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Your data matches 16 different statistics following compositions of up to 3 maps.
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Matching statistic: St001176
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,1]
=> [1]
=> 0
[1,2,3] => [1,1,1]
=> [1,1]
=> 1
[1,3,2] => [2,1]
=> [1]
=> 0
[2,1,3] => [2,1]
=> [1]
=> 0
[3,2,1] => [2,1]
=> [1]
=> 0
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 2
[1,2,4,3] => [2,1,1]
=> [1,1]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> 1
[1,3,4,2] => [3,1]
=> [1]
=> 0
[1,4,2,3] => [3,1]
=> [1]
=> 0
[1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> 0
[2,3,1,4] => [3,1]
=> [1]
=> 0
[2,4,3,1] => [3,1]
=> [1]
=> 0
[3,1,2,4] => [3,1]
=> [1]
=> 0
[3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[3,2,4,1] => [3,1]
=> [1]
=> 0
[3,4,1,2] => [2,2]
=> [2]
=> 0
[4,1,3,2] => [3,1]
=> [1]
=> 0
[4,2,1,3] => [3,1]
=> [1]
=> 0
[4,2,3,1] => [2,1,1]
=> [1,1]
=> 1
[4,3,2,1] => [2,2]
=> [2]
=> 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 3
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> 1
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> 1
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> 1
[1,3,4,5,2] => [4,1]
=> [1]
=> 0
[1,3,5,2,4] => [4,1]
=> [1]
=> 0
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 1
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> 1
[1,4,2,5,3] => [4,1]
=> [1]
=> 0
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> 1
[1,4,5,3,2] => [4,1]
=> [1]
=> 0
[1,5,2,3,4] => [4,1]
=> [1]
=> 0
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,5,4,2,3] => [4,1]
=> [1]
=> 0
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> 2
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> 1
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> 1
Description
The size of a partition minus its first part.
This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St000228
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,1]
=> [1]
=> []
=> 0
[1,2,3] => [1,1,1]
=> [1,1]
=> [1]
=> 1
[1,3,2] => [2,1]
=> [1]
=> []
=> 0
[2,1,3] => [2,1]
=> [1]
=> []
=> 0
[3,2,1] => [2,1]
=> [1]
=> []
=> 0
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[1,3,4,2] => [3,1]
=> [1]
=> []
=> 0
[1,4,2,3] => [3,1]
=> [1]
=> []
=> 0
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> []
=> 0
[2,3,1,4] => [3,1]
=> [1]
=> []
=> 0
[2,4,3,1] => [3,1]
=> [1]
=> []
=> 0
[3,1,2,4] => [3,1]
=> [1]
=> []
=> 0
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,4,1] => [3,1]
=> [1]
=> []
=> 0
[3,4,1,2] => [2,2]
=> [2]
=> []
=> 0
[4,1,3,2] => [3,1]
=> [1]
=> []
=> 0
[4,2,1,3] => [3,1]
=> [1]
=> []
=> 0
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[4,3,2,1] => [2,2]
=> [2]
=> []
=> 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,3,4,5,2] => [4,1]
=> [1]
=> []
=> 0
[1,3,5,2,4] => [4,1]
=> [1]
=> []
=> 0
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,4,2,5,3] => [4,1]
=> [1]
=> []
=> 0
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[1,4,5,3,2] => [4,1]
=> [1]
=> []
=> 0
[1,5,2,3,4] => [4,1]
=> [1]
=> []
=> 0
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[1,5,4,2,3] => [4,1]
=> [1]
=> []
=> 0
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> [1]
=> 1
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St000738
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 91% ●values known / values provided: 100%●distinct values known / distinct values provided: 91%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 91% ●values known / values provided: 100%●distinct values known / distinct values provided: 91%
Values
[1,2] => [1,1]
=> [1]
=> [[1]]
=> 1 = 0 + 1
[1,2,3] => [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[1,3,2] => [2,1]
=> [1]
=> [[1]]
=> 1 = 0 + 1
[2,1,3] => [2,1]
=> [1]
=> [[1]]
=> 1 = 0 + 1
[3,2,1] => [2,1]
=> [1]
=> [[1]]
=> 1 = 0 + 1
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 2 + 1
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[1,3,4,2] => [3,1]
=> [1]
=> [[1]]
=> 1 = 0 + 1
[1,4,2,3] => [3,1]
=> [1]
=> [[1]]
=> 1 = 0 + 1
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[2,1,4,3] => [2,2]
=> [2]
=> [[1,2]]
=> 1 = 0 + 1
[2,3,1,4] => [3,1]
=> [1]
=> [[1]]
=> 1 = 0 + 1
[2,4,3,1] => [3,1]
=> [1]
=> [[1]]
=> 1 = 0 + 1
[3,1,2,4] => [3,1]
=> [1]
=> [[1]]
=> 1 = 0 + 1
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[3,2,4,1] => [3,1]
=> [1]
=> [[1]]
=> 1 = 0 + 1
[3,4,1,2] => [2,2]
=> [2]
=> [[1,2]]
=> 1 = 0 + 1
[4,1,3,2] => [3,1]
=> [1]
=> [[1]]
=> 1 = 0 + 1
[4,2,1,3] => [3,1]
=> [1]
=> [[1]]
=> 1 = 0 + 1
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[4,3,2,1] => [2,2]
=> [2]
=> [[1,2]]
=> 1 = 0 + 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4 = 3 + 1
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 2 + 1
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 2 + 1
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 2 + 1
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 2 + 1
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[1,3,4,5,2] => [4,1]
=> [1]
=> [[1]]
=> 1 = 0 + 1
[1,3,5,2,4] => [4,1]
=> [1]
=> [[1]]
=> 1 = 0 + 1
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[1,4,2,5,3] => [4,1]
=> [1]
=> [[1]]
=> 1 = 0 + 1
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 2 + 1
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[1,4,5,3,2] => [4,1]
=> [1]
=> [[1]]
=> 1 = 0 + 1
[1,5,2,3,4] => [4,1]
=> [1]
=> [[1]]
=> 1 = 0 + 1
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 2 + 1
[1,5,4,2,3] => [4,1]
=> [1]
=> [[1]]
=> 1 = 0 + 1
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 2 + 1
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[1,2,3,4,5,6,7,8,9,10,11,12] => [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 10 + 1
Description
The first entry in the last row of a standard tableau.
For the last entry in the first row, see [[St000734]].
Matching statistic: St000371
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000371: Permutations ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 91%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000371: Permutations ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 91%
Values
[1,2] => [1,1]
=> [[1],[2]]
=> [2,1] => 0
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[1,3,2] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 0
[2,1,3] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 0
[3,2,1] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 0
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 2
[1,2,4,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 1
[1,3,2,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 1
[1,3,4,2] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 0
[1,4,2,3] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 0
[1,4,3,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 1
[2,1,3,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 1
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 0
[2,3,1,4] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 0
[2,4,3,1] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 0
[3,1,2,4] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 0
[3,2,1,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 1
[3,2,4,1] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 0
[3,4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 0
[4,1,3,2] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 0
[4,2,1,3] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 0
[4,2,3,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 1
[4,3,2,1] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 3
[1,2,3,5,4] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 2
[1,2,4,3,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 2
[1,2,4,5,3] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 1
[1,2,5,3,4] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 1
[1,2,5,4,3] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 2
[1,3,2,4,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 2
[1,3,2,5,4] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 1
[1,3,4,2,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 1
[1,3,4,5,2] => [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 0
[1,3,5,2,4] => [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 0
[1,3,5,4,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 1
[1,4,2,3,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 1
[1,4,2,5,3] => [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 0
[1,4,3,2,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 2
[1,4,3,5,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 1
[1,4,5,2,3] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 1
[1,4,5,3,2] => [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 0
[1,5,2,3,4] => [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 0
[1,5,2,4,3] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 1
[1,5,3,2,4] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 1
[1,5,3,4,2] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 2
[1,5,4,2,3] => [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 0
[1,5,4,3,2] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 1
[2,1,3,4,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 2
[2,1,3,5,4] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 1
[2,1,4,3,5] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 1
[2,1,4,3,6,5,8,7,10,9,12,11] => [2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> [11,12,9,10,7,8,5,6,3,4,1,2] => ? = 8
[6,5,4,3,2,1,12,11,10,9,8,7] => [2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> [11,12,9,10,7,8,5,6,3,4,1,2] => ? = 8
[12,11,10,9,8,7,6,5,4,3,2,1] => [2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> [11,12,9,10,7,8,5,6,3,4,1,2] => ? = 8
[1,2,3,4,5,6,7,8,9,11,10] => [2,1,1,1,1,1,1,1,1,1]
=> [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? => ? = 8
[2,1,3,4,5,6,7,8,9,10,11] => [2,1,1,1,1,1,1,1,1,1]
=> [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? => ? = 8
[11,1,2,3,4,5,6,10,7,8,9] => [9,2]
=> [[1,2,5,6,7,8,9,10,11],[3,4]]
=> ? => ? = 0
[10,1,2,3,4,5,11,6,7,8,9] => [8,3]
=> [[1,2,3,7,8,9,10,11],[4,5,6]]
=> ? => ? = 0
[11,1,2,3,4,10,5,6,7,8,9] => [8,3]
=> [[1,2,3,7,8,9,10,11],[4,5,6]]
=> ? => ? = 0
[11,10,1,2,3,4,5,6,7,8,9] => [6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> ? => ? = 0
[10,9,8,7,6,5,4,3,2,1,11] => [2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 7
[1,11,10,9,8,7,6,5,4,3,2] => [2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 7
[2,4,6,8,11,12,1,3,5,7,9,10] => [9,3]
=> [[1,2,3,7,8,9,10,11,12],[4,5,6]]
=> ? => ? = 0
[2,4,6,9,10,12,1,3,5,7,8,11] => [7,5]
=> [[1,2,3,4,5,11,12],[6,7,8,9,10]]
=> ? => ? = 0
[2,4,6,10,11,12,1,3,5,7,8,9] => [7,5]
=> [[1,2,3,4,5,11,12],[6,7,8,9,10]]
=> ? => ? = 0
[2,4,7,8,11,12,1,3,5,6,9,10] => [6,3,3]
=> [[1,2,3,10,11,12],[4,5,6],[7,8,9]]
=> ? => ? = 3
[2,4,7,9,11,12,1,3,5,6,8,10] => [9,3]
=> [[1,2,3,7,8,9,10,11,12],[4,5,6]]
=> ? => ? = 0
[2,4,8,9,10,12,1,3,5,6,7,11] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> ? => ? = 0
[2,4,8,9,11,12,1,3,5,6,7,10] => [7,3,2]
=> [[1,2,5,9,10,11,12],[3,4,8],[6,7]]
=> ? => ? = 2
[2,4,8,10,11,12,1,3,5,6,7,9] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> ? => ? = 0
[2,5,6,8,10,12,1,3,4,7,9,11] => [7,5]
=> [[1,2,3,4,5,11,12],[6,7,8,9,10]]
=> ? => ? = 0
[2,5,6,9,10,12,1,3,4,7,8,11] => [5,5,2]
=> [[1,2,5,6,7],[3,4,10,11,12],[8,9]]
=> ? => ? = 2
[2,5,6,9,11,12,1,3,4,7,8,10] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> ? => ? = 0
[2,5,7,8,11,12,1,3,4,6,9,10] => [9,3]
=> [[1,2,3,7,8,9,10,11,12],[4,5,6]]
=> ? => ? = 0
[2,5,7,9,10,12,1,3,4,6,8,11] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> ? => ? = 0
[2,5,7,9,11,12,1,3,4,6,8,10] => [7,3,2]
=> [[1,2,5,9,10,11,12],[3,4,8],[6,7]]
=> ? => ? = 2
[2,5,7,10,11,12,1,3,4,6,8,9] => [7,5]
=> [[1,2,3,4,5,11,12],[6,7,8,9,10]]
=> ? => ? = 0
[2,5,8,9,10,12,1,3,4,6,7,11] => [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ? => ? = 2
[2,5,8,9,11,12,1,3,4,6,7,10] => [5,3,2,2]
=> [[1,2,7,11,12],[3,4,10],[5,6],[8,9]]
=> [8,9,5,6,3,4,10,1,2,7,11,12] => ? = 4
[2,5,8,10,11,12,1,3,4,6,7,9] => [5,5,2]
=> [[1,2,5,6,7],[3,4,10,11,12],[8,9]]
=> ? => ? = 2
[2,5,9,10,11,12,1,3,4,6,7,8] => [7,5]
=> [[1,2,3,4,5,11,12],[6,7,8,9,10]]
=> ? => ? = 0
[2,6,7,8,10,12,1,3,4,5,9,11] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> ? => ? = 0
[2,6,7,9,10,12,1,3,4,5,8,11] => [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ? => ? = 2
[2,6,7,9,11,12,1,3,4,5,8,10] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> ? => ? = 0
[2,6,8,9,10,12,1,3,4,5,7,11] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? => ? = 4
[2,6,8,9,11,12,1,3,4,5,7,10] => [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ? => ? = 2
[2,6,8,10,11,12,1,3,4,5,7,9] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> ? => ? = 0
[2,7,8,9,10,12,1,3,4,5,6,11] => [3,3,2,2,2]
=> [[1,2,9],[3,4,12],[5,6],[7,8],[10,11]]
=> ? => ? = 6
[2,7,8,9,11,12,1,3,4,5,6,10] => [5,3,2,2]
=> [[1,2,7,11,12],[3,4,10],[5,6],[8,9]]
=> [8,9,5,6,3,4,10,1,2,7,11,12] => ? = 4
[2,7,8,10,11,12,1,3,4,5,6,9] => [7,3,2]
=> [[1,2,5,9,10,11,12],[3,4,8],[6,7]]
=> ? => ? = 2
[2,7,9,10,11,12,1,3,4,5,6,8] => [9,3]
=> [[1,2,3,7,8,9,10,11,12],[4,5,6]]
=> ? => ? = 0
[3,4,6,8,10,12,1,2,5,7,9,11] => [9,3]
=> [[1,2,3,7,8,9,10,11,12],[4,5,6]]
=> ? => ? = 0
[3,4,6,8,11,12,1,2,5,7,9,10] => [6,3,3]
=> [[1,2,3,10,11,12],[4,5,6],[7,8,9]]
=> ? => ? = 3
[3,4,7,8,10,12,1,2,5,6,9,11] => [6,3,3]
=> [[1,2,3,10,11,12],[4,5,6],[7,8,9]]
=> ? => ? = 3
[3,4,7,8,11,12,1,2,5,6,9,10] => [3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> ? => ? = 6
[3,4,7,9,10,12,1,2,5,6,8,11] => [9,3]
=> [[1,2,3,7,8,9,10,11,12],[4,5,6]]
=> ? => ? = 0
[3,4,7,9,11,12,1,2,5,6,8,10] => [6,3,3]
=> [[1,2,3,10,11,12],[4,5,6],[7,8,9]]
=> ? => ? = 3
[3,4,7,10,11,12,1,2,5,6,8,9] => [9,3]
=> [[1,2,3,7,8,9,10,11,12],[4,5,6]]
=> ? => ? = 0
[3,4,8,9,11,12,1,2,5,6,7,10] => [9,3]
=> [[1,2,3,7,8,9,10,11,12],[4,5,6]]
=> ? => ? = 0
[3,5,6,9,10,12,1,2,4,7,8,11] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> ? => ? = 0
[3,5,6,9,11,12,1,2,4,7,8,10] => [6,4,2]
=> [[1,2,5,6,11,12],[3,4,9,10],[7,8]]
=> [7,8,3,4,9,10,1,2,5,6,11,12] => ? = 2
Description
The number of mid points of decreasing subsequences of length 3 in a permutation.
For a permutation π of {1,…,n}, this is the number of indices j such that there exist indices i,k with i<j<k and π(i)>π(j)>π(k). In other words, this is the number of indices that are neither left-to-right maxima nor right-to-left minima.
This statistic can also be expressed as the number of occurrences of the mesh pattern ([3,2,1], {(0,2),(0,3),(2,0),(3,0)}): the shading fixes the first and the last element of the decreasing subsequence.
See also [[St000119]].
Matching statistic: St001330
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 64%
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 64%
Values
[1,2] => {{1},{2}}
=> [1,1] => ([(0,1)],2)
=> 2 = 0 + 2
[1,2,3] => {{1},{2},{3}}
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[1,3,2] => {{1},{2,3}}
=> [1,2] => ([(1,2)],3)
=> 2 = 0 + 2
[2,1,3] => {{1,2},{3}}
=> [2,1] => ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[3,2,1] => {{1,3},{2}}
=> [2,1] => ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[1,2,3,4] => {{1},{2},{3},{4}}
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 2 + 2
[1,2,4,3] => {{1},{2},{3,4}}
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[1,3,2,4] => {{1},{2,3},{4}}
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 2
[1,3,4,2] => {{1},{2,3,4}}
=> [1,3] => ([(2,3)],4)
=> 2 = 0 + 2
[1,4,2,3] => {{1},{2,3,4}}
=> [1,3] => ([(2,3)],4)
=> 2 = 0 + 2
[1,4,3,2] => {{1},{2,4},{3}}
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 2
[2,1,3,4] => {{1,2},{3},{4}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 2
[2,1,4,3] => {{1,2},{3,4}}
=> [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[2,3,1,4] => {{1,2,3},{4}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[2,4,3,1] => {{1,2,4},{3}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[3,1,2,4] => {{1,2,3},{4}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[3,2,1,4] => {{1,3},{2},{4}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 2
[3,2,4,1] => {{1,3,4},{2}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[3,4,1,2] => {{1,3},{2,4}}
=> [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[4,1,3,2] => {{1,2,4},{3}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[4,2,1,3] => {{1,3,4},{2}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[4,2,3,1] => {{1,4},{2},{3}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 2
[4,3,2,1] => {{1,4},{2,3}}
=> [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 3 + 2
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 2
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[1,2,5,3,4] => {{1},{2},{3,4,5}}
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 2
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 2
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> [1,4] => ([(3,4)],5)
=> 2 = 0 + 2
[1,3,5,2,4] => {{1},{2,3,4,5}}
=> [1,4] => ([(3,4)],5)
=> 2 = 0 + 2
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,4,2,3,5] => {{1},{2,3,4},{5}}
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,4,2,5,3] => {{1},{2,3,4,5}}
=> [1,4] => ([(3,4)],5)
=> 2 = 0 + 2
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 2
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,4,5,3,2] => {{1},{2,3,4,5}}
=> [1,4] => ([(3,4)],5)
=> 2 = 0 + 2
[1,5,2,3,4] => {{1},{2,3,4,5}}
=> [1,4] => ([(3,4)],5)
=> 2 = 0 + 2
[1,5,2,4,3] => {{1},{2,3,5},{4}}
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,5,3,2,4] => {{1},{2,4,5},{3}}
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,5,3,4,2] => {{1},{2,5},{3},{4}}
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 2
[1,5,4,2,3] => {{1},{2,3,4,5}}
=> [1,4] => ([(3,4)],5)
=> 2 = 0 + 2
[1,5,4,3,2] => {{1},{2,5},{3,4}}
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 2
[2,1,3,5,4] => {{1,2},{3},{4,5}}
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[2,1,4,3,5] => {{1,2},{3,4},{5}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[2,1,4,5,3] => {{1,2},{3,4,5}}
=> [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,1,5,3,4] => {{1,2},{3,4,5}}
=> [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,1,5,4,3] => {{1,2},{3,5},{4}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[2,3,1,4,5] => {{1,2,3},{4},{5}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[2,3,1,5,4] => {{1,2,3},{4,5}}
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,3,4,1,5] => {{1,2,3,4},{5}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,3,5,4,1] => {{1,2,3,5},{4}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,4,1,3,5] => {{1,2,3,4},{5}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,4,3,1,5] => {{1,2,4},{3},{5}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[2,4,3,5,1] => {{1,2,4,5},{3}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,4,5,1,3] => {{1,2,4},{3,5}}
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,5,1,4,3] => {{1,2,3,5},{4}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,5,3,1,4] => {{1,2,4,5},{3}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,5,3,4,1] => {{1,2,5},{3},{4}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[2,5,4,3,1] => {{1,2,5},{3,4}}
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[3,1,2,4,5] => {{1,2,3},{4},{5}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[3,1,2,5,4] => {{1,2,3},{4,5}}
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[3,1,4,2,5] => {{1,2,3,4},{5}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[3,1,5,4,2] => {{1,2,3,5},{4}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 2
[3,2,1,5,4] => {{1,3},{2},{4,5}}
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[3,2,4,1,5] => {{1,3,4},{2},{5}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[3,2,4,5,1] => {{1,3,4,5},{2}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[3,2,5,1,4] => {{1,3,4,5},{2}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[3,2,5,4,1] => {{1,3,5},{2},{4}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[3,4,1,2,5] => {{1,3},{2,4},{5}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[3,4,1,5,2] => {{1,3},{2,4,5}}
=> [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[3,4,2,1,5] => {{1,2,3,4},{5}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[3,4,5,2,1] => {{1,3,5},{2,4}}
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[3,5,1,2,4] => {{1,3},{2,4,5}}
=> [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[3,5,1,4,2] => {{1,3},{2,5},{4}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[3,5,2,4,1] => {{1,2,3,5},{4}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[3,5,4,1,2] => {{1,3,4},{2,5}}
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[4,1,3,2,5] => {{1,2,4},{3},{5}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[4,2,1,3,5] => {{1,3,4},{2},{5}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[4,2,3,1,5] => {{1,4},{2},{3},{5}}
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 2
[4,2,3,5,1] => {{1,4,5},{2},{3}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[4,2,5,1,3] => {{1,4},{2},{3,5}}
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[4,3,2,1,5] => {{1,4},{2,3},{5}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[4,5,3,1,2] => {{1,4},{2,5},{3}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[5,1,3,4,2] => {{1,2,5},{3},{4}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[5,2,1,4,3] => {{1,3,5},{2},{4}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[5,2,3,1,4] => {{1,4,5},{2},{3}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[5,2,3,4,1] => {{1,5},{2},{3},{4}}
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 2
[5,2,4,3,1] => {{1,5},{2},{3,4}}
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[5,3,2,4,1] => {{1,5},{2,3},{4}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[5,4,3,2,1] => {{1,5},{2,4},{3}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,2,3,5,4,6] => {{1},{2},{3},{4,5},{6}}
=> [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 2
[1,2,3,6,5,4] => {{1},{2},{3},{4,6},{5}}
=> [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 2
[1,2,4,3,5,6] => {{1},{2},{3,4},{5},{6}}
=> [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 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: St001232
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 55%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 55%
Values
[1,2] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[2,1,3,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[2,1,4,3] => [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 0 + 1
[2,3,1,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[2,4,3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[3,1,2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[3,2,4,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[3,4,1,2] => [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 0 + 1
[4,1,3,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[4,2,1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[4,3,2,1] => [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 0 + 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 3 + 1
[1,2,3,5,4] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 2 + 1
[1,2,4,3,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 2 + 1
[1,2,4,5,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,2,5,3,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 2 + 1
[1,3,2,4,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 2 + 1
[1,3,2,5,4] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[1,3,4,2,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,3,4,5,2] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,3,5,2,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,4,2,3,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,4,2,5,3] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,4,3,2,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 2 + 1
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,4,5,2,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[1,4,5,3,2] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,5,2,3,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,5,2,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,5,3,2,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 2 + 1
[1,5,4,2,3] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,5,4,3,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[2,1,3,4,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 2 + 1
[2,1,3,5,4] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[2,1,4,3,5] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[2,1,4,5,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 1
[2,1,5,3,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 1
[2,1,5,4,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[2,3,1,4,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[2,3,1,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 1
[2,3,4,1,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[2,3,5,4,1] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[2,4,1,3,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[2,4,3,1,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[2,4,3,5,1] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[2,4,5,1,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 1
[2,5,1,4,3] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[2,5,3,1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[2,5,4,3,1] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 1
[3,1,2,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 1
[3,2,1,5,4] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[3,4,1,2,5] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[3,4,1,5,2] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 1
[3,4,5,2,1] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 1
[3,5,1,2,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 1
[3,5,1,4,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[3,5,4,1,2] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 1
[4,1,5,2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 1
[4,2,5,1,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[4,3,2,1,5] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[4,3,2,5,1] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 1
[4,3,5,1,2] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 1
[4,5,1,3,2] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 1
[4,5,2,1,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 1
[4,5,3,1,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[5,1,4,3,2] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 1
[5,2,4,3,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[5,3,2,1,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 1
[5,3,2,4,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[5,3,4,2,1] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 1
[5,4,1,2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 1
[5,4,2,3,1] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 1
[5,4,3,2,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[1,2,4,3,6,5] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ? = 2 + 1
[1,2,5,6,3,4] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ? = 2 + 1
[1,2,6,5,4,3] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ? = 2 + 1
[1,3,2,4,6,5] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ? = 2 + 1
[1,3,2,5,4,6] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ? = 2 + 1
[1,3,2,5,6,4] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ? = 1 + 1
[1,3,2,6,4,5] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ? = 1 + 1
[1,3,2,6,5,4] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ? = 2 + 1
[1,3,4,2,6,5] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ? = 1 + 1
[1,3,5,6,2,4] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ? = 1 + 1
[1,3,6,5,4,2] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ? = 1 + 1
[1,4,2,3,6,5] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ? = 1 + 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000864
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000864: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 45%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000864: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 45%
Values
[1,2] => [1,1]
=> [[1],[2]]
=> [2,1] => 1 = 0 + 1
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 2 = 1 + 1
[1,3,2] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 1 = 0 + 1
[2,1,3] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 1 = 0 + 1
[3,2,1] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 1 = 0 + 1
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 3 = 2 + 1
[1,2,4,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2 = 1 + 1
[1,3,2,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2 = 1 + 1
[1,3,4,2] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1 = 0 + 1
[1,4,2,3] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1 = 0 + 1
[1,4,3,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2 = 1 + 1
[2,1,3,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2 = 1 + 1
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 1 = 0 + 1
[2,3,1,4] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1 = 0 + 1
[2,4,3,1] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1 = 0 + 1
[3,1,2,4] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1 = 0 + 1
[3,2,1,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2 = 1 + 1
[3,2,4,1] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1 = 0 + 1
[3,4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 1 = 0 + 1
[4,1,3,2] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1 = 0 + 1
[4,2,1,3] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1 = 0 + 1
[4,2,3,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2 = 1 + 1
[4,3,2,1] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 1 = 0 + 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 4 = 3 + 1
[1,2,3,5,4] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 3 = 2 + 1
[1,2,4,3,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 3 = 2 + 1
[1,2,4,5,3] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2 = 1 + 1
[1,2,5,3,4] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2 = 1 + 1
[1,2,5,4,3] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 3 = 2 + 1
[1,3,2,4,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 3 = 2 + 1
[1,3,2,5,4] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 2 = 1 + 1
[1,3,4,2,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2 = 1 + 1
[1,3,4,5,2] => [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 1 = 0 + 1
[1,3,5,2,4] => [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 1 = 0 + 1
[1,3,5,4,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2 = 1 + 1
[1,4,2,3,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2 = 1 + 1
[1,4,2,5,3] => [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 1 = 0 + 1
[1,4,3,2,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 3 = 2 + 1
[1,4,3,5,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2 = 1 + 1
[1,4,5,2,3] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 2 = 1 + 1
[1,4,5,3,2] => [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 1 = 0 + 1
[1,5,2,3,4] => [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 1 = 0 + 1
[1,5,2,4,3] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2 = 1 + 1
[1,5,3,2,4] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2 = 1 + 1
[1,5,3,4,2] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 3 = 2 + 1
[1,5,4,2,3] => [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 1 = 0 + 1
[1,5,4,3,2] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 2 = 1 + 1
[2,1,3,4,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 3 = 2 + 1
[2,1,3,5,4] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 2 = 1 + 1
[2,1,4,3,5] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 2 = 1 + 1
[1,2,3,4,5,6,7] => [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ? = 5 + 1
[1,2,3,4,5,7,6] => [2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => ? = 4 + 1
[1,2,3,4,6,5,7] => [2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => ? = 4 + 1
[1,2,3,4,6,7,5] => [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => ? = 3 + 1
[1,2,3,4,7,5,6] => [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => ? = 3 + 1
[1,2,3,4,7,6,5] => [2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => ? = 4 + 1
[1,2,3,5,4,6,7] => [2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => ? = 4 + 1
[1,2,3,5,4,7,6] => [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5] => ? = 3 + 1
[1,2,3,5,6,4,7] => [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => ? = 3 + 1
[1,2,3,5,6,7,4] => [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 2 + 1
[1,2,3,5,7,4,6] => [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 2 + 1
[1,2,3,5,7,6,4] => [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => ? = 3 + 1
[1,2,3,6,4,5,7] => [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => ? = 3 + 1
[1,2,3,6,4,7,5] => [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 2 + 1
[1,2,3,6,5,4,7] => [2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => ? = 4 + 1
[1,2,3,6,5,7,4] => [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => ? = 3 + 1
[1,2,3,6,7,4,5] => [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5] => ? = 3 + 1
[1,2,3,6,7,5,4] => [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 2 + 1
[1,2,3,7,4,5,6] => [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 2 + 1
[1,2,3,7,4,6,5] => [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => ? = 3 + 1
[1,2,3,7,5,4,6] => [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => ? = 3 + 1
[1,2,3,7,5,6,4] => [2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => ? = 4 + 1
[1,2,3,7,6,4,5] => [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 2 + 1
[1,2,3,7,6,5,4] => [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5] => ? = 3 + 1
[1,2,4,3,5,6,7] => [2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => ? = 4 + 1
[1,2,4,3,5,7,6] => [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5] => ? = 3 + 1
[1,2,4,3,6,5,7] => [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5] => ? = 3 + 1
[1,2,4,3,6,7,5] => [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? = 2 + 1
[1,2,4,3,7,5,6] => [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? = 2 + 1
[1,2,4,3,7,6,5] => [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5] => ? = 3 + 1
[1,2,4,5,3,6,7] => [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => ? = 3 + 1
[1,2,4,5,3,7,6] => [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? = 2 + 1
[1,2,4,5,6,3,7] => [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 2 + 1
[1,2,4,5,6,7,3] => [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 1 + 1
[1,2,4,5,7,3,6] => [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 1 + 1
[1,2,4,5,7,6,3] => [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 2 + 1
[1,2,4,6,3,5,7] => [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 2 + 1
[1,2,4,6,3,7,5] => [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 1 + 1
[1,2,4,6,5,3,7] => [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => ? = 3 + 1
[1,2,4,6,5,7,3] => [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 2 + 1
[1,2,4,6,7,3,5] => [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? = 2 + 1
[1,2,4,6,7,5,3] => [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 1 + 1
[1,2,4,7,3,5,6] => [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 1 + 1
[1,2,4,7,3,6,5] => [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 2 + 1
[1,2,4,7,5,3,6] => [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 2 + 1
[1,2,4,7,5,6,3] => [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => ? = 3 + 1
[1,2,4,7,6,3,5] => [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 1 + 1
[1,2,4,7,6,5,3] => [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? = 2 + 1
[1,2,5,3,4,6,7] => [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => ? = 3 + 1
[1,2,5,3,4,7,6] => [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? = 2 + 1
Description
The number of circled entries of the shifted recording tableau of a permutation.
The diagram of a strict partition λ1<λ2<⋯<λℓ of n is a tableau with ℓ rows, the i-th row being indented by i cells. A shifted standard Young tableau is a filling of such a diagram, where entries in rows and columns are strictly increasing.
The shifted Robinson-Schensted algorithm [1] associates to a permutation a pair (P,Q) of standard shifted Young tableaux of the same shape, where off-diagonal entries in Q may be circled.
This statistic records the number of circled entries in Q.
Matching statistic: St001960
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001960: Permutations ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 36%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001960: Permutations ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 36%
Values
[1,2] => [1,1]
=> [[1],[2]]
=> [2,1] => 0
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[1,3,2] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[2,1,3] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[3,2,1] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 2
[1,2,4,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[1,3,2,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[1,3,4,2] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[1,4,2,3] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[1,4,3,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[2,1,3,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 0
[2,3,1,4] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[2,4,3,1] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[3,1,2,4] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[3,2,1,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[3,2,4,1] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[3,4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 0
[4,1,3,2] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[4,2,1,3] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[4,2,3,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[4,3,2,1] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 3
[1,2,3,5,4] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 2
[1,2,4,3,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 2
[1,2,4,5,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,2,5,3,4] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,2,5,4,3] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 2
[1,3,2,4,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 2
[1,3,2,5,4] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 1
[1,3,4,2,5] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,3,4,5,2] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 0
[1,3,5,2,4] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 0
[1,3,5,4,2] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,4,2,3,5] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,4,2,5,3] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 0
[1,4,3,2,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 2
[1,4,3,5,2] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,4,5,2,3] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 1
[1,4,5,3,2] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 0
[1,5,2,3,4] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 0
[1,5,2,4,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,5,3,2,4] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,5,3,4,2] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 2
[1,5,4,2,3] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 0
[1,5,4,3,2] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 1
[2,1,3,4,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 2
[2,1,3,5,4] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 1
[2,1,4,3,5] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 1
[1,2,3,4,5,6] => [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => ? = 4
[1,2,3,4,6,5] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => ? = 3
[1,2,3,5,4,6] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => ? = 3
[1,2,3,5,6,4] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => ? = 2
[1,2,3,6,4,5] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => ? = 2
[1,2,3,6,5,4] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => ? = 3
[1,2,4,3,5,6] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => ? = 3
[1,2,4,3,6,5] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => ? = 2
[1,2,4,5,3,6] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => ? = 2
[1,2,4,5,6,3] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => ? = 1
[1,2,4,6,3,5] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => ? = 1
[1,2,4,6,5,3] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => ? = 2
[1,2,5,3,4,6] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => ? = 2
[1,2,5,3,6,4] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => ? = 1
[1,2,5,4,3,6] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => ? = 3
[1,2,5,4,6,3] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => ? = 2
[1,2,5,6,3,4] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => ? = 2
[1,2,5,6,4,3] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => ? = 1
[1,2,6,3,4,5] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => ? = 1
[1,2,6,3,5,4] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => ? = 2
[1,2,6,4,3,5] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => ? = 2
[1,2,6,4,5,3] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => ? = 3
[1,2,6,5,3,4] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => ? = 1
[1,2,6,5,4,3] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => ? = 2
[1,3,2,4,5,6] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => ? = 3
[1,3,2,4,6,5] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => ? = 2
[1,3,2,5,4,6] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => ? = 2
[1,3,2,5,6,4] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => ? = 1
[1,3,2,6,4,5] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => ? = 1
[1,3,2,6,5,4] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => ? = 2
[1,3,4,2,5,6] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => ? = 2
[1,3,4,2,6,5] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => ? = 1
[1,3,4,5,2,6] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => ? = 1
[1,3,4,5,6,2] => [5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ? = 0
[1,3,4,6,2,5] => [5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ? = 0
[1,3,4,6,5,2] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => ? = 1
[1,3,5,2,4,6] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => ? = 1
[1,3,5,2,6,4] => [5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ? = 0
[1,3,5,4,2,6] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => ? = 2
[1,3,5,4,6,2] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => ? = 1
[1,3,5,6,2,4] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => ? = 1
[1,3,5,6,4,2] => [5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ? = 0
[1,3,6,2,4,5] => [5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ? = 0
[1,3,6,2,5,4] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => ? = 1
[1,3,6,4,2,5] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => ? = 1
[1,3,6,4,5,2] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => ? = 2
[1,3,6,5,2,4] => [5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ? = 0
[1,3,6,5,4,2] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => ? = 1
[1,4,2,3,5,6] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => ? = 2
[1,4,2,3,6,5] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => ? = 1
Description
The number of descents of a permutation minus one if its first entry is not one.
This statistic appears in [1, Theorem 2.3] in a gamma-positivity result, see also [2].
Matching statistic: St001169
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001169: Dyck paths ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 36%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001169: Dyck paths ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 36%
Values
[1,2] => [1,2] => [1,0,1,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[3,2,1] => [2,3,1] => [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,4,3,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 0 + 1
[2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 0 + 1
[2,4,3,1] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 0 + 1
[3,2,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2 = 1 + 1
[3,2,4,1] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 0 + 1
[3,4,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[4,1,3,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[4,2,1,3] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 0 + 1
[4,2,3,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2 = 1 + 1
[4,3,2,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 4 = 3 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,2,4,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,2,5,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,2,5,4,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,3,4,2,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,3,4,5,2] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,3,5,2,4] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,3,5,4,2] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 2 = 1 + 1
[1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,4,2,5,3] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,4,3,2,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,4,3,5,2] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,4,5,2,3] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 2 = 1 + 1
[1,4,5,3,2] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,5,2,3,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,5,2,4,3] => [1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 2 = 1 + 1
[1,5,3,2,4] => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,5,3,4,2] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,5,4,2,3] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,5,4,3,2] => [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 2 = 1 + 1
[2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4 + 1
[1,2,3,4,6,5] => [1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 3 + 1
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 3 + 1
[1,2,3,5,6,4] => [1,2,3,6,4,5] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 2 + 1
[1,2,3,6,4,5] => [1,2,3,6,5,4] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 2 + 1
[1,2,3,6,5,4] => [1,2,3,5,6,4] => [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 3 + 1
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 3 + 1
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 2 + 1
[1,2,4,5,3,6] => [1,2,5,3,4,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> ? = 2 + 1
[1,2,4,5,6,3] => [1,2,6,3,4,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,2,4,6,3,5] => [1,2,6,5,3,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,2,4,6,5,3] => [1,2,5,6,3,4] => [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 2 + 1
[1,2,5,3,4,6] => [1,2,5,4,3,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> ? = 2 + 1
[1,2,5,3,6,4] => [1,2,6,4,3,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,2,5,4,3,6] => [1,2,4,5,3,6] => [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> ? = 3 + 1
[1,2,5,4,6,3] => [1,2,4,6,3,5] => [1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 2 + 1
[1,2,5,6,3,4] => [1,2,5,3,6,4] => [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 2 + 1
[1,2,5,6,4,3] => [1,2,6,3,5,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,2,6,3,4,5] => [1,2,6,5,4,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,2,6,3,5,4] => [1,2,5,6,4,3] => [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 2 + 1
[1,2,6,4,3,5] => [1,2,4,6,5,3] => [1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 2 + 1
[1,2,6,4,5,3] => [1,2,4,5,6,3] => [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 3 + 1
[1,2,6,5,3,4] => [1,2,6,4,5,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,2,6,5,4,3] => [1,2,5,4,6,3] => [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 2 + 1
[1,3,2,4,5,6] => [1,3,2,4,5,6] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> ? = 2 + 1
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 2 + 1
[1,3,2,5,6,4] => [1,3,2,6,4,5] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> ? = 1 + 1
[1,3,2,6,4,5] => [1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> ? = 1 + 1
[1,3,2,6,5,4] => [1,3,2,5,6,4] => [1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> ? = 2 + 1
[1,3,4,2,5,6] => [1,4,2,3,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 2 + 1
[1,3,4,2,6,5] => [1,4,2,3,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 1 + 1
[1,3,4,5,2,6] => [1,5,2,3,4,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 1 + 1
[1,3,4,5,6,2] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[1,3,4,6,2,5] => [1,6,5,2,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[1,3,4,6,5,2] => [1,5,6,2,3,4] => [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1 + 1
[1,3,5,2,4,6] => [1,5,4,2,3,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 1 + 1
[1,3,5,2,6,4] => [1,6,4,2,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[1,3,5,4,2,6] => [1,4,5,2,3,6] => [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 2 + 1
[1,3,5,4,6,2] => [1,4,6,2,3,5] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,3,5,6,2,4] => [1,5,2,3,6,4] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 1 + 1
[1,3,5,6,4,2] => [1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[1,3,6,2,4,5] => [1,6,5,4,2,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[1,3,6,2,5,4] => [1,5,6,4,2,3] => [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1 + 1
[1,3,6,4,2,5] => [1,4,6,5,2,3] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,3,6,4,5,2] => [1,4,5,6,2,3] => [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 2 + 1
[1,3,6,5,2,4] => [1,6,4,5,2,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[1,3,6,5,4,2] => [1,5,4,6,2,3] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 1 + 1
[1,4,2,3,5,6] => [1,4,3,2,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 2 + 1
[1,4,2,3,6,5] => [1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 1 + 1
Description
Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra.
Matching statistic: St000015
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000015: Dyck paths ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 36%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000015: Dyck paths ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 36%
Values
[1,2] => [1,2] => [1,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 0 + 2
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 0 + 2
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 0 + 2
[3,2,1] => [2,3,1] => [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 0 + 2
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4 = 2 + 2
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 1 + 2
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 1 + 2
[1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 0 + 2
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 0 + 2
[1,4,3,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 1 + 2
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3 = 1 + 2
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2 = 0 + 2
[2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 0 + 2
[2,4,3,1] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2 = 0 + 2
[3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 0 + 2
[3,2,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 1 + 2
[3,2,4,1] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 0 + 2
[3,4,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2 = 0 + 2
[4,1,3,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2 = 0 + 2
[4,2,1,3] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 0 + 2
[4,2,3,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3 = 1 + 2
[4,3,2,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2 = 0 + 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 5 = 3 + 2
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 4 = 2 + 2
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 4 = 2 + 2
[1,2,4,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 3 = 1 + 2
[1,2,5,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 3 = 1 + 2
[1,2,5,4,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 4 = 2 + 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 4 = 2 + 2
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 3 = 1 + 2
[1,3,4,2,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 3 = 1 + 2
[1,3,4,5,2] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[1,3,5,2,4] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[1,3,5,4,2] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 3 = 1 + 2
[1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 3 = 1 + 2
[1,4,2,5,3] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[1,4,3,2,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 4 = 2 + 2
[1,4,3,5,2] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 3 = 1 + 2
[1,4,5,2,3] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
[1,4,5,3,2] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[1,5,2,3,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[1,5,2,4,3] => [1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 3 = 1 + 2
[1,5,3,2,4] => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 3 = 1 + 2
[1,5,3,4,2] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 4 = 2 + 2
[1,5,4,2,3] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[1,5,4,3,2] => [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
[2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 4 = 2 + 2
[2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 3 = 1 + 2
[2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 3 = 1 + 2
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4 + 2
[1,2,3,4,6,5] => [1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 3 + 2
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 3 + 2
[1,2,3,5,6,4] => [1,2,3,6,4,5] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 2 + 2
[1,2,3,6,4,5] => [1,2,3,6,5,4] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 2 + 2
[1,2,3,6,5,4] => [1,2,3,5,6,4] => [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 3 + 2
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 3 + 2
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 2 + 2
[1,2,4,5,3,6] => [1,2,5,3,4,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> ? = 2 + 2
[1,2,4,5,6,3] => [1,2,6,3,4,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
[1,2,4,6,3,5] => [1,2,6,5,3,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
[1,2,4,6,5,3] => [1,2,5,6,3,4] => [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 2 + 2
[1,2,5,3,4,6] => [1,2,5,4,3,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> ? = 2 + 2
[1,2,5,3,6,4] => [1,2,6,4,3,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
[1,2,5,4,3,6] => [1,2,4,5,3,6] => [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> ? = 3 + 2
[1,2,5,4,6,3] => [1,2,4,6,3,5] => [1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 2 + 2
[1,2,5,6,3,4] => [1,2,5,3,6,4] => [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 2 + 2
[1,2,5,6,4,3] => [1,2,6,3,5,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
[1,2,6,3,4,5] => [1,2,6,5,4,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
[1,2,6,3,5,4] => [1,2,5,6,4,3] => [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 2 + 2
[1,2,6,4,3,5] => [1,2,4,6,5,3] => [1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 2 + 2
[1,2,6,4,5,3] => [1,2,4,5,6,3] => [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 3 + 2
[1,2,6,5,3,4] => [1,2,6,4,5,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
[1,2,6,5,4,3] => [1,2,5,4,6,3] => [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 2 + 2
[1,3,2,4,5,6] => [1,3,2,4,5,6] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 + 2
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> ? = 2 + 2
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 2 + 2
[1,3,2,5,6,4] => [1,3,2,6,4,5] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> ? = 1 + 2
[1,3,2,6,4,5] => [1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> ? = 1 + 2
[1,3,2,6,5,4] => [1,3,2,5,6,4] => [1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> ? = 2 + 2
[1,3,4,2,5,6] => [1,4,2,3,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 2 + 2
[1,3,4,2,6,5] => [1,4,2,3,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 1 + 2
[1,3,4,5,2,6] => [1,5,2,3,4,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 1 + 2
[1,3,4,5,6,2] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[1,3,4,6,2,5] => [1,6,5,2,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[1,3,4,6,5,2] => [1,5,6,2,3,4] => [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1 + 2
[1,3,5,2,4,6] => [1,5,4,2,3,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 1 + 2
[1,3,5,2,6,4] => [1,6,4,2,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[1,3,5,4,2,6] => [1,4,5,2,3,6] => [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 2 + 2
[1,3,5,4,6,2] => [1,4,6,2,3,5] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 1 + 2
[1,3,5,6,2,4] => [1,5,2,3,6,4] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 1 + 2
[1,3,5,6,4,2] => [1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[1,3,6,2,4,5] => [1,6,5,4,2,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[1,3,6,2,5,4] => [1,5,6,4,2,3] => [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1 + 2
[1,3,6,4,2,5] => [1,4,6,5,2,3] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 1 + 2
[1,3,6,4,5,2] => [1,4,5,6,2,3] => [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 2 + 2
[1,3,6,5,2,4] => [1,6,4,5,2,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[1,3,6,5,4,2] => [1,5,4,6,2,3] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 1 + 2
[1,4,2,3,5,6] => [1,4,3,2,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 2 + 2
[1,4,2,3,6,5] => [1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 1 + 2
Description
The number of peaks of a Dyck path.
The following 6 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000702The number of weak deficiencies of a permutation. St000991The number of right-to-left minima of a permutation. St001769The reflection length of a signed permutation. St000942The number of critical left to right maxima of the parking functions. St001712The number of natural descents of a standard Young tableau. St001200The number of simple modules in eAe with projective dimension at most 2 in the corresponding Nakayama algebra A with minimal faithful projective-injective module eA.
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