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Your data matches 79 different statistics following compositions of up to 3 maps.
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Matching statistic: St000228
(load all 105 compositions to match this statistic)
(load all 105 compositions to match this statistic)
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1
[2]
=> 2
[1,1]
=> 2
[3]
=> 3
[2,1]
=> 3
[1,1,1]
=> 3
[4]
=> 4
[3,1]
=> 4
[2,2]
=> 4
[2,1,1]
=> 4
[1,1,1,1]
=> 4
[5]
=> 5
[4,1]
=> 5
[3,2]
=> 5
[3,1,1]
=> 5
[2,2,1]
=> 5
[2,1,1,1]
=> 5
[1,1,1,1,1]
=> 5
[6]
=> 6
[5,1]
=> 6
[4,2]
=> 6
[4,1,1]
=> 6
[3,3]
=> 6
[3,2,1]
=> 6
[3,1,1,1]
=> 6
[2,2,2]
=> 6
[2,2,1,1]
=> 6
[2,1,1,1,1]
=> 6
[1,1,1,1,1,1]
=> 6
[7]
=> 7
[6,1]
=> 7
[5,2]
=> 7
[5,1,1]
=> 7
[4,3]
=> 7
[4,2,1]
=> 7
[4,1,1,1]
=> 7
[3,3,1]
=> 7
[3,2,2]
=> 7
[3,2,1,1]
=> 7
[3,1,1,1,1]
=> 7
[2,2,2,1]
=> 7
[2,2,1,1,1]
=> 7
[2,1,1,1,1,1]
=> 7
[1,1,1,1,1,1,1]
=> 7
[8]
=> 8
[7,1]
=> 8
[6,2]
=> 8
[6,1,1]
=> 8
[5,3]
=> 8
[5,2,1]
=> 8
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St001034
(load all 30 compositions to match this statistic)
(load all 30 compositions to match this statistic)
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%
St001034: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> 1
[2]
=> [1,0,1,0]
=> 2
[1,1]
=> [1,1,0,0]
=> 2
[3]
=> [1,0,1,0,1,0]
=> 3
[2,1]
=> [1,0,1,1,0,0]
=> 3
[1,1,1]
=> [1,1,0,1,0,0]
=> 3
[4]
=> [1,0,1,0,1,0,1,0]
=> 4
[3,1]
=> [1,0,1,0,1,1,0,0]
=> 4
[2,2]
=> [1,1,1,0,0,0]
=> 4
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 4
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 4
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 5
[3,2]
=> [1,0,1,1,1,0,0,0]
=> 5
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 5
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 5
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 5
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 5
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 6
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 6
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 6
[3,3]
=> [1,1,1,0,1,0,0,0]
=> 6
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 6
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 6
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 6
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 6
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 6
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 6
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 7
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 7
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 7
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 7
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> 7
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 7
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> 7
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 7
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 7
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 7
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> 7
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> 7
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 7
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> 7
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 7
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 8
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 8
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 8
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 8
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 8
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> 8
Description
The area of the parallelogram polyomino associated with the Dyck path.
The (bivariate) generating function is given in [1].
Matching statistic: St000018
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
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%
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]
=> [1,0,1,0]
=> [2,1] => 1
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 3
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 4
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 4
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 4
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 4
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 5
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 5
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 5
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 5
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 5
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 5
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 5
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => 6
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => 6
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => 6
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 6
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 6
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 6
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 6
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 6
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 6
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => 6
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 6
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7] => 7
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => 7
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => 7
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => 7
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 7
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 7
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 7
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => 7
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => 7
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 7
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => 7
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 7
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => 7
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => 7
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => 7
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => 8
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [8,2,1,3,4,5,6,7] => 8
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [7,3,1,2,4,5,6] => 8
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [7,2,3,1,4,5,6] => 8
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => 8
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,4,5] => 8
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 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
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%
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]
=> [1,0,1,0]
=> [1,2] => 1
[2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 2
[1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 3
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 3
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 4
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 4
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 4
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 4
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => 5
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => 5
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 5
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 5
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 5
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => 5
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => 5
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => 6
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,3,4,2,1,6] => 6
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => 6
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 6
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 6
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 6
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 6
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 6
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 6
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,6,4,5,3,2] => 6
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => 6
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,6,5,4,3,2,1,8] => 7
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [6,5,3,4,2,1,7] => 7
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,3,2,4,1,6] => 7
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,2,4,3,1,6] => 7
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 7
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 7
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 7
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 7
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 7
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 7
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,6,4,3,5,2] => 7
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 7
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,6,3,5,4,2] => 7
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,7,6,4,5,3,2] => 7
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => 7
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,7,6,5,4,3,2,1,9] => 8
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [7,6,4,5,3,2,1,8] => 8
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [6,4,3,5,2,1,7] => 8
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [6,3,5,4,2,1,7] => 8
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,3,2,5,1,6] => 8
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [5,2,3,4,1,6] => 8
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: St000459
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000459: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000459: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [[1]]
=> [1] => [1]
=> 1
[2]
=> [[1,2]]
=> [1,2] => [1,1]
=> 2
[1,1]
=> [[1],[2]]
=> [2,1] => [2]
=> 2
[3]
=> [[1,2,3]]
=> [1,2,3] => [1,1,1]
=> 3
[2,1]
=> [[1,2],[3]]
=> [3,1,2] => [2,1]
=> 3
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => [3]
=> 3
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => [1,1,1,1]
=> 4
[3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => [2,1,1]
=> 4
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [2,1,1]
=> 4
[2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => [3,1]
=> 4
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [4]
=> 4
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => [2,1,1,1]
=> 5
[3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => [2,1,1,1]
=> 5
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [3,1,1]
=> 5
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [3,1,1]
=> 5
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [4,1]
=> 5
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5]
=> 5
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,1,1,1,1,1]
=> 6
[5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [2,1,1,1,1]
=> 6
[4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => [2,1,1,1,1]
=> 6
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => [3,1,1,1]
=> 6
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [2,1,1,1,1]
=> 6
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => [3,1,1,1]
=> 6
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => [4,1,1]
=> 6
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [3,1,1,1]
=> 6
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => [4,1,1]
=> 6
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => [5,1]
=> 6
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [6]
=> 6
[7]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => [1,1,1,1,1,1,1]
=> 7
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => [2,1,1,1,1,1]
=> 7
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => [2,1,1,1,1,1]
=> 7
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => [3,1,1,1,1]
=> 7
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => [2,1,1,1,1,1]
=> 7
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => [3,1,1,1,1]
=> 7
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => [4,1,1,1]
=> 7
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => [3,1,1,1,1]
=> 7
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => [3,1,1,1,1]
=> 7
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => [4,1,1,1]
=> 7
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => [5,1,1]
=> 7
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => [4,1,1,1]
=> 7
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => [5,1,1]
=> 7
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => [6,1]
=> 7
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => [7]
=> 7
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> 8
[7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => [2,1,1,1,1,1,1]
=> 8
[6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => [2,1,1,1,1,1,1]
=> 8
[6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> [8,7,1,2,3,4,5,6] => [3,1,1,1,1,1]
=> 8
[5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [6,7,8,1,2,3,4,5] => [2,1,1,1,1,1,1]
=> 8
[5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> [8,6,7,1,2,3,4,5] => [3,1,1,1,1,1]
=> 8
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: St000293
(load all 22 compositions to match this statistic)
(load all 22 compositions to match this statistic)
Mp00095: Integer partitions —to binary word⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 91% ●values known / values provided: 99%●distinct values known / distinct values provided: 91%
St000293: Binary words ⟶ ℤResult quality: 91% ●values known / values provided: 99%●distinct values known / distinct values provided: 91%
Values
[1]
=> 10 => 1
[2]
=> 100 => 2
[1,1]
=> 110 => 2
[3]
=> 1000 => 3
[2,1]
=> 1010 => 3
[1,1,1]
=> 1110 => 3
[4]
=> 10000 => 4
[3,1]
=> 10010 => 4
[2,2]
=> 1100 => 4
[2,1,1]
=> 10110 => 4
[1,1,1,1]
=> 11110 => 4
[5]
=> 100000 => 5
[4,1]
=> 100010 => 5
[3,2]
=> 10100 => 5
[3,1,1]
=> 100110 => 5
[2,2,1]
=> 11010 => 5
[2,1,1,1]
=> 101110 => 5
[1,1,1,1,1]
=> 111110 => 5
[6]
=> 1000000 => 6
[5,1]
=> 1000010 => 6
[4,2]
=> 100100 => 6
[4,1,1]
=> 1000110 => 6
[3,3]
=> 11000 => 6
[3,2,1]
=> 101010 => 6
[3,1,1,1]
=> 1001110 => 6
[2,2,2]
=> 11100 => 6
[2,2,1,1]
=> 110110 => 6
[2,1,1,1,1]
=> 1011110 => 6
[1,1,1,1,1,1]
=> 1111110 => 6
[7]
=> 10000000 => 7
[6,1]
=> 10000010 => 7
[5,2]
=> 1000100 => 7
[5,1,1]
=> 10000110 => 7
[4,3]
=> 101000 => 7
[4,2,1]
=> 1001010 => 7
[4,1,1,1]
=> 10001110 => 7
[3,3,1]
=> 110010 => 7
[3,2,2]
=> 101100 => 7
[3,2,1,1]
=> 1010110 => 7
[3,1,1,1,1]
=> 10011110 => 7
[2,2,2,1]
=> 111010 => 7
[2,2,1,1,1]
=> 1101110 => 7
[2,1,1,1,1,1]
=> 10111110 => 7
[1,1,1,1,1,1,1]
=> 11111110 => 7
[8]
=> 100000000 => 8
[7,1]
=> 100000010 => 8
[6,2]
=> 10000100 => 8
[6,1,1]
=> 100000110 => 8
[5,3]
=> 1001000 => 8
[5,2,1]
=> 10001010 => 8
[]
=> => ? = 0
Description
The number of inversions of a binary word.
Matching statistic: St000460
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000460: Integer partitions ⟶ ℤResult quality: 91% ●values known / values provided: 99%●distinct values known / distinct values provided: 91%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000460: Integer partitions ⟶ ℤResult quality: 91% ●values known / values provided: 99%●distinct values known / distinct values provided: 91%
Values
[1]
=> [[1]]
=> [1] => [1]
=> 1
[2]
=> [[1,2]]
=> [1,2] => [1,1]
=> 2
[1,1]
=> [[1],[2]]
=> [2,1] => [2]
=> 2
[3]
=> [[1,2,3]]
=> [1,2,3] => [1,1,1]
=> 3
[2,1]
=> [[1,2],[3]]
=> [3,1,2] => [2,1]
=> 3
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => [3]
=> 3
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => [1,1,1,1]
=> 4
[3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => [2,1,1]
=> 4
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [2,1,1]
=> 4
[2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => [3,1]
=> 4
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [4]
=> 4
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => [2,1,1,1]
=> 5
[3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => [2,1,1,1]
=> 5
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [3,1,1]
=> 5
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [3,1,1]
=> 5
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [4,1]
=> 5
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5]
=> 5
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,1,1,1,1,1]
=> 6
[5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [2,1,1,1,1]
=> 6
[4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => [2,1,1,1,1]
=> 6
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => [3,1,1,1]
=> 6
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [2,1,1,1,1]
=> 6
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => [3,1,1,1]
=> 6
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => [4,1,1]
=> 6
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [3,1,1,1]
=> 6
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => [4,1,1]
=> 6
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => [5,1]
=> 6
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [6]
=> 6
[7]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => [1,1,1,1,1,1,1]
=> 7
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => [2,1,1,1,1,1]
=> 7
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => [2,1,1,1,1,1]
=> 7
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => [3,1,1,1,1]
=> 7
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => [2,1,1,1,1,1]
=> 7
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => [3,1,1,1,1]
=> 7
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => [4,1,1,1]
=> 7
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => [3,1,1,1,1]
=> 7
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => [3,1,1,1,1]
=> 7
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => [4,1,1,1]
=> 7
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => [5,1,1]
=> 7
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => [4,1,1,1]
=> 7
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => [5,1,1]
=> 7
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => [6,1]
=> 7
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => [7]
=> 7
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> 8
[7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => [2,1,1,1,1,1,1]
=> 8
[6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => [2,1,1,1,1,1,1]
=> 8
[6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> [8,7,1,2,3,4,5,6] => [3,1,1,1,1,1]
=> 8
[5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [6,7,8,1,2,3,4,5] => [2,1,1,1,1,1,1]
=> 8
[5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> [8,6,7,1,2,3,4,5] => [3,1,1,1,1,1]
=> 8
[]
=> []
=> [] => []
=> ? = 0
Description
The hook length of the last cell along the main diagonal of an integer partition.
Matching statistic: St000734
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00153: Standard tableaux —inverse promotion⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 91% ●values known / values provided: 99%●distinct values known / distinct values provided: 91%
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00153: Standard tableaux —inverse promotion⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 91% ●values known / values provided: 99%●distinct values known / distinct values provided: 91%
Values
[1]
=> [[1]]
=> [[1]]
=> [[1]]
=> 1
[2]
=> [[1,2]]
=> [[1,2]]
=> [[1,2]]
=> 2
[1,1]
=> [[1],[2]]
=> [[1,2]]
=> [[1,2]]
=> 2
[3]
=> [[1,2,3]]
=> [[1,2,3]]
=> [[1,2,3]]
=> 3
[2,1]
=> [[1,2],[3]]
=> [[1,2,3]]
=> [[1,2,3]]
=> 3
[1,1,1]
=> [[1],[2],[3]]
=> [[1,2],[3]]
=> [[1,3],[2]]
=> 3
[4]
=> [[1,2,3,4]]
=> [[1,2,3,4]]
=> [[1,2,3,4]]
=> 4
[3,1]
=> [[1,2,3],[4]]
=> [[1,2,3,4]]
=> [[1,2,3,4]]
=> 4
[2,2]
=> [[1,2],[3,4]]
=> [[1,2,3,4]]
=> [[1,2,3,4]]
=> 4
[2,1,1]
=> [[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> [[1,2,4],[3]]
=> 4
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2],[3],[4]]
=> [[1,4],[2],[3]]
=> 4
[5]
=> [[1,2,3,4,5]]
=> [[1,2,3,4,5]]
=> [[1,2,3,4,5]]
=> 5
[4,1]
=> [[1,2,3,4],[5]]
=> [[1,2,3,4,5]]
=> [[1,2,3,4,5]]
=> 5
[3,2]
=> [[1,2,3],[4,5]]
=> [[1,2,3,4,5]]
=> [[1,2,3,4,5]]
=> 5
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> [[1,2,3,5],[4]]
=> 5
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,2,3,4],[5]]
=> [[1,2,3,5],[4]]
=> 5
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,2,3],[4],[5]]
=> [[1,2,5],[3],[4]]
=> 5
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,2],[3],[4],[5]]
=> [[1,5],[2],[3],[4]]
=> 5
[6]
=> [[1,2,3,4,5,6]]
=> [[1,2,3,4,5,6]]
=> [[1,2,3,4,5,6]]
=> 6
[5,1]
=> [[1,2,3,4,5],[6]]
=> [[1,2,3,4,5,6]]
=> [[1,2,3,4,5,6]]
=> 6
[4,2]
=> [[1,2,3,4],[5,6]]
=> [[1,2,3,4,5,6]]
=> [[1,2,3,4,5,6]]
=> 6
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [[1,2,3,4,5],[6]]
=> [[1,2,3,4,6],[5]]
=> 6
[3,3]
=> [[1,2,3],[4,5,6]]
=> [[1,2,3,4,5,6]]
=> [[1,2,3,4,5,6]]
=> 6
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [[1,2,3,4,5],[6]]
=> [[1,2,3,4,6],[5]]
=> 6
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [[1,2,3,4],[5],[6]]
=> [[1,2,3,6],[4],[5]]
=> 6
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [[1,2,3,4],[5,6]]
=> [[1,2,3,6],[4,5]]
=> 6
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [[1,2,3,4],[5],[6]]
=> [[1,2,3,6],[4],[5]]
=> 6
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [[1,2,3],[4],[5],[6]]
=> [[1,2,6],[3],[4],[5]]
=> 6
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [[1,2],[3],[4],[5],[6]]
=> [[1,6],[2],[3],[4],[5]]
=> 6
[7]
=> [[1,2,3,4,5,6,7]]
=> [[1,2,3,4,5,6,7]]
=> [[1,2,3,4,5,6,7]]
=> 7
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [[1,2,3,4,5,6,7]]
=> [[1,2,3,4,5,6,7]]
=> 7
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [[1,2,3,4,5,6,7]]
=> [[1,2,3,4,5,6,7]]
=> 7
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [[1,2,3,4,5,6],[7]]
=> [[1,2,3,4,5,7],[6]]
=> 7
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [[1,2,3,4,5,6,7]]
=> [[1,2,3,4,5,6,7]]
=> 7
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [[1,2,3,4,5,6],[7]]
=> [[1,2,3,4,5,7],[6]]
=> 7
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [[1,2,3,4,5],[6],[7]]
=> [[1,2,3,4,7],[5],[6]]
=> 7
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [[1,2,3,4,5,6],[7]]
=> [[1,2,3,4,5,7],[6]]
=> 7
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [[1,2,3,4,5],[6,7]]
=> [[1,2,3,4,7],[5,6]]
=> 7
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [[1,2,3,4,5],[6],[7]]
=> [[1,2,3,4,7],[5],[6]]
=> 7
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [[1,2,3,4],[5],[6],[7]]
=> [[1,2,3,7],[4],[5],[6]]
=> 7
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [[1,2,3,4],[5,6],[7]]
=> [[1,2,3,7],[4,5],[6]]
=> 7
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [[1,2,3,4],[5],[6],[7]]
=> [[1,2,3,7],[4],[5],[6]]
=> 7
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [[1,2,3],[4],[5],[6],[7]]
=> [[1,2,7],[3],[4],[5],[6]]
=> 7
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [[1,7],[2],[3],[4],[5],[6]]
=> 7
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [[1,2,3,4,5,6,7,8]]
=> [[1,2,3,4,5,6,7,8]]
=> 8
[7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [[1,2,3,4,5,6,7,8]]
=> [[1,2,3,4,5,6,7,8]]
=> 8
[6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [[1,2,3,4,5,6,7,8]]
=> [[1,2,3,4,5,6,7,8]]
=> 8
[6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> [[1,2,3,4,5,6,7],[8]]
=> [[1,2,3,4,5,6,8],[7]]
=> 8
[5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [[1,2,3,4,5,6,7,8]]
=> [[1,2,3,4,5,6,7,8]]
=> 8
[5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> [[1,2,3,4,5,6,7],[8]]
=> [[1,2,3,4,5,6,8],[7]]
=> 8
[]
=> []
=> []
=> []
=> ? = 0
Description
The last entry in the first row of a standard tableau.
Matching statistic: St000870
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000870: Integer partitions ⟶ ℤResult quality: 91% ●values known / values provided: 99%●distinct values known / distinct values provided: 91%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000870: Integer partitions ⟶ ℤResult quality: 91% ●values known / values provided: 99%●distinct values known / distinct values provided: 91%
Values
[1]
=> [[1]]
=> [1] => [1]
=> 1
[2]
=> [[1,2]]
=> [1,2] => [1,1]
=> 2
[1,1]
=> [[1],[2]]
=> [2,1] => [2]
=> 2
[3]
=> [[1,2,3]]
=> [1,2,3] => [1,1,1]
=> 3
[2,1]
=> [[1,2],[3]]
=> [3,1,2] => [2,1]
=> 3
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => [3]
=> 3
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => [1,1,1,1]
=> 4
[3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => [2,1,1]
=> 4
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [2,1,1]
=> 4
[2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => [3,1]
=> 4
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [4]
=> 4
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => [2,1,1,1]
=> 5
[3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => [2,1,1,1]
=> 5
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [3,1,1]
=> 5
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [3,1,1]
=> 5
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [4,1]
=> 5
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5]
=> 5
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,1,1,1,1,1]
=> 6
[5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [2,1,1,1,1]
=> 6
[4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => [2,1,1,1,1]
=> 6
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => [3,1,1,1]
=> 6
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [2,1,1,1,1]
=> 6
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => [3,1,1,1]
=> 6
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => [4,1,1]
=> 6
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [3,1,1,1]
=> 6
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => [4,1,1]
=> 6
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => [5,1]
=> 6
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [6]
=> 6
[7]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => [1,1,1,1,1,1,1]
=> 7
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => [2,1,1,1,1,1]
=> 7
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => [2,1,1,1,1,1]
=> 7
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => [3,1,1,1,1]
=> 7
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => [2,1,1,1,1,1]
=> 7
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => [3,1,1,1,1]
=> 7
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => [4,1,1,1]
=> 7
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => [3,1,1,1,1]
=> 7
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => [3,1,1,1,1]
=> 7
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => [4,1,1,1]
=> 7
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => [5,1,1]
=> 7
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => [4,1,1,1]
=> 7
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => [5,1,1]
=> 7
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => [6,1]
=> 7
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => [7]
=> 7
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> 8
[7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => [2,1,1,1,1,1,1]
=> 8
[6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => [2,1,1,1,1,1,1]
=> 8
[6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> [8,7,1,2,3,4,5,6] => [3,1,1,1,1,1]
=> 8
[5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [6,7,8,1,2,3,4,5] => [2,1,1,1,1,1,1]
=> 8
[5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> [8,6,7,1,2,3,4,5] => [3,1,1,1,1,1]
=> 8
[]
=> []
=> [] => []
=> ? = 0
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: St000294
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St000294: Binary words ⟶ ℤResult quality: 82% ●values known / values provided: 99%●distinct values known / distinct values provided: 82%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St000294: Binary words ⟶ ℤResult quality: 82% ●values known / values provided: 99%●distinct values known / distinct values provided: 82%
Values
[1]
=> [[1]]
=> [1] => => ? = 1
[2]
=> [[1,2]]
=> [1,2] => 1 => 2
[1,1]
=> [[1],[2]]
=> [2,1] => 0 => 2
[3]
=> [[1,2,3]]
=> [1,2,3] => 11 => 3
[2,1]
=> [[1,2],[3]]
=> [3,1,2] => 00 => 3
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 00 => 3
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 111 => 4
[3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 000 => 4
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 000 => 4
[2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 000 => 4
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 000 => 4
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 1111 => 5
[4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 0000 => 5
[3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 0000 => 5
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 0000 => 5
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 0000 => 5
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 0000 => 5
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 0000 => 5
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => 11111 => 6
[5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => 00000 => 6
[4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => 00000 => 6
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 00000 => 6
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => 00000 => 6
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 00000 => 6
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => 00000 => 6
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 00000 => 6
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => 00000 => 6
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => 00000 => 6
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => 00000 => 6
[7]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => 111111 => 7
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => 000000 => 7
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => 000000 => 7
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => 000000 => 7
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => 000000 => 7
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => 000000 => 7
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => 000000 => 7
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => 000000 => 7
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => 000000 => 7
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => 000000 => 7
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => 000000 => 7
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => 000000 => 7
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => 000000 => 7
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => 000000 => 7
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => 000000 => 7
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => 1111111 => 8
[7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => 0000000 => 8
[6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => 0000000 => 8
[6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> [8,7,1,2,3,4,5,6] => 0000000 => 8
[5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [6,7,8,1,2,3,4,5] => 0000000 => 8
[5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> [8,6,7,1,2,3,4,5] => 0000000 => 8
[5,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8]]
=> [8,7,6,1,2,3,4,5] => 0000000 => 8
[]
=> []
=> [] => => ? = 0
Description
The number of distinct factors of a binary word.
This is also known as the subword complexity of a binary word, see [1].
The following 69 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000518The number of distinct subsequences in a binary word. St000296The length of the symmetric border of a binary word. St000393The number of strictly increasing runs in a binary word. St000627The exponent of a binary word. St000876The number of factors in the Catalan decomposition of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000922The minimal number such that all substrings of this length are unique. St000982The length of the longest constant subword. St001267The length of the Lyndon factorization of the binary word. St001415The length of the longest palindromic prefix of a binary word. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001437The flex of a binary word. St001884The number of borders of a binary word. St000295The length of the border of a binary word. St000519The largest length of a factor maximising the subword complexity. St000290The major index of a binary word. St001759The Rajchgot index of a permutation. St000395The sum of the heights of the peaks of a Dyck path. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St001397Number of pairs of incomparable elements in a finite poset. St000004The major index of a permutation. St000057The Shynar inversion number of a standard tableau. St000067The inversion number of the alternating sign matrix. St000332The positive inversions of an alternating sign matrix. St000795The mad of a permutation. St001428The number of B-inversions of a signed permutation. St000229Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001523The degree of symmetry of a Dyck path. St001641The number of ascent tops in the flattened set partition such that all smaller elements appear before. St000719The number of alignments in a perfect matching. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001342The number of vertices in the center of a graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St001622The number of join-irreducible elements of a lattice. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001430The number of positive entries in a signed permutation. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St000029The depth of a permutation. St000224The sorting index of a permutation. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001726The number of visible inversions of a permutation. St000197The number of entries equal to positive one in the alternating sign matrix. St001717The largest size of an interval in a poset. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001245The cyclic maximal difference between two consecutive entries of a permutation. St001958The degree of the polynomial interpolating the values of a permutation. St000189The number of elements in the poset. St000656The number of cuts of a poset. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St000060The greater neighbor of the maximum. St001246The maximal difference between two consecutive entries of a permutation. St001772The number of occurrences of the signed pattern 12 in a signed permutation. 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. St001925The minimal number of zeros in a row of an alternating sign matrix. St000890The number of nonzero entries in an alternating sign matrix. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001401The number of distinct entries in a semistandard tableau. St001875The number of simple modules with projective dimension at most 1. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000136The dinv of a parking function. St000194The number of primary dinversion pairs of a labelled dyck path corresponding to a parking function.
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