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Your data matches 12 different statistics following compositions of up to 3 maps.
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Matching statistic: St000074
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St000074: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St000074: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0,1,0]
=> [2,1]
=> [[1,3],[2]]
=> [[2,1,0],[1,1],[1]]
=> 0
[1,0,1,1,0,0]
=> [1,1]
=> [[1],[2]]
=> [[1,1],[1]]
=> 0
[1,1,0,0,1,0]
=> [2]
=> [[1,2]]
=> [[2,0],[1]]
=> 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [[3,2,1,0,0,0],[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> 0
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> 0
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[3,1,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [[2,1,1,0],[1,1,1],[1,1],[1]]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [[1,2,5],[3,4]]
=> [[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [[1,3,4],[2]]
=> [[3,1,0,0],[2,1,0],[1,1],[1]]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [[1,3],[2]]
=> [[2,1,0],[1,1],[1]]
=> 0
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1],[2]]
=> [[1,1],[1]]
=> 0
[1,1,1,0,0,0,1,0]
=> [3]
=> [[1,2,3]]
=> [[3,0,0],[2,0],[1]]
=> 2
[1,1,1,0,0,1,0,0]
=> [2]
=> [[1,2]]
=> [[2,0],[1]]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [[2,2,2,1,0,0,0],[2,2,1,1,0,0],[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [[3,2,1,1,0,0,0],[2,2,1,1,0,0],[2,1,1,1,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [[2,2,1,1,0,0],[2,1,1,1,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [[4,1,1,1,0,0,0],[3,1,1,1,0,0],[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [[3,1,1,1,0,0],[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,1,1,1],[1,1,1],[1,1],[1]]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [[3,2,2,0,0,0,0],[2,2,2,0,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [[2,2,2,0,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [[3,3,1,0,0,0,0],[3,2,1,0,0,0],[3,1,1,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [[4,2,1,0,0,0,0],[3,2,1,0,0,0],[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [[3,2,1,0,0,0],[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [[4,1,1,0,0,0],[3,1,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[3,1,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [[2,1,1,0],[1,1,1],[1,1],[1]]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [[1,2,3,7],[4,5,6]]
=> [[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 4
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> [[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 4
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> [[4,2,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> 3
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [[1,2,5],[3,4]]
=> [[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> 2
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [[1,3,4,5],[2]]
=> [[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [[1,3,4],[2]]
=> [[3,1,0,0],[2,1,0],[1,1],[1]]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [[1,3],[2]]
=> [[2,1,0],[1,1],[1]]
=> 0
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [[1],[2]]
=> [[1,1],[1]]
=> 0
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [[1,2,3,4]]
=> [[4,0,0,0],[3,0,0],[2,0],[1]]
=> 3
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [[1,2,3]]
=> [[3,0,0],[2,0],[1]]
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [[1,2]]
=> [[2,0],[1]]
=> 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [[2,2,1,1,1,0,0],[2,1,1,1,1,0],[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> 0
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [[3,1,1,1,1,0,0],[2,1,1,1,1,0],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [[2,1,1,1,1,0],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> 0
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [[2,2,2,1,0,0,0],[2,2,1,1,0,0],[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> 0
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [[3,2,1,1,0,0,0],[2,2,1,1,0,0],[2,1,1,1,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> 0
Description
The number of special entries.
An entry $a_{i,j}$ of a Gelfand-Tsetlin pattern is special if $a_{i-1,j-i} > a_{i,j} > a_{i-1,j}$. That is, it is neither boxed nor circled.
Matching statistic: St000441
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000441: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000441: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0,1,0]
=> [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 0
[1,0,1,1,0,0]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[1,1,0,0,1,0]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => 0
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 0
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 0
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 2
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 0
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[1,1,1,0,0,0,1,0]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 2
[1,1,1,0,0,1,0,0]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 3
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 0
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => 4
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => 4
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => 3
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 2
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 2
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 0
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 3
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 2
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5] => 0
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 0
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => 0
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => 0
Description
The number of successions of a permutation.
A succession of a permutation $\pi$ is an index $i$ such that $\pi(i)+1 = \pi(i+1)$. Successions are also known as ''small ascents'' or ''1-rises''.
Matching statistic: St000502
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000502: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000502: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0,1,0]
=> [2,1]
=> [[1,3],[2]]
=> {{1,3},{2}}
=> 0
[1,0,1,1,0,0]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0
[1,1,0,0,1,0]
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> {{1,3,6},{2,5},{4}}
=> 0
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> {{1,3},{2,5},{4}}
=> 0
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> 0
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> 2
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [[1,3],[2]]
=> {{1,3},{2}}
=> 0
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0
[1,1,1,0,0,0,1,0]
=> [3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 2
[1,1,1,0,0,1,0,0]
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> {{1,3},{2,5},{4,7},{6}}
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> {{1,4,7},{2,6},{3},{5}}
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> {{1,4},{2,6},{3},{5}}
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> {{1,5,6,7},{2},{3},{4}}
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> {{1,5,6},{2},{3},{4}}
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> {{1,5},{2},{3},{4}}
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> {{1,2,7},{3,4},{5,6}}
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> {{1,2},{3,4},{5,6}}
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> {{1,3,4},{2,6,7},{5}}
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> {{1,3,6,7},{2,5},{4}}
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> {{1,3,6},{2,5},{4}}
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> {{1,3},{2,5},{4}}
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> {{1,4,5,6},{2},{3}}
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [[1,2,3,7],[4,5,6]]
=> {{1,2,3,7},{4,5,6}}
=> 4
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> {{1,2,3},{4,5,6}}
=> 4
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> {{1,2,5,6},{3,4}}
=> 3
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 2
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [[1,3],[2]]
=> {{1,3},{2}}
=> 0
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [[1,2,3,4]]
=> {{1,2,3,4}}
=> 3
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> {{1,5},{2,7},{3},{4},{6}}
=> 0
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> {{1,6,7},{2},{3},{4},{5}}
=> 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> {{1,6},{2},{3},{4},{5}}
=> 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> 0
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> {{1,3},{2,5},{4,7},{6}}
=> 0
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> {{1,4,7},{2,6},{3},{5}}
=> 0
Description
The number of successions of a set partitions.
This is the number of indices $i$ such that $i$ and $i+1$ belonging to the same block.
Matching statistic: St000355
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000355: Permutations ⟶ ℤResult quality: 62% ●values known / values provided: 87%●distinct values known / distinct values provided: 62%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000355: Permutations ⟶ ℤResult quality: 62% ●values known / values provided: 87%●distinct values known / distinct values provided: 62%
Values
[1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[1,1,0,0,1,0]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 0
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 3
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 0
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 4
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 4
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => 3
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,2,1,5] => 2
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 3
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,4,6,5,3,2] => 0
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,5,4,6,3,2] => 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5,6,4,3,2] => 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => 0
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 0
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => ? = 0
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => ? = 0
[1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [4,6,5,3,2,1,7] => ? = 4
[1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [5,4,6,3,2,1,7] => ? = 5
[1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [5,6,4,3,2,1,7] => ? = 4
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => ? = 5
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => ? = 0
[1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => ? = 0
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => ? = 0
[1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0]
=> [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [4,6,5,3,2,1,7] => ? = 4
[1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> [6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [5,4,6,3,2,1,7] => ? = 5
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,7,5,4,3,2,1,8] => ? = 5
[1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [5,6,4,3,2,1,7] => ? = 4
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,6,5,4,3,2,1,8] => ? = 6
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => ? = 5
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,7,6,5,4,3,2,1,9] => ? = 7
[1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => ? = 0
[1,1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => ? = 0
[1,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => ? = 0
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0,0]
=> [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [4,6,5,3,2,1,7] => ? = 4
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0,0]
=> [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,7,5,4,3,2,1,8] => ? = 5
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0,0]
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [5,6,4,3,2,1,7] => ? = 4
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,6,5,4,3,2,1,8] => ? = 6
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => ? = 5
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,0]
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,7,6,5,4,3,2,1,9] => ? = 7
[1,1,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => ? = 0
[1,1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => ? = 0
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0,0]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,6,5,4,3,2,1,8] => ? = 6
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0,0]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => ? = 5
[1,1,1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => ? = 0
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0,0,0]
=> [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,7,5,4,3,2,1,8] => ? = 5
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0,0,0]
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [5,6,4,3,2,1,7] => ? = 4
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0,0,0,0]
=> [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,7,5,4,3,2,1,8] => ? = 5
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0,0,0,0]
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [5,6,4,3,2,1,7] => ? = 4
Description
The number of occurrences of the pattern 21-3.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $21\!\!-\!\!3$.
Matching statistic: St000586
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00112: Set partitions —complement⟶ Set partitions
St000586: Set partitions ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 88%
Mp00112: Set partitions —complement⟶ Set partitions
St000586: Set partitions ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 88%
Values
[1,0,1,0,1,0]
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> 0
[1,0,1,1,0,0]
=> {{1},{2,3}}
=> {{1,2},{3}}
=> 0
[1,1,0,0,1,0]
=> {{1,2},{3}}
=> {{1},{2,3}}
=> 1
[1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 0
[1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> {{1,2},{3},{4}}
=> 0
[1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> 1
[1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> {{1,3},{2},{4}}
=> 0
[1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> {{1,2,3},{4}}
=> 0
[1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> {{1},{2},{3,4}}
=> 2
[1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> {{1,2},{3,4}}
=> 2
[1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> 1
[1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> 0
[1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> {{1,2,4},{3}}
=> 0
[1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> {{1},{2,3,4}}
=> 2
[1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> {{1,4},{2,3}}
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> {{1,2,3},{4},{5}}
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> {{1,4},{2},{3},{5}}
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> {{1,2,4},{3},{5}}
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> {{1},{2,3,4},{5}}
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> {{1,4},{2,3},{5}}
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> {{1,3,4},{2},{5}}
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> {{1,2,3,4},{5}}
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> {{1,3},{2},{4,5}}
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> {{1,2,3},{4,5}}
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> {{1,2},{3,5},{4}}
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> {{1},{2,5},{3},{4}}
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> {{1,5},{2},{3},{4}}
=> {{1,5},{2},{3},{4}}
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> {{1,4,5},{2},{3}}
=> {{1,2,5},{3},{4}}
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> {{1},{2,3,5},{4}}
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> {{1,5},{2},{3,4}}
=> {{1,5},{2,3},{4}}
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> {{1,3,5},{2},{4}}
=> {{1,3,5},{2},{4}}
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> {{1,2,3,5},{4}}
=> 0
[1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> {{1},{2},{3,4,5}}
=> 4
[1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> {{1,2},{3,4,5}}
=> 4
[1,1,1,0,0,1,0,0,1,0]
=> {{1,4},{2,3},{5}}
=> {{1},{2,5},{3,4}}
=> 3
[1,1,1,0,0,1,0,1,0,0]
=> {{1,5},{2,3},{4}}
=> {{1,5},{2},{3,4}}
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> {{1,4,5},{2,3}}
=> {{1,2,5},{3,4}}
=> 2
[1,1,1,0,1,0,0,0,1,0]
=> {{1,2,4},{3},{5}}
=> {{1},{2,4,5},{3}}
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> {{1,5},{2,4},{3}}
=> {{1,5},{2,4},{3}}
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> {{1,2,5},{3},{4}}
=> {{1,4,5},{2},{3}}
=> 0
[1,1,1,0,1,1,0,0,0,0]
=> {{1,2,4,5},{3}}
=> {{1,2,4,5},{3}}
=> 0
[1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> {{1},{2,3,4,5}}
=> 3
[1,1,1,1,0,0,0,1,0,0]
=> {{1,5},{2,3,4}}
=> {{1,5},{2,3,4}}
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> {{1,2,5},{3,4}}
=> {{1,4,5},{2,3}}
=> 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> {{1},{2,3,5,6},{4}}
=> {{1,2,4,5},{3},{6}}
=> 0
[1,0,1,1,1,1,0,0,1,0,0,0]
=> {{1},{2,3,6},{4,5}}
=> {{1,4,5},{2,3},{6}}
=> 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> {{1},{2,3,4,6},{5}}
=> {{1,3,4,5},{2},{6}}
=> 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2,3,4,5,6}}
=> {{1,2,3,4,5},{6}}
=> 0
[1,1,0,1,0,1,1,1,0,0,0,0]
=> {{1,4,5,6},{2},{3}}
=> {{1,2,3,6},{4},{5}}
=> 0
[1,1,0,1,1,0,1,0,1,0,0,0]
=> {{1,3,6},{2},{4},{5}}
=> {{1,4,6},{2},{3},{5}}
=> 0
[1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> {{1,3,4,5,6,8},{2},{7}}
=> {{1,3,4,5,6,8},{2},{7}}
=> ? = 0
[1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> {{1,2,4,5,7,8},{3},{6}}
=> {{1,2,4,5,7,8},{3},{6}}
=> ? = 0
[1,1,1,0,1,1,1,1,0,0,1,0,0,0,0,0]
=> {{1,2,4,5,8},{3},{6,7}}
=> {{1,4,5,7,8},{2,3},{6}}
=> ? = 1
[1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]
=> {{1,2,4,5,6,8},{3},{7}}
=> {{1,3,4,5,7,8},{2},{6}}
=> ? = 0
[1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> {{1,2,3,6,7,8},{4},{5}}
=> {{1,2,3,6,7,8},{4},{5}}
=> ? = 0
[1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0]
=> {{1,2,3,5,8},{4},{6},{7}}
=> {{1,4,6,7,8},{2},{3},{5}}
=> ? = 0
[1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> {{1,2,3,5,7,8},{4},{6}}
=> {{1,2,4,6,7,8},{3},{5}}
=> ? = 0
[1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0]
=> {{1,2,3,8},{4},{5,6,7}}
=> {{1,6,7,8},{2,3,4},{5}}
=> ? = 2
[1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0]
=> {{1,2,3,5,8},{4},{6,7}}
=> {{1,4,6,7,8},{2,3},{5}}
=> ? = 1
[1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
=> {{1,2,3,5,6,8},{4},{7}}
=> {{1,3,4,6,7,8},{2},{5}}
=> ? = 0
[1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0]
=> {{1,2,3,6,8},{4,5},{7}}
=> {{1,3,6,7,8},{2},{4,5}}
=> ? = 3
[1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0]
=> {{1,2,3,7,8},{4,6},{5}}
=> {{1,2,6,7,8},{3,5},{4}}
=> ? = 2
[1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
=> {{1,2,3,8},{4,7},{5},{6}}
=> {{1,6,7,8},{2,5},{3},{4}}
=> ? = 1
[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5},{6},{7}}
=> {{1,5,6,7,8},{2},{3},{4}}
=> ? = 0
[1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> {{1,2,3,4,7,8},{5},{6}}
=> {{1,2,5,6,7,8},{3},{4}}
=> ? = 0
[1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0]
=> {{1,8},{2,3,4,6,7},{5}}
=> {{1,8},{2,3,5,6,7},{4}}
=> ? = 4
[1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0]
=> {{1,2,8},{3,4,6,7},{5}}
=> {{1,7,8},{2,3,5,6},{4}}
=> ? = 3
[1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
=> {{1,2,3,8},{4,6,7},{5}}
=> {{1,6,7,8},{2,3,5},{4}}
=> ? = 2
[1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5},{6,7}}
=> {{1,5,6,7,8},{2,3},{4}}
=> ? = 1
[1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> {{1,2,3,8},{4,5,6},{7}}
=> {{1,6,7,8},{2},{3,4,5}}
=> ? = 4
[1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> {{1,2,3,7,8},{4,5,6}}
=> {{1,2,6,7,8},{3,4,5}}
=> ? = 4
[1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> {{1,8},{2,3,4,7},{5,6}}
=> {{1,8},{2,5,6,7},{3,4}}
=> ? = 5
[1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0]
=> {{1,2,8},{3,4,7},{5,6}}
=> {{1,7,8},{2,5,6},{3,4}}
=> ? = 4
[1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> {{1,2,3,8},{4,7},{5,6}}
=> {{1,6,7,8},{2,5},{3,4}}
=> ? = 3
[1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5,6},{7}}
=> {{1,5,6,7,8},{2},{3,4}}
=> ? = 2
[1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
=> {{1,8},{2,3,4,5,7},{6}}
=> {{1,8},{2,4,5,6,7},{3}}
=> ? = 4
[1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0]
=> {{1,2,8},{3,4,5,7},{6}}
=> {{1,7,8},{2,4,5,6},{3}}
=> ? = 3
[1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> {{1,2,3,8},{4,5,7},{6}}
=> {{1,6,7,8},{2,4,5},{3}}
=> ? = 2
[1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5,7},{6}}
=> {{1,5,6,7,8},{2,4},{3}}
=> ? = 1
[1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> {{1,2,8},{3,4,5,6,7}}
=> {{1,7,8},{2,3,4,5,6}}
=> ? = 4
[1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> {{1,2,3,8},{4,5,6,7}}
=> {{1,6,7,8},{2,3,4,5}}
=> ? = 3
[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5,6,7}}
=> {{1,5,6,7,8},{2,3,4}}
=> ? = 2
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,6,7,8},{9}}
=> {{1},{2,3,4,5,6,7,8,9}}
=> ? = 7
[1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> {{1,3,4,5,6,7,8,9},{2}}
=> {{1,2,3,4,5,6,7,9},{8}}
=> ? = 0
[1,1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> {{1,2,4,5,6,7,9},{3},{8}}
=> ?
=> ? = 0
[1,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> {{1,2,4,5,6,7,8,9},{3}}
=> {{1,2,3,4,5,6,8,9},{7}}
=> ? = 0
[1,1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> {{1,2,3,5,6,8,9},{4},{7}}
=> {{1,2,4,5,7,8,9},{3},{6}}
=> ? = 0
[1,1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> {{1,2,3,5,6,7,9},{4},{8}}
=> ?
=> ? = 0
[1,1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> {{1,2,3,5,6,7,8,9},{4}}
=> {{1,2,3,4,5,7,8,9},{6}}
=> ? = 0
[1,1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0]
=> {{1,2,3,4,7,8,9},{5},{6}}
=> ?
=> ? = 0
[1,1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0,0]
=> {{1,2,3,4,6,9},{5},{7},{8}}
=> ?
=> ? = 0
[1,1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0,0]
=> {{1,2,3,4,6,8,9},{5},{7}}
=> {{1,2,4,6,7,8,9},{3},{5}}
=> ? = 0
[1,1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0,0]
=> {{1,2,3,4,9},{5},{6,7,8}}
=> {{1,6,7,8,9},{2,3,4},{5}}
=> ? = 2
[1,1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0,0]
=> {{1,2,3,4,6,9},{5},{7,8}}
=> ?
=> ? = 1
[1,1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0,0]
=> {{1,2,3,4,6,7,9},{5},{8}}
=> {{1,3,4,6,7,8,9},{2},{5}}
=> ? = 0
[1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0]
=> {{1,2,3,4,6,7,8,9},{5}}
=> {{1,2,3,4,6,7,8,9},{5}}
=> ? = 0
[1,1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0,0]
=> {{1,2,3,4,7,9},{5,6},{8}}
=> {{1,3,6,7,8,9},{2},{4,5}}
=> ? = 3
[1,1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0,0]
=> {{1,2,3,4,7,8,9},{5,6}}
=> ?
=> ? = 3
[1,1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0,0]
=> {{1,2,3,4,8,9},{5,7},{6}}
=> {{1,2,6,7,8,9},{3,5},{4}}
=> ? = 2
[1,1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0,0]
=> {{1,2,3,4,9},{5,8},{6},{7}}
=> ?
=> ? = 1
Description
The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal.
Matching statistic: St000359
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000359: Permutations ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
St000359: Permutations ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => 0
[1,0,1,1,0,0]
=> [2,3,1] => [3,1,2] => 0
[1,1,0,0,1,0]
=> [3,1,2] => [2,3,1] => 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,3,2,1] => 0
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [4,3,1,2] => 0
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [4,2,3,1] => 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [4,2,1,3] => 0
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,1,2,3] => 0
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [3,4,2,1] => 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,4,1,2] => 2
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [3,2,4,1] => 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1,4] => 0
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,1,2,4] => 0
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,3,4,1] => 2
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,3,1,4] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [5,4,1,2,3] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [5,3,2,1,4] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [5,3,1,2,4] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [5,2,3,4,1] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [5,2,3,1,4] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [5,2,1,3,4] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [4,5,2,1,3] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [4,5,1,2,3] => 3
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [4,3,5,1,2] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [4,3,2,5,1] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,3,2,1,5] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [4,3,1,2,5] => 0
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [4,2,3,5,1] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [4,2,3,1,5] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [4,2,1,3,5] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => 0
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [3,4,5,2,1] => 4
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [3,4,5,1,2] => 4
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [3,4,2,5,1] => 3
[1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [3,4,2,1,5] => 2
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,4,1,2,5] => 2
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [3,2,4,5,1] => 2
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => [3,2,4,1,5] => 1
[1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [3,2,1,4,5] => 0
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => 0
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [2,3,4,5,1] => 3
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [2,3,4,1,5] => 2
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [2,3,1,4,5] => 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => [6,3,1,2,4,5] => 0
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => [6,2,3,1,4,5] => 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => [6,2,1,3,4,5] => 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [6,1,2,3,4,5] => 0
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [3,4,5,2,1,6] => [5,4,1,2,3,6] => 0
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [4,3,2,5,1,6] => [5,3,2,1,4,6] => 0
[1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,1,7] => [6,3,1,2,4,5,7] => ? = 0
[1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [4,2,3,5,6,1,7] => [6,2,3,1,4,5,7] => ? = 1
[1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,1,7] => [6,2,1,3,4,5,7] => ? = 0
[1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,1,6,7] => [5,4,1,2,3,6,7] => ? = 0
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [4,3,2,5,1,6,7] => [5,3,2,1,4,6,7] => ? = 0
[1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,1,6,7] => [5,3,1,2,4,6,7] => ? = 0
[1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [5,2,3,4,1,6,7] => [5,2,3,4,1,6,7] => ? = 2
[1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [4,2,3,5,1,6,7] => [5,2,3,1,4,6,7] => ? = 1
[1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,1,6,7] => [5,2,1,3,4,6,7] => ? = 0
[1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [4,3,5,1,2,6,7] => [4,5,2,1,3,6,7] => ? = 3
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [3,4,5,1,2,6,7] => [4,5,1,2,3,6,7] => ? = 3
[1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [4,5,2,1,3,6,7] => [4,3,5,1,2,6,7] => ? = 2
[1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [5,3,2,1,4,6,7] => [4,3,2,5,1,6,7] => ? = 1
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [3,4,2,1,5,6,7] => [4,3,1,2,5,6,7] => ? = 0
[1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [6,2,3,1,4,5,7] => [4,2,3,5,6,1,7] => ? = 3
[1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [5,2,3,1,4,6,7] => [4,2,3,5,1,6,7] => ? = 2
[1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [4,2,3,1,5,6,7] => [4,2,3,1,5,6,7] => ? = 1
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [3,2,4,1,5,6,7] => [4,2,1,3,5,6,7] => ? = 0
[1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [5,4,1,2,3,6,7] => [3,4,5,2,1,6,7] => ? = 4
[1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [4,5,1,2,3,6,7] => [3,4,5,1,2,6,7] => ? = 4
[1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [6,3,1,2,4,5,7] => [3,4,2,5,6,1,7] => ? = 4
[1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [5,3,1,2,4,6,7] => [3,4,2,5,1,6,7] => ? = 3
[1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [4,3,1,2,5,6,7] => [3,4,2,1,5,6,7] => ? = 2
[1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,2,1,3,4,5,7] => [3,2,4,5,6,1,7] => ? = 3
[1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,2,1,3,4,6,7] => [3,2,4,5,1,6,7] => ? = 2
[1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,3,5,6,7] => [3,2,4,1,5,6,7] => ? = 1
[1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,4,5,6,7,1,8] => [7,2,1,3,4,5,6,8] => ? = 0
[1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> [3,4,2,5,6,1,7,8] => [6,3,1,2,4,5,7,8] => ? = 0
[1,1,1,0,1,1,1,1,0,0,1,0,0,0,0,0]
=> [4,2,3,5,6,1,7,8] => [6,2,3,1,4,5,7,8] => ? = 1
[1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,4,5,6,1,7,8] => [6,2,1,3,4,5,7,8] => ? = 0
[1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> [3,4,5,2,1,6,7,8] => [5,4,1,2,3,6,7,8] => ? = 0
[1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0]
=> [4,3,2,5,1,6,7,8] => [5,3,2,1,4,6,7,8] => ? = 0
[1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> [3,4,2,5,1,6,7,8] => [5,3,1,2,4,6,7,8] => ? = 0
[1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0]
=> [5,2,3,4,1,6,7,8] => [5,2,3,4,1,6,7,8] => ? = 2
[1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0]
=> [4,2,3,5,1,6,7,8] => [5,2,3,1,4,6,7,8] => ? = 1
[1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,4,5,1,6,7,8] => [5,2,1,3,4,6,7,8] => ? = 0
[1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0]
=> [4,3,5,1,2,6,7,8] => [4,5,2,1,3,6,7,8] => ? = 3
[1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
=> [3,4,5,1,2,6,7,8] => [4,5,1,2,3,6,7,8] => ? = 3
[1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0]
=> [4,5,2,1,3,6,7,8] => [4,3,5,1,2,6,7,8] => ? = 2
[1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
=> [5,3,2,1,4,6,7,8] => [4,3,2,5,1,6,7,8] => ? = 1
[1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> [3,4,2,1,5,6,7,8] => [4,3,1,2,5,6,7,8] => ? = 0
[1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0]
=> [7,2,3,1,4,5,6,8] => [4,2,3,5,6,7,1,8] => ? = 4
[1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0]
=> [6,2,3,1,4,5,7,8] => [4,2,3,5,6,1,7,8] => ? = 3
[1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
=> [5,2,3,1,4,6,7,8] => [4,2,3,5,1,6,7,8] => ? = 2
[1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> [4,2,3,1,5,6,7,8] => [4,2,3,1,5,6,7,8] => ? = 1
[1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [3,2,4,1,5,6,7,8] => [4,2,1,3,5,6,7,8] => ? = 0
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,1,5,6,7,8] => [4,1,2,3,5,6,7,8] => ? = 0
[1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [5,4,1,2,3,6,7,8] => [3,4,5,2,1,6,7,8] => ? = 4
[1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> [4,5,1,2,3,6,7,8] => [3,4,5,1,2,6,7,8] => ? = 4
[1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> [7,3,1,2,4,5,6,8] => [3,4,2,5,6,7,1,8] => ? = 5
Description
The number of occurrences of the pattern 23-1.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $23\!\!-\!\!1$.
Matching statistic: St001377
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St001377: Permutations ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 62%
Mp00069: Permutations —complement⟶ Permutations
St001377: Permutations ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 62%
Values
[1,0,1,0,1,0]
=> [3,2,1] => [1,2,3] => 0
[1,0,1,1,0,0]
=> [2,3,1] => [2,1,3] => 0
[1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [2,1,3,4] => 0
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,3,2,4] => 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [2,3,1,4] => 0
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [3,2,1,4] => 0
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,2,4,3] => 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,4,3] => 2
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,3,4,2] => 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [2,3,4,1] => 0
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,2,4,1] => 0
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,4,3,2] => 2
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,4,3,1] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,2,1,4,5] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [2,3,4,1,5] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,2,4,1,5] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,4,3,2,5] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [2,4,3,1,5] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,4,2,1,5] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [4,3,2,1,5] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [2,3,1,5,4] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [3,2,1,5,4] => 3
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [2,1,4,5,3] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,3,4,5,2] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [2,3,4,5,1] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,2,4,5,1] => 0
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,4,3,5,2] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [2,4,3,5,1] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [3,4,2,5,1] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [4,3,2,5,1] => 0
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,2,5,4,3] => 4
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [2,1,5,4,3] => 4
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,3,5,4,2] => 3
[1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [2,3,5,4,1] => 2
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,2,5,4,1] => 2
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [1,4,5,3,2] => 2
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => [2,4,5,3,1] => 1
[1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [3,4,5,2,1] => 0
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [4,3,5,2,1] => 0
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [1,5,4,3,2] => 3
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [2,5,4,3,1] => 2
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [3,5,4,2,1] => 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => [4,3,5,2,1,6] => 0
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => [3,5,4,2,1,6] => 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => [4,5,3,2,1,6] => 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [5,4,3,2,1,6] => 0
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [3,4,5,2,1,6] => [4,3,2,5,6,1] => 0
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [4,3,2,5,1,6] => [3,4,5,2,6,1] => 0
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => [5,6,4,3,2,1,7] => ? = 0
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [6,5,4,3,2,1,7] => ? = 0
[1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,1,7] => [5,4,6,3,2,7,1] => ? = 0
[1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [4,2,3,5,6,1,7] => [4,6,5,3,2,7,1] => ? = 1
[1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,1,7] => [5,6,4,3,2,7,1] => ? = 0
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,1,7] => [6,5,4,3,2,7,1] => ? = 0
[1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,1,6,7] => [5,4,3,6,7,2,1] => ? = 0
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [4,3,2,5,1,6,7] => [4,5,6,3,7,2,1] => ? = 0
[1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,1,6,7] => [5,4,6,3,7,2,1] => ? = 0
[1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [5,2,3,4,1,6,7] => [3,6,5,4,7,2,1] => ? = 2
[1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [4,2,3,5,1,6,7] => [4,6,5,3,7,2,1] => ? = 1
[1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,1,6,7] => [5,6,4,3,7,2,1] => ? = 0
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,1,6,7] => [6,5,4,3,7,2,1] => ? = 0
[1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [4,3,5,1,2,6,7] => [4,5,3,7,6,2,1] => ? = 3
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [3,4,5,1,2,6,7] => [5,4,3,7,6,2,1] => ? = 3
[1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [4,5,2,1,3,6,7] => [4,3,6,7,5,2,1] => ? = 2
[1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [5,3,2,1,4,6,7] => [3,5,6,7,4,2,1] => ? = 1
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [4,3,2,1,5,6,7] => [4,5,6,7,3,2,1] => ? = 0
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [3,4,2,1,5,6,7] => [5,4,6,7,3,2,1] => ? = 0
[1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [7,2,3,1,4,5,6] => [1,6,5,7,4,3,2] => ? = 4
[1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [6,2,3,1,4,5,7] => [2,6,5,7,4,3,1] => ? = 3
[1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [5,2,3,1,4,6,7] => [3,6,5,7,4,2,1] => ? = 2
[1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [4,2,3,1,5,6,7] => [4,6,5,7,3,2,1] => ? = 1
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [3,2,4,1,5,6,7] => [5,6,4,7,3,2,1] => ? = 0
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [2,3,4,1,5,6,7] => [6,5,4,7,3,2,1] => ? = 0
[1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [5,4,1,2,3,6,7] => [3,4,7,6,5,2,1] => ? = 4
[1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [4,5,1,2,3,6,7] => [4,3,7,6,5,2,1] => ? = 4
[1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [7,3,1,2,4,5,6] => [1,5,7,6,4,3,2] => ? = 5
[1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [6,3,1,2,4,5,7] => [2,5,7,6,4,3,1] => ? = 4
[1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [5,3,1,2,4,6,7] => [3,5,7,6,4,2,1] => ? = 3
[1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [4,3,1,2,5,6,7] => [4,5,7,6,3,2,1] => ? = 2
[1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [3,4,1,2,5,6,7] => [5,4,7,6,3,2,1] => ? = 2
[1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => [1,6,7,5,4,3,2] => ? = 4
[1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,2,1,3,4,5,7] => [2,6,7,5,4,3,1] => ? = 3
[1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,2,1,3,4,6,7] => [3,6,7,5,4,2,1] => ? = 2
[1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,3,5,6,7] => [4,6,7,5,3,2,1] => ? = 1
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [3,2,1,4,5,6,7] => [5,6,7,4,3,2,1] => ? = 0
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [2,3,1,4,5,6,7] => [6,5,7,4,3,2,1] => ? = 0
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => [1,7,6,5,4,3,2] => ? = 5
[1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [6,1,2,3,4,5,7] => [2,7,6,5,4,3,1] => ? = 4
[1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [5,1,2,3,4,6,7] => [3,7,6,5,4,2,1] => ? = 3
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [4,1,2,3,5,6,7] => [4,7,6,5,3,2,1] => ? = 2
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [3,1,2,4,5,6,7] => [5,7,6,4,3,2,1] => ? = 1
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => [7,6,5,4,3,2,1,8] => ? = 0
[1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,4,5,6,7,1,8] => [6,7,5,4,3,2,8,1] => ? = 0
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1,8] => [7,6,5,4,3,2,8,1] => ? = 0
[1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> [3,4,2,5,6,1,7,8] => [6,5,7,4,3,8,2,1] => ? = 0
[1,1,1,0,1,1,1,1,0,0,1,0,0,0,0,0]
=> [4,2,3,5,6,1,7,8] => [5,7,6,4,3,8,2,1] => ? = 1
[1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,4,5,6,1,7,8] => [6,7,5,4,3,8,2,1] => ? = 0
[1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,1,7,8] => [7,6,5,4,3,8,2,1] => ? = 0
Description
The major index minus the number of inversions of a permutation.
This is, the difference between [[St000004]] and [[St000018]].
Matching statistic: St001330
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 19%●distinct values known / distinct values provided: 12%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 19%●distinct values known / distinct values provided: 12%
Values
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 2
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 0 + 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 0 + 2
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 0 + 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 2
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 0 + 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2 = 0 + 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 0 + 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 2 = 0 + 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 3 + 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 1 + 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2 = 0 + 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 0 + 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 2 + 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2 = 0 + 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 + 2
[1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 + 2
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 2
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 2 + 2
[1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 2
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 2
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 0 + 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2 = 0 + 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 3 + 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [4,3,1,7,2,5,6] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5)],7)
=> ? = 1 + 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 2 = 0 + 2
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 2 = 0 + 2
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 2 = 0 + 2
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 2 = 0 + 2
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 2 = 0 + 2
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [7,3,1,5,2,4,6] => ([(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 2 + 2
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [2,7,4,1,3,5,6] => ([(0,6),(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 2 = 0 + 2
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,7,1,3,6,4,5] => ([(0,4),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 3 + 2
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,1,2,5,3,4,6] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 2
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,1,4,2,3,7,5] => ([(0,4),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 2 + 2
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,6,4,1,3,7,5] => ([(0,4),(1,3),(2,5),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2 = 0 + 2
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2 = 0 + 2
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [7,3,1,6,2,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 2
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [2,7,5,1,3,4,6] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [6,3,1,2,4,7,5] => ([(0,4),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 1 + 2
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 2 = 0 + 2
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 2 = 0 + 2
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,7,1,6,3,4,5] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 4 + 2
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,1,5,2,3,4,6] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 2
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,3,7,1,2,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 4 + 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [7,3,1,2,6,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 3 + 2
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,7,1,5,3,4,6] => ([(0,4),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 2 + 2
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,1,4,2,3,5,6] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 2
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,6,7,1,3,4,5] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3 + 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,4,1,2,3,7,5] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 2 + 2
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,1,2,7,4,6] => ([(0,4),(1,4),(1,6),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
=> ? = 1 + 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 2 = 0 + 2
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 2 = 0 + 2
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 2
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,7,1,2,3,4,6] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3 + 2
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,4,1,2,3,5,6] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 2
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [7,3,1,2,4,5,6] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [2,3,8,1,4,5,6,7] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 2 = 0 + 2
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,1,2,3,4,5,8,6] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> 2 = 0 + 2
[1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [3,1,7,2,4,5,8,6] => ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> 2 = 0 + 2
[1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [2,8,4,1,3,5,6,7] => ([(0,7),(1,7),(2,7),(3,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 + 2
[1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [2,7,1,3,4,5,8,6] => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> 2 = 0 + 2
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,1,2,3,4,8,5,7] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> 2 = 0 + 2
[1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [4,1,2,6,3,8,5,7] => ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
=> 2 = 0 + 2
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [2,4,1,7,3,5,8,6] => ([(0,6),(1,5),(2,7),(3,4),(3,5),(4,7),(6,7)],8)
=> 2 = 0 + 2
[1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [3,1,6,2,4,8,5,7] => ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> 2 = 0 + 2
[1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [2,8,5,1,3,4,6,7] => ([(0,7),(1,7),(2,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 2
[1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [2,6,1,3,4,8,5,7] => ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> 2 = 0 + 2
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,1,2,3,8,4,6,7] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(5,7)],8)
=> 2 = 0 + 2
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,8,5,7] => ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> 2 = 0 + 2
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [3,1,5,2,8,4,6,7] => ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
=> 2 = 0 + 2
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,5,1,3,8,4,6,7] => ([(0,7),(1,6),(2,6),(3,4),(4,7),(5,6),(5,7)],8)
=> 2 = 0 + 2
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [4,1,2,8,3,5,6,7] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(5,7)],8)
=> 2 = 0 + 2
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,4,1,8,3,5,6,7] => ([(0,7),(1,7),(2,7),(3,4),(4,6),(5,6),(5,7)],8)
=> 2 = 0 + 2
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [3,1,8,2,4,5,6,7] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> 2 = 0 + 2
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: St001811
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St001811: Permutations ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 62%
Mp00069: Permutations —complement⟶ Permutations
St001811: Permutations ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 62%
Values
[1,0,1,0,1,0]
=> [3,2,1] => [1,2,3] => 0
[1,0,1,1,0,0]
=> [2,3,1] => [2,1,3] => 0
[1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [2,1,3,4] => 0
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,3,2,4] => 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [2,3,1,4] => 0
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [3,2,1,4] => 0
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,2,4,3] => 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,4,3] => 2
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,3,4,2] => 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [2,3,4,1] => 0
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,2,4,1] => 0
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,4,3,2] => 2
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,4,3,1] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,2,1,4,5] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [2,3,4,1,5] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,2,4,1,5] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,4,3,2,5] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [2,4,3,1,5] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,4,2,1,5] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [4,3,2,1,5] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [2,3,1,5,4] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [3,2,1,5,4] => 3
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [2,1,4,5,3] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,3,4,5,2] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [2,3,4,5,1] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,2,4,5,1] => 0
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,4,3,5,2] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [2,4,3,5,1] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [3,4,2,5,1] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [4,3,2,5,1] => 0
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,2,5,4,3] => 4
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [2,1,5,4,3] => 4
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,3,5,4,2] => 3
[1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [2,3,5,4,1] => 2
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,2,5,4,1] => 2
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [1,4,5,3,2] => 2
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => [2,4,5,3,1] => 1
[1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [3,4,5,2,1] => 0
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [4,3,5,2,1] => 0
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [1,5,4,3,2] => 3
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [2,5,4,3,1] => 2
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [3,5,4,2,1] => 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => [4,3,5,2,1,6] => ? = 0
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => [3,5,4,2,1,6] => ? = 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => [4,5,3,2,1,6] => ? = 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [5,4,3,2,1,6] => ? = 0
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [3,4,5,2,1,6] => [4,3,2,5,6,1] => ? = 0
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [4,3,2,5,1,6] => [3,4,5,2,6,1] => ? = 0
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,1,6] => [4,3,5,2,6,1] => ? = 0
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [5,2,3,4,1,6] => [2,5,4,3,6,1] => ? = 2
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,1,6] => [3,5,4,2,6,1] => ? = 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,1,6] => [4,5,3,2,6,1] => ? = 0
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,1,6] => [5,4,3,2,6,1] => ? = 0
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [4,3,5,1,2,6] => [3,4,2,6,5,1] => ? = 3
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,4,5,1,2,6] => [4,3,2,6,5,1] => ? = 3
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [4,5,2,1,3,6] => [3,2,5,6,4,1] => ? = 2
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [5,3,2,1,4,6] => [2,4,5,6,3,1] => ? = 1
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [4,3,2,1,5,6] => [3,4,5,6,2,1] => ? = 0
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [3,4,2,1,5,6] => [4,3,5,6,2,1] => ? = 0
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => [1,5,4,6,3,2] => ? = 3
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [5,2,3,1,4,6] => [2,5,4,6,3,1] => ? = 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [4,2,3,1,5,6] => [3,5,4,6,2,1] => ? = 1
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [3,2,4,1,5,6] => [4,5,3,6,2,1] => ? = 0
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [2,3,4,1,5,6] => [5,4,3,6,2,1] => ? = 0
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [5,4,1,2,3,6] => [2,3,6,5,4,1] => ? = 4
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [4,5,1,2,3,6] => [3,2,6,5,4,1] => ? = 4
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => [1,4,6,5,3,2] => ? = 4
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [5,3,1,2,4,6] => [2,4,6,5,3,1] => ? = 3
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [4,3,1,2,5,6] => [3,4,6,5,2,1] => ? = 2
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [3,4,1,2,5,6] => [4,3,6,5,2,1] => ? = 2
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => [1,5,6,4,3,2] => ? = 3
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [5,2,1,3,4,6] => [2,5,6,4,3,1] => ? = 2
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,3,5,6] => [3,5,6,4,2,1] => ? = 1
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [3,2,1,4,5,6] => [4,5,6,3,2,1] => ? = 0
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [2,3,1,4,5,6] => [5,4,6,3,2,1] => ? = 0
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => [1,6,5,4,3,2] => ? = 4
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,1,2,3,4,6] => [2,6,5,4,3,1] => ? = 3
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [4,1,2,3,5,6] => [3,6,5,4,2,1] => ? = 2
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,2,4,5,6] => [4,6,5,3,2,1] => ? = 1
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => [5,6,4,3,2,1,7] => ? = 0
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [6,5,4,3,2,1,7] => ? = 0
[1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,1,7] => [5,4,6,3,2,7,1] => ? = 0
[1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [4,2,3,5,6,1,7] => [4,6,5,3,2,7,1] => ? = 1
[1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,1,7] => [5,6,4,3,2,7,1] => ? = 0
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,1,7] => [6,5,4,3,2,7,1] => ? = 0
[1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,1,6,7] => [5,4,3,6,7,2,1] => ? = 0
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [4,3,2,5,1,6,7] => [4,5,6,3,7,2,1] => ? = 0
[1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,1,6,7] => [5,4,6,3,7,2,1] => ? = 0
[1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [5,2,3,4,1,6,7] => [3,6,5,4,7,2,1] => ? = 2
[1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [4,2,3,5,1,6,7] => [4,6,5,3,7,2,1] => ? = 1
[1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,1,6,7] => [5,6,4,3,7,2,1] => ? = 0
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,1,6,7] => [6,5,4,3,7,2,1] => ? = 0
Description
The Castelnuovo-Mumford regularity of a permutation.
The ''Castelnuovo-Mumford regularity'' of a permutation $\sigma$ is the ''Castelnuovo-Mumford regularity'' of the ''matrix Schubert variety'' $X_\sigma$.
Equivalently, it is the difference between the degrees of the ''Grothendieck polynomial'' and the ''Schubert polynomial'' for $\sigma$. It can be computed by subtracting the ''Coxeter length'' [[St000018]] from the ''Rajchgot index'' [[St001759]].
Matching statistic: St001816
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001816: Standard tableaux ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 25%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001816: Standard tableaux ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 25%
Values
[1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[1,0,1,1,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0
[1,1,0,0,1,0]
=> [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> ? = 0
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> ? = 0
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> ? = 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> ? = 0
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> ? = 0
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> ? = 2
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> ? = 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> ? = 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 2
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> ? = 0
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[1,3,4,6,8],[2,5,7,9,10]]
=> ? = 0
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[1,3,4,6,7],[2,5,8,9,10]]
=> ? = 0
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[1,3,4,5,9],[2,6,7,8,10]]
=> ? = 2
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 0
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[1,2,5,6,8],[3,4,7,9,10]]
=> ? = 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[1,2,5,6,7],[3,4,8,9,10]]
=> ? = 3
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[1,2,4,7,8],[3,5,6,9,10]]
=> ? = 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[1,2,4,6,9],[3,5,7,8,10]]
=> ? = 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> ? = 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> ? = 0
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> ? = 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> ? = 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> ? = 0
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[1,2,3,7,9],[4,5,6,8,10]]
=> ? = 4
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> ? = 4
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? = 3
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> ? = 2
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> ? = 2
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 2
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> ? = 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 3
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 2
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [[1,3,4,5,7,8],[2,6,9,10,11,12]]
=> ? = 0
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [[1,3,4,5,6,9],[2,7,8,10,11,12]]
=> ? = 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[1,3,4,5,6,8],[2,7,9,10,11,12]]
=> ? = 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,4,5,6,7],[2,8,9,10,11,12]]
=> ? = 0
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> ? = 0
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[1,3,4,6,8],[2,5,7,9,10]]
=> ? = 0
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[1,3,4,6,7],[2,5,8,9,10]]
=> ? = 0
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[1,3,4,5,9],[2,6,7,8,10]]
=> ? = 2
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 0
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[1,2,5,6,8],[3,4,7,9,10]]
=> ? = 3
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[1,2,5,6,7],[3,4,8,9,10]]
=> ? = 3
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[1,2,4,7,8],[3,5,6,9,10]]
=> ? = 2
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[1,2,4,6,9],[3,5,7,8,10]]
=> ? = 1
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0
[1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0
[1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[1,1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[1,1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0
Description
Eigenvalues of the top-to-random operator acting on a simple module.
These eigenvalues are given in [1] and [3].
The simple module of the symmetric group indexed by a partition $\lambda$ has dimension equal to the number of standard tableaux of shape $\lambda$. Hence, the eigenvalues of any linear operator defined on this module can be indexed by standard tableaux of shape $\lambda$; this statistic gives all the eigenvalues of the operator acting on the module.
This statistic bears different names, such as the type in [2] or eig in [3].
Similarly, the eigenvalues of the random-to-random operator acting on a simple module is [[St000508]].
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