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Your data matches 23 different statistics following compositions of up to 3 maps.
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Matching statistic: St000229
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(load all 2 compositions to match this statistic)
St000229: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> 1
{{1,2}}
=> 2
{{1},{2}}
=> 2
{{1,2,3}}
=> 3
{{1,2},{3}}
=> 3
{{1,3},{2}}
=> 4
{{1},{2,3}}
=> 3
{{1},{2},{3}}
=> 3
{{1,2,3,4}}
=> 4
{{1,2,3},{4}}
=> 4
{{1,2,4},{3}}
=> 5
{{1,2},{3,4}}
=> 4
{{1,2},{3},{4}}
=> 4
{{1,3,4},{2}}
=> 5
{{1,3},{2,4}}
=> 6
{{1,3},{2},{4}}
=> 5
{{1,4},{2,3}}
=> 6
{{1},{2,3,4}}
=> 4
{{1},{2,3},{4}}
=> 4
{{1,4},{2},{3}}
=> 6
{{1},{2,4},{3}}
=> 5
{{1},{2},{3,4}}
=> 4
{{1},{2},{3},{4}}
=> 4
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: St001034
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(load all 4 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001034: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ 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}}
=> [2,1] => [1,1,0,0]
=> 2
{{1},{2}}
=> [1,2] => [1,0,1,0]
=> 2
{{1,2,3}}
=> [2,3,1] => [1,1,0,1,0,0]
=> 3
{{1,2},{3}}
=> [2,1,3] => [1,1,0,0,1,0]
=> 3
{{1,3},{2}}
=> [3,2,1] => [1,1,1,0,0,0]
=> 4
{{1},{2,3}}
=> [1,3,2] => [1,0,1,1,0,0]
=> 3
{{1},{2},{3}}
=> [1,2,3] => [1,0,1,0,1,0]
=> 3
{{1,2,3,4}}
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 4
{{1,2,3},{4}}
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 4
{{1,2,4},{3}}
=> [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 5
{{1,2},{3,4}}
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 4
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 4
{{1,3,4},{2}}
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 5
{{1,3},{2,4}}
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 6
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 5
{{1,4},{2,3}}
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 6
{{1},{2,3,4}}
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 4
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 4
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 6
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 5
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 4
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4
Description
The area of the parallelogram polyomino associated with the Dyck path.
The (bivariate) generating function is given in [1].
Matching statistic: St000395
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(load all 4 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
St000395: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
St000395: 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}}
=> [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 2
{{1},{2}}
=> [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 2
{{1,2,3}}
=> [2,3,1] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 3
{{1,2},{3}}
=> [2,1,3] => [[.,.],[.,.]]
=> [1,1,1,0,0,0]
=> 3
{{1,3},{2}}
=> [3,2,1] => [[[.,.],.],.]
=> [1,1,0,1,0,0]
=> 4
{{1},{2,3}}
=> [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 3
{{1},{2},{3}}
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 3
{{1,2,3,4}}
=> [2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,0,0,1,0,1,0]
=> 4
{{1,2,3},{4}}
=> [2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,1,0,1,0,0,0]
=> 6
{{1,2,4},{3}}
=> [2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,0,1,1,0,0]
=> 4
{{1,2},{3,4}}
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,1,0,0,1,0,0]
=> 5
{{1,2},{3},{4}}
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0]
=> 4
{{1,3,4},{2}}
=> [3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,1,0,1,1,0,0,0]
=> 5
{{1,3},{2,4}}
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,1,0,1,0,0,0]
=> 6
{{1,3},{2},{4}}
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,1,0,0,0,0]
=> 4
{{1,4},{2,3}}
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,0,1,0,1,0,0]
=> 6
{{1},{2,3,4}}
=> [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,0,1,0]
=> 4
{{1},{2,3},{4}}
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 4
{{1,4},{2},{3}}
=> [4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,0,1,1,0,0,0]
=> 5
{{1},{2,4},{3}}
=> [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,0]
=> 5
{{1},{2},{3,4}}
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 4
{{1},{2},{3},{4}}
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 4
Description
The sum of the heights of the peaks of a Dyck path.
Matching statistic: St001018
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(load all 3 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
St001018: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
St001018: 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}}
=> [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 2
{{1},{2}}
=> [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 2
{{1,2,3}}
=> [2,3,1] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 3
{{1,2},{3}}
=> [2,1,3] => [[.,.],[.,.]]
=> [1,1,1,0,0,0]
=> 3
{{1,3},{2}}
=> [3,2,1] => [[[.,.],.],.]
=> [1,1,0,1,0,0]
=> 4
{{1},{2,3}}
=> [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 3
{{1},{2},{3}}
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 3
{{1,2,3,4}}
=> [2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,0,0,1,0,1,0]
=> 4
{{1,2,3},{4}}
=> [2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,1,0,1,0,0,0]
=> 6
{{1,2,4},{3}}
=> [2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,0,1,1,0,0]
=> 4
{{1,2},{3,4}}
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,1,0,0,1,0,0]
=> 5
{{1,2},{3},{4}}
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0]
=> 4
{{1,3,4},{2}}
=> [3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,1,0,1,1,0,0,0]
=> 5
{{1,3},{2,4}}
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,1,0,1,0,0,0]
=> 6
{{1,3},{2},{4}}
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,1,0,0,0,0]
=> 4
{{1,4},{2,3}}
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,0,1,0,1,0,0]
=> 5
{{1},{2,3,4}}
=> [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,0,1,0]
=> 4
{{1},{2,3},{4}}
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 4
{{1,4},{2},{3}}
=> [4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,0,1,1,0,0,0]
=> 5
{{1},{2,4},{3}}
=> [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,0]
=> 6
{{1},{2},{3,4}}
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 4
{{1},{2},{3},{4}}
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 4
Description
Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001348
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Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001348: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001348: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => [1,0]
=> 1
{{1,2}}
=> [2] => [1,1] => [1,0,1,0]
=> 2
{{1},{2}}
=> [1,1] => [2] => [1,1,0,0]
=> 2
{{1,2,3}}
=> [3] => [1,1,1] => [1,0,1,0,1,0]
=> 3
{{1,2},{3}}
=> [2,1] => [2,1] => [1,1,0,0,1,0]
=> 3
{{1,3},{2}}
=> [2,1] => [2,1] => [1,1,0,0,1,0]
=> 3
{{1},{2,3}}
=> [1,2] => [1,2] => [1,0,1,1,0,0]
=> 4
{{1},{2},{3}}
=> [1,1,1] => [3] => [1,1,1,0,0,0]
=> 3
{{1,2,3,4}}
=> [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
{{1,2,3},{4}}
=> [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 4
{{1,2,4},{3}}
=> [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 4
{{1,2},{3,4}}
=> [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 6
{{1,2},{3},{4}}
=> [2,1,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 4
{{1,3,4},{2}}
=> [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 4
{{1,3},{2,4}}
=> [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 6
{{1,3},{2},{4}}
=> [2,1,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 4
{{1,4},{2,3}}
=> [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 6
{{1},{2,3,4}}
=> [1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 5
{{1},{2,3},{4}}
=> [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 5
{{1,4},{2},{3}}
=> [2,1,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 4
{{1},{2,4},{3}}
=> [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 5
{{1},{2},{3,4}}
=> [1,1,2] => [1,3] => [1,0,1,1,1,0,0,0]
=> 5
{{1},{2},{3},{4}}
=> [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 4
Description
The bounce of the parallelogram polyomino associated with the Dyck path.
A bijection due to Delest and Viennot [1] associates a Dyck path with a parallelogram polyomino. The bounce statistic is defined in [2].
Matching statistic: St001894
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00167: Signed permutations —inverse Kreweras complement⟶ Signed permutations
St001894: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00167: Signed permutations —inverse Kreweras complement⟶ Signed permutations
St001894: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => [-1] => 1
{{1,2}}
=> [2,1] => [2,1] => [1,-2] => 2
{{1},{2}}
=> [1,2] => [1,2] => [2,-1] => 2
{{1,2,3}}
=> [2,3,1] => [2,3,1] => [1,2,-3] => 3
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => [1,3,-2] => 3
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => [2,1,-3] => 4
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => [3,2,-1] => 3
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [2,3,-1] => 3
{{1,2,3,4}}
=> [2,3,4,1] => [2,3,4,1] => [1,2,3,-4] => 4
{{1,2,3},{4}}
=> [2,3,1,4] => [2,3,1,4] => [1,2,4,-3] => 4
{{1,2,4},{3}}
=> [2,4,3,1] => [2,4,3,1] => [1,3,2,-4] => 5
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => [1,4,3,-2] => 4
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => [1,3,4,-2] => 4
{{1,3,4},{2}}
=> [3,2,4,1] => [3,2,4,1] => [2,1,3,-4] => 5
{{1,3},{2,4}}
=> [3,4,1,2] => [3,4,1,2] => [4,1,2,-3] => 6
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => [2,1,4,-3] => 5
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => [3,2,1,-4] => 6
{{1},{2,3,4}}
=> [1,3,4,2] => [1,3,4,2] => [4,2,3,-1] => 4
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => [3,2,4,-1] => 4
{{1,4},{2},{3}}
=> [4,2,3,1] => [4,2,3,1] => [2,3,1,-4] => 6
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => [4,3,2,-1] => 5
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => [2,4,3,-1] => 4
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [2,3,4,-1] => 4
Description
The depth of a signed permutation.
The depth of a positive root is its rank in the root poset. The depth of an element of a Coxeter group is the minimal sum of depths for any representation as product of reflections.
Matching statistic: St000018
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000018: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000018: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => [1] => 0 = 1 - 1
{{1,2}}
=> [2,1] => [1,2] => [2,1] => 1 = 2 - 1
{{1},{2}}
=> [1,2] => [1,2] => [2,1] => 1 = 2 - 1
{{1,2,3}}
=> [2,3,1] => [1,2,3] => [2,3,1] => 2 = 3 - 1
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => [2,3,1] => 2 = 3 - 1
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => [3,2,1] => 3 = 4 - 1
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => [2,3,1] => 2 = 3 - 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [2,3,1] => 2 = 3 - 1
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => [2,3,4,1] => 3 = 4 - 1
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => [2,3,4,1] => 3 = 4 - 1
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => [2,4,3,1] => 4 = 5 - 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => [2,3,4,1] => 3 = 4 - 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => [2,3,4,1] => 3 = 4 - 1
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => [4,2,3,1] => 5 = 6 - 1
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => [3,2,4,1] => 4 = 5 - 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => [3,2,4,1] => 4 = 5 - 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => [3,4,2,1] => 5 = 6 - 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => [2,3,4,1] => 3 = 4 - 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => [2,3,4,1] => 3 = 4 - 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => [3,4,2,1] => 5 = 6 - 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => [2,4,3,1] => 4 = 5 - 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => [2,3,4,1] => 3 = 4 - 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 3 = 4 - 1
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: St000070
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000070: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000070: Posets ⟶ 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}}
=> [2,1] => [1,2] => ([(0,1)],2)
=> 3 = 2 + 1
{{1},{2}}
=> [1,2] => [1,2] => ([(0,1)],2)
=> 3 = 2 + 1
{{1,2,3}}
=> [2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 4 = 3 + 1
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 4 = 3 + 1
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => ([(0,1),(0,2)],3)
=> 5 = 4 + 1
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> 4 = 3 + 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 4 = 3 + 1
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 5 = 4 + 1
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 5 = 4 + 1
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 6 = 5 + 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 5 = 4 + 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 5 = 4 + 1
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 7 = 6 + 1
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 6 = 5 + 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 6 = 5 + 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 7 = 6 + 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 5 = 4 + 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 5 = 4 + 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 7 = 6 + 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 6 = 5 + 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 5 = 4 + 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 5 = 4 + 1
Description
The number of antichains in a poset.
An antichain in a poset $P$ is a subset of elements of $P$ which are pairwise incomparable.
An order ideal is a subset $I$ of $P$ such that $a\in I$ and $b \leq_P a$ implies $b \in I$. Since there is a one-to-one correspondence between antichains and order ideals, this statistic is also the number of order ideals in a poset.
Matching statistic: St000494
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000494: Permutations ⟶ ℤResult quality: 83% ●values known / values provided: 96%●distinct values known / distinct values provided: 83%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000494: Permutations ⟶ ℤResult quality: 83% ●values known / values provided: 96%●distinct values known / distinct values provided: 83%
Values
{{1}}
=> [1] => [1] => [1] => ? = 1 - 1
{{1,2}}
=> [2,1] => [1,2] => [2,1] => 1 = 2 - 1
{{1},{2}}
=> [1,2] => [1,2] => [2,1] => 1 = 2 - 1
{{1,2,3}}
=> [2,3,1] => [1,2,3] => [2,3,1] => 2 = 3 - 1
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => [2,3,1] => 2 = 3 - 1
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => [3,2,1] => 3 = 4 - 1
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => [2,3,1] => 2 = 3 - 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [2,3,1] => 2 = 3 - 1
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => [2,3,4,1] => 3 = 4 - 1
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => [2,3,4,1] => 3 = 4 - 1
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => [2,4,3,1] => 4 = 5 - 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => [2,3,4,1] => 3 = 4 - 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => [2,3,4,1] => 3 = 4 - 1
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => [4,2,3,1] => 5 = 6 - 1
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => [3,2,4,1] => 4 = 5 - 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => [3,2,4,1] => 4 = 5 - 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => [3,4,2,1] => 5 = 6 - 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => [2,3,4,1] => 3 = 4 - 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => [2,3,4,1] => 3 = 4 - 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => [3,4,2,1] => 5 = 6 - 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => [2,4,3,1] => 4 = 5 - 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => [2,3,4,1] => 3 = 4 - 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 3 = 4 - 1
Description
The number of inversions of distance at most 3 of a permutation.
An inversion of a permutation $\pi$ is a pair $i < j$ such that $\sigma(i) > \sigma(j)$. Let $j-i$ be the distance of such an inversion. Then inversions of distance at most 1 are then exactly the descents of $\pi$, see [[St000021]]. This statistic counts the number of inversions of distance at most 3.
Matching statistic: St001875
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 50% ●values known / values provided: 57%●distinct values known / distinct values provided: 50%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 50% ●values known / values provided: 57%●distinct values known / distinct values provided: 50%
Values
{{1}}
=> [1] => ([],1)
=> ([(0,1)],2)
=> ? = 1 + 1
{{1,2}}
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
{{1},{2}}
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
{{1,2,3}}
=> [2,3,1] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4 = 3 + 1
{{1,2},{3}}
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 4 = 3 + 1
{{1,3},{2}}
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 4 + 1
{{1},{2,3}}
=> [1,3,2] => ([(0,1),(0,2)],3)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 4 = 3 + 1
{{1},{2},{3}}
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
{{1,2,3,4}}
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {4,5,5,5,5,6,6,6} + 1
{{1,2,3},{4}}
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 5 = 4 + 1
{{1,2,4},{3}}
=> [2,4,3,1] => ([(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ? ∊ {4,5,5,5,5,6,6,6} + 1
{{1,2},{3,4}}
=> [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 5 = 4 + 1
{{1,2},{3},{4}}
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 5 = 4 + 1
{{1,3,4},{2}}
=> [3,2,4,1] => ([(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> ? ∊ {4,5,5,5,5,6,6,6} + 1
{{1,3},{2,4}}
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {4,5,5,5,5,6,6,6} + 1
{{1,3},{2},{4}}
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> ? ∊ {4,5,5,5,5,6,6,6} + 1
{{1,4},{2,3}}
=> [4,3,2,1] => ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? ∊ {4,5,5,5,5,6,6,6} + 1
{{1},{2,3,4}}
=> [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 5 = 4 + 1
{{1},{2,3},{4}}
=> [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 5 = 4 + 1
{{1,4},{2},{3}}
=> [4,2,3,1] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? ∊ {4,5,5,5,5,6,6,6} + 1
{{1},{2,4},{3}}
=> [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
=> ? ∊ {4,5,5,5,5,6,6,6} + 1
{{1},{2},{3,4}}
=> [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 5 = 4 + 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
Description
The number of simple modules with projective dimension at most 1.
The following 13 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001645The pebbling number of a connected graph. St001926Sparre Andersen's position of the maximum of a signed permutation. St000080The rank of the poset. St000307The number of rowmotion orbits of a poset. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St000739The first entry in the last row of a semistandard tableau. St001401The number of distinct entries in a semistandard tableau. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St000101The cocharge of a semistandard tableau. St000454The largest eigenvalue of a graph if it is integral.
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