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Your data matches 18 different statistics following compositions of up to 3 maps.
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Matching statistic: St000228
(load all 20 compositions to match this statistic)
(load all 20 compositions to match this statistic)
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%
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => []
=> 0
[-1] => [1]
=> 1
[1,2] => []
=> 0
[1,-2] => [1]
=> 1
[-1,2] => [1]
=> 1
[-1,-2] => [1,1]
=> 2
[2,1] => []
=> 0
[2,-1] => [2]
=> 2
[-2,1] => [2]
=> 2
[-2,-1] => []
=> 0
[1,2,3] => []
=> 0
[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] => []
=> 0
[1,3,-2] => [2]
=> 2
[1,-3,2] => [2]
=> 2
[1,-3,-2] => []
=> 0
[-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] => []
=> 0
[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] => []
=> 0
[-2,-1,-3] => [1]
=> 1
[2,3,1] => []
=> 0
[2,3,-1] => [3]
=> 3
[2,-3,1] => [3]
=> 3
[2,-3,-1] => []
=> 0
[-2,3,1] => [3]
=> 3
[-2,3,-1] => []
=> 0
[-2,-3,1] => []
=> 0
[-2,-3,-1] => [3]
=> 3
[3,1,2] => []
=> 0
[3,1,-2] => [3]
=> 3
[3,-1,2] => [3]
=> 3
[3,-1,-2] => []
=> 0
[-3,1,2] => [3]
=> 3
[-3,1,-2] => []
=> 0
[-3,-1,2] => []
=> 0
[-3,-1,-2] => [3]
=> 3
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St001034
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
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%
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] => []
=> []
=> 0
[-1] => [1]
=> [1,0]
=> 1
[1,2] => []
=> []
=> 0
[1,-2] => [1]
=> [1,0]
=> 1
[-1,2] => [1]
=> [1,0]
=> 1
[-1,-2] => [1,1]
=> [1,1,0,0]
=> 2
[2,1] => []
=> []
=> 0
[2,-1] => [2]
=> [1,0,1,0]
=> 2
[-2,1] => [2]
=> [1,0,1,0]
=> 2
[-2,-1] => []
=> []
=> 0
[1,2,3] => []
=> []
=> 0
[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] => []
=> []
=> 0
[1,3,-2] => [2]
=> [1,0,1,0]
=> 2
[1,-3,2] => [2]
=> [1,0,1,0]
=> 2
[1,-3,-2] => []
=> []
=> 0
[-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] => []
=> []
=> 0
[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] => []
=> []
=> 0
[-2,-1,-3] => [1]
=> [1,0]
=> 1
[2,3,1] => []
=> []
=> 0
[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] => []
=> []
=> 0
[-2,3,1] => [3]
=> [1,0,1,0,1,0]
=> 3
[-2,3,-1] => []
=> []
=> 0
[-2,-3,1] => []
=> []
=> 0
[-2,-3,-1] => [3]
=> [1,0,1,0,1,0]
=> 3
[3,1,2] => []
=> []
=> 0
[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] => []
=> []
=> 0
[-3,1,2] => [3]
=> [1,0,1,0,1,0]
=> 3
[-3,1,-2] => []
=> []
=> 0
[-3,-1,2] => []
=> []
=> 0
[-3,-1,-2] => [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: St000018
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%
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] => []
=> []
=> [] => 0
[-1] => [1]
=> [1,0,1,0]
=> [2,1] => 1
[1,2] => []
=> []
=> [] => 0
[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] => []
=> []
=> [] => 0
[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
[-2,-1] => []
=> []
=> [] => 0
[1,2,3] => []
=> []
=> [] => 0
[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] => []
=> []
=> [] => 0
[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] => []
=> []
=> [] => 0
[-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] => []
=> []
=> [] => 0
[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] => []
=> []
=> [] => 0
[-2,-1,-3] => [1]
=> [1,0,1,0]
=> [2,1] => 1
[2,3,1] => []
=> []
=> [] => 0
[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] => []
=> []
=> [] => 0
[-2,3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[-2,3,-1] => []
=> []
=> [] => 0
[-2,-3,1] => []
=> []
=> [] => 0
[-2,-3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[3,1,2] => []
=> []
=> [] => 0
[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] => []
=> []
=> [] => 0
[-3,1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[-3,1,-2] => []
=> []
=> [] => 0
[-3,-1,2] => []
=> []
=> [] => 0
[-3,-1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 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: St000246
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
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%
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] => []
=> []
=> [] => 0
[-1] => [1]
=> [1,0,1,0]
=> [1,2] => 1
[1,2] => []
=> []
=> [] => 0
[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] => []
=> []
=> [] => 0
[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
[-2,-1] => []
=> []
=> [] => 0
[1,2,3] => []
=> []
=> [] => 0
[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] => []
=> []
=> [] => 0
[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] => []
=> []
=> [] => 0
[-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] => []
=> []
=> [] => 0
[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] => []
=> []
=> [] => 0
[-2,-1,-3] => [1]
=> [1,0,1,0]
=> [1,2] => 1
[2,3,1] => []
=> []
=> [] => 0
[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] => []
=> []
=> [] => 0
[-2,3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[-2,3,-1] => []
=> []
=> [] => 0
[-2,-3,1] => []
=> []
=> [] => 0
[-2,-3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[3,1,2] => []
=> []
=> [] => 0
[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] => []
=> []
=> [] => 0
[-3,1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[-3,1,-2] => []
=> []
=> [] => 0
[-3,-1,2] => []
=> []
=> [] => 0
[-3,-1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 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: St000293
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 89%
Mp00095: Integer partitions —to binary word⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 89%
Values
[1] => []
=> => ? = 0
[-1] => [1]
=> 10 => 1
[1,2] => []
=> => ? = 0
[1,-2] => [1]
=> 10 => 1
[-1,2] => [1]
=> 10 => 1
[-1,-2] => [1,1]
=> 110 => 2
[2,1] => []
=> => ? = 0
[2,-1] => [2]
=> 100 => 2
[-2,1] => [2]
=> 100 => 2
[-2,-1] => []
=> => ? = 0
[1,2,3] => []
=> => ? = 0
[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] => []
=> => ? = 0
[1,3,-2] => [2]
=> 100 => 2
[1,-3,2] => [2]
=> 100 => 2
[1,-3,-2] => []
=> => ? = 0
[-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] => []
=> => ? = 0
[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] => []
=> => ? = 0
[-2,-1,-3] => [1]
=> 10 => 1
[2,3,1] => []
=> => ? = 0
[2,3,-1] => [3]
=> 1000 => 3
[2,-3,1] => [3]
=> 1000 => 3
[2,-3,-1] => []
=> => ? = 0
[-2,3,1] => [3]
=> 1000 => 3
[-2,3,-1] => []
=> => ? = 0
[-2,-3,1] => []
=> => ? = 0
[-2,-3,-1] => [3]
=> 1000 => 3
[3,1,2] => []
=> => ? = 0
[3,1,-2] => [3]
=> 1000 => 3
[3,-1,2] => [3]
=> 1000 => 3
[3,-1,-2] => []
=> => ? = 0
[-3,1,2] => [3]
=> 1000 => 3
[-3,1,-2] => []
=> => ? = 0
[-3,-1,2] => []
=> => ? = 0
[-3,-1,-2] => [3]
=> 1000 => 3
[3,2,1] => []
=> => ? = 0
[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] => []
=> => ? = 0
[-3,-2,1] => [2,1]
=> 1010 => 3
[-3,-2,-1] => [1]
=> 10 => 1
[1,2,3,4] => []
=> => ? = 0
[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
[1,2,4,3] => []
=> => ? = 0
[1,2,-4,-3] => []
=> => ? = 0
[1,3,2,4] => []
=> => ? = 0
[1,-3,-2,4] => []
=> => ? = 0
[1,3,4,2] => []
=> => ? = 0
[1,3,-4,-2] => []
=> => ? = 0
[1,-3,4,-2] => []
=> => ? = 0
[1,-3,-4,2] => []
=> => ? = 0
[1,4,2,3] => []
=> => ? = 0
[1,4,-2,-3] => []
=> => ? = 0
[1,-4,2,-3] => []
=> => ? = 0
[1,-4,-2,3] => []
=> => ? = 0
[1,4,3,2] => []
=> => ? = 0
[1,-4,3,-2] => []
=> => ? = 0
[2,1,3,4] => []
=> => ? = 0
[-2,-1,3,4] => []
=> => ? = 0
[2,1,4,3] => []
=> => ? = 0
[2,1,-4,-3] => []
=> => ? = 0
[-2,-1,4,3] => []
=> => ? = 0
[-2,-1,-4,-3] => []
=> => ? = 0
[2,3,1,4] => []
=> => ? = 0
[2,-3,-1,4] => []
=> => ? = 0
[-2,3,-1,4] => []
=> => ? = 0
[-2,-3,1,4] => []
=> => ? = 0
[2,3,4,1] => []
=> => ? = 0
[2,3,-4,-1] => []
=> => ? = 0
[2,-3,4,-1] => []
=> => ? = 0
[2,-3,-4,1] => []
=> => ? = 0
[-2,3,4,-1] => []
=> => ? = 0
[-2,3,-4,1] => []
=> => ? = 0
Description
The number of inversions of a binary word.
Matching statistic: St000290
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: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 89%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
St000290: Binary words ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 89%
Values
[1] => []
=> => ? => ? = 0
[-1] => [1]
=> 10 => 10 => 1
[1,2] => []
=> => ? => ? = 0
[1,-2] => [1]
=> 10 => 10 => 1
[-1,2] => [1]
=> 10 => 10 => 1
[-1,-2] => [1,1]
=> 110 => 110 => 2
[2,1] => []
=> => ? => ? = 0
[2,-1] => [2]
=> 100 => 010 => 2
[-2,1] => [2]
=> 100 => 010 => 2
[-2,-1] => []
=> => ? => ? = 0
[1,2,3] => []
=> => ? => ? = 0
[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] => []
=> => ? => ? = 0
[1,3,-2] => [2]
=> 100 => 010 => 2
[1,-3,2] => [2]
=> 100 => 010 => 2
[1,-3,-2] => []
=> => ? => ? = 0
[-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] => []
=> => ? => ? = 0
[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] => []
=> => ? => ? = 0
[-2,-1,-3] => [1]
=> 10 => 10 => 1
[2,3,1] => []
=> => ? => ? = 0
[2,3,-1] => [3]
=> 1000 => 0010 => 3
[2,-3,1] => [3]
=> 1000 => 0010 => 3
[2,-3,-1] => []
=> => ? => ? = 0
[-2,3,1] => [3]
=> 1000 => 0010 => 3
[-2,3,-1] => []
=> => ? => ? = 0
[-2,-3,1] => []
=> => ? => ? = 0
[-2,-3,-1] => [3]
=> 1000 => 0010 => 3
[3,1,2] => []
=> => ? => ? = 0
[3,1,-2] => [3]
=> 1000 => 0010 => 3
[3,-1,2] => [3]
=> 1000 => 0010 => 3
[3,-1,-2] => []
=> => ? => ? = 0
[-3,1,2] => [3]
=> 1000 => 0010 => 3
[-3,1,-2] => []
=> => ? => ? = 0
[-3,-1,2] => []
=> => ? => ? = 0
[-3,-1,-2] => [3]
=> 1000 => 0010 => 3
[3,2,1] => []
=> => ? => ? = 0
[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] => []
=> => ? => ? = 0
[-3,-2,1] => [2,1]
=> 1010 => 0110 => 3
[-3,-2,-1] => [1]
=> 10 => 10 => 1
[1,2,3,4] => []
=> => ? => ? = 0
[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
[1,2,4,3] => []
=> => ? => ? = 0
[1,2,-4,-3] => []
=> => ? => ? = 0
[1,3,2,4] => []
=> => ? => ? = 0
[1,-3,-2,4] => []
=> => ? => ? = 0
[1,3,4,2] => []
=> => ? => ? = 0
[1,3,-4,-2] => []
=> => ? => ? = 0
[1,-3,4,-2] => []
=> => ? => ? = 0
[1,-3,-4,2] => []
=> => ? => ? = 0
[1,4,2,3] => []
=> => ? => ? = 0
[1,4,-2,-3] => []
=> => ? => ? = 0
[1,-4,2,-3] => []
=> => ? => ? = 0
[1,-4,-2,3] => []
=> => ? => ? = 0
[1,4,3,2] => []
=> => ? => ? = 0
[1,-4,3,-2] => []
=> => ? => ? = 0
[2,1,3,4] => []
=> => ? => ? = 0
[-2,-1,3,4] => []
=> => ? => ? = 0
[2,1,4,3] => []
=> => ? => ? = 0
[2,1,-4,-3] => []
=> => ? => ? = 0
[-2,-1,4,3] => []
=> => ? => ? = 0
[-2,-1,-4,-3] => []
=> => ? => ? = 0
[2,3,1,4] => []
=> => ? => ? = 0
[2,-3,-1,4] => []
=> => ? => ? = 0
[-2,3,-1,4] => []
=> => ? => ? = 0
[-2,-3,1,4] => []
=> => ? => ? = 0
[2,3,4,1] => []
=> => ? => ? = 0
[2,3,-4,-1] => []
=> => ? => ? = 0
[2,-3,4,-1] => []
=> => ? => ? = 0
[2,-3,-4,1] => []
=> => ? => ? = 0
[-2,3,4,-1] => []
=> => ? => ? = 0
[-2,3,-4,1] => []
=> => ? => ? = 0
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
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000395: Dyck paths ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 89%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000395: Dyck paths ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 89%
Values
[1] => []
=> []
=> []
=> ? = 0
[-1] => [1]
=> [1,0]
=> [1,0]
=> 1
[1,2] => []
=> []
=> []
=> ? = 0
[1,-2] => [1]
=> [1,0]
=> [1,0]
=> 1
[-1,2] => [1]
=> [1,0]
=> [1,0]
=> 1
[-1,-2] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[2,1] => []
=> []
=> []
=> ? = 0
[2,-1] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[-2,1] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[-2,-1] => []
=> []
=> []
=> ? = 0
[1,2,3] => []
=> []
=> []
=> ? = 0
[1,2,-3] => [1]
=> [1,0]
=> [1,0]
=> 1
[1,-2,3] => [1]
=> [1,0]
=> [1,0]
=> 1
[1,-2,-3] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[-1,2,3] => [1]
=> [1,0]
=> [1,0]
=> 1
[-1,2,-3] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[-1,-2,3] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[-1,-2,-3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3
[1,3,2] => []
=> []
=> []
=> ? = 0
[1,3,-2] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[1,-3,2] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[1,-3,-2] => []
=> []
=> []
=> ? = 0
[-1,3,2] => [1]
=> [1,0]
=> [1,0]
=> 1
[-1,3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[-1,-3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[-1,-3,-2] => [1]
=> [1,0]
=> [1,0]
=> 1
[2,1,3] => []
=> []
=> []
=> ? = 0
[2,1,-3] => [1]
=> [1,0]
=> [1,0]
=> 1
[2,-1,3] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[2,-1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[-2,1,3] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[-2,1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[-2,-1,3] => []
=> []
=> []
=> ? = 0
[-2,-1,-3] => [1]
=> [1,0]
=> [1,0]
=> 1
[2,3,1] => []
=> []
=> []
=> ? = 0
[2,3,-1] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[2,-3,1] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[2,-3,-1] => []
=> []
=> []
=> ? = 0
[-2,3,1] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[-2,3,-1] => []
=> []
=> []
=> ? = 0
[-2,-3,1] => []
=> []
=> []
=> ? = 0
[-2,-3,-1] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[3,1,2] => []
=> []
=> []
=> ? = 0
[3,1,-2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[3,-1,2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[3,-1,-2] => []
=> []
=> []
=> ? = 0
[-3,1,2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[-3,1,-2] => []
=> []
=> []
=> ? = 0
[-3,-1,2] => []
=> []
=> []
=> ? = 0
[-3,-1,-2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[3,2,1] => []
=> []
=> []
=> ? = 0
[3,2,-1] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[3,-2,1] => [1]
=> [1,0]
=> [1,0]
=> 1
[3,-2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[-3,2,1] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[-3,2,-1] => []
=> []
=> []
=> ? = 0
[-3,-2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[-3,-2,-1] => [1]
=> [1,0]
=> [1,0]
=> 1
[1,2,3,4] => []
=> []
=> []
=> ? = 0
[1,2,3,-4] => [1]
=> [1,0]
=> [1,0]
=> 1
[1,2,-3,4] => [1]
=> [1,0]
=> [1,0]
=> 1
[1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,-2,3,4] => [1]
=> [1,0]
=> [1,0]
=> 1
[1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,-2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3
[-1,2,3,4] => [1]
=> [1,0]
=> [1,0]
=> 1
[-1,2,3,-4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[-1,2,-3,4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[-1,2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3
[1,2,4,3] => []
=> []
=> []
=> ? = 0
[1,2,-4,-3] => []
=> []
=> []
=> ? = 0
[1,3,2,4] => []
=> []
=> []
=> ? = 0
[1,-3,-2,4] => []
=> []
=> []
=> ? = 0
[1,3,4,2] => []
=> []
=> []
=> ? = 0
[1,3,-4,-2] => []
=> []
=> []
=> ? = 0
[1,-3,4,-2] => []
=> []
=> []
=> ? = 0
[1,-3,-4,2] => []
=> []
=> []
=> ? = 0
[1,4,2,3] => []
=> []
=> []
=> ? = 0
[1,4,-2,-3] => []
=> []
=> []
=> ? = 0
[1,-4,2,-3] => []
=> []
=> []
=> ? = 0
[1,-4,-2,3] => []
=> []
=> []
=> ? = 0
[1,4,3,2] => []
=> []
=> []
=> ? = 0
[1,-4,3,-2] => []
=> []
=> []
=> ? = 0
[2,1,3,4] => []
=> []
=> []
=> ? = 0
[-2,-1,3,4] => []
=> []
=> []
=> ? = 0
[2,1,4,3] => []
=> []
=> []
=> ? = 0
[2,1,-4,-3] => []
=> []
=> []
=> ? = 0
[-2,-1,4,3] => []
=> []
=> []
=> ? = 0
[-2,-1,-4,-3] => []
=> []
=> []
=> ? = 0
[2,3,1,4] => []
=> []
=> []
=> ? = 0
[2,-3,-1,4] => []
=> []
=> []
=> ? = 0
[-2,3,-1,4] => []
=> []
=> []
=> ? = 0
[-2,-3,1,4] => []
=> []
=> []
=> ? = 0
[2,3,4,1] => []
=> []
=> []
=> ? = 0
[2,3,-4,-1] => []
=> []
=> []
=> ? = 0
[2,-3,4,-1] => []
=> []
=> []
=> ? = 0
[2,-3,-4,1] => []
=> []
=> []
=> ? = 0
[-2,3,4,-1] => []
=> []
=> []
=> ? = 0
[-2,3,-4,1] => []
=> []
=> []
=> ? = 0
Description
The sum of the heights of the peaks of a Dyck path.
Matching statistic: St001641
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St001641: Set partitions ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 78%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St001641: Set partitions ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 78%
Values
[1] => []
=> []
=> {}
=> ? = 0 - 1
[-1] => [1]
=> [[1]]
=> {{1}}
=> 0 = 1 - 1
[1,2] => []
=> []
=> {}
=> ? = 0 - 1
[1,-2] => [1]
=> [[1]]
=> {{1}}
=> 0 = 1 - 1
[-1,2] => [1]
=> [[1]]
=> {{1}}
=> 0 = 1 - 1
[-1,-2] => [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 1 = 2 - 1
[2,1] => []
=> []
=> {}
=> ? = 0 - 1
[2,-1] => [2]
=> [[1,2]]
=> {{1,2}}
=> 1 = 2 - 1
[-2,1] => [2]
=> [[1,2]]
=> {{1,2}}
=> 1 = 2 - 1
[-2,-1] => []
=> []
=> {}
=> ? = 0 - 1
[1,2,3] => []
=> []
=> {}
=> ? = 0 - 1
[1,2,-3] => [1]
=> [[1]]
=> {{1}}
=> 0 = 1 - 1
[1,-2,3] => [1]
=> [[1]]
=> {{1}}
=> 0 = 1 - 1
[1,-2,-3] => [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 1 = 2 - 1
[-1,2,3] => [1]
=> [[1]]
=> {{1}}
=> 0 = 1 - 1
[-1,2,-3] => [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 1 = 2 - 1
[-1,-2,3] => [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 1 = 2 - 1
[-1,-2,-3] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 2 = 3 - 1
[1,3,2] => []
=> []
=> {}
=> ? = 0 - 1
[1,3,-2] => [2]
=> [[1,2]]
=> {{1,2}}
=> 1 = 2 - 1
[1,-3,2] => [2]
=> [[1,2]]
=> {{1,2}}
=> 1 = 2 - 1
[1,-3,-2] => []
=> []
=> {}
=> ? = 0 - 1
[-1,3,2] => [1]
=> [[1]]
=> {{1}}
=> 0 = 1 - 1
[-1,3,-2] => [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 2 = 3 - 1
[-1,-3,2] => [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 2 = 3 - 1
[-1,-3,-2] => [1]
=> [[1]]
=> {{1}}
=> 0 = 1 - 1
[2,1,3] => []
=> []
=> {}
=> ? = 0 - 1
[2,1,-3] => [1]
=> [[1]]
=> {{1}}
=> 0 = 1 - 1
[2,-1,3] => [2]
=> [[1,2]]
=> {{1,2}}
=> 1 = 2 - 1
[2,-1,-3] => [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 2 = 3 - 1
[-2,1,3] => [2]
=> [[1,2]]
=> {{1,2}}
=> 1 = 2 - 1
[-2,1,-3] => [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 2 = 3 - 1
[-2,-1,3] => []
=> []
=> {}
=> ? = 0 - 1
[-2,-1,-3] => [1]
=> [[1]]
=> {{1}}
=> 0 = 1 - 1
[2,3,1] => []
=> []
=> {}
=> ? = 0 - 1
[2,3,-1] => [3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 2 = 3 - 1
[2,-3,1] => [3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 2 = 3 - 1
[2,-3,-1] => []
=> []
=> {}
=> ? = 0 - 1
[-2,3,1] => [3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 2 = 3 - 1
[-2,3,-1] => []
=> []
=> {}
=> ? = 0 - 1
[-2,-3,1] => []
=> []
=> {}
=> ? = 0 - 1
[-2,-3,-1] => [3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 2 = 3 - 1
[3,1,2] => []
=> []
=> {}
=> ? = 0 - 1
[3,1,-2] => [3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 2 = 3 - 1
[3,-1,2] => [3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 2 = 3 - 1
[3,-1,-2] => []
=> []
=> {}
=> ? = 0 - 1
[-3,1,2] => [3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 2 = 3 - 1
[-3,1,-2] => []
=> []
=> {}
=> ? = 0 - 1
[-3,-1,2] => []
=> []
=> {}
=> ? = 0 - 1
[-3,-1,-2] => [3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 2 = 3 - 1
[3,2,1] => []
=> []
=> {}
=> ? = 0 - 1
[3,2,-1] => [2]
=> [[1,2]]
=> {{1,2}}
=> 1 = 2 - 1
[3,-2,1] => [1]
=> [[1]]
=> {{1}}
=> 0 = 1 - 1
[3,-2,-1] => [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 2 = 3 - 1
[-3,2,1] => [2]
=> [[1,2]]
=> {{1,2}}
=> 1 = 2 - 1
[-3,2,-1] => []
=> []
=> {}
=> ? = 0 - 1
[-3,-2,1] => [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 2 = 3 - 1
[-3,-2,-1] => [1]
=> [[1]]
=> {{1}}
=> 0 = 1 - 1
[1,2,3,4] => []
=> []
=> {}
=> ? = 0 - 1
[1,2,3,-4] => [1]
=> [[1]]
=> {{1}}
=> 0 = 1 - 1
[1,2,-3,4] => [1]
=> [[1]]
=> {{1}}
=> 0 = 1 - 1
[1,2,-3,-4] => [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 1 = 2 - 1
[1,-2,3,4] => [1]
=> [[1]]
=> {{1}}
=> 0 = 1 - 1
[1,-2,3,-4] => [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 1 = 2 - 1
[1,-2,-3,4] => [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 1 = 2 - 1
[1,-2,-3,-4] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 2 = 3 - 1
[-1,2,3,4] => [1]
=> [[1]]
=> {{1}}
=> 0 = 1 - 1
[-1,2,3,-4] => [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 1 = 2 - 1
[-1,2,-3,4] => [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 1 = 2 - 1
[-1,2,-3,-4] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 2 = 3 - 1
[1,2,4,3] => []
=> []
=> {}
=> ? = 0 - 1
[1,2,-4,-3] => []
=> []
=> {}
=> ? = 0 - 1
[1,3,2,4] => []
=> []
=> {}
=> ? = 0 - 1
[1,-3,-2,4] => []
=> []
=> {}
=> ? = 0 - 1
[1,3,4,2] => []
=> []
=> {}
=> ? = 0 - 1
[1,3,-4,-2] => []
=> []
=> {}
=> ? = 0 - 1
[1,-3,4,-2] => []
=> []
=> {}
=> ? = 0 - 1
[1,-3,-4,2] => []
=> []
=> {}
=> ? = 0 - 1
[1,4,2,3] => []
=> []
=> {}
=> ? = 0 - 1
[1,4,-2,-3] => []
=> []
=> {}
=> ? = 0 - 1
[1,-4,2,-3] => []
=> []
=> {}
=> ? = 0 - 1
[1,-4,-2,3] => []
=> []
=> {}
=> ? = 0 - 1
[1,4,3,2] => []
=> []
=> {}
=> ? = 0 - 1
[1,-4,3,-2] => []
=> []
=> {}
=> ? = 0 - 1
[2,1,3,4] => []
=> []
=> {}
=> ? = 0 - 1
[-2,-1,3,4] => []
=> []
=> {}
=> ? = 0 - 1
[2,1,4,3] => []
=> []
=> {}
=> ? = 0 - 1
[2,1,-4,-3] => []
=> []
=> {}
=> ? = 0 - 1
[-2,-1,4,3] => []
=> []
=> {}
=> ? = 0 - 1
[-2,-1,-4,-3] => []
=> []
=> {}
=> ? = 0 - 1
[2,3,1,4] => []
=> []
=> {}
=> ? = 0 - 1
[2,-3,-1,4] => []
=> []
=> {}
=> ? = 0 - 1
[-2,3,-1,4] => []
=> []
=> {}
=> ? = 0 - 1
[-2,-3,1,4] => []
=> []
=> {}
=> ? = 0 - 1
[2,3,4,1] => []
=> []
=> {}
=> ? = 0 - 1
[2,3,-4,-1] => []
=> []
=> {}
=> ? = 0 - 1
[2,-3,4,-1] => []
=> []
=> {}
=> ? = 0 - 1
[2,-3,-4,1] => []
=> []
=> {}
=> ? = 0 - 1
[-2,3,4,-1] => []
=> []
=> {}
=> ? = 0 - 1
[-2,3,-4,1] => []
=> []
=> {}
=> ? = 0 - 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 $P$.
Matching statistic: St000189
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: 67% ●values known / values provided: 74%●distinct values known / distinct values provided: 67%
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000189: Posets ⟶ ℤResult quality: 67% ●values known / values provided: 74%●distinct values known / distinct values provided: 67%
Values
[1] => []
=> [[],[]]
=> ([],0)
=> ? = 0
[-1] => [1]
=> [[1],[]]
=> ([],1)
=> 1
[1,2] => []
=> [[],[]]
=> ([],0)
=> ? = 0
[1,-2] => [1]
=> [[1],[]]
=> ([],1)
=> 1
[-1,2] => [1]
=> [[1],[]]
=> ([],1)
=> 1
[-1,-2] => [1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> 2
[2,1] => []
=> [[],[]]
=> ([],0)
=> ? = 0
[2,-1] => [2]
=> [[2],[]]
=> ([(0,1)],2)
=> 2
[-2,1] => [2]
=> [[2],[]]
=> ([(0,1)],2)
=> 2
[-2,-1] => []
=> [[],[]]
=> ([],0)
=> ? = 0
[1,2,3] => []
=> [[],[]]
=> ([],0)
=> ? = 0
[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] => []
=> [[],[]]
=> ([],0)
=> ? = 0
[1,3,-2] => [2]
=> [[2],[]]
=> ([(0,1)],2)
=> 2
[1,-3,2] => [2]
=> [[2],[]]
=> ([(0,1)],2)
=> 2
[1,-3,-2] => []
=> [[],[]]
=> ([],0)
=> ? = 0
[-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] => []
=> [[],[]]
=> ([],0)
=> ? = 0
[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] => []
=> [[],[]]
=> ([],0)
=> ? = 0
[-2,-1,-3] => [1]
=> [[1],[]]
=> ([],1)
=> 1
[2,3,1] => []
=> [[],[]]
=> ([],0)
=> ? = 0
[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] => []
=> [[],[]]
=> ([],0)
=> ? = 0
[-2,3,1] => [3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 3
[-2,3,-1] => []
=> [[],[]]
=> ([],0)
=> ? = 0
[-2,-3,1] => []
=> [[],[]]
=> ([],0)
=> ? = 0
[-2,-3,-1] => [3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 3
[3,1,2] => []
=> [[],[]]
=> ([],0)
=> ? = 0
[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] => []
=> [[],[]]
=> ([],0)
=> ? = 0
[-3,1,2] => [3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 3
[-3,1,-2] => []
=> [[],[]]
=> ([],0)
=> ? = 0
[-3,-1,2] => []
=> [[],[]]
=> ([],0)
=> ? = 0
[-3,-1,-2] => [3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 3
[3,2,1] => []
=> [[],[]]
=> ([],0)
=> ? = 0
[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] => []
=> [[],[]]
=> ([],0)
=> ? = 0
[-3,-2,1] => [2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> 3
[-3,-2,-1] => [1]
=> [[1],[]]
=> ([],1)
=> 1
[1,2,3,4] => []
=> [[],[]]
=> ([],0)
=> ? = 0
[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
[1,2,4,3] => []
=> [[],[]]
=> ([],0)
=> ? = 0
[1,2,-4,-3] => []
=> [[],[]]
=> ([],0)
=> ? = 0
[1,3,2,4] => []
=> [[],[]]
=> ([],0)
=> ? = 0
[1,-3,-2,4] => []
=> [[],[]]
=> ([],0)
=> ? = 0
[1,3,4,2] => []
=> [[],[]]
=> ([],0)
=> ? = 0
[1,3,-4,-2] => []
=> [[],[]]
=> ([],0)
=> ? = 0
[1,-3,4,-2] => []
=> [[],[]]
=> ([],0)
=> ? = 0
[1,-3,-4,2] => []
=> [[],[]]
=> ([],0)
=> ? = 0
[1,4,2,3] => []
=> [[],[]]
=> ([],0)
=> ? = 0
[1,4,-2,-3] => []
=> [[],[]]
=> ([],0)
=> ? = 0
[1,-4,2,-3] => []
=> [[],[]]
=> ([],0)
=> ? = 0
[1,-4,-2,3] => []
=> [[],[]]
=> ([],0)
=> ? = 0
[1,4,3,2] => []
=> [[],[]]
=> ([],0)
=> ? = 0
[1,-4,3,-2] => []
=> [[],[]]
=> ([],0)
=> ? = 0
[2,1,3,4] => []
=> [[],[]]
=> ([],0)
=> ? = 0
[-2,-1,3,4] => []
=> [[],[]]
=> ([],0)
=> ? = 0
[2,1,4,3] => []
=> [[],[]]
=> ([],0)
=> ? = 0
[2,1,-4,-3] => []
=> [[],[]]
=> ([],0)
=> ? = 0
[-2,-1,4,3] => []
=> [[],[]]
=> ([],0)
=> ? = 0
[-2,-1,-4,-3] => []
=> [[],[]]
=> ([],0)
=> ? = 0
[2,3,1,4] => []
=> [[],[]]
=> ([],0)
=> ? = 0
[2,-3,-1,4] => []
=> [[],[]]
=> ([],0)
=> ? = 0
[-2,3,-1,4] => []
=> [[],[]]
=> ([],0)
=> ? = 0
[-2,-3,1,4] => []
=> [[],[]]
=> ([],0)
=> ? = 0
[2,3,4,1] => []
=> [[],[]]
=> ([],0)
=> ? = 0
[2,3,-4,-1] => []
=> [[],[]]
=> ([],0)
=> ? = 0
[2,-3,4,-1] => []
=> [[],[]]
=> ([],0)
=> ? = 0
[2,-3,-4,1] => []
=> [[],[]]
=> ([],0)
=> ? = 0
[-2,3,4,-1] => []
=> [[],[]]
=> ([],0)
=> ? = 0
[-2,3,-4,1] => []
=> [[],[]]
=> ([],0)
=> ? = 0
Description
The number of elements in the poset.
Matching statistic: St000229
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: 67% ●values known / values provided: 74%●distinct values known / distinct values provided: 67%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000229: Set partitions ⟶ ℤResult quality: 67% ●values known / values provided: 74%●distinct values known / distinct values provided: 67%
Values
[1] => []
=> []
=> {}
=> ? = 0
[-1] => [1]
=> [[1]]
=> {{1}}
=> 1
[1,2] => []
=> []
=> {}
=> ? = 0
[1,-2] => [1]
=> [[1]]
=> {{1}}
=> 1
[-1,2] => [1]
=> [[1]]
=> {{1}}
=> 1
[-1,-2] => [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 2
[2,1] => []
=> []
=> {}
=> ? = 0
[2,-1] => [2]
=> [[1,2]]
=> {{1,2}}
=> 2
[-2,1] => [2]
=> [[1,2]]
=> {{1,2}}
=> 2
[-2,-1] => []
=> []
=> {}
=> ? = 0
[1,2,3] => []
=> []
=> {}
=> ? = 0
[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] => []
=> []
=> {}
=> ? = 0
[1,3,-2] => [2]
=> [[1,2]]
=> {{1,2}}
=> 2
[1,-3,2] => [2]
=> [[1,2]]
=> {{1,2}}
=> 2
[1,-3,-2] => []
=> []
=> {}
=> ? = 0
[-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] => []
=> []
=> {}
=> ? = 0
[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] => []
=> []
=> {}
=> ? = 0
[-2,-1,-3] => [1]
=> [[1]]
=> {{1}}
=> 1
[2,3,1] => []
=> []
=> {}
=> ? = 0
[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] => []
=> []
=> {}
=> ? = 0
[-2,3,1] => [3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 3
[-2,3,-1] => []
=> []
=> {}
=> ? = 0
[-2,-3,1] => []
=> []
=> {}
=> ? = 0
[-2,-3,-1] => [3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 3
[3,1,2] => []
=> []
=> {}
=> ? = 0
[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] => []
=> []
=> {}
=> ? = 0
[-3,1,2] => [3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 3
[-3,1,-2] => []
=> []
=> {}
=> ? = 0
[-3,-1,2] => []
=> []
=> {}
=> ? = 0
[-3,-1,-2] => [3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 3
[3,2,1] => []
=> []
=> {}
=> ? = 0
[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] => []
=> []
=> {}
=> ? = 0
[-3,-2,1] => [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 3
[-3,-2,-1] => [1]
=> [[1]]
=> {{1}}
=> 1
[1,2,3,4] => []
=> []
=> {}
=> ? = 0
[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
[1,2,4,3] => []
=> []
=> {}
=> ? = 0
[1,2,-4,-3] => []
=> []
=> {}
=> ? = 0
[1,3,2,4] => []
=> []
=> {}
=> ? = 0
[1,-3,-2,4] => []
=> []
=> {}
=> ? = 0
[1,3,4,2] => []
=> []
=> {}
=> ? = 0
[1,3,-4,-2] => []
=> []
=> {}
=> ? = 0
[1,-3,4,-2] => []
=> []
=> {}
=> ? = 0
[1,-3,-4,2] => []
=> []
=> {}
=> ? = 0
[1,4,2,3] => []
=> []
=> {}
=> ? = 0
[1,4,-2,-3] => []
=> []
=> {}
=> ? = 0
[1,-4,2,-3] => []
=> []
=> {}
=> ? = 0
[1,-4,-2,3] => []
=> []
=> {}
=> ? = 0
[1,4,3,2] => []
=> []
=> {}
=> ? = 0
[1,-4,3,-2] => []
=> []
=> {}
=> ? = 0
[2,1,3,4] => []
=> []
=> {}
=> ? = 0
[-2,-1,3,4] => []
=> []
=> {}
=> ? = 0
[2,1,4,3] => []
=> []
=> {}
=> ? = 0
[2,1,-4,-3] => []
=> []
=> {}
=> ? = 0
[-2,-1,4,3] => []
=> []
=> {}
=> ? = 0
[-2,-1,-4,-3] => []
=> []
=> {}
=> ? = 0
[2,3,1,4] => []
=> []
=> {}
=> ? = 0
[2,-3,-1,4] => []
=> []
=> {}
=> ? = 0
[-2,3,-1,4] => []
=> []
=> {}
=> ? = 0
[-2,-3,1,4] => []
=> []
=> {}
=> ? = 0
[2,3,4,1] => []
=> []
=> {}
=> ? = 0
[2,3,-4,-1] => []
=> []
=> {}
=> ? = 0
[2,-3,4,-1] => []
=> []
=> {}
=> ? = 0
[2,-3,-4,1] => []
=> []
=> {}
=> ? = 0
[-2,3,4,-1] => []
=> []
=> {}
=> ? = 0
[-2,3,-4,1] => []
=> []
=> {}
=> ? = 0
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]
The following 8 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001759The Rajchgot index of a permutation. St000719The number of alignments in a perfect matching. St000186The sum of the first row in a Gelfand-Tsetlin pattern. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. 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.
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