},{3}}
 => 3
[-3,-2,-1] => [1]
 => [[1]]
 => {{1}}
 => 1
[1,2,3,-4] => [1]
 => [[1]]
 => {{1}}
 => 1
[1,2,-3,4] => [1]
 => [[1]]
 => {{1}}
 => 1
[1,2,-3,-4] => [1,1]
 => [[1],[2]]
 => {{1},{2}}
 => 2
[1,-2,3,4] => [1]
 => [[1]]
 => {{1}}
 => 1
[1,-2,3,-4] => [1,1]
 => [[1],[2]]
 => {{1},{2}}
 => 2
[1,-2,-3,4] => [1,1]
 => [[1],[2]]
 => {{1},{2}}
 => 2
[1,-2,-3,-4] => [1,1,1]
 => [[1],[2],[3]]
 => {{1},{2},{3}}
 => 3
[-1,2,3,4] => [1]
 => [[1]]
 => {{1}}
 => 1
[-1,2,3,-4] => [1,1]
 => [[1],[2]]
 => {{1},{2}}
 => 2
[-1,2,-3,4] => [1,1]
 => [[1],[2]]
 => {{1},{2}}
 => 2
[-1,2,-3,-4] => [1,1,1]
 => [[1],[2],[3]]
 => {{1},{2},{3}}
 => 3
Description
Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. This is, for a set partition $P = \{B_1,\ldots,B_k\}$ of $\{1,\ldots,n\}$, the statistic is $$d(P) = \sum_i \big(\operatorname{max}(B_i)-\operatorname{min}(B_i)+1\big).$$ This statistic is called ''dimension index'' in [2]
Matching statistic: St000246
Mapping
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing 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
[1,-2] => [1]
 => [1,0,1,0]
 => [1,2] => 1
[-1,2] => [1]
 => [1,0,1,0]
 => [1,2] => 1
[-1,-2] => [1,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 2
[2,-1] => [2]
 => [1,1,0,0,1,0]
 => [2,1,3] => 2
[-2,1] => [2]
 => [1,1,0,0,1,0]
 => [2,1,3] => 2
[1,2,-3] => [1]
 => [1,0,1,0]
 => [1,2] => 1
[1,-2,3] => [1]
 => [1,0,1,0]
 => [1,2] => 1
[1,-2,-3] => [1,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 2
[-1,2,3] => [1]
 => [1,0,1,0]
 => [1,2] => 1
[-1,2,-3] => [1,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 2
[-1,-2,3] => [1,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 2
[-1,-2,-3] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 3
[1,3,-2] => [2]
 => [1,1,0,0,1,0]
 => [2,1,3] => 2
[1,-3,2] => [2]
 => [1,1,0,0,1,0]
 => [2,1,3] => 2
[-1,3,2] => [1]
 => [1,0,1,0]
 => [1,2] => 1
[-1,3,-2] => [2,1]
 => [1,0,1,0,1,0]
 => [1,2,3] => 3
[-1,-3,2] => [2,1]
 => [1,0,1,0,1,0]
 => [1,2,3] => 3
[-1,-3,-2] => [1]
 => [1,0,1,0]
 => [1,2] => 1
[2,1,-3] => [1]
 => [1,0,1,0]
 => [1,2] => 1
[2,-1,3] => [2]
 => [1,1,0,0,1,0]
 => [2,1,3] => 2
[2,-1,-3] => [2,1]
 => [1,0,1,0,1,0]
 => [1,2,3] => 3
[-2,1,3] => [2]
 => [1,1,0,0,1,0]
 => [2,1,3] => 2
[-2,1,-3] => [2,1]
 => [1,0,1,0,1,0]
 => [1,2,3] => 3
[-2,-1,-3] => [1]
 => [1,0,1,0]
 => [1,2] => 1
[2,3,-1] => [3]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 3
[2,-3,1] => [3]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 3
[-2,3,1] => [3]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 3
[-2,-3,-1] => [3]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 3
[3,1,-2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 3
[3,-1,2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 3
[-3,1,2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 3
[-3,-1,-2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 3
[3,2,-1] => [2]
 => [1,1,0,0,1,0]
 => [2,1,3] => 2
[3,-2,1] => [1]
 => [1,0,1,0]
 => [1,2] => 1
[3,-2,-1] => [2,1]
 => [1,0,1,0,1,0]
 => [1,2,3] => 3
[-3,2,1] => [2]
 => [1,1,0,0,1,0]
 => [2,1,3] => 2
[-3,-2,1] => [2,1]
 => [1,0,1,0,1,0]
 => [1,2,3] => 3
[-3,-2,-1] => [1]
 => [1,0,1,0]
 => [1,2] => 1
[1,2,3,-4] => [1]
 => [1,0,1,0]
 => [1,2] => 1
[1,2,-3,4] => [1]
 => [1,0,1,0]
 => [1,2] => 1
[1,2,-3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 2
[1,-2,3,4] => [1]
 => [1,0,1,0]
 => [1,2] => 1
[1,-2,3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 2
[1,-2,-3,4] => [1,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 2
[1,-2,-3,-4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 3
[-1,2,3,4] => [1]
 => [1,0,1,0]
 => [1,2] => 1
[-1,2,3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 2
[-1,2,-3,4] => [1,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 2
[-1,2,-3,-4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 3
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
Mp00169: Signed permutations —odd 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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1] => [1]
 => 10 => 10 => 1
[1,-2] => [1]
 => 10 => 10 => 1
[-1,2] => [1]
 => 10 => 10 => 1
[-1,-2] => [1,1]
 => 110 => 110 => 2
[2,-1] => [2]
 => 100 => 010 => 2
[-2,1] => [2]
 => 100 => 010 => 2
[1,2,-3] => [1]
 => 10 => 10 => 1
[1,-2,3] => [1]
 => 10 => 10 => 1
[1,-2,-3] => [1,1]
 => 110 => 110 => 2
[-1,2,3] => [1]
 => 10 => 10 => 1
[-1,2,-3] => [1,1]
 => 110 => 110 => 2
[-1,-2,3] => [1,1]
 => 110 => 110 => 2
[-1,-2,-3] => [1,1,1]
 => 1110 => 1110 => 3
[1,3,-2] => [2]
 => 100 => 010 => 2
[1,-3,2] => [2]
 => 100 => 010 => 2
[-1,3,2] => [1]
 => 10 => 10 => 1
[-1,3,-2] => [2,1]
 => 1010 => 0110 => 3
[-1,-3,2] => [2,1]
 => 1010 => 0110 => 3
[-1,-3,-2] => [1]
 => 10 => 10 => 1
[2,1,-3] => [1]
 => 10 => 10 => 1
[2,-1,3] => [2]
 => 100 => 010 => 2
[2,-1,-3] => [2,1]
 => 1010 => 0110 => 3
[-2,1,3] => [2]
 => 100 => 010 => 2
[-2,1,-3] => [2,1]
 => 1010 => 0110 => 3
[-2,-1,-3] => [1]
 => 10 => 10 => 1
[2,3,-1] => [3]
 => 1000 => 0010 => 3
[2,-3,1] => [3]
 => 1000 => 0010 => 3
[-2,3,1] => [3]
 => 1000 => 0010 => 3
[-2,-3,-1] => [3]
 => 1000 => 0010 => 3
[3,1,-2] => [3]
 => 1000 => 0010 => 3
[3,-1,2] => [3]
 => 1000 => 0010 => 3
[-3,1,2] => [3]
 => 1000 => 0010 => 3
[-3,-1,-2] => [3]
 => 1000 => 0010 => 3
[3,2,-1] => [2]
 => 100 => 010 => 2
[3,-2,1] => [1]
 => 10 => 10 => 1
[3,-2,-1] => [2,1]
 => 1010 => 0110 => 3
[-3,2,1] => [2]
 => 100 => 010 => 2
[-3,-2,1] => [2,1]
 => 1010 => 0110 => 3
[-3,-2,-1] => [1]
 => 10 => 10 => 1
[1,2,3,-4] => [1]
 => 10 => 10 => 1
[1,2,-3,4] => [1]
 => 10 => 10 => 1
[1,2,-3,-4] => [1,1]
 => 110 => 110 => 2
[1,-2,3,4] => [1]
 => 10 => 10 => 1
[1,-2,3,-4] => [1,1]
 => 110 => 110 => 2
[1,-2,-3,4] => [1,1]
 => 110 => 110 => 2
[1,-2,-3,-4] => [1,1,1]
 => 1110 => 1110 => 3
[-1,2,3,4] => [1]
 => 10 => 10 => 1
[-1,2,3,-4] => [1,1]
 => 110 => 110 => 2
[-1,2,-3,4] => [1,1]
 => 110 => 110 => 2
[-1,2,-3,-4] => [1,1,1]
 => 1110 => 1110 => 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 $\binom{n}{a}_q$.
The following 24 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000395The sum of the heights of the peaks of a Dyck path. St000719The number of alignments in a perfect matching. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001759The Rajchgot index of a permutation. St001065Number of indecomposable reflexive modules 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. St001641The number of ascent tops in the flattened set partition such that all smaller elements appear before. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St000806The semiperimeter of the associated bargraph. St000519The largest length of a factor maximising the subword complexity. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. 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. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. 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 $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.