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Your data matches 11 different statistics following compositions of up to 3 maps.
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Matching statistic: St000248
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
St000248: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> 2
{{1},{2}}
=> 0
{{1,2,3}}
=> 3
{{1,2},{3}}
=> 1
{{1,3},{2}}
=> 1
{{1},{2,3}}
=> 1
{{1},{2},{3}}
=> 0
{{1,2,3,4}}
=> 4
{{1,2,3},{4}}
=> 2
{{1,2,4},{3}}
=> 2
{{1,2},{3,4}}
=> 2
{{1,2},{3},{4}}
=> 1
{{1,3,4},{2}}
=> 2
{{1,3},{2,4}}
=> 0
{{1,3},{2},{4}}
=> 0
{{1,4},{2,3}}
=> 2
{{1},{2,3,4}}
=> 2
{{1},{2,3},{4}}
=> 1
{{1,4},{2},{3}}
=> 1
{{1},{2,4},{3}}
=> 0
{{1},{2},{3,4}}
=> 1
{{1},{2},{3},{4}}
=> 0
{{1,2,3,4,5}}
=> 5
{{1,2,3,4},{5}}
=> 3
{{1,2,3,5},{4}}
=> 3
{{1,2,3},{4,5}}
=> 3
{{1,2,3},{4},{5}}
=> 2
{{1,2,4,5},{3}}
=> 3
{{1,2,4},{3,5}}
=> 1
{{1,2,4},{3},{5}}
=> 1
{{1,2,5},{3,4}}
=> 3
{{1,2},{3,4,5}}
=> 3
{{1,2},{3,4},{5}}
=> 2
{{1,2,5},{3},{4}}
=> 2
{{1,2},{3,5},{4}}
=> 1
{{1,2},{3},{4,5}}
=> 2
{{1,2},{3},{4},{5}}
=> 1
{{1,3,4,5},{2}}
=> 3
{{1,3,4},{2,5}}
=> 1
{{1,3,4},{2},{5}}
=> 1
{{1,3,5},{2,4}}
=> 1
{{1,3},{2,4,5}}
=> 1
{{1,3},{2,4},{5}}
=> 0
{{1,3,5},{2},{4}}
=> 1
{{1,3},{2,5},{4}}
=> 0
{{1,3},{2},{4,5}}
=> 1
{{1,3},{2},{4},{5}}
=> 0
{{1,4,5},{2,3}}
=> 3
{{1,4},{2,3,5}}
=> 1
{{1,4},{2,3},{5}}
=> 1
Description
The number of anti-singletons of a set partition.
An anti-singleton of a set partition $S$ is an index $i$ such that $i$ and $i+1$ (considered cyclically) are both in the same block of $S$.
For noncrossing set partitions, this is also the number of singletons of the image of $S$ under the Kreweras complement.
Matching statistic: St000241
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
St000241: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000241: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => 2
{{1},{2}}
=> [1,2] => 0
{{1,2,3}}
=> [2,3,1] => 3
{{1,2},{3}}
=> [2,1,3] => 1
{{1,3},{2}}
=> [3,2,1] => 1
{{1},{2,3}}
=> [1,3,2] => 1
{{1},{2},{3}}
=> [1,2,3] => 0
{{1,2,3,4}}
=> [2,3,4,1] => 4
{{1,2,3},{4}}
=> [2,3,1,4] => 2
{{1,2,4},{3}}
=> [2,4,3,1] => 2
{{1,2},{3,4}}
=> [2,1,4,3] => 2
{{1,2},{3},{4}}
=> [2,1,3,4] => 1
{{1,3,4},{2}}
=> [3,2,4,1] => 2
{{1,3},{2,4}}
=> [3,4,1,2] => 0
{{1,3},{2},{4}}
=> [3,2,1,4] => 0
{{1,4},{2,3}}
=> [4,3,2,1] => 2
{{1},{2,3,4}}
=> [1,3,4,2] => 2
{{1},{2,3},{4}}
=> [1,3,2,4] => 1
{{1,4},{2},{3}}
=> [4,2,3,1] => 1
{{1},{2,4},{3}}
=> [1,4,3,2] => 0
{{1},{2},{3,4}}
=> [1,2,4,3] => 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => 5
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => 3
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => 3
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => 3
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => 2
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => 3
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => 3
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => 3
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => 2
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => 2
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => 2
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => 3
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => 0
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => 0
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => 0
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => 3
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => 1
Description
The number of cyclical small excedances.
A cyclical small excedance is an index $i$ such that $\pi_i = i+1$ considered cyclically.
Matching statistic: St000247
(load all 39 compositions to match this statistic)
(load all 39 compositions to match this statistic)
Mp00221: Set partitions —conjugate⟶ Set partitions
St000247: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000247: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> {{1},{2}}
=> 2
{{1},{2}}
=> {{1,2}}
=> 0
{{1,2,3}}
=> {{1},{2},{3}}
=> 3
{{1,2},{3}}
=> {{1,2},{3}}
=> 1
{{1,3},{2}}
=> {{1},{2,3}}
=> 1
{{1},{2,3}}
=> {{1,3},{2}}
=> 1
{{1},{2},{3}}
=> {{1,2,3}}
=> 0
{{1,2,3,4}}
=> {{1},{2},{3},{4}}
=> 4
{{1,2,3},{4}}
=> {{1,2},{3},{4}}
=> 2
{{1,2,4},{3}}
=> {{1},{2,3},{4}}
=> 2
{{1,2},{3,4}}
=> {{1,3},{2},{4}}
=> 2
{{1,2},{3},{4}}
=> {{1,2,3},{4}}
=> 1
{{1,3,4},{2}}
=> {{1},{2},{3,4}}
=> 2
{{1,3},{2,4}}
=> {{1,3},{2,4}}
=> 0
{{1,3},{2},{4}}
=> {{1,2},{3,4}}
=> 0
{{1,4},{2,3}}
=> {{1},{2,4},{3}}
=> 2
{{1},{2,3,4}}
=> {{1,4},{2},{3}}
=> 2
{{1},{2,3},{4}}
=> {{1,2,4},{3}}
=> 1
{{1,4},{2},{3}}
=> {{1},{2,3,4}}
=> 1
{{1},{2,4},{3}}
=> {{1,4},{2,3}}
=> 0
{{1},{2},{3,4}}
=> {{1,3,4},{2}}
=> 1
{{1},{2},{3},{4}}
=> {{1,2,3,4}}
=> 0
{{1,2,3,4,5}}
=> {{1},{2},{3},{4},{5}}
=> 5
{{1,2,3,4},{5}}
=> {{1,2},{3},{4},{5}}
=> 3
{{1,2,3,5},{4}}
=> {{1},{2,3},{4},{5}}
=> 3
{{1,2,3},{4,5}}
=> {{1,3},{2},{4},{5}}
=> 3
{{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> 2
{{1,2,4,5},{3}}
=> {{1},{2},{3,4},{5}}
=> 3
{{1,2,4},{3,5}}
=> {{1,3},{2,4},{5}}
=> 1
{{1,2,4},{3},{5}}
=> {{1,2},{3,4},{5}}
=> 1
{{1,2,5},{3,4}}
=> {{1},{2,4},{3},{5}}
=> 3
{{1,2},{3,4,5}}
=> {{1,4},{2},{3},{5}}
=> 3
{{1,2},{3,4},{5}}
=> {{1,2,4},{3},{5}}
=> 2
{{1,2,5},{3},{4}}
=> {{1},{2,3,4},{5}}
=> 2
{{1,2},{3,5},{4}}
=> {{1,4},{2,3},{5}}
=> 1
{{1,2},{3},{4,5}}
=> {{1,3,4},{2},{5}}
=> 2
{{1,2},{3},{4},{5}}
=> {{1,2,3,4},{5}}
=> 1
{{1,3,4,5},{2}}
=> {{1},{2},{3},{4,5}}
=> 3
{{1,3,4},{2,5}}
=> {{1,4},{2,5},{3}}
=> 1
{{1,3,4},{2},{5}}
=> {{1,2},{3},{4,5}}
=> 1
{{1,3,5},{2,4}}
=> {{1},{2,4},{3,5}}
=> 1
{{1,3},{2,4,5}}
=> {{1,4},{2},{3,5}}
=> 1
{{1,3},{2,4},{5}}
=> {{1,2,4},{3,5}}
=> 0
{{1,3,5},{2},{4}}
=> {{1},{2,3},{4,5}}
=> 1
{{1,3},{2,5},{4}}
=> {{1,4},{2,3,5}}
=> 0
{{1,3},{2},{4,5}}
=> {{1,3},{2},{4,5}}
=> 1
{{1,3},{2},{4},{5}}
=> {{1,2,3},{4,5}}
=> 0
{{1,4,5},{2,3}}
=> {{1},{2},{3,5},{4}}
=> 3
{{1,4},{2,3,5}}
=> {{1,3},{2,5},{4}}
=> 1
{{1,4},{2,3},{5}}
=> {{1,2},{3,5},{4}}
=> 1
Description
The number of singleton blocks of a set partition.
Matching statistic: St000022
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00221: Set partitions —conjugate⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00080: Set partitions —to permutation⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> {{1},{2}}
=> [1,2] => 2
{{1},{2}}
=> {{1,2}}
=> [2,1] => 0
{{1,2,3}}
=> {{1},{2},{3}}
=> [1,2,3] => 3
{{1,2},{3}}
=> {{1,2},{3}}
=> [2,1,3] => 1
{{1,3},{2}}
=> {{1},{2,3}}
=> [1,3,2] => 1
{{1},{2,3}}
=> {{1,3},{2}}
=> [3,2,1] => 1
{{1},{2},{3}}
=> {{1,2,3}}
=> [2,3,1] => 0
{{1,2,3,4}}
=> {{1},{2},{3},{4}}
=> [1,2,3,4] => 4
{{1,2,3},{4}}
=> {{1,2},{3},{4}}
=> [2,1,3,4] => 2
{{1,2,4},{3}}
=> {{1},{2,3},{4}}
=> [1,3,2,4] => 2
{{1,2},{3,4}}
=> {{1,3},{2},{4}}
=> [3,2,1,4] => 2
{{1,2},{3},{4}}
=> {{1,2,3},{4}}
=> [2,3,1,4] => 1
{{1,3,4},{2}}
=> {{1},{2},{3,4}}
=> [1,2,4,3] => 2
{{1,3},{2,4}}
=> {{1,3},{2,4}}
=> [3,4,1,2] => 0
{{1,3},{2},{4}}
=> {{1,2},{3,4}}
=> [2,1,4,3] => 0
{{1,4},{2,3}}
=> {{1},{2,4},{3}}
=> [1,4,3,2] => 2
{{1},{2,3,4}}
=> {{1,4},{2},{3}}
=> [4,2,3,1] => 2
{{1},{2,3},{4}}
=> {{1,2,4},{3}}
=> [2,4,3,1] => 1
{{1,4},{2},{3}}
=> {{1},{2,3,4}}
=> [1,3,4,2] => 1
{{1},{2,4},{3}}
=> {{1,4},{2,3}}
=> [4,3,2,1] => 0
{{1},{2},{3,4}}
=> {{1,3,4},{2}}
=> [3,2,4,1] => 1
{{1},{2},{3},{4}}
=> {{1,2,3,4}}
=> [2,3,4,1] => 0
{{1,2,3,4,5}}
=> {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => 5
{{1,2,3,4},{5}}
=> {{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => 3
{{1,2,3,5},{4}}
=> {{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => 3
{{1,2,3},{4,5}}
=> {{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => 3
{{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> [2,3,1,4,5] => 2
{{1,2,4,5},{3}}
=> {{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => 3
{{1,2,4},{3,5}}
=> {{1,3},{2,4},{5}}
=> [3,4,1,2,5] => 1
{{1,2,4},{3},{5}}
=> {{1,2},{3,4},{5}}
=> [2,1,4,3,5] => 1
{{1,2,5},{3,4}}
=> {{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => 3
{{1,2},{3,4,5}}
=> {{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => 3
{{1,2},{3,4},{5}}
=> {{1,2,4},{3},{5}}
=> [2,4,3,1,5] => 2
{{1,2,5},{3},{4}}
=> {{1},{2,3,4},{5}}
=> [1,3,4,2,5] => 2
{{1,2},{3,5},{4}}
=> {{1,4},{2,3},{5}}
=> [4,3,2,1,5] => 1
{{1,2},{3},{4,5}}
=> {{1,3,4},{2},{5}}
=> [3,2,4,1,5] => 2
{{1,2},{3},{4},{5}}
=> {{1,2,3,4},{5}}
=> [2,3,4,1,5] => 1
{{1,3,4,5},{2}}
=> {{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => 3
{{1,3,4},{2,5}}
=> {{1,4},{2,5},{3}}
=> [4,5,3,1,2] => 1
{{1,3,4},{2},{5}}
=> {{1,2},{3},{4,5}}
=> [2,1,3,5,4] => 1
{{1,3,5},{2,4}}
=> {{1},{2,4},{3,5}}
=> [1,4,5,2,3] => 1
{{1,3},{2,4,5}}
=> {{1,4},{2},{3,5}}
=> [4,2,5,1,3] => 1
{{1,3},{2,4},{5}}
=> {{1,2,4},{3,5}}
=> [2,4,5,1,3] => 0
{{1,3,5},{2},{4}}
=> {{1},{2,3},{4,5}}
=> [1,3,2,5,4] => 1
{{1,3},{2,5},{4}}
=> {{1,4},{2,3,5}}
=> [4,3,5,1,2] => 0
{{1,3},{2},{4,5}}
=> {{1,3},{2},{4,5}}
=> [3,2,1,5,4] => 1
{{1,3},{2},{4},{5}}
=> {{1,2,3},{4,5}}
=> [2,3,1,5,4] => 0
{{1,4,5},{2,3}}
=> {{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => 3
{{1,4},{2,3,5}}
=> {{1,3},{2,5},{4}}
=> [3,5,1,4,2] => 1
{{1,4},{2,3},{5}}
=> {{1,2},{3,5},{4}}
=> [2,1,5,4,3] => 1
Description
The number of fixed points of a permutation.
Matching statistic: St000475
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00221: Set partitions —conjugate⟶ Set partitions
Mp00079: Set partitions —shape⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00079: Set partitions —shape⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> {{1},{2}}
=> [1,1]
=> 2
{{1},{2}}
=> {{1,2}}
=> [2]
=> 0
{{1,2,3}}
=> {{1},{2},{3}}
=> [1,1,1]
=> 3
{{1,2},{3}}
=> {{1,2},{3}}
=> [2,1]
=> 1
{{1,3},{2}}
=> {{1},{2,3}}
=> [2,1]
=> 1
{{1},{2,3}}
=> {{1,3},{2}}
=> [2,1]
=> 1
{{1},{2},{3}}
=> {{1,2,3}}
=> [3]
=> 0
{{1,2,3,4}}
=> {{1},{2},{3},{4}}
=> [1,1,1,1]
=> 4
{{1,2,3},{4}}
=> {{1,2},{3},{4}}
=> [2,1,1]
=> 2
{{1,2,4},{3}}
=> {{1},{2,3},{4}}
=> [2,1,1]
=> 2
{{1,2},{3,4}}
=> {{1,3},{2},{4}}
=> [2,1,1]
=> 2
{{1,2},{3},{4}}
=> {{1,2,3},{4}}
=> [3,1]
=> 1
{{1,3,4},{2}}
=> {{1},{2},{3,4}}
=> [2,1,1]
=> 2
{{1,3},{2,4}}
=> {{1,3},{2,4}}
=> [2,2]
=> 0
{{1,3},{2},{4}}
=> {{1,2},{3,4}}
=> [2,2]
=> 0
{{1,4},{2,3}}
=> {{1},{2,4},{3}}
=> [2,1,1]
=> 2
{{1},{2,3,4}}
=> {{1,4},{2},{3}}
=> [2,1,1]
=> 2
{{1},{2,3},{4}}
=> {{1,2,4},{3}}
=> [3,1]
=> 1
{{1,4},{2},{3}}
=> {{1},{2,3,4}}
=> [3,1]
=> 1
{{1},{2,4},{3}}
=> {{1,4},{2,3}}
=> [2,2]
=> 0
{{1},{2},{3,4}}
=> {{1,3,4},{2}}
=> [3,1]
=> 1
{{1},{2},{3},{4}}
=> {{1,2,3,4}}
=> [4]
=> 0
{{1,2,3,4,5}}
=> {{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> 5
{{1,2,3,4},{5}}
=> {{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> 3
{{1,2,3,5},{4}}
=> {{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> 3
{{1,2,3},{4,5}}
=> {{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> 3
{{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> [3,1,1]
=> 2
{{1,2,4,5},{3}}
=> {{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> 3
{{1,2,4},{3,5}}
=> {{1,3},{2,4},{5}}
=> [2,2,1]
=> 1
{{1,2,4},{3},{5}}
=> {{1,2},{3,4},{5}}
=> [2,2,1]
=> 1
{{1,2,5},{3,4}}
=> {{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> 3
{{1,2},{3,4,5}}
=> {{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> 3
{{1,2},{3,4},{5}}
=> {{1,2,4},{3},{5}}
=> [3,1,1]
=> 2
{{1,2,5},{3},{4}}
=> {{1},{2,3,4},{5}}
=> [3,1,1]
=> 2
{{1,2},{3,5},{4}}
=> {{1,4},{2,3},{5}}
=> [2,2,1]
=> 1
{{1,2},{3},{4,5}}
=> {{1,3,4},{2},{5}}
=> [3,1,1]
=> 2
{{1,2},{3},{4},{5}}
=> {{1,2,3,4},{5}}
=> [4,1]
=> 1
{{1,3,4,5},{2}}
=> {{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> 3
{{1,3,4},{2,5}}
=> {{1,4},{2,5},{3}}
=> [2,2,1]
=> 1
{{1,3,4},{2},{5}}
=> {{1,2},{3},{4,5}}
=> [2,2,1]
=> 1
{{1,3,5},{2,4}}
=> {{1},{2,4},{3,5}}
=> [2,2,1]
=> 1
{{1,3},{2,4,5}}
=> {{1,4},{2},{3,5}}
=> [2,2,1]
=> 1
{{1,3},{2,4},{5}}
=> {{1,2,4},{3,5}}
=> [3,2]
=> 0
{{1,3,5},{2},{4}}
=> {{1},{2,3},{4,5}}
=> [2,2,1]
=> 1
{{1,3},{2,5},{4}}
=> {{1,4},{2,3,5}}
=> [3,2]
=> 0
{{1,3},{2},{4,5}}
=> {{1,3},{2},{4,5}}
=> [2,2,1]
=> 1
{{1,3},{2},{4},{5}}
=> {{1,2,3},{4,5}}
=> [3,2]
=> 0
{{1,4,5},{2,3}}
=> {{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> 3
{{1,4},{2,3,5}}
=> {{1,3},{2,5},{4}}
=> [2,2,1]
=> 1
{{1,4},{2,3},{5}}
=> {{1,2},{3,5},{4}}
=> [2,2,1]
=> 1
Description
The number of parts equal to 1 in a partition.
Matching statistic: St000215
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000215: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000215: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => [2,1] => [2,1] => 2
{{1},{2}}
=> [1,2] => [1,2] => [1,2] => 0
{{1,2,3}}
=> [2,3,1] => [3,1,2] => [3,2,1] => 3
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => [2,3,1] => 1
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => [1,3,2] => 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => [4,3,2,1] => 4
{{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => [3,2,1,4] => 2
{{1,2,4},{3}}
=> [2,4,3,1] => [4,1,3,2] => [3,4,2,1] => 2
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
{{1,3,4},{2}}
=> [3,2,4,1] => [4,2,1,3] => [2,4,3,1] => 2
{{1,3},{2,4}}
=> [3,4,1,2] => [3,4,1,2] => [3,1,4,2] => 0
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => [2,3,1,4] => 0
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => [3,2,4,1] => 2
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => [1,4,3,2] => 2
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [4,2,3,1] => [2,3,4,1] => 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => [1,3,4,2] => 0
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [5,1,2,3,4] => [5,4,3,2,1] => 5
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,1,2,3,5] => [4,3,2,1,5] => 3
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [5,1,2,4,3] => [4,5,3,2,1] => 3
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,1,2,5,4] => [3,2,1,5,4] => 3
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,1,2,4,5] => [3,2,1,4,5] => 2
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [5,1,3,2,4] => [3,5,4,2,1] => 3
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,1,5,2,3] => [4,2,1,5,3] => 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [4,1,3,2,5] => [3,4,2,1,5] => 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [5,1,4,3,2] => [4,3,5,2,1] => 3
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,3,4] => [2,1,5,4,3] => 3
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 2
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [5,1,3,4,2] => [3,4,5,2,1] => 2
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,5,4,3] => [2,1,4,5,3] => 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 2
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [5,2,1,3,4] => [2,5,4,3,1] => 3
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [4,5,1,3,2] => [4,3,1,5,2] => 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [4,2,1,3,5] => [2,4,3,1,5] => 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [5,4,1,2,3] => [4,2,5,3,1] => 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,5,1,2,4] => [3,1,5,4,2] => 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,4,1,2,5] => [3,1,4,2,5] => 0
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [5,2,1,4,3] => [2,4,5,3,1] => 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,5,1,4,2] => [3,1,4,5,2] => 0
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,1,5,4] => [2,3,1,5,4] => 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4,5] => [2,3,1,4,5] => 0
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [5,3,2,1,4] => [3,2,5,4,1] => 3
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,5,2,1,3] => [4,1,5,3,2] => 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [4,3,2,1,5] => [3,2,4,1,5] => 1
Description
The number of adjacencies of a permutation, zero appended.
An adjacency is a descent of the form $(e+1,e)$ in the word corresponding to the permutation in one-line notation. This statistic, $\operatorname{adj_0}$, counts adjacencies in the word with a zero appended.
$(\operatorname{adj_0}, \operatorname{des})$ and $(\operatorname{fix}, \operatorname{exc})$ are equidistributed, see [1].
Matching statistic: St000445
Mp00221: Set partitions —conjugate⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> {{1},{2}}
=> [1,1] => [1,0,1,0]
=> 2
{{1},{2}}
=> {{1,2}}
=> [2] => [1,1,0,0]
=> 0
{{1,2,3}}
=> {{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> 3
{{1,2},{3}}
=> {{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> 1
{{1,3},{2}}
=> {{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> 1
{{1},{2,3}}
=> {{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> 1
{{1},{2},{3}}
=> {{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> 0
{{1,2,3,4}}
=> {{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
{{1,2,3},{4}}
=> {{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
{{1,2,4},{3}}
=> {{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
{{1,2},{3,4}}
=> {{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
{{1,2},{3},{4}}
=> {{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
{{1,3,4},{2}}
=> {{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
{{1,3},{2,4}}
=> {{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 0
{{1,3},{2},{4}}
=> {{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 0
{{1,4},{2,3}}
=> {{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
{{1},{2,3,4}}
=> {{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
{{1},{2,3},{4}}
=> {{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
{{1,4},{2},{3}}
=> {{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 1
{{1},{2,4},{3}}
=> {{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 0
{{1},{2},{3,4}}
=> {{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
{{1},{2},{3},{4}}
=> {{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> 0
{{1,2,3,4,5}}
=> {{1},{2},{3},{4},{5}}
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 5
{{1,2,3,4},{5}}
=> {{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 3
{{1,2,3,5},{4}}
=> {{1},{2,3},{4},{5}}
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3
{{1,2,3},{4,5}}
=> {{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 3
{{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2
{{1,2,4,5},{3}}
=> {{1},{2},{3,4},{5}}
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 3
{{1,2,4},{3,5}}
=> {{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
{{1,2,4},{3},{5}}
=> {{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
{{1,2,5},{3,4}}
=> {{1},{2,4},{3},{5}}
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3
{{1,2},{3,4,5}}
=> {{1,4},{2},{3},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 3
{{1,2},{3,4},{5}}
=> {{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2
{{1,2,5},{3},{4}}
=> {{1},{2,3,4},{5}}
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
{{1,2},{3,5},{4}}
=> {{1,4},{2,3},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
{{1,2},{3},{4,5}}
=> {{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2
{{1,2},{3},{4},{5}}
=> {{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
{{1,3,4,5},{2}}
=> {{1},{2},{3},{4,5}}
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 3
{{1,3,4},{2,5}}
=> {{1,4},{2,5},{3}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
{{1,3,4},{2},{5}}
=> {{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
{{1,3,5},{2,4}}
=> {{1},{2,4},{3,5}}
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
{{1,3},{2,4,5}}
=> {{1,4},{2},{3,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
{{1,3},{2,4},{5}}
=> {{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
{{1,3,5},{2},{4}}
=> {{1},{2,3},{4,5}}
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
{{1,3},{2,5},{4}}
=> {{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
{{1,3},{2},{4,5}}
=> {{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
{{1,3},{2},{4},{5}}
=> {{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
{{1,4,5},{2,3}}
=> {{1},{2},{3,5},{4}}
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 3
{{1,4},{2,3,5}}
=> {{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
{{1,4},{2,3},{5}}
=> {{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
Description
The number of rises of length 1 of a Dyck path.
Matching statistic: St000674
Mp00221: Set partitions —conjugate⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000674: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000674: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> {{1},{2}}
=> [1,1] => [1,0,1,0]
=> 2
{{1},{2}}
=> {{1,2}}
=> [2] => [1,1,0,0]
=> 0
{{1,2,3}}
=> {{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> 3
{{1,2},{3}}
=> {{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> 1
{{1,3},{2}}
=> {{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> 1
{{1},{2,3}}
=> {{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> 1
{{1},{2},{3}}
=> {{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> 0
{{1,2,3,4}}
=> {{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
{{1,2,3},{4}}
=> {{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
{{1,2,4},{3}}
=> {{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
{{1,2},{3,4}}
=> {{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
{{1,2},{3},{4}}
=> {{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
{{1,3,4},{2}}
=> {{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
{{1,3},{2,4}}
=> {{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 0
{{1,3},{2},{4}}
=> {{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 0
{{1,4},{2,3}}
=> {{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
{{1},{2,3,4}}
=> {{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
{{1},{2,3},{4}}
=> {{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
{{1,4},{2},{3}}
=> {{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 1
{{1},{2,4},{3}}
=> {{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 0
{{1},{2},{3,4}}
=> {{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
{{1},{2},{3},{4}}
=> {{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> 0
{{1,2,3,4,5}}
=> {{1},{2},{3},{4},{5}}
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 5
{{1,2,3,4},{5}}
=> {{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 3
{{1,2,3,5},{4}}
=> {{1},{2,3},{4},{5}}
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3
{{1,2,3},{4,5}}
=> {{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 3
{{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2
{{1,2,4,5},{3}}
=> {{1},{2},{3,4},{5}}
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 3
{{1,2,4},{3,5}}
=> {{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
{{1,2,4},{3},{5}}
=> {{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
{{1,2,5},{3,4}}
=> {{1},{2,4},{3},{5}}
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3
{{1,2},{3,4,5}}
=> {{1,4},{2},{3},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 3
{{1,2},{3,4},{5}}
=> {{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2
{{1,2,5},{3},{4}}
=> {{1},{2,3,4},{5}}
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
{{1,2},{3,5},{4}}
=> {{1,4},{2,3},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
{{1,2},{3},{4,5}}
=> {{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2
{{1,2},{3},{4},{5}}
=> {{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
{{1,3,4,5},{2}}
=> {{1},{2},{3},{4,5}}
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 3
{{1,3,4},{2,5}}
=> {{1,4},{2,5},{3}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
{{1,3,4},{2},{5}}
=> {{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
{{1,3,5},{2,4}}
=> {{1},{2,4},{3,5}}
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
{{1,3},{2,4,5}}
=> {{1,4},{2},{3,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
{{1,3},{2,4},{5}}
=> {{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
{{1,3,5},{2},{4}}
=> {{1},{2,3},{4,5}}
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
{{1,3},{2,5},{4}}
=> {{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
{{1,3},{2},{4,5}}
=> {{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
{{1,3},{2},{4},{5}}
=> {{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
{{1,4,5},{2,3}}
=> {{1},{2},{3,5},{4}}
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 3
{{1,4},{2,3,5}}
=> {{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
{{1,4},{2,3},{5}}
=> {{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
Description
The number of hills of a Dyck path.
A hill is a peak with up step starting and down step ending at height zero.
Matching statistic: St000894
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000894: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000894: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => [1,2] => [[1,0],[0,1]]
=> 2
{{1},{2}}
=> [1,2] => [2,1] => [[0,1],[1,0]]
=> 0
{{1,2,3}}
=> [2,3,1] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 3
{{1,2},{3}}
=> [2,1,3] => [1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> 1
{{1,3},{2}}
=> [3,2,1] => [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> 1
{{1},{2,3}}
=> [1,3,2] => [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> 1
{{1},{2},{3}}
=> [1,2,3] => [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 4
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 2
{{1,2,4},{3}}
=> [2,4,3,1] => [1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 2
{{1,2},{3,4}}
=> [2,1,4,3] => [1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 2
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 1
{{1,3,4},{2}}
=> [3,2,4,1] => [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 2
{{1,3},{2,4}}
=> [3,4,1,2] => [4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> 0
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 0
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 2
{{1},{2,3,4}}
=> [1,3,4,2] => [4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> 2
{{1},{2,3},{4}}
=> [1,3,2,4] => [3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 0
{{1},{2},{3,4}}
=> [1,2,4,3] => [2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 5
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,5,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 3
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,4,3,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 3
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,5,4,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> 3
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,4,5,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 2
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,3,2,4,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 3
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,5,2,3,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
=> 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,4,3,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> 3
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,5,3,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> 3
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,4,3,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> 2
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,3,4,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 2
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,5,4,3,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,3,5,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> 2
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,3,4,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [2,1,3,4,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 3
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [5,1,3,2,4] => [[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0]]
=> 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [2,1,3,5,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [4,1,2,3,5] => [[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
=> 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [5,1,2,4,3] => [[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0]]
=> 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [4,1,2,5,3] => [[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0]]
=> 0
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [2,1,4,3,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [5,1,4,2,3] => [[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0]]
=> 0
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [2,1,5,4,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [2,1,4,5,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 0
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [3,2,1,4,5] => [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 3
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [5,2,1,3,4] => [[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0]]
=> 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [3,2,1,5,4] => [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 1
Description
The trace of an alternating sign matrix.
Matching statistic: St000895
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000895: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000895: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => [1,2] => [[1,0],[0,1]]
=> 2
{{1},{2}}
=> [1,2] => [2,1] => [[0,1],[1,0]]
=> 0
{{1,2,3}}
=> [2,3,1] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 3
{{1,2},{3}}
=> [2,1,3] => [1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> 1
{{1,3},{2}}
=> [3,2,1] => [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> 1
{{1},{2,3}}
=> [1,3,2] => [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> 1
{{1},{2},{3}}
=> [1,2,3] => [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 4
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 2
{{1,2,4},{3}}
=> [2,4,3,1] => [1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 2
{{1,2},{3,4}}
=> [2,1,4,3] => [1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 2
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 1
{{1,3,4},{2}}
=> [3,2,4,1] => [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 2
{{1,3},{2,4}}
=> [3,4,1,2] => [4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> 0
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 0
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 2
{{1},{2,3,4}}
=> [1,3,4,2] => [4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> 2
{{1},{2,3},{4}}
=> [1,3,2,4] => [3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 0
{{1},{2},{3,4}}
=> [1,2,4,3] => [2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 5
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,5,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 3
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,4,3,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 3
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,5,4,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> 3
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,4,5,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 2
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,3,2,4,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 3
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,5,2,3,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
=> 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,4,3,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> 3
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,5,3,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> 3
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,4,3,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> 2
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,3,4,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 2
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,5,4,3,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,3,5,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> 2
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,3,4,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [2,1,3,4,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 3
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [5,1,3,2,4] => [[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0]]
=> 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [2,1,3,5,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [4,1,2,3,5] => [[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
=> 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [5,1,2,4,3] => [[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0]]
=> 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [4,1,2,5,3] => [[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0]]
=> 0
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [2,1,4,3,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [5,1,4,2,3] => [[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0]]
=> 0
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [2,1,5,4,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [2,1,4,5,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 0
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [3,2,1,4,5] => [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 3
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [5,2,1,3,4] => [[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0]]
=> 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [3,2,1,5,4] => [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 1
Description
The number of ones on the main diagonal of an alternating sign matrix.
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St001903The number of fixed points of a parking function.
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