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Your data matches 28 different statistics following compositions of up to 3 maps.
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Matching statistic: St000010
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1]
=> 1
[1,0,1,0]
=> [1,2] => [1,2] => [1,1]
=> 2
[1,1,0,0]
=> [2,1] => [1,2] => [1,1]
=> 2
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,1,1]
=> 3
[1,0,1,1,0,0]
=> [1,3,2] => [1,2,3] => [1,1,1]
=> 3
[1,1,0,0,1,0]
=> [2,1,3] => [1,2,3] => [1,1,1]
=> 3
[1,1,0,1,0,0]
=> [2,3,1] => [1,2,3] => [1,1,1]
=> 3
[1,1,1,0,0,0]
=> [3,2,1] => [1,3,2] => [2,1]
=> 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3,4] => [1,1,1,1]
=> 4
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,2,3,4] => [1,1,1,1]
=> 4
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,2,3,4] => [1,1,1,1]
=> 4
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,2,4,3] => [2,1,1]
=> 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,2,3,4] => [1,1,1,1]
=> 4
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,2,3,4] => [1,1,1,1]
=> 4
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,2,3,4] => [1,1,1,1]
=> 4
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,3,4] => [1,1,1,1]
=> 4
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,2,4,3] => [2,1,1]
=> 3
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [1,3,2,4] => [2,1,1]
=> 3
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,3,4,2] => [2,1,1]
=> 3
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [1,3,2,4] => [2,1,1]
=> 3
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,4,2,3] => [2,1,1]
=> 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,3,5,4] => [2,1,1,1]
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,2,3,5,4] => [2,1,1,1]
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,2,4,3,5] => [2,1,1,1]
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,2,4,5,3] => [2,1,1,1]
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,2,4,3,5] => [2,1,1,1]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,2,5,3,4] => [2,1,1,1]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,2,3,5,4] => [2,1,1,1]
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,2,3,5,4] => [2,1,1,1]
=> 4
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [1,2,4,3,5] => [2,1,1,1]
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,2,4,5,3] => [2,1,1,1]
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [1,2,4,3,5] => [2,1,1,1]
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,2,5,3,4] => [2,1,1,1]
=> 4
Description
The length of the partition.
Matching statistic: St000384
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000384: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000384: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1]
=> 1
[1,0,1,0]
=> [1,2] => [1,2] => [1,1]
=> 2
[1,1,0,0]
=> [2,1] => [1,2] => [1,1]
=> 2
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,1,1]
=> 3
[1,0,1,1,0,0]
=> [1,3,2] => [1,2,3] => [1,1,1]
=> 3
[1,1,0,0,1,0]
=> [2,1,3] => [1,2,3] => [1,1,1]
=> 3
[1,1,0,1,0,0]
=> [2,3,1] => [1,2,3] => [1,1,1]
=> 3
[1,1,1,0,0,0]
=> [3,2,1] => [1,3,2] => [2,1]
=> 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3,4] => [1,1,1,1]
=> 4
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,2,3,4] => [1,1,1,1]
=> 4
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,2,3,4] => [1,1,1,1]
=> 4
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,2,4,3] => [2,1,1]
=> 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,2,3,4] => [1,1,1,1]
=> 4
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,2,3,4] => [1,1,1,1]
=> 4
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,2,3,4] => [1,1,1,1]
=> 4
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,3,4] => [1,1,1,1]
=> 4
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,2,4,3] => [2,1,1]
=> 3
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [1,3,2,4] => [2,1,1]
=> 3
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,3,4,2] => [2,1,1]
=> 3
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [1,3,2,4] => [2,1,1]
=> 3
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,4,2,3] => [2,1,1]
=> 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,3,5,4] => [2,1,1,1]
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,2,3,5,4] => [2,1,1,1]
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,2,4,3,5] => [2,1,1,1]
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,2,4,5,3] => [2,1,1,1]
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,2,4,3,5] => [2,1,1,1]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,2,5,3,4] => [2,1,1,1]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,2,3,5,4] => [2,1,1,1]
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,2,3,5,4] => [2,1,1,1]
=> 4
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [1,2,4,3,5] => [2,1,1,1]
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,2,4,5,3] => [2,1,1,1]
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [1,2,4,3,5] => [2,1,1,1]
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,2,5,3,4] => [2,1,1,1]
=> 4
Description
The maximal part of the shifted composition of an integer partition.
A partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ is shifted into a composition by adding $i-1$ to the $i$-th part.
The statistic is then $\operatorname{max}_i\{ \lambda_i + i - 1 \}$.
See also [[St000380]].
Matching statistic: St000784
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000784: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000784: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1]
=> 1
[1,0,1,0]
=> [1,2] => [1,2] => [1,1]
=> 2
[1,1,0,0]
=> [2,1] => [1,2] => [1,1]
=> 2
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,1,1]
=> 3
[1,0,1,1,0,0]
=> [1,3,2] => [1,2,3] => [1,1,1]
=> 3
[1,1,0,0,1,0]
=> [2,1,3] => [1,2,3] => [1,1,1]
=> 3
[1,1,0,1,0,0]
=> [2,3,1] => [1,2,3] => [1,1,1]
=> 3
[1,1,1,0,0,0]
=> [3,2,1] => [1,3,2] => [2,1]
=> 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3,4] => [1,1,1,1]
=> 4
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,2,3,4] => [1,1,1,1]
=> 4
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,2,3,4] => [1,1,1,1]
=> 4
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,2,4,3] => [2,1,1]
=> 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,2,3,4] => [1,1,1,1]
=> 4
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,2,3,4] => [1,1,1,1]
=> 4
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,2,3,4] => [1,1,1,1]
=> 4
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,3,4] => [1,1,1,1]
=> 4
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,2,4,3] => [2,1,1]
=> 3
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [1,3,2,4] => [2,1,1]
=> 3
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,3,4,2] => [2,1,1]
=> 3
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [1,3,2,4] => [2,1,1]
=> 3
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,4,2,3] => [2,1,1]
=> 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,3,5,4] => [2,1,1,1]
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,2,3,5,4] => [2,1,1,1]
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,2,4,3,5] => [2,1,1,1]
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,2,4,5,3] => [2,1,1,1]
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,2,4,3,5] => [2,1,1,1]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,2,5,3,4] => [2,1,1,1]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,2,3,5,4] => [2,1,1,1]
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,2,3,5,4] => [2,1,1,1]
=> 4
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [1,2,4,3,5] => [2,1,1,1]
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,2,4,5,3] => [2,1,1,1]
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [1,2,4,3,5] => [2,1,1,1]
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,2,5,3,4] => [2,1,1,1]
=> 4
Description
The maximum of the length and the largest part of the integer partition.
This is the side length of the smallest square the Ferrers diagram of the partition fits into. It is also the minimal number of colours required to colour the cells of the Ferrers diagram such that no two cells in a column or in a row have the same colour, see [1].
See also [[St001214]].
Matching statistic: St000393
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000393: Binary words ⟶ ℤResult quality: 77% ●values known / values provided: 100%●distinct values known / distinct values provided: 77%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000393: Binary words ⟶ ℤResult quality: 77% ●values known / values provided: 100%●distinct values known / distinct values provided: 77%
Values
[1,0]
=> [1] => [1] => => ? = 1 - 1
[1,0,1,0]
=> [1,2] => [1,2] => 0 => 1 = 2 - 1
[1,1,0,0]
=> [2,1] => [1,2] => 0 => 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 00 => 2 = 3 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,2,3] => 00 => 2 = 3 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [1,2,3] => 00 => 2 = 3 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [1,2,3] => 00 => 2 = 3 - 1
[1,1,1,0,0,0]
=> [3,2,1] => [1,3,2] => 01 => 1 = 2 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 000 => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3,4] => 000 => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,2,3,4] => 000 => 3 = 4 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,2,3,4] => 000 => 3 = 4 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,2,4,3] => 001 => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,2,3,4] => 000 => 3 = 4 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,2,3,4] => 000 => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,2,3,4] => 000 => 3 = 4 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,3,4] => 000 => 3 = 4 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,2,4,3] => 001 => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [1,3,2,4] => 010 => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,3,4,2] => 001 => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [1,3,2,4] => 010 => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,4,2,3] => 010 => 2 = 3 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0000 => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4,5] => 0000 => 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,3,4,5] => 0000 => 4 = 5 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,3,4,5] => 0000 => 4 = 5 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,3,5,4] => 0001 => 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,2,3,4,5] => 0000 => 4 = 5 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,2,3,4,5] => 0000 => 4 = 5 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,2,3,4,5] => 0000 => 4 = 5 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,2,3,4,5] => 0000 => 4 = 5 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,2,3,5,4] => 0001 => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,2,4,3,5] => 0010 => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,2,4,5,3] => 0001 => 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,2,4,3,5] => 0010 => 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,2,5,3,4] => 0010 => 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [1,2,3,4,5] => 0000 => 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,2,3,4,5] => 0000 => 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,2,3,4,5] => 0000 => 4 = 5 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,2,3,4,5] => 0000 => 4 = 5 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,2,3,5,4] => 0001 => 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,2,3,4,5] => 0000 => 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,2,3,4,5] => 0000 => 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,2,3,4,5] => 0000 => 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 0000 => 4 = 5 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,2,3,5,4] => 0001 => 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [1,2,4,3,5] => 0010 => 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,2,4,5,3] => 0001 => 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [1,2,4,3,5] => 0010 => 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,2,5,3,4] => 0010 => 3 = 4 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [1,3,2,4,5] => 0100 => 3 = 4 - 1
[1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,8,7,6,5,4,2,1] => [1,3,7,2,8,4,6,5] => ? => ? = 5 - 1
[1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0]
=> [4,7,6,5,8,3,2,1] => [1,4,5,8,2,7,3,6] => ? => ? = 6 - 1
[1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
=> [4,7,8,6,5,3,2,1] => [1,4,6,3,8,2,7,5] => ? => ? = 5 - 1
[]
=> [] => [] => ? => ? = 0 - 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,9,10,11,12,2] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => ? = 12 - 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => ? = 12 - 1
[1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
=> [3,2,4,1,7,6,8,5,11,10,12,9] => [1,3,4,2,5,7,8,6,9,11,12,10] => 00100010001 => ? = 9 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,3,2,5,4,7,6,9,8,11,12,10] => [1,2,3,4,5,6,7,8,9,10,11,12] => 00000000000 => ? = 12 - 1
Description
The number of strictly increasing runs in a binary word.
Matching statistic: St000507
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 92% ●values known / values provided: 96%●distinct values known / distinct values provided: 92%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 92% ●values known / values provided: 96%●distinct values known / distinct values provided: 92%
Values
[1,0]
=> [1] => [1] => [[1]]
=> 1
[1,0,1,0]
=> [1,2] => [1,2] => [[1,2]]
=> 2
[1,1,0,0]
=> [2,1] => [1,2] => [[1,2]]
=> 2
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [[1,2,3]]
=> 3
[1,0,1,1,0,0]
=> [1,3,2] => [1,2,3] => [[1,2,3]]
=> 3
[1,1,0,0,1,0]
=> [2,1,3] => [1,2,3] => [[1,2,3]]
=> 3
[1,1,0,1,0,0]
=> [2,3,1] => [1,2,3] => [[1,2,3]]
=> 3
[1,1,1,0,0,0]
=> [3,2,1] => [1,3,2] => [[1,2],[3]]
=> 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3,4] => [[1,2,3,4]]
=> 4
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,2,3,4] => [[1,2,3,4]]
=> 4
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,2,3,4] => [[1,2,3,4]]
=> 4
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,2,4,3] => [[1,2,3],[4]]
=> 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 4
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,2,3,4] => [[1,2,3,4]]
=> 4
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,2,3,4] => [[1,2,3,4]]
=> 4
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,3,4] => [[1,2,3,4]]
=> 4
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,2,4,3] => [[1,2,3],[4]]
=> 3
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [1,3,2,4] => [[1,2,4],[3]]
=> 3
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,3,4,2] => [[1,2,3],[4]]
=> 3
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [1,3,2,4] => [[1,2,4],[3]]
=> 3
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,4,2,3] => [[1,2,4],[3]]
=> 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 5
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 5
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 5
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 5
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,2,4,5,3] => [[1,2,3,4],[5]]
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,2,5,3,4] => [[1,2,3,5],[4]]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 5
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 5
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 5
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 5
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 5
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 5
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 4
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,2,4,5,3] => [[1,2,3,4],[5]]
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,2,5,3,4] => [[1,2,3,5],[4]]
=> 4
[1,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [1,4,3,6,5,7,2,8] => [1,2,4,6,7,3,5,8] => ?
=> ? = 7
[1,0,1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [1,4,3,7,8,6,5,2] => [1,2,4,7,5,8,3,6] => ?
=> ? = 6
[1,0,1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [1,4,5,6,3,7,2,8] => [1,2,4,6,7,3,5,8] => ?
=> ? = 7
[1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,4,6,8,7,5,3,2] => [1,2,4,8,3,6,5,7] => ?
=> ? = 6
[1,0,1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,5,6,7,4,3,2,8] => [1,2,5,4,7,3,6,8] => ?
=> ? = 6
[1,1,0,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [2,4,3,6,5,7,1,8] => [1,2,4,6,7,3,5,8] => ?
=> ? = 7
[1,1,0,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [2,4,5,6,3,7,1,8] => [1,2,4,6,7,3,5,8] => ?
=> ? = 7
[1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [2,4,5,6,8,7,3,1] => [1,2,4,6,7,3,5,8] => ?
=> ? = 7
[1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [2,4,6,8,7,5,3,1] => [1,2,4,8,3,6,5,7] => ?
=> ? = 6
[1,1,0,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [2,5,6,7,4,3,1,8] => [1,2,5,4,7,3,6,8] => ?
=> ? = 6
[1,1,0,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [2,5,6,7,4,8,3,1] => [1,2,5,4,7,3,6,8] => ?
=> ? = 6
[1,1,1,0,0,1,0,1,0,0,1,1,1,0,0,0]
=> [3,2,4,5,1,8,7,6] => [1,3,4,5,2,6,8,7] => ?
=> ? = 6
[1,1,1,0,0,1,0,1,1,0,1,0,0,0,1,0]
=> [3,2,4,6,7,5,1,8] => [1,3,4,6,5,7,2,8] => ?
=> ? = 6
[1,1,1,0,0,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,5,4,7,8,6,1] => [1,3,5,7,6,8,2,4] => ?
=> ? = 6
[1,1,1,0,0,1,1,0,1,0,0,0,1,1,0,0]
=> [3,2,5,6,4,1,8,7] => [1,3,5,4,6,2,7,8] => ?
=> ? = 6
[1,1,1,0,0,1,1,0,1,0,0,1,0,1,0,0]
=> [3,2,5,6,4,7,8,1] => [1,3,5,4,6,7,8,2] => ?
=> ? = 6
[1,1,1,0,0,1,1,0,1,1,0,0,0,1,0,0]
=> [3,2,5,7,6,4,8,1] => [1,3,5,6,4,7,8,2] => ?
=> ? = 6
[1,1,1,0,1,0,1,0,0,1,1,0,1,0,0,0]
=> [3,4,5,2,7,8,6,1] => [1,3,5,7,6,8,2,4] => ?
=> ? = 6
[1,1,1,0,1,1,0,1,0,0,0,1,1,0,0,0]
=> [3,5,6,4,2,8,7,1] => [1,3,6,8,2,5,4,7] => ?
=> ? = 6
[1,1,1,0,1,1,0,1,0,0,1,0,0,1,0,0]
=> [3,5,6,4,7,2,8,1] => [1,3,6,2,5,7,8,4] => ?
=> ? = 6
[1,1,1,0,1,1,0,1,0,0,1,0,1,0,0,0]
=> [3,5,6,4,7,8,2,1] => [1,3,6,8,2,5,7,4] => ?
=> ? = 6
[1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0]
=> [3,5,6,7,4,8,2,1] => [1,3,6,8,2,5,4,7] => ?
=> ? = 6
[1,1,1,0,1,1,1,0,0,0,0,0,1,0,1,0]
=> [3,6,5,4,2,1,7,8] => [1,3,5,2,6,4,7,8] => ?
=> ? = 6
[1,1,1,0,1,1,1,0,0,0,0,0,1,1,0,0]
=> [3,6,5,4,2,1,8,7] => [1,3,5,2,6,4,7,8] => ?
=> ? = 6
[1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> [3,6,5,4,2,7,8,1] => [1,3,5,2,6,7,8,4] => ?
=> ? = 6
[1,1,1,0,1,1,1,0,0,0,1,0,0,1,0,0]
=> [3,6,5,4,7,2,8,1] => [1,3,5,7,8,2,6,4] => ?
=> ? = 6
[1,1,1,0,1,1,1,0,0,0,1,0,1,0,0,0]
=> [3,6,5,4,7,8,2,1] => [1,3,5,7,2,6,8,4] => ?
=> ? = 6
[1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [4,6,7,5,3,2,1,8] => [1,4,5,3,7,2,6,8] => ?
=> ? = 6
[1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0]
=> [4,7,6,5,8,3,2,1] => [1,4,5,8,2,7,3,6] => ?
=> ? = 6
[1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
=> [4,7,8,6,5,3,2,1] => [1,4,6,3,8,2,7,5] => ?
=> ? = 5
[1,1,1,1,1,0,1,0,1,0,0,0,1,0,0,0]
=> [5,6,7,4,3,8,2,1] => [1,5,3,7,2,6,8,4] => ?
=> ? = 5
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,4,5,6,8,9,7,1] => [1,2,3,4,5,6,8,7,9] => [[1,2,3,4,5,6,7,9],[8]]
=> ? = 8
[1,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,4,5,6,9,8,7,1] => [1,2,3,4,5,6,9,7,8] => [[1,2,3,4,5,6,7,9],[8]]
=> ? = 8
[1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [2,3,4,5,7,8,6,9,1] => [1,2,3,4,5,7,6,8,9] => [[1,2,3,4,5,6,8,9],[7]]
=> ? = 8
[1,1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [2,3,4,5,7,9,8,6,1] => [1,2,3,4,5,7,8,6,9] => [[1,2,3,4,5,6,7,9],[8]]
=> ? = 8
[1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [2,3,4,6,7,5,8,9,1] => [1,2,3,4,6,5,7,8,9] => [[1,2,3,4,5,7,8,9],[6]]
=> ? = 8
[1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [2,3,5,6,4,7,8,9,1] => [1,2,3,5,4,6,7,8,9] => [[1,2,3,4,6,7,8,9],[5]]
=> ? = 8
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9,10,11] => [1,2,3,4,5,6,7,8,9,10,11] => [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? = 11
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,8,9,11,10] => [1,2,3,4,5,6,7,8,9,10,11] => [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? = 11
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,6,7,8,10,11,9] => [1,2,3,4,5,6,7,8,9,10,11] => [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? = 11
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,4,5,8,9,7,6] => [1,2,3,4,5,6,8,7,9] => [[1,2,3,4,5,6,7,9],[8]]
=> ? = 8
[1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,4,5,9,8,7,6] => [1,2,3,4,5,6,9,7,8] => [[1,2,3,4,5,6,7,9],[8]]
=> ? = 8
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,9,10,11,2] => [1,2,3,4,5,6,7,8,9,10,11] => [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? = 11
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => [1,2,3,4,5,6,7,8,9,10,11] => [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? = 11
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,5,6,7,8,11,10,9] => [1,2,3,4,5,6,7,8,9,11,10] => [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? = 10
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,8,9,10,11,1] => [1,3,4,5,6,7,8,9,10,11,2] => [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? = 10
[1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,9,8,7,6,5] => [1,2,3,4,5,9,6,8,7] => [[1,2,3,4,5,6,8],[7],[9]]
=> ? = 7
[1,0,1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,2,3,5,4,9,8,7,6] => [1,2,3,4,5,6,9,7,8] => [[1,2,3,4,5,6,7,9],[8]]
=> ? = 8
[1,0,1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,4,3,9,8,7,6,5] => [1,2,3,4,5,9,6,8,7] => [[1,2,3,4,5,6,8],[7],[9]]
=> ? = 7
[1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,4,3,2,5,6,7,8,9] => [1,2,4,3,5,6,7,8,9] => [[1,2,3,5,6,7,8,9],[4]]
=> ? = 8
Description
The number of ascents of a standard tableau.
Entry $i$ of a standard Young tableau is an '''ascent''' if $i+1$ appears to the right or above $i$ in the tableau (with respect to the English notation for tableaux).
Matching statistic: St000245
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 77%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 77%
Values
[1,0]
=> [1] => [1] => [1] => 0 = 1 - 1
[1,0,1,0]
=> [1,2] => [1,2] => [1,2] => 1 = 2 - 1
[1,1,0,0]
=> [2,1] => [1,2] => [1,2] => 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 2 = 3 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,2,3] => [1,2,3] => 2 = 3 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [1,2,3] => [1,2,3] => 2 = 3 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [1,2,3] => [1,2,3] => 2 = 3 - 1
[1,1,1,0,0,0]
=> [3,2,1] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 3 = 4 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 3 = 4 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 3 = 4 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 3 = 4 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 3 = 4 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [1,3,2,4] => [1,3,2,4] => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,3,4,2] => [1,4,2,3] => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [1,3,2,4] => [1,3,2,4] => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,4,2,3] => [1,3,4,2] => 2 = 3 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 4 = 5 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 4 = 5 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 4 = 5 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 4 = 5 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 4 = 5 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 4 = 5 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,3,4] => 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,2,4,3,5] => [1,2,4,3,5] => 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,2,5,3,4] => [1,2,4,5,3] => 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 4 = 5 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 4 = 5 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 4 = 5 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [1,2,4,3,5] => [1,2,4,3,5] => 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,2,4,5,3] => [1,2,5,3,4] => 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [1,2,4,3,5] => [1,2,4,3,5] => 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,2,5,3,4] => [1,2,4,5,3] => 3 = 4 - 1
[1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [3,2,4,5,1,6,7] => [1,3,4,5,2,6,7] => [1,5,2,3,4,6,7] => ? = 6 - 1
[1,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> [3,2,4,5,1,7,6] => [1,3,4,5,2,6,7] => [1,5,2,3,4,6,7] => ? = 6 - 1
[1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [3,2,4,5,6,1,7] => [1,3,4,5,6,2,7] => [1,6,2,3,4,5,7] => ? = 6 - 1
[1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [3,2,4,6,5,1,7] => [1,3,4,6,2,5,7] => [1,5,6,2,3,4,7] => ? = 6 - 1
[1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [3,2,4,6,7,5,1] => [1,3,4,6,5,7,2] => [1,7,2,3,4,6,5] => ? = 5 - 1
[1,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> [3,2,5,6,4,1,7] => [1,3,5,4,6,2,7] => [1,6,2,3,5,4,7] => ? = 5 - 1
[1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [3,2,5,6,4,7,1] => [1,3,5,4,6,7,2] => [1,7,2,3,5,4,6] => ? = 5 - 1
[1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [3,2,5,7,6,4,1] => [1,3,5,6,4,7,2] => [1,7,2,3,6,4,5] => ? = 5 - 1
[1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [3,2,6,5,7,4,1] => [1,3,6,4,5,7,2] => [1,7,2,3,5,6,4] => ? = 5 - 1
[1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [3,2,6,7,5,4,1] => [1,3,6,4,7,2,5] => [1,5,7,2,3,6,4] => ? = 5 - 1
[1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [3,2,7,6,5,4,1] => [1,3,7,2,4,6,5] => [1,4,6,5,7,2,3] => ? = 5 - 1
[1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [3,4,6,7,5,2,1] => [1,3,6,2,4,7,5] => [1,4,7,5,6,2,3] => ? = 5 - 1
[1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [3,4,7,6,5,2,1] => [1,3,7,2,4,6,5] => [1,4,6,5,7,2,3] => ? = 5 - 1
[1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [3,5,4,6,2,1,7] => [1,3,4,6,2,5,7] => [1,5,6,2,3,4,7] => ? = 6 - 1
[1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [3,5,4,6,7,2,1] => [1,3,4,6,2,5,7] => [1,5,6,2,3,4,7] => ? = 6 - 1
[1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [3,5,6,4,2,1,7] => [1,3,6,2,5,4,7] => [1,5,4,6,2,3,7] => ? = 5 - 1
[1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [3,5,6,4,2,7,1] => [1,3,6,7,2,5,4] => [1,5,4,7,2,3,6] => ? = 5 - 1
[1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [3,5,6,4,7,2,1] => [1,3,6,2,5,7,4] => [1,7,4,5,6,2,3] => ? = 5 - 1
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [3,5,6,7,4,2,1] => [1,3,6,2,5,4,7] => [1,5,4,6,2,3,7] => ? = 5 - 1
[1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [3,5,7,6,4,2,1] => [1,3,7,2,5,4,6] => [1,5,4,6,7,2,3] => ? = 5 - 1
[1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [3,6,5,4,2,1,7] => [1,3,5,2,6,4,7] => [1,6,4,5,2,3,7] => ? = 5 - 1
[1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [3,6,5,4,2,7,1] => [1,3,5,2,6,7,4] => [1,7,4,5,2,3,6] => ? = 5 - 1
[1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [3,6,5,4,7,2,1] => [1,3,5,7,2,6,4] => [1,6,4,7,2,3,5] => ? = 5 - 1
[1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [3,6,5,7,4,2,1] => [1,3,5,4,7,2,6] => [1,6,7,2,3,5,4] => ? = 5 - 1
[1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [3,6,7,5,4,2,1] => [1,3,7,2,6,4,5] => [1,5,6,4,7,2,3] => ? = 5 - 1
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [3,7,6,5,4,2,1] => [1,3,6,2,7,4,5] => [1,5,7,4,6,2,3] => ? = 5 - 1
[1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [4,5,3,2,1,6,7] => [1,4,2,5,3,6,7] => [1,5,3,4,2,6,7] => ? = 5 - 1
[1,1,1,1,0,1,0,0,0,0,1,1,0,0]
=> [4,5,3,2,1,7,6] => [1,4,2,5,3,6,7] => [1,5,3,4,2,6,7] => ? = 5 - 1
[1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [4,5,3,2,6,1,7] => [1,4,2,5,6,3,7] => [1,6,3,4,2,5,7] => ? = 5 - 1
[1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [4,5,3,2,6,7,1] => [1,4,2,5,6,7,3] => [1,7,3,4,2,5,6] => ? = 5 - 1
[1,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> [4,5,3,2,7,6,1] => [1,4,2,5,7,3,6] => [1,6,7,3,4,2,5] => ? = 5 - 1
[1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [4,5,3,6,2,1,7] => [1,4,6,2,5,3,7] => [1,5,3,6,2,4,7] => ? = 5 - 1
[1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [4,5,3,6,2,7,1] => [1,4,6,7,2,5,3] => [1,5,3,7,2,4,6] => ? = 5 - 1
[1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [4,5,3,6,7,2,1] => [1,4,6,2,5,7,3] => [1,7,3,5,6,2,4] => ? = 5 - 1
[1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [4,5,3,7,6,2,1] => [1,4,7,2,5,6,3] => [1,6,3,5,7,2,4] => ? = 5 - 1
[1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [4,5,6,3,2,1,7] => [1,4,3,6,2,5,7] => [1,5,6,2,4,3,7] => ? = 5 - 1
[1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [4,5,6,3,2,7,1] => [1,4,3,6,7,2,5] => [1,5,7,2,4,3,6] => ? = 5 - 1
[1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [4,5,6,3,7,2,1] => [1,4,3,6,2,5,7] => [1,5,6,2,4,3,7] => ? = 5 - 1
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => [1,4,7,2,5,3,6] => [1,5,3,6,7,2,4] => ? = 5 - 1
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [4,5,7,6,3,2,1] => [1,4,6,2,5,3,7] => [1,5,3,6,2,4,7] => ? = 5 - 1
[1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [4,6,5,3,2,1,7] => [1,4,3,5,2,6,7] => [1,5,2,4,3,6,7] => ? = 5 - 1
[1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [4,6,5,3,2,7,1] => [1,4,3,5,2,6,7] => [1,5,2,4,3,6,7] => ? = 5 - 1
[1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [4,6,5,3,7,2,1] => [1,4,3,5,7,2,6] => [1,6,7,2,4,3,5] => ? = 5 - 1
[1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [4,6,5,7,3,2,1] => [1,4,7,2,6,3,5] => [1,5,6,3,7,2,4] => ? = 5 - 1
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [4,6,7,5,3,2,1] => [1,4,5,3,7,2,6] => [1,6,7,2,5,3,4] => ? = 5 - 1
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [4,7,6,5,3,2,1] => [1,4,5,3,6,2,7] => [1,6,2,5,3,4,7] => ? = 5 - 1
[1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [5,4,3,6,2,1,7] => [1,5,2,4,6,3,7] => [1,6,3,4,5,2,7] => ? = 5 - 1
[1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [5,4,3,6,2,7,1] => [1,5,2,4,6,7,3] => [1,7,3,4,5,2,6] => ? = 5 - 1
[1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [5,4,3,6,7,2,1] => [1,5,7,2,4,6,3] => [1,6,3,4,7,2,5] => ? = 5 - 1
[1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [5,4,3,7,6,2,1] => [1,5,6,2,4,7,3] => [1,7,3,4,6,2,5] => ? = 5 - 1
Description
The number of ascents of a permutation.
Matching statistic: St000702
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000702: Permutations ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 46%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000702: Permutations ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 46%
Values
[1,0]
=> [1] => [1] => [1] => ? = 1
[1,0,1,0]
=> [1,2] => [1,2] => [1,2] => 2
[1,1,0,0]
=> [2,1] => [1,2] => [1,2] => 2
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 3
[1,0,1,1,0,0]
=> [1,3,2] => [1,2,3] => [1,2,3] => 3
[1,1,0,0,1,0]
=> [2,1,3] => [1,2,3] => [1,2,3] => 3
[1,1,0,1,0,0]
=> [2,3,1] => [1,2,3] => [1,2,3] => 3
[1,1,1,0,0,0]
=> [3,2,1] => [1,3,2] => [1,3,2] => 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 4
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 4
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 4
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 4
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 4
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 4
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 4
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 3
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [1,3,2,4] => [1,3,2,4] => 3
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,3,4,2] => [1,4,3,2] => 3
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [1,3,2,4] => [1,3,2,4] => 3
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,4,2,3] => [1,4,2,3] => 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,4,3] => 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,2,4,3,5] => [1,2,4,3,5] => 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,3,4] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => 4
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => 4
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [1,2,4,3,5] => [1,2,4,3,5] => 4
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,2,4,5,3] => [1,2,5,4,3] => 4
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [1,2,4,3,5] => [1,2,4,3,5] => 4
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,2,5,3,4] => [1,2,5,3,4] => 4
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 4
[1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [3,2,4,5,1,6,7] => [1,3,4,5,2,6,7] => [1,5,3,4,2,6,7] => ? = 6
[1,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> [3,2,4,5,1,7,6] => [1,3,4,5,2,6,7] => [1,5,3,4,2,6,7] => ? = 6
[1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [3,2,4,5,6,1,7] => [1,3,4,5,6,2,7] => [1,6,3,4,5,2,7] => ? = 6
[1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,1] => [1,3,4,5,6,7,2] => [1,7,3,4,5,6,2] => ? = 6
[1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [3,2,4,5,7,6,1] => [1,3,4,5,7,2,6] => [1,7,3,4,5,2,6] => ? = 6
[1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [3,2,4,6,5,1,7] => [1,3,4,6,2,5,7] => [1,6,3,4,2,5,7] => ? = 6
[1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [3,2,4,6,5,7,1] => [1,3,4,6,7,2,5] => [1,7,3,4,2,6,5] => ? = 6
[1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [3,2,4,6,7,5,1] => [1,3,4,6,5,7,2] => [1,7,3,4,6,5,2] => ? = 5
[1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [3,2,4,7,6,5,1] => [1,3,4,7,2,5,6] => [1,7,3,4,2,5,6] => ? = 6
[1,1,1,0,0,1,1,0,0,0,1,0,1,0]
=> [3,2,5,4,1,6,7] => [1,3,5,2,4,6,7] => [1,5,3,2,4,6,7] => ? = 6
[1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [3,2,5,4,1,7,6] => [1,3,5,2,4,6,7] => [1,5,3,2,4,6,7] => ? = 6
[1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [3,2,5,4,6,1,7] => [1,3,5,6,2,4,7] => [1,6,3,2,5,4,7] => ? = 6
[1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [3,2,5,4,6,7,1] => [1,3,5,6,7,2,4] => [1,7,3,2,5,6,4] => ? = 6
[1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,7,6,1] => [1,3,5,7,2,4,6] => [1,7,3,2,5,4,6] => ? = 6
[1,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> [3,2,5,6,4,1,7] => [1,3,5,4,6,2,7] => [1,6,3,5,4,2,7] => ? = 5
[1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [3,2,5,6,4,7,1] => [1,3,5,4,6,7,2] => [1,7,3,5,4,6,2] => ? = 5
[1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [3,2,5,6,7,4,1] => [1,3,5,7,2,4,6] => [1,7,3,2,5,4,6] => ? = 6
[1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [3,2,5,7,6,4,1] => [1,3,5,6,4,7,2] => [1,7,3,6,5,4,2] => ? = 5
[1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [3,2,6,5,4,1,7] => [1,3,6,2,4,5,7] => [1,6,3,2,4,5,7] => ? = 6
[1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [3,2,6,5,4,7,1] => [1,3,6,7,2,4,5] => [1,7,3,2,4,6,5] => ? = 6
[1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [3,2,6,5,7,4,1] => [1,3,6,4,5,7,2] => [1,7,3,6,4,5,2] => ? = 5
[1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [3,2,6,7,5,4,1] => [1,3,6,4,7,2,5] => [1,7,3,6,4,2,5] => ? = 5
[1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [3,2,7,6,5,4,1] => [1,3,7,2,4,6,5] => [1,6,3,2,7,4,5] => ? = 5
[1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [3,4,5,2,1,6,7] => [1,3,5,2,4,6,7] => [1,5,3,2,4,6,7] => ? = 6
[1,1,1,0,1,0,1,0,0,0,1,1,0,0]
=> [3,4,5,2,1,7,6] => [1,3,5,2,4,6,7] => [1,5,3,2,4,6,7] => ? = 6
[1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [3,4,5,2,6,1,7] => [1,3,5,6,2,4,7] => [1,6,3,2,5,4,7] => ? = 6
[1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [3,4,5,2,6,7,1] => [1,3,5,6,7,2,4] => [1,7,3,2,5,6,4] => ? = 6
[1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [3,4,5,2,7,6,1] => [1,3,5,7,2,4,6] => [1,7,3,2,5,4,6] => ? = 6
[1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [3,4,5,6,2,1,7] => [1,3,5,2,4,6,7] => [1,5,3,2,4,6,7] => ? = 6
[1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [3,4,5,6,2,7,1] => [1,3,5,2,4,6,7] => [1,5,3,2,4,6,7] => ? = 6
[1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [3,4,5,6,7,2,1] => [1,3,5,7,2,4,6] => [1,7,3,2,5,4,6] => ? = 6
[1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [3,4,5,7,6,2,1] => [1,3,5,6,2,4,7] => [1,6,3,2,5,4,7] => ? = 6
[1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [3,4,6,5,2,1,7] => [1,3,6,2,4,5,7] => [1,6,3,2,4,5,7] => ? = 6
[1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [3,4,6,5,2,7,1] => [1,3,6,7,2,4,5] => [1,7,3,2,4,6,5] => ? = 6
[1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [3,4,6,5,7,2,1] => [1,3,6,2,4,5,7] => [1,6,3,2,4,5,7] => ? = 6
[1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [3,4,6,7,5,2,1] => [1,3,6,2,4,7,5] => [1,7,3,2,6,4,5] => ? = 5
[1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [3,4,7,6,5,2,1] => [1,3,7,2,4,6,5] => [1,6,3,2,7,4,5] => ? = 5
[1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [3,5,4,6,2,1,7] => [1,3,4,6,2,5,7] => [1,6,3,4,2,5,7] => ? = 6
[1,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> [3,5,4,6,2,7,1] => [1,3,4,6,7,2,5] => [1,7,3,4,2,6,5] => ? = 6
[1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [3,5,4,6,7,2,1] => [1,3,4,6,2,5,7] => [1,6,3,4,2,5,7] => ? = 6
[1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,5,4,7,6,2,1] => [1,3,4,7,2,5,6] => [1,7,3,4,2,5,6] => ? = 6
[1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [3,5,6,4,2,1,7] => [1,3,6,2,5,4,7] => [1,5,3,6,2,4,7] => ? = 5
[1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [3,5,6,4,2,7,1] => [1,3,6,7,2,5,4] => [1,5,3,7,2,6,4] => ? = 5
[1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [3,5,6,4,7,2,1] => [1,3,6,2,5,7,4] => [1,7,3,6,2,5,4] => ? = 5
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [3,5,6,7,4,2,1] => [1,3,6,2,5,4,7] => [1,5,3,6,2,4,7] => ? = 5
[1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [3,5,7,6,4,2,1] => [1,3,7,2,5,4,6] => [1,5,3,7,2,4,6] => ? = 5
[1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [3,6,5,4,2,1,7] => [1,3,5,2,6,4,7] => [1,6,3,5,2,4,7] => ? = 5
[1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [3,6,5,4,2,7,1] => [1,3,5,2,6,7,4] => [1,7,3,5,2,6,4] => ? = 5
[1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [3,6,5,4,7,2,1] => [1,3,5,7,2,6,4] => [1,6,3,7,5,2,4] => ? = 5
Description
The number of weak deficiencies of a permutation.
This is defined as
$$\operatorname{wdec}(\sigma)=\#\{i:\sigma(i) \leq i\}.$$
The number of weak exceedances is [[St000213]], the number of deficiencies is [[St000703]].
Matching statistic: St001330
Values
[1,0]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
[1,0,1,0]
=> ([(0,1)],2)
=> ([],2)
=> ([(0,1)],2)
=> 2
[1,1,0,0]
=> ([(0,1)],2)
=> ([],2)
=> ([(0,1)],2)
=> 2
[1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2
[1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
[1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
[1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
[1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
[1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
[1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
[1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
[1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
[1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
[1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
[1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
[1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
[1,1,1,0,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(3,6),(4,5)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,5),(1,6),(1,7),(2,3),(2,4),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,5),(1,6),(1,7),(2,3),(2,4),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,1,1,1,1,0,0,0,0,0]
=> ([(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)
=> ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3
[1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,0,1,0,1,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,0,1,0,1,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,0,1,0,1,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,0,1,0,1,0,1,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[1,0,1,0,1,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,0,1,0,1,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,0,1,0,1,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,0,1,0,1,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,0,1,0,1,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[1,0,1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[1,0,1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[1,0,1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ([(4,7),(5,6),(6,7)],8)
=> ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,0,1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ([(4,7),(5,6),(6,7)],8)
=> ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,0,1,1,0,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,0,1,1,0,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,0,1,1,0,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,0,1,1,0,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,0,1,1,0,0,1,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[1,0,1,1,0,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,0,1,1,0,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,0,1,1,0,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,0,1,1,0,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[1,0,1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[1,0,1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[1,0,1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ([(4,7),(5,6),(6,7)],8)
=> ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,0,1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ([(4,7),(5,6),(6,7)],8)
=> ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,0,1,1,1,0,0,0,1,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[1,0,1,1,1,0,0,0,1,1,0,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[1,0,1,1,1,0,0,1,0,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[1,0,1,1,1,0,0,1,0,1,0,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[1,0,1,1,1,0,0,1,1,0,0,0]
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ([(4,7),(5,6)],8)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,1,0,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,1,0,0,1,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,1,0,0,1,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St000213
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000213: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 46%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000213: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 46%
Values
[1,0]
=> [1] => [1] => [1] => 1
[1,0,1,0]
=> [1,2] => [1,2] => [1,2] => 2
[1,1,0,0]
=> [2,1] => [1,2] => [1,2] => 2
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 3
[1,0,1,1,0,0]
=> [1,3,2] => [1,2,3] => [1,2,3] => 3
[1,1,0,0,1,0]
=> [2,1,3] => [1,2,3] => [1,2,3] => 3
[1,1,0,1,0,0]
=> [2,3,1] => [1,2,3] => [1,2,3] => 3
[1,1,1,0,0,0]
=> [3,2,1] => [1,3,2] => [1,3,2] => 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 4
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 4
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 4
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 4
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 4
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 4
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 4
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 3
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [1,3,2,4] => [1,3,2,4] => 3
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,3,4,2] => [1,4,3,2] => 3
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [1,3,2,4] => [1,3,2,4] => 3
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,4,2,3] => [1,3,4,2] => 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,4,3] => 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,2,4,3,5] => [1,2,4,3,5] => 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,2,5,3,4] => [1,2,4,5,3] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => 4
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => 4
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [1,2,4,3,5] => [1,2,4,3,5] => 4
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,2,4,5,3] => [1,2,5,4,3] => 4
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [1,2,4,3,5] => [1,2,4,3,5] => 4
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,2,5,3,4] => [1,2,4,5,3] => 4
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 7
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 7
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,6,5,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 7
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 7
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => ? = 6
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,3,5,4,6,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 7
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,5,4,7,6] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 7
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,3,5,6,4,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 7
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 7
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,3,5,7,6,4] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => ? = 6
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,5,4,7] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => ? = 6
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,5,7,4] => [1,2,3,4,6,7,5] => [1,2,3,4,7,6,5] => ? = 6
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,6,7,5,4] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => ? = 6
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,6,5,4] => [1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => ? = 6
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,4,3,5,6,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 7
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,4,3,5,7,6] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 7
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 7
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,4,3,6,7,5] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 7
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,7,6,5] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => ? = 6
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,2,4,5,3,6,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 7
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,2,4,5,3,7,6] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 7
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,4,5,6,3,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 7
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 7
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,2,4,5,7,6,3] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => ? = 6
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,4,6,5,3,7] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => ? = 6
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,4,6,5,7,3] => [1,2,3,4,6,7,5] => [1,2,3,4,7,6,5] => ? = 6
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,4,6,7,5,3] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => ? = 6
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,4,7,6,5,3] => [1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => ? = 6
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,5,4,3,6,7] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ? = 6
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,5,4,3,7,6] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ? = 6
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,2,5,4,6,3,7] => [1,2,3,5,6,4,7] => [1,2,3,6,5,4,7] => ? = 6
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,4,6,7,3] => [1,2,3,5,6,7,4] => [1,2,3,7,5,6,4] => ? = 6
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,2,5,4,7,6,3] => [1,2,3,5,7,4,6] => [1,2,3,6,5,7,4] => ? = 6
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,5,6,4,3,7] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ? = 6
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,5,6,4,7,3] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ? = 6
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,5,6,7,4,3] => [1,2,3,5,7,4,6] => [1,2,3,6,5,7,4] => ? = 6
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,5,7,6,4,3] => [1,2,3,5,6,4,7] => [1,2,3,6,5,4,7] => ? = 6
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,5,4,3,7] => [1,2,3,6,4,5,7] => [1,2,3,5,6,4,7] => ? = 6
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,6,5,4,7,3] => [1,2,3,6,7,4,5] => [1,2,3,5,7,6,4] => ? = 6
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,6,5,7,4,3] => [1,2,3,6,4,5,7] => [1,2,3,5,6,4,7] => ? = 6
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,6,7,5,4,3] => [1,2,3,6,4,7,5] => [1,2,3,6,7,4,5] => ? = 5
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,6,5,4,3] => [1,2,3,7,4,6,5] => [1,2,3,6,7,5,4] => ? = 5
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 7
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,2,4,5,7,6] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 7
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,6,5,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 7
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,3,2,4,6,7,5] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 7
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,2,4,7,6,5] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => ? = 6
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,4,6,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 7
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 7
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,2,5,6,4,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 7
Description
The number of weak exceedances (also weak excedences) of a permutation.
This is defined as
$$\operatorname{wex}(\sigma)=\#\{i:\sigma(i) \geq i\}.$$
The number of weak exceedances is given by the number of exceedances (see [[St000155]]) plus the number of fixed points (see [[St000022]]) of $\sigma$.
Matching statistic: St000325
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000325: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 46%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000325: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 46%
Values
[1,0]
=> [1] => [1] => [1] => 1
[1,0,1,0]
=> [1,2] => [1,2] => [2,1] => 2
[1,1,0,0]
=> [2,1] => [1,2] => [2,1] => 2
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [3,2,1] => 3
[1,0,1,1,0,0]
=> [1,3,2] => [1,2,3] => [3,2,1] => 3
[1,1,0,0,1,0]
=> [2,1,3] => [1,2,3] => [3,2,1] => 3
[1,1,0,1,0,0]
=> [2,3,1] => [1,2,3] => [3,2,1] => 3
[1,1,1,0,0,0]
=> [3,2,1] => [1,3,2] => [2,3,1] => 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3,4] => [4,3,2,1] => 4
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,2,3,4] => [4,3,2,1] => 4
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,2,3,4] => [4,3,2,1] => 4
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,2,4,3] => [3,4,2,1] => 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,2,3,4] => [4,3,2,1] => 4
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,2,3,4] => [4,3,2,1] => 4
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,2,3,4] => [4,3,2,1] => 4
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,3,4] => [4,3,2,1] => 4
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,2,4,3] => [3,4,2,1] => 3
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [1,3,2,4] => [4,2,3,1] => 3
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,3,4,2] => [2,4,3,1] => 3
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [1,3,2,4] => [4,2,3,1] => 3
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,4,2,3] => [3,2,4,1] => 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,3,5,4] => [4,5,3,2,1] => 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,2,3,5,4] => [4,5,3,2,1] => 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,2,4,3,5] => [5,3,4,2,1] => 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,2,4,5,3] => [3,5,4,2,1] => 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,2,4,3,5] => [5,3,4,2,1] => 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,2,5,3,4] => [4,3,5,2,1] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,2,3,5,4] => [4,5,3,2,1] => 4
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,2,3,5,4] => [4,5,3,2,1] => 4
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [1,2,4,3,5] => [5,3,4,2,1] => 4
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,2,4,5,3] => [3,5,4,2,1] => 4
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [1,2,4,3,5] => [5,3,4,2,1] => 4
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,2,5,3,4] => [4,3,5,2,1] => 4
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 7
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 7
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,6,5,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 7
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 7
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => ? = 6
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,3,5,4,6,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 7
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,5,4,7,6] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 7
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,3,5,6,4,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 7
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 7
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,3,5,7,6,4] => [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => ? = 6
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,5,4,7] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => ? = 6
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,5,7,4] => [1,2,3,4,6,7,5] => [5,7,6,4,3,2,1] => ? = 6
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,6,7,5,4] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => ? = 6
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,6,5,4] => [1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => ? = 6
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,4,3,5,6,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 7
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,4,3,5,7,6] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 7
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 7
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,4,3,6,7,5] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 7
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,7,6,5] => [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => ? = 6
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,2,4,5,3,6,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 7
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,2,4,5,3,7,6] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 7
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,4,5,6,3,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 7
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 7
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,2,4,5,7,6,3] => [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => ? = 6
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,4,6,5,3,7] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => ? = 6
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,4,6,5,7,3] => [1,2,3,4,6,7,5] => [5,7,6,4,3,2,1] => ? = 6
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,4,6,7,5,3] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => ? = 6
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,4,7,6,5,3] => [1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => ? = 6
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,5,4,3,6,7] => [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => ? = 6
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,5,4,3,7,6] => [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => ? = 6
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,2,5,4,6,3,7] => [1,2,3,5,6,4,7] => [7,4,6,5,3,2,1] => ? = 6
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,4,6,7,3] => [1,2,3,5,6,7,4] => [4,7,6,5,3,2,1] => ? = 6
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,2,5,4,7,6,3] => [1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => ? = 6
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,5,6,4,3,7] => [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => ? = 6
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,5,6,4,7,3] => [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => ? = 6
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,5,6,7,4,3] => [1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => ? = 6
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,5,7,6,4,3] => [1,2,3,5,6,4,7] => [7,4,6,5,3,2,1] => ? = 6
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,5,4,3,7] => [1,2,3,6,4,5,7] => [7,5,4,6,3,2,1] => ? = 6
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,6,5,4,7,3] => [1,2,3,6,7,4,5] => [5,4,7,6,3,2,1] => ? = 6
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,6,5,7,4,3] => [1,2,3,6,4,5,7] => [7,5,4,6,3,2,1] => ? = 6
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,6,7,5,4,3] => [1,2,3,6,4,7,5] => [5,7,4,6,3,2,1] => ? = 5
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,6,5,4,3] => [1,2,3,7,4,6,5] => [5,6,4,7,3,2,1] => ? = 5
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 7
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,2,4,5,7,6] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 7
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,6,5,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 7
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,3,2,4,6,7,5] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 7
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,2,4,7,6,5] => [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => ? = 6
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,4,6,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 7
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 7
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,2,5,6,4,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 7
Description
The width of the tree associated to a permutation.
A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1].
The width of the tree is given by the number of leaves of this tree.
Note that, due to the construction of this tree, the width of the tree is always one more than the number of descents [[St000021]]. This also matches the number of runs in a permutation [[St000470]].
See also [[St000308]] for the height of this tree.
The following 18 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000470The number of runs in a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000619The number of cyclic descents of a permutation. St000454The largest eigenvalue of a graph if it is integral. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. 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. St001626The number of maximal proper sublattices of a lattice. St001960The number of descents of a permutation minus one if its first entry is not one. St001863The number of weak excedances of a signed permutation. St000717The number of ordinal summands of a poset. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St000216The absolute length of a permutation. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001668The number of points of the poset minus the width of the poset. St001935The number of ascents in a parking function. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
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