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Your data matches 94 different statistics following compositions of up to 3 maps.
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Matching statistic: St001175
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Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1]
=> 0
[1,0,1,0]
=> [1,2] => [2]
=> 0
[1,1,0,0]
=> [2,1] => [1,1]
=> 0
[1,0,1,0,1,0]
=> [1,2,3] => [3]
=> 0
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> 0
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> 0
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> 0
[1,1,1,0,0,0]
=> [3,2,1] => [1,1,1]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [3,1]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,1]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,1]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [2,1,1]
=> 0
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [3,1]
=> 0
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1]
=> 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [3,1]
=> 0
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,1,1]
=> 0
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,1,1]
=> 0
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,1,1]
=> 0
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,1,1]
=> 0
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [4,1]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [4,1]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [4,1]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,1,1]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [4,1]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,2]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [4,1]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [4,1]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [3,1,1]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [3,1,1]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,1,1]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [3,1,1]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,1,1,1]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [4,1]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [3,2]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [3,2]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,2]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,2,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [4,1]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [4,1]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [3,1,1]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,1,1]
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,1,1]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,1,1]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,1,1,1]
=> 0
Description
The size of a partition minus the hook length of the base cell.
This is, the number of boxes in the diagram of a partition that are neither in the first row nor in the first column.
Matching statistic: St001033
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Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001033: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001033: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1]
=> [1,0]
=> 0
[1,0,1,0]
=> [1,2] => [2]
=> [1,0,1,0]
=> 0
[1,1,0,0]
=> [2,1] => [1,1]
=> [1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> [1,2,3] => [3]
=> [1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 0
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 0
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> [1,0,1,1,0,0]
=> 0
[1,1,1,0,0,0]
=> [3,2,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> [1,1,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
Description
The normalized area of the parallelogram polyomino associated with the Dyck path.
The area of the smallest parallelogram polyomino equals the semilength of the Dyck path. This statistic is therefore the area of the parallelogram polyomino minus the semilength of the Dyck path.
The area itself is equidistributed with [[St001034]] and with [[St000395]].
Matching statistic: St001596
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Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001596: Skew partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001596: Skew partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1]
=> [[1],[]]
=> 0
[1,0,1,0]
=> [1,2] => [1,1]
=> [[1,1],[]]
=> 0
[1,1,0,0]
=> [2,1] => [2]
=> [[2],[]]
=> 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> [[1,1,1],[]]
=> 0
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> [[2,1],[]]
=> 0
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> [[2,1],[]]
=> 0
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> [[2,1],[]]
=> 0
[1,1,1,0,0,0]
=> [3,2,1] => [3]
=> [[3],[]]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> [[1,1,1,1],[]]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> [[2,1,1],[]]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> [[2,1,1],[]]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,1,1]
=> [[2,1,1],[]]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [3,1]
=> [[3,1],[]]
=> 0
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> [[2,1,1],[]]
=> 0
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> [[2,2],[]]
=> 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,1,1]
=> [[2,1,1],[]]
=> 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,1,1]
=> [[2,1,1],[]]
=> 0
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,1]
=> [[3,1],[]]
=> 0
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,1]
=> [[3,1],[]]
=> 0
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,1]
=> [[3,1],[]]
=> 0
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [3,1]
=> [[3,1],[]]
=> 0
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4]
=> [[4],[]]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> [[2,1,1,1],[]]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> [[2,1,1,1],[]]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,1,1,1]
=> [[2,1,1,1],[]]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,1,1]
=> [[3,1,1],[]]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> [[2,1,1,1],[]]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> [[2,2,1],[]]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> [[2,1,1,1],[]]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> [[2,1,1,1],[]]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [3,1,1]
=> [[3,1,1],[]]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [3,1,1]
=> [[3,1,1],[]]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,1,1]
=> [[3,1,1],[]]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [3,1,1]
=> [[3,1,1],[]]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [4,1]
=> [[4,1],[]]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> [[2,1,1,1],[]]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> [[2,2,1],[]]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> [[2,2,1],[]]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,2,1]
=> [[2,2,1],[]]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [3,2]
=> [[3,2],[]]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> [[2,1,1,1],[]]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> [[2,2,1],[]]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> [[2,1,1,1],[]]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> [[2,1,1,1],[]]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [3,1,1]
=> [[3,1,1],[]]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,1,1]
=> [[3,1,1],[]]
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,1,1]
=> [[3,1,1],[]]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,1,1]
=> [[3,1,1],[]]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [4,1]
=> [[4,1],[]]
=> 0
Description
The number of two-by-two squares inside a skew partition.
This is, the number of cells $(i,j)$ in a skew partition for which the box $(i+1,j+1)$ is also a cell inside the skew partition.
Matching statistic: St000345
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000345: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000345: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1]
=> []
=> 1 = 0 + 1
[1,0,1,0]
=> [1,2] => [1,1]
=> [1]
=> 1 = 0 + 1
[1,1,0,0]
=> [2,1] => [2]
=> []
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> [1]
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> [1]
=> 1 = 0 + 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> [1]
=> 1 = 0 + 1
[1,1,1,0,0,0]
=> [3,2,1] => [3]
=> []
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [3,1]
=> [1]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> [2]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,1]
=> [1]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,1]
=> [1]
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,1]
=> [1]
=> 1 = 0 + 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [3,1]
=> [1]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4]
=> []
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> [2,1]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [4,1]
=> [1]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> [2,1]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> [2,1]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,2,1]
=> [2,1]
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [3,2]
=> [2]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> [2,1]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [4,1]
=> [1]
=> 1 = 0 + 1
Description
The number of refinements of a partition.
A partition $\lambda$ refines a partition $\mu$ if the parts of $\mu$ can be subdivided to obtain the parts of $\lambda$.
Matching statistic: St000935
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000935: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000935: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1]
=> []
=> 1 = 0 + 1
[1,0,1,0]
=> [1,2] => [1,1]
=> [1]
=> 1 = 0 + 1
[1,1,0,0]
=> [2,1] => [2]
=> []
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> [1]
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> [1]
=> 1 = 0 + 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> [1]
=> 1 = 0 + 1
[1,1,1,0,0,0]
=> [3,2,1] => [3]
=> []
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [3,1]
=> [1]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> [2]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,1]
=> [1]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,1]
=> [1]
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,1]
=> [1]
=> 1 = 0 + 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [3,1]
=> [1]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4]
=> []
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> [2,1]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [4,1]
=> [1]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> [2,1]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> [2,1]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,2,1]
=> [2,1]
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [3,2]
=> [2]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> [2,1]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [4,1]
=> [1]
=> 1 = 0 + 1
Description
The number of ordered refinements of an integer partition.
This is, for an integer partition $\mu = (\mu_1,\ldots,\mu_n)$ the number of integer partition $\lambda = (\lambda_1,\ldots,\lambda_m)$ such that there are indices $1 = a_0 < \ldots < a_n = m$ with $\mu_j = \lambda_{a_{j-1}} + \ldots + \lambda_{a_j-1}$.
Matching statistic: St001389
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001389: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001389: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1]
=> []
=> ? = 0 + 1
[1,0,1,0]
=> [1,2] => [1,1]
=> [1]
=> 1 = 0 + 1
[1,1,0,0]
=> [2,1] => [2]
=> []
=> ? = 0 + 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> [1]
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> [1]
=> 1 = 0 + 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> [1]
=> 1 = 0 + 1
[1,1,1,0,0,0]
=> [3,2,1] => [3]
=> []
=> ? = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [3,1]
=> [1]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> [2]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,1]
=> [1]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,1]
=> [1]
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,1]
=> [1]
=> 1 = 0 + 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [3,1]
=> [1]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4]
=> []
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> [2,1]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [4,1]
=> [1]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> [2,1]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> [2,1]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,2,1]
=> [2,1]
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [3,2]
=> [2]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> [2,1]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [4,1]
=> [1]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [3,2]
=> [2]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [5]
=> []
=> ? = 0 + 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,5,4,3,2,1] => [6]
=> []
=> ? = 0 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7,6,5,4,3,2,1] => [7]
=> []
=> ? = 0 + 1
Description
The number of partitions of the same length below the given integer partition.
For a partition $\lambda_1 \geq \dots \lambda_k > 0$, this number is
$$ \det\left( \binom{\lambda_{k+1-i}}{j-i+1} \right)_{1 \le i,j \le k}.$$
Matching statistic: St001624
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
St001624: Lattices ⟶ ℤResult quality: 67% ●values known / values provided: 92%●distinct values known / distinct values provided: 67%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
St001624: Lattices ⟶ ℤResult quality: 67% ●values known / values provided: 92%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1] => ([],1)
=> ([],1)
=> 1 = 0 + 1
[1,0,1,0]
=> [2,1] => ([],2)
=> ([],1)
=> 1 = 0 + 1
[1,1,0,0]
=> [1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [3,2,1] => ([],3)
=> ([],1)
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [2,3,1] => ([(1,2)],3)
=> ([(0,1)],2)
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> 1 = 0 + 1
[1,1,0,1,0,0]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1 = 0 + 1
[1,1,1,0,0,0]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => ([],4)
=> ([],1)
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => ([(2,3)],4)
=> ([(0,1)],2)
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => ([(2,3)],4)
=> ([(0,1)],2)
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => ([(2,3)],4)
=> ([(0,1)],2)
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1 = 0 + 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => ([],5)
=> ([],1)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => ([(3,4)],5)
=> ([(0,1)],2)
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => ([(3,4)],5)
=> ([(0,1)],2)
=> 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => ([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => ([(3,4)],5)
=> ([(0,1)],2)
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => ([(3,4)],5)
=> ([(0,1)],2)
=> 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,1,2] => ([(0,5),(1,4),(2,3)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2 + 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => ([(0,5),(1,3),(4,2),(5,4)],6)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,5,6,1,2,3] => ([(0,5),(1,4),(4,2),(5,3)],6)
=> ([(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 + 1
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => ([(0,5),(1,3),(4,2),(5,4)],6)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,2,3,1] => ([(1,6),(2,5),(3,4)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2 + 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,2,3,1] => ([(1,6),(2,4),(5,3),(6,5)],7)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [5,6,7,2,3,4,1] => ([(1,6),(2,5),(5,3),(6,4)],7)
=> ([(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 + 1
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [6,7,2,3,4,5,1] => ([(1,6),(2,4),(5,3),(6,5)],7)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,3,1,2] => ([(1,6),(2,5),(3,4)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2 + 1
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,1,2] => ([(1,6),(2,4),(5,3),(6,5)],7)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [6,7,5,3,4,1,2] => ([(1,6),(2,5),(3,4)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2 + 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,6,3,4,1,2] => ([(1,6),(2,5),(3,4)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2 + 1
[1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [6,5,7,3,4,1,2] => ([(0,6),(1,6),(2,5),(3,4)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2 + 1
[1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,4,1,2] => ([(0,5),(1,4),(2,6),(6,3)],7)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? = 2 + 1
[1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [6,7,4,3,5,1,2] => ([(0,6),(1,6),(2,5),(3,4)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2 + 1
[1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,7,1,2] => ([(0,6),(1,3),(2,4),(4,5),(5,6)],7)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,5,1,2] => ([(0,5),(1,4),(2,6),(6,3)],7)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? = 2 + 1
[1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,7,1,2] => ([(0,5),(1,4),(2,3),(4,6),(5,6)],7)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> ? = 2 + 1
[1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,7,1,2] => ([(0,6),(1,3),(2,4),(4,6),(6,5)],7)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [7,3,4,5,6,1,2] => ([(1,6),(2,4),(5,3),(6,5)],7)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [6,3,4,5,7,1,2] => ([(0,6),(1,3),(2,4),(4,5),(5,6)],7)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [5,3,4,6,7,1,2] => ([(0,6),(1,3),(2,4),(4,6),(6,5)],7)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,7,1,2] => ([(0,6),(1,6),(2,3),(4,5),(6,4)],7)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,1,2] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 1 + 1
[1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,2,1,3] => ([(0,6),(1,6),(2,5),(3,4)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2 + 1
[1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,2,1,3] => ([(0,3),(1,6),(2,6),(3,5),(5,4)],7)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [5,6,7,2,3,1,4] => ([(0,6),(1,3),(2,4),(3,5),(4,6)],7)
=> ([(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 + 1
[1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [6,7,2,3,4,1,5] => ([(0,6),(1,3),(2,4),(4,5),(5,6)],7)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,1,2,3] => ([(1,6),(2,5),(5,3),(6,4)],7)
=> ([(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 + 1
[1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,1,2,3] => ([(0,5),(1,4),(2,6),(6,3)],7)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? = 2 + 1
[1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,1,2,3] => ([(0,6),(1,3),(2,4),(3,5),(4,6)],7)
=> ([(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 + 1
[1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,1,2,3] => ([(1,6),(2,5),(5,3),(6,4)],7)
=> ([(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 + 1
[1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,1,2,3] => ([(0,6),(1,3),(2,4),(3,5),(4,6)],7)
=> ([(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 + 1
[1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,1,2,3] => ([(0,6),(1,6),(2,3),(3,5),(6,4)],7)
=> ([(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 + 1
[1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,1,2,3] => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 2 + 1
[1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,1,2,4] => ([(0,6),(1,3),(2,4),(3,5),(4,6)],7)
=> ([(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 + 1
[1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,1,2,5] => ([(0,5),(1,4),(2,3),(4,6),(5,6)],7)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> ? = 2 + 1
[1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,1,2,7] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> ? = 2 + 1
[1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,1,2,7] => ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1 + 1
[1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> [5,6,7,2,1,3,4] => ([(0,6),(1,6),(2,3),(3,5),(6,4)],7)
=> ([(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 + 1
[1,1,1,0,1,1,0,0,0,0,1,1,0,0]
=> [6,7,2,3,1,4,5] => ([(0,6),(1,3),(2,4),(4,6),(6,5)],7)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [6,7,5,1,2,3,4] => ([(1,6),(2,4),(5,3),(6,5)],7)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [7,5,6,1,2,3,4] => ([(1,6),(2,4),(5,3),(6,5)],7)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,1,1,1,0,0,0,0,1,1,0,1,0,0]
=> [6,5,7,1,2,3,4] => ([(0,3),(1,6),(2,6),(3,5),(5,4)],7)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [5,6,7,1,2,3,4] => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 2 + 1
[1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> [6,7,4,1,2,3,5] => ([(0,6),(1,3),(2,4),(4,5),(5,6)],7)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [4,5,6,1,2,3,7] => ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 + 1
[1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [6,7,3,1,2,4,5] => ([(0,6),(1,3),(2,4),(4,6),(6,5)],7)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [3,4,5,1,2,6,7] => ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 1 + 1
[1,1,1,1,0,1,0,0,0,0,1,1,0,0]
=> [6,7,2,1,3,4,5] => ([(0,6),(1,6),(2,3),(4,5),(6,4)],7)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
Description
The breadth of a lattice.
The '''breadth''' of a lattice is the least integer $b$ such that any join $x_1\vee x_2\vee\cdots\vee x_n$, with $n > b$, can be expressed as a join over a proper subset of $\{x_1,x_2,\ldots,x_n\}$.
Matching statistic: St001878
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 67% ●values known / values provided: 81%●distinct values known / distinct values provided: 67%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 67% ●values known / values provided: 81%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1] => ([],1)
=> ([],1)
=> ? = 0 + 1
[1,0,1,0]
=> [2,1] => ([],2)
=> ([],1)
=> ? = 0 + 1
[1,1,0,0]
=> [1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 0 + 1
[1,0,1,0,1,0]
=> [3,2,1] => ([],3)
=> ([],1)
=> ? = 0 + 1
[1,0,1,1,0,0]
=> [2,3,1] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 0 + 1
[1,1,0,0,1,0]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 0 + 1
[1,1,0,1,0,0]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? = 0 + 1
[1,1,1,0,0,0]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => ([],4)
=> ([],1)
=> ? = 0 + 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 0 + 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 0 + 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 0 + 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 0 + 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 0 + 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 0 + 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => ([],5)
=> ([],1)
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => ([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => ([],6)
=> ([],1)
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => ([(4,5)],6)
=> ([(0,1)],2)
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,2,1] => ([(4,5)],6)
=> ([(0,1)],2)
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,6,3,2,1] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => ([(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,2,1] => ([(4,5)],6)
=> ([(0,1)],2)
=> ? = 0 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,5,2,1] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 0 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,6,2,1] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 0 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,5,3,6,2,1] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,2,1] => ([(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,3,4,6,2,1] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [4,3,5,6,2,1] => ([(2,5),(3,5),(5,4)],6)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => ([(2,3),(3,5),(5,4)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,3,1] => ([(4,5)],6)
=> ([(0,1)],2)
=> ? = 0 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,3,1] => ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,3,1] => ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,6,2,3,1] => ([(1,5),(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,5,6,2,3,1] => ([(1,3),(2,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,4,1] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 0 + 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,5,1] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 0 + 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,6,1] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 0 + 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,2] => ([(4,5)],6)
=> ([(0,1)],2)
=> ? = 0 + 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,1,2] => ([(0,5),(1,4),(2,3)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2 + 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => ([(0,5),(1,3),(4,2),(5,4)],6)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1,3] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 0 + 1
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1,4] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 0 + 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,1,5] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 0 + 1
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1,6] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 0 + 1
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,5,6,1,2,3] => ([(0,5),(1,4),(4,2),(5,3)],6)
=> ([(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 + 1
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => ([(0,5),(1,3),(4,2),(5,4)],6)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1] => ([],7)
=> ([],1)
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,3,2,1] => ([(5,6)],7)
=> ([(0,1)],2)
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,3,2,1] => ([(5,6)],7)
=> ([(0,1)],2)
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => ([(4,6),(5,6)],7)
=> ([(0,1)],2)
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,3,2,1] => ([(5,6)],7)
=> ([(0,1)],2)
=> ? = 0 + 1
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
Matching statistic: St001687
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St001687: Permutations ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St001687: Permutations ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 0
[1,0,1,0]
=> [1,2] => [2,1] => [2,1] => 0
[1,1,0,0]
=> [2,1] => [1,2] => [1,2] => 0
[1,0,1,0,1,0]
=> [1,2,3] => [3,2,1] => [3,2,1] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [2,3,1] => [1,3,2] => 0
[1,1,0,0,1,0]
=> [2,1,3] => [3,1,2] => [3,1,2] => 0
[1,1,0,1,0,0]
=> [2,3,1] => [1,3,2] => [1,3,2] => 0
[1,1,1,0,0,0]
=> [3,2,1] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [3,4,2,1] => [1,4,3,2] => 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [4,2,3,1] => [4,1,3,2] => 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,3,1] => [1,4,3,2] => 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [2,3,4,1] => [1,2,4,3] => 0
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [4,3,1,2] => [4,3,1,2] => 0
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,4,1,2] => [2,4,1,3] => 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [4,1,3,2] => [4,1,3,2] => 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,4,3,2] => [1,4,3,2] => 0
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,3,4,2] => [1,2,4,3] => 0
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [4,1,2,3] => [4,1,2,3] => 0
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,4,2,3] => [1,4,2,3] => 0
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [1,2,4,3] => [1,2,4,3] => 0
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [4,5,3,2,1] => [1,5,4,3,2] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [5,3,4,2,1] => [5,1,4,3,2] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [3,5,4,2,1] => [1,5,4,3,2] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,4,5,2,1] => [1,2,5,4,3] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [5,4,2,3,1] => [5,4,1,3,2] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [4,5,2,3,1] => [2,5,1,4,3] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [5,2,4,3,1] => [5,1,4,3,2] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,5,4,3,1] => [1,5,4,3,2] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [2,4,5,3,1] => [1,2,5,4,3] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [5,2,3,4,1] => [5,1,2,4,3] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [2,5,3,4,1] => [1,5,2,4,3] => 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [2,3,5,4,1] => [1,2,5,4,3] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,3,4,5,1] => [1,2,3,5,4] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [5,4,3,1,2] => [5,4,3,1,2] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [4,5,3,1,2] => [2,5,4,1,3] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [5,3,4,1,2] => [5,2,4,1,3] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,5,4,1,2] => [2,5,4,1,3] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [3,4,5,1,2] => [1,3,5,2,4] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [5,4,1,3,2] => [5,4,1,3,2] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [4,5,1,3,2] => [2,5,1,4,3] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [5,1,4,3,2] => [5,1,4,3,2] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,5,4,3,2] => [1,5,4,3,2] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,4,5,3,2] => [1,2,5,4,3] => 0
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [5,1,3,4,2] => [5,1,2,4,3] => 0
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,5,3,4,2] => [1,5,2,4,3] => 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [1,3,5,4,2] => [1,2,5,4,3] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,3,4,5,2] => [1,2,3,5,4] => 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => [1,7,6,5,4,3,2] => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [7,1,6,5,4,3,2] => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => [5,7,6,4,3,2,1] => [1,7,6,5,4,3,2] => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => [7,6,1,5,4,3,2] => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => [2,7,1,6,5,4,3] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,3,5,6,4,7] => [7,4,6,5,3,2,1] => [7,1,6,5,4,3,2] => ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => [4,7,6,5,3,2,1] => [1,7,6,5,4,3,2] => ? = 0
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,5,4,7] => [7,4,5,6,3,2,1] => [7,1,2,6,5,4,3] => ? = 0
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,5,7,4] => [4,7,5,6,3,2,1] => [1,7,2,6,5,4,3] => ? = 0
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,4,3,5,6,7] => [7,6,5,3,4,2,1] => [7,6,5,1,4,3,2] => ? = 0
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,4,3,5,7,6] => [6,7,5,3,4,2,1] => [2,7,6,1,5,4,3] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5,7] => [7,5,6,3,4,2,1] => [7,2,6,1,5,4,3] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,4,3,6,7,5] => [5,7,6,3,4,2,1] => [2,7,6,1,5,4,3] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,2,4,5,3,6,7] => [7,6,3,5,4,2,1] => [7,6,1,5,4,3,2] => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,2,4,5,3,7,6] => [6,7,3,5,4,2,1] => [2,7,1,6,5,4,3] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,4,5,6,3,7] => [7,3,6,5,4,2,1] => [7,1,6,5,4,3,2] => ? = 0
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => [3,7,6,5,4,2,1] => [1,7,6,5,4,3,2] => ? = 0
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,4,6,5,3,7] => [7,3,5,6,4,2,1] => [7,1,2,6,5,4,3] => ? = 0
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,4,6,5,7,3] => [3,7,5,6,4,2,1] => [1,7,2,6,5,4,3] => ? = 0
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,5,4,3,6,7] => [7,6,3,4,5,2,1] => [7,6,1,2,5,4,3] => ? = 0
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,5,4,3,7,6] => [6,7,3,4,5,2,1] => [3,7,1,2,6,5,4] => ? = 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,2,5,4,6,3,7] => [7,3,6,4,5,2,1] => [7,1,6,2,5,4,3] => ? = 0
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,4,6,7,3] => [3,7,6,4,5,2,1] => [1,7,6,2,5,4,3] => ? = 0
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,5,6,4,3,7] => [7,3,4,6,5,2,1] => [7,1,2,6,5,4,3] => ? = 0
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,5,6,4,7,3] => [3,7,4,6,5,2,1] => [1,7,2,6,5,4,3] => ? = 0
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,5,4,3,7] => [7,3,4,5,6,2,1] => [7,1,2,3,6,5,4] => ? = 0
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,6,5,4,7,3] => [3,7,4,5,6,2,1] => [1,7,2,3,6,5,4] => ? = 0
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6,7] => [7,6,5,4,2,3,1] => [7,6,5,4,1,3,2] => ? = 0
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,2,4,5,7,6] => [6,7,5,4,2,3,1] => [2,7,6,5,1,4,3] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,6,5,7] => [7,5,6,4,2,3,1] => [7,2,6,5,1,4,3] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,3,2,4,6,7,5] => [5,7,6,4,2,3,1] => [2,7,6,5,1,4,3] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,4,6,7] => [7,6,4,5,2,3,1] => [7,6,2,5,1,4,3] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6] => [6,7,4,5,2,3,1] => [3,7,2,6,1,5,4] => ? = 2
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,2,5,6,4,7] => [7,4,6,5,2,3,1] => [7,2,6,5,1,4,3] => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,3,2,5,6,7,4] => [4,7,6,5,2,3,1] => [2,7,6,5,1,4,3] => ? = 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,3,2,6,5,4,7] => [7,4,5,6,2,3,1] => [7,1,3,6,2,5,4] => ? = 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,3,2,6,5,7,4] => [4,7,5,6,2,3,1] => [1,7,3,6,2,5,4] => ? = 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,3,4,2,5,6,7] => [7,6,5,2,4,3,1] => [7,6,5,1,4,3,2] => ? = 0
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5,7,6] => [6,7,5,2,4,3,1] => [2,7,6,1,5,4,3] => ? = 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,3,4,2,6,5,7] => [7,5,6,2,4,3,1] => [7,2,6,1,5,4,3] => ? = 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,3,4,2,6,7,5] => [5,7,6,2,4,3,1] => [2,7,6,1,5,4,3] => ? = 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,3,4,5,2,6,7] => [7,6,2,5,4,3,1] => [7,6,1,5,4,3,2] => ? = 0
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,3,4,5,2,7,6] => [6,7,2,5,4,3,1] => [2,7,1,6,5,4,3] => ? = 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,3,4,5,6,2,7] => [7,2,6,5,4,3,1] => [7,1,6,5,4,3,2] => ? = 0
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,2] => [2,7,6,5,4,3,1] => [1,7,6,5,4,3,2] => ? = 0
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,3,4,6,5,2,7] => [7,2,5,6,4,3,1] => [7,1,2,6,5,4,3] => ? = 0
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,3,4,6,5,7,2] => [2,7,5,6,4,3,1] => [1,7,2,6,5,4,3] => ? = 0
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,3,5,4,2,6,7] => [7,6,2,4,5,3,1] => [7,6,1,2,5,4,3] => ? = 0
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,3,5,4,2,7,6] => [6,7,2,4,5,3,1] => [3,7,1,2,6,5,4] => ? = 1
Description
The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation.
Matching statistic: St000779
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000779: Permutations ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000779: Permutations ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => ? = 0
[1,0,1,0]
=> [1,2] => [2,1] => [2,1] => 0
[1,1,0,0]
=> [2,1] => [1,2] => [1,2] => 0
[1,0,1,0,1,0]
=> [1,2,3] => [3,2,1] => [3,2,1] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [2,3,1] => [1,3,2] => 0
[1,1,0,0,1,0]
=> [2,1,3] => [3,1,2] => [3,1,2] => 0
[1,1,0,1,0,0]
=> [2,3,1] => [1,3,2] => [1,3,2] => 0
[1,1,1,0,0,0]
=> [3,2,1] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [3,4,2,1] => [1,4,3,2] => 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [4,2,3,1] => [4,1,3,2] => 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,3,1] => [1,4,3,2] => 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [2,3,4,1] => [1,2,4,3] => 0
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [4,3,1,2] => [4,3,1,2] => 0
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,4,1,2] => [2,4,1,3] => 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [4,1,3,2] => [4,1,3,2] => 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,4,3,2] => [1,4,3,2] => 0
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,3,4,2] => [1,2,4,3] => 0
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [4,1,2,3] => [4,1,2,3] => 0
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,4,2,3] => [1,4,2,3] => 0
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [1,2,4,3] => [1,2,4,3] => 0
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [4,5,3,2,1] => [1,5,4,3,2] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [5,3,4,2,1] => [5,1,4,3,2] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [3,5,4,2,1] => [1,5,4,3,2] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,4,5,2,1] => [1,2,5,4,3] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [5,4,2,3,1] => [5,4,1,3,2] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [4,5,2,3,1] => [2,5,1,4,3] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [5,2,4,3,1] => [5,1,4,3,2] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,5,4,3,1] => [1,5,4,3,2] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [2,4,5,3,1] => [1,2,5,4,3] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [5,2,3,4,1] => [5,1,2,4,3] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [2,5,3,4,1] => [1,5,2,4,3] => 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [2,3,5,4,1] => [1,2,5,4,3] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,3,4,5,1] => [1,2,3,5,4] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [5,4,3,1,2] => [5,4,3,1,2] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [4,5,3,1,2] => [2,5,4,1,3] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [5,3,4,1,2] => [5,2,4,1,3] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,5,4,1,2] => [2,5,4,1,3] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [3,4,5,1,2] => [1,3,5,2,4] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [5,4,1,3,2] => [5,4,1,3,2] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [4,5,1,3,2] => [2,5,1,4,3] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [5,1,4,3,2] => [5,1,4,3,2] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,5,4,3,2] => [1,5,4,3,2] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,4,5,3,2] => [1,2,5,4,3] => 0
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [5,1,3,4,2] => [5,1,2,4,3] => 0
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,5,3,4,2] => [1,5,2,4,3] => 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [1,3,5,4,2] => [1,2,5,4,3] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,3,4,5,2] => [1,2,3,5,4] => 0
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [5,4,1,2,3] => [5,4,1,2,3] => 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => [1,7,6,5,4,3,2] => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [7,1,6,5,4,3,2] => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => [5,7,6,4,3,2,1] => [1,7,6,5,4,3,2] => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => [7,6,1,5,4,3,2] => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => [2,7,1,6,5,4,3] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,3,5,6,4,7] => [7,4,6,5,3,2,1] => [7,1,6,5,4,3,2] => ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => [4,7,6,5,3,2,1] => [1,7,6,5,4,3,2] => ? = 0
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,5,4,7] => [7,4,5,6,3,2,1] => [7,1,2,6,5,4,3] => ? = 0
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,5,7,4] => [4,7,5,6,3,2,1] => [1,7,2,6,5,4,3] => ? = 0
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,4,3,5,6,7] => [7,6,5,3,4,2,1] => [7,6,5,1,4,3,2] => ? = 0
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,4,3,5,7,6] => [6,7,5,3,4,2,1] => [2,7,6,1,5,4,3] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5,7] => [7,5,6,3,4,2,1] => [7,2,6,1,5,4,3] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,4,3,6,7,5] => [5,7,6,3,4,2,1] => [2,7,6,1,5,4,3] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,2,4,5,3,6,7] => [7,6,3,5,4,2,1] => [7,6,1,5,4,3,2] => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,2,4,5,3,7,6] => [6,7,3,5,4,2,1] => [2,7,1,6,5,4,3] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,4,5,6,3,7] => [7,3,6,5,4,2,1] => [7,1,6,5,4,3,2] => ? = 0
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => [3,7,6,5,4,2,1] => [1,7,6,5,4,3,2] => ? = 0
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,4,6,5,3,7] => [7,3,5,6,4,2,1] => [7,1,2,6,5,4,3] => ? = 0
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,4,6,5,7,3] => [3,7,5,6,4,2,1] => [1,7,2,6,5,4,3] => ? = 0
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,5,4,3,6,7] => [7,6,3,4,5,2,1] => [7,6,1,2,5,4,3] => ? = 0
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,5,4,3,7,6] => [6,7,3,4,5,2,1] => [3,7,1,2,6,5,4] => ? = 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,2,5,4,6,3,7] => [7,3,6,4,5,2,1] => [7,1,6,2,5,4,3] => ? = 0
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,4,6,7,3] => [3,7,6,4,5,2,1] => [1,7,6,2,5,4,3] => ? = 0
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,5,6,4,3,7] => [7,3,4,6,5,2,1] => [7,1,2,6,5,4,3] => ? = 0
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,5,6,4,7,3] => [3,7,4,6,5,2,1] => [1,7,2,6,5,4,3] => ? = 0
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,5,4,3,7] => [7,3,4,5,6,2,1] => [7,1,2,3,6,5,4] => ? = 0
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,6,5,4,7,3] => [3,7,4,5,6,2,1] => [1,7,2,3,6,5,4] => ? = 0
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6,7] => [7,6,5,4,2,3,1] => [7,6,5,4,1,3,2] => ? = 0
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,2,4,5,7,6] => [6,7,5,4,2,3,1] => [2,7,6,5,1,4,3] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,6,5,7] => [7,5,6,4,2,3,1] => [7,2,6,5,1,4,3] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,3,2,4,6,7,5] => [5,7,6,4,2,3,1] => [2,7,6,5,1,4,3] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,4,6,7] => [7,6,4,5,2,3,1] => [7,6,2,5,1,4,3] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6] => [6,7,4,5,2,3,1] => [3,7,2,6,1,5,4] => ? = 2
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,2,5,6,4,7] => [7,4,6,5,2,3,1] => [7,2,6,5,1,4,3] => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,3,2,5,6,7,4] => [4,7,6,5,2,3,1] => [2,7,6,5,1,4,3] => ? = 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,3,2,6,5,4,7] => [7,4,5,6,2,3,1] => [7,1,3,6,2,5,4] => ? = 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,3,2,6,5,7,4] => [4,7,5,6,2,3,1] => [1,7,3,6,2,5,4] => ? = 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,3,4,2,5,6,7] => [7,6,5,2,4,3,1] => [7,6,5,1,4,3,2] => ? = 0
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5,7,6] => [6,7,5,2,4,3,1] => [2,7,6,1,5,4,3] => ? = 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,3,4,2,6,5,7] => [7,5,6,2,4,3,1] => [7,2,6,1,5,4,3] => ? = 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,3,4,2,6,7,5] => [5,7,6,2,4,3,1] => [2,7,6,1,5,4,3] => ? = 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,3,4,5,2,6,7] => [7,6,2,5,4,3,1] => [7,6,1,5,4,3,2] => ? = 0
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,3,4,5,2,7,6] => [6,7,2,5,4,3,1] => [2,7,1,6,5,4,3] => ? = 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,3,4,5,6,2,7] => [7,2,6,5,4,3,1] => [7,1,6,5,4,3,2] => ? = 0
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,2] => [2,7,6,5,4,3,1] => [1,7,6,5,4,3,2] => ? = 0
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,3,4,6,5,2,7] => [7,2,5,6,4,3,1] => [7,1,2,6,5,4,3] => ? = 0
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,3,4,6,5,7,2] => [2,7,5,6,4,3,1] => [1,7,2,6,5,4,3] => ? = 0
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,3,5,4,2,6,7] => [7,6,2,4,5,3,1] => [7,6,1,2,5,4,3] => ? = 0
Description
The tier of a permutation.
This is the number of elements $i$ such that $[i+1,k,i]$ is an occurrence of the pattern $[2,3,1]$. For example, $[3,5,6,1,2,4]$ has tier $2$, with witnesses $[3,5,2]$ (or $[3,6,2]$) and $[5,6,4]$.
According to [1], this is the number of passes minus one needed to sort the permutation using a single stack. The generating function for this statistic appears as [[OEIS:A122890]] and [[OEIS:A158830]] in the form of triangles read by rows, see [sec. 4, 1].
The following 84 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001083The number of boxed occurrences of 132 in a permutation. St001621The number of atoms of a lattice. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001577The minimal number of edges to add or remove to make a graph a cograph. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001545The second Elser number of a connected graph. St001330The hat guessing number of a graph. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St000360The number of occurrences of the pattern 32-1. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000314The number of left-to-right-maxima of a permutation. St001737The number of descents of type 2 in a permutation. St000632The jump number of the poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000640The rank of the largest boolean interval in a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000068The number of minimal elements in a poset. St000181The number of connected components of the Hasse diagram for the poset. St001868The number of alignments of type NE of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001890The maximum magnitude of the Möbius function of a poset. St001866The nesting alignments of a signed permutation. St000455The second largest eigenvalue of a graph if it is integral. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St000741The Colin de Verdière graph invariant. St001301The first Betti number of the order complex associated with the poset. St000908The length of the shortest maximal antichain in a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001550The number of inversions between exceedances where the greater exceedance is linked. St000914The sum of the values of the Möbius function of a poset. St001867The number of alignments of type EN of a signed permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000461The rix statistic of a permutation. St000488The number of cycles of a permutation of length at most 2. St000516The number of stretching pairs of a permutation. St000629The defect of a binary word. St000666The number of right tethers of a permutation. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000877The depth of the binary word interpreted as a path. St000951The dimension of $Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001549The number of restricted non-inversions between exceedances. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001811The Castelnuovo-Mumford regularity of a permutation. St001847The number of occurrences of the pattern 1432 in a permutation. St000335The difference of lower and upper interactions. St000570The Edelman-Greene number of a permutation. St000805The number of peaks of the associated bargraph. St000876The number of factors in the Catalan decomposition of a binary word. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St001162The minimum jump of a permutation. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001344The neighbouring number of a permutation. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001722The number of minimal chains with small intervals between a binary word and the top element. St001845The number of join irreducibles minus the rank of a lattice. St001613The binary logarithm of the size of the center of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001720The minimal length of a chain of small intervals in a lattice.
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