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Your data matches 152 different statistics following compositions of up to 3 maps.
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Matching statistic: St001175
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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%
St001175: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 0
[1,2] => [2]
=> 0
[2,1] => [1,1]
=> 0
[1,2,3] => [3]
=> 0
[1,3,2] => [2,1]
=> 0
[2,1,3] => [2,1]
=> 0
[2,3,1] => [2,1]
=> 0
[3,1,2] => [2,1]
=> 0
[3,2,1] => [1,1,1]
=> 0
[1,2,3,4] => [4]
=> 0
[1,2,4,3] => [3,1]
=> 0
[1,3,2,4] => [3,1]
=> 0
[1,3,4,2] => [3,1]
=> 0
[1,4,2,3] => [3,1]
=> 0
[1,4,3,2] => [2,1,1]
=> 0
[2,1,3,4] => [3,1]
=> 0
[2,1,4,3] => [2,2]
=> 1
[2,3,1,4] => [3,1]
=> 0
[2,3,4,1] => [3,1]
=> 0
[2,4,1,3] => [2,2]
=> 1
[2,4,3,1] => [2,1,1]
=> 0
[3,1,2,4] => [3,1]
=> 0
[3,1,4,2] => [2,2]
=> 1
[3,2,1,4] => [2,1,1]
=> 0
[3,2,4,1] => [2,1,1]
=> 0
[3,4,1,2] => [2,2]
=> 1
[3,4,2,1] => [2,1,1]
=> 0
[4,1,2,3] => [3,1]
=> 0
[4,1,3,2] => [2,1,1]
=> 0
[4,2,1,3] => [2,1,1]
=> 0
[4,2,3,1] => [2,1,1]
=> 0
[4,3,1,2] => [2,1,1]
=> 0
[4,3,2,1] => [1,1,1,1]
=> 0
[1,2,3,4,5] => [5]
=> 0
[1,2,3,5,4] => [4,1]
=> 0
[1,2,4,3,5] => [4,1]
=> 0
[1,2,4,5,3] => [4,1]
=> 0
[1,2,5,3,4] => [4,1]
=> 0
[1,2,5,4,3] => [3,1,1]
=> 0
[1,3,2,4,5] => [4,1]
=> 0
[1,3,2,5,4] => [3,2]
=> 1
[1,3,4,2,5] => [4,1]
=> 0
[1,3,4,5,2] => [4,1]
=> 0
[1,3,5,2,4] => [3,2]
=> 1
[1,3,5,4,2] => [3,1,1]
=> 0
[1,4,2,3,5] => [4,1]
=> 0
[1,4,2,5,3] => [3,2]
=> 1
[1,4,3,2,5] => [3,1,1]
=> 0
[1,4,3,5,2] => [3,1,1]
=> 0
[1,4,5,2,3] => [3,2]
=> 1
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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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%
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] => [1]
=> [1,0]
=> 0
[1,2] => [2]
=> [1,0,1,0]
=> 0
[2,1] => [1,1]
=> [1,1,0,0]
=> 0
[1,2,3] => [3]
=> [1,0,1,0,1,0]
=> 0
[1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 0
[2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 0
[2,3,1] => [2,1]
=> [1,0,1,1,0,0]
=> 0
[3,1,2] => [2,1]
=> [1,0,1,1,0,0]
=> 0
[3,2,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,2,3,4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[1,3,2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[1,3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[2,1,3,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[2,1,4,3] => [2,2]
=> [1,1,1,0,0,0]
=> 1
[2,3,1,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[2,3,4,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[2,4,1,3] => [2,2]
=> [1,1,1,0,0,0]
=> 1
[2,4,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[3,1,2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[3,1,4,2] => [2,2]
=> [1,1,1,0,0,0]
=> 1
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[3,2,4,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[3,4,1,2] => [2,2]
=> [1,1,1,0,0,0]
=> 1
[3,4,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[4,1,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[4,1,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[4,2,1,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[4,3,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[4,3,2,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,2,3,4,5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,3,5,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,2,4,3,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,2,4,5,3] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,2,5,3,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,2,5,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,3,2,4,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,3,2,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,3,4,2,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,3,4,5,2] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,3,5,2,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,4,2,3,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,4,2,5,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,4,3,2,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,4,5,2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
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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Mp00060: Permutations —Robinson-Schensted tableau shape⟶ 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%
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] => [1]
=> [[1],[]]
=> 0
[1,2] => [2]
=> [[2],[]]
=> 0
[2,1] => [1,1]
=> [[1,1],[]]
=> 0
[1,2,3] => [3]
=> [[3],[]]
=> 0
[1,3,2] => [2,1]
=> [[2,1],[]]
=> 0
[2,1,3] => [2,1]
=> [[2,1],[]]
=> 0
[2,3,1] => [2,1]
=> [[2,1],[]]
=> 0
[3,1,2] => [2,1]
=> [[2,1],[]]
=> 0
[3,2,1] => [1,1,1]
=> [[1,1,1],[]]
=> 0
[1,2,3,4] => [4]
=> [[4],[]]
=> 0
[1,2,4,3] => [3,1]
=> [[3,1],[]]
=> 0
[1,3,2,4] => [3,1]
=> [[3,1],[]]
=> 0
[1,3,4,2] => [3,1]
=> [[3,1],[]]
=> 0
[1,4,2,3] => [3,1]
=> [[3,1],[]]
=> 0
[1,4,3,2] => [2,1,1]
=> [[2,1,1],[]]
=> 0
[2,1,3,4] => [3,1]
=> [[3,1],[]]
=> 0
[2,1,4,3] => [2,2]
=> [[2,2],[]]
=> 1
[2,3,1,4] => [3,1]
=> [[3,1],[]]
=> 0
[2,3,4,1] => [3,1]
=> [[3,1],[]]
=> 0
[2,4,1,3] => [2,2]
=> [[2,2],[]]
=> 1
[2,4,3,1] => [2,1,1]
=> [[2,1,1],[]]
=> 0
[3,1,2,4] => [3,1]
=> [[3,1],[]]
=> 0
[3,1,4,2] => [2,2]
=> [[2,2],[]]
=> 1
[3,2,1,4] => [2,1,1]
=> [[2,1,1],[]]
=> 0
[3,2,4,1] => [2,1,1]
=> [[2,1,1],[]]
=> 0
[3,4,1,2] => [2,2]
=> [[2,2],[]]
=> 1
[3,4,2,1] => [2,1,1]
=> [[2,1,1],[]]
=> 0
[4,1,2,3] => [3,1]
=> [[3,1],[]]
=> 0
[4,1,3,2] => [2,1,1]
=> [[2,1,1],[]]
=> 0
[4,2,1,3] => [2,1,1]
=> [[2,1,1],[]]
=> 0
[4,2,3,1] => [2,1,1]
=> [[2,1,1],[]]
=> 0
[4,3,1,2] => [2,1,1]
=> [[2,1,1],[]]
=> 0
[4,3,2,1] => [1,1,1,1]
=> [[1,1,1,1],[]]
=> 0
[1,2,3,4,5] => [5]
=> [[5],[]]
=> 0
[1,2,3,5,4] => [4,1]
=> [[4,1],[]]
=> 0
[1,2,4,3,5] => [4,1]
=> [[4,1],[]]
=> 0
[1,2,4,5,3] => [4,1]
=> [[4,1],[]]
=> 0
[1,2,5,3,4] => [4,1]
=> [[4,1],[]]
=> 0
[1,2,5,4,3] => [3,1,1]
=> [[3,1,1],[]]
=> 0
[1,3,2,4,5] => [4,1]
=> [[4,1],[]]
=> 0
[1,3,2,5,4] => [3,2]
=> [[3,2],[]]
=> 1
[1,3,4,2,5] => [4,1]
=> [[4,1],[]]
=> 0
[1,3,4,5,2] => [4,1]
=> [[4,1],[]]
=> 0
[1,3,5,2,4] => [3,2]
=> [[3,2],[]]
=> 1
[1,3,5,4,2] => [3,1,1]
=> [[3,1,1],[]]
=> 0
[1,4,2,3,5] => [4,1]
=> [[4,1],[]]
=> 0
[1,4,2,5,3] => [3,2]
=> [[3,2],[]]
=> 1
[1,4,3,2,5] => [3,1,1]
=> [[3,1,1],[]]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [[3,1,1],[]]
=> 0
[1,4,5,2,3] => [3,2]
=> [[3,2],[]]
=> 1
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
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ 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%
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] => [1]
=> []
=> 1 = 0 + 1
[1,2] => [2]
=> []
=> 1 = 0 + 1
[2,1] => [1,1]
=> [1]
=> 1 = 0 + 1
[1,2,3] => [3]
=> []
=> 1 = 0 + 1
[1,3,2] => [2,1]
=> [1]
=> 1 = 0 + 1
[2,1,3] => [2,1]
=> [1]
=> 1 = 0 + 1
[2,3,1] => [2,1]
=> [1]
=> 1 = 0 + 1
[3,1,2] => [2,1]
=> [1]
=> 1 = 0 + 1
[3,2,1] => [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,2,3,4] => [4]
=> []
=> 1 = 0 + 1
[1,2,4,3] => [3,1]
=> [1]
=> 1 = 0 + 1
[1,3,2,4] => [3,1]
=> [1]
=> 1 = 0 + 1
[1,3,4,2] => [3,1]
=> [1]
=> 1 = 0 + 1
[1,4,2,3] => [3,1]
=> [1]
=> 1 = 0 + 1
[1,4,3,2] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[2,1,3,4] => [3,1]
=> [1]
=> 1 = 0 + 1
[2,1,4,3] => [2,2]
=> [2]
=> 2 = 1 + 1
[2,3,1,4] => [3,1]
=> [1]
=> 1 = 0 + 1
[2,3,4,1] => [3,1]
=> [1]
=> 1 = 0 + 1
[2,4,1,3] => [2,2]
=> [2]
=> 2 = 1 + 1
[2,4,3,1] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[3,1,2,4] => [3,1]
=> [1]
=> 1 = 0 + 1
[3,1,4,2] => [2,2]
=> [2]
=> 2 = 1 + 1
[3,2,1,4] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[3,2,4,1] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[3,4,1,2] => [2,2]
=> [2]
=> 2 = 1 + 1
[3,4,2,1] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[4,1,2,3] => [3,1]
=> [1]
=> 1 = 0 + 1
[4,1,3,2] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[4,2,1,3] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[4,2,3,1] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[4,3,1,2] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,2,3,4,5] => [5]
=> []
=> 1 = 0 + 1
[1,2,3,5,4] => [4,1]
=> [1]
=> 1 = 0 + 1
[1,2,4,3,5] => [4,1]
=> [1]
=> 1 = 0 + 1
[1,2,4,5,3] => [4,1]
=> [1]
=> 1 = 0 + 1
[1,2,5,3,4] => [4,1]
=> [1]
=> 1 = 0 + 1
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,3,2,4,5] => [4,1]
=> [1]
=> 1 = 0 + 1
[1,3,2,5,4] => [3,2]
=> [2]
=> 2 = 1 + 1
[1,3,4,2,5] => [4,1]
=> [1]
=> 1 = 0 + 1
[1,3,4,5,2] => [4,1]
=> [1]
=> 1 = 0 + 1
[1,3,5,2,4] => [3,2]
=> [2]
=> 2 = 1 + 1
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,4,2,3,5] => [4,1]
=> [1]
=> 1 = 0 + 1
[1,4,2,5,3] => [3,2]
=> [2]
=> 2 = 1 + 1
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,4,5,2,3] => [3,2]
=> [2]
=> 2 = 1 + 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
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ 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%
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] => [1]
=> []
=> 1 = 0 + 1
[1,2] => [2]
=> []
=> 1 = 0 + 1
[2,1] => [1,1]
=> [1]
=> 1 = 0 + 1
[1,2,3] => [3]
=> []
=> 1 = 0 + 1
[1,3,2] => [2,1]
=> [1]
=> 1 = 0 + 1
[2,1,3] => [2,1]
=> [1]
=> 1 = 0 + 1
[2,3,1] => [2,1]
=> [1]
=> 1 = 0 + 1
[3,1,2] => [2,1]
=> [1]
=> 1 = 0 + 1
[3,2,1] => [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,2,3,4] => [4]
=> []
=> 1 = 0 + 1
[1,2,4,3] => [3,1]
=> [1]
=> 1 = 0 + 1
[1,3,2,4] => [3,1]
=> [1]
=> 1 = 0 + 1
[1,3,4,2] => [3,1]
=> [1]
=> 1 = 0 + 1
[1,4,2,3] => [3,1]
=> [1]
=> 1 = 0 + 1
[1,4,3,2] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[2,1,3,4] => [3,1]
=> [1]
=> 1 = 0 + 1
[2,1,4,3] => [2,2]
=> [2]
=> 2 = 1 + 1
[2,3,1,4] => [3,1]
=> [1]
=> 1 = 0 + 1
[2,3,4,1] => [3,1]
=> [1]
=> 1 = 0 + 1
[2,4,1,3] => [2,2]
=> [2]
=> 2 = 1 + 1
[2,4,3,1] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[3,1,2,4] => [3,1]
=> [1]
=> 1 = 0 + 1
[3,1,4,2] => [2,2]
=> [2]
=> 2 = 1 + 1
[3,2,1,4] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[3,2,4,1] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[3,4,1,2] => [2,2]
=> [2]
=> 2 = 1 + 1
[3,4,2,1] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[4,1,2,3] => [3,1]
=> [1]
=> 1 = 0 + 1
[4,1,3,2] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[4,2,1,3] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[4,2,3,1] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[4,3,1,2] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,2,3,4,5] => [5]
=> []
=> 1 = 0 + 1
[1,2,3,5,4] => [4,1]
=> [1]
=> 1 = 0 + 1
[1,2,4,3,5] => [4,1]
=> [1]
=> 1 = 0 + 1
[1,2,4,5,3] => [4,1]
=> [1]
=> 1 = 0 + 1
[1,2,5,3,4] => [4,1]
=> [1]
=> 1 = 0 + 1
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,3,2,4,5] => [4,1]
=> [1]
=> 1 = 0 + 1
[1,3,2,5,4] => [3,2]
=> [2]
=> 2 = 1 + 1
[1,3,4,2,5] => [4,1]
=> [1]
=> 1 = 0 + 1
[1,3,4,5,2] => [4,1]
=> [1]
=> 1 = 0 + 1
[1,3,5,2,4] => [3,2]
=> [2]
=> 2 = 1 + 1
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,4,2,3,5] => [4,1]
=> [1]
=> 1 = 0 + 1
[1,4,2,5,3] => [3,2]
=> [2]
=> 2 = 1 + 1
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,4,5,2,3] => [3,2]
=> [2]
=> 2 = 1 + 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: St000204
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000204: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000204: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [[1]]
=> [1] => [.,.]
=> 0
[1,2] => [[1,2]]
=> [1,2] => [.,[.,.]]
=> 0
[2,1] => [[1],[2]]
=> [2,1] => [[.,.],.]
=> 0
[1,2,3] => [[1,2,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> 0
[1,3,2] => [[1,2],[3]]
=> [3,1,2] => [[.,.],[.,.]]
=> 0
[2,1,3] => [[1,3],[2]]
=> [2,1,3] => [[.,.],[.,.]]
=> 0
[2,3,1] => [[1,3],[2]]
=> [2,1,3] => [[.,.],[.,.]]
=> 0
[3,1,2] => [[1,2],[3]]
=> [3,1,2] => [[.,.],[.,.]]
=> 0
[3,2,1] => [[1],[2],[3]]
=> [3,2,1] => [[[.,.],.],.]
=> 0
[1,2,3,4] => [[1,2,3,4]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[1,2,4,3] => [[1,2,3],[4]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> 0
[1,3,2,4] => [[1,2,4],[3]]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> 0
[1,3,4,2] => [[1,2,4],[3]]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> 0
[1,4,2,3] => [[1,2,3],[4]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> 0
[1,4,3,2] => [[1,2],[3],[4]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> 0
[2,1,3,4] => [[1,3,4],[2]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> 0
[2,1,4,3] => [[1,3],[2,4]]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> 1
[2,3,1,4] => [[1,3,4],[2]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> 0
[2,3,4,1] => [[1,3,4],[2]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> 0
[2,4,1,3] => [[1,3],[2,4]]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> 1
[2,4,3,1] => [[1,3],[2],[4]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> 0
[3,1,2,4] => [[1,2,4],[3]]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> 0
[3,1,4,2] => [[1,2],[3,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> 1
[3,2,1,4] => [[1,4],[2],[3]]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> 0
[3,2,4,1] => [[1,4],[2],[3]]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> 0
[3,4,1,2] => [[1,2],[3,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> 1
[3,4,2,1] => [[1,4],[2],[3]]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> 0
[4,1,2,3] => [[1,2,3],[4]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> 0
[4,1,3,2] => [[1,2],[3],[4]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> 0
[4,2,1,3] => [[1,3],[2],[4]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> 0
[4,2,3,1] => [[1,3],[2],[4]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> 0
[4,3,1,2] => [[1,2],[3],[4]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> 0
[4,3,2,1] => [[1],[2],[3],[4]]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> 0
[1,2,3,4,5] => [[1,2,3,4,5]]
=> [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,2,3,5,4] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => [[.,.],[.,[.,[.,.]]]]
=> 0
[1,2,4,3,5] => [[1,2,3,5],[4]]
=> [4,1,2,3,5] => [[.,.],[.,[.,[.,.]]]]
=> 0
[1,2,4,5,3] => [[1,2,3,5],[4]]
=> [4,1,2,3,5] => [[.,.],[.,[.,[.,.]]]]
=> 0
[1,2,5,3,4] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => [[.,.],[.,[.,[.,.]]]]
=> 0
[1,2,5,4,3] => [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [[[.,.],.],[.,[.,.]]]
=> 0
[1,3,2,4,5] => [[1,2,4,5],[3]]
=> [3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> 0
[1,3,2,5,4] => [[1,2,4],[3,5]]
=> [3,5,1,2,4] => [[.,[.,.]],[.,[.,.]]]
=> 1
[1,3,4,2,5] => [[1,2,4,5],[3]]
=> [3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> 0
[1,3,4,5,2] => [[1,2,4,5],[3]]
=> [3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> 0
[1,3,5,2,4] => [[1,2,4],[3,5]]
=> [3,5,1,2,4] => [[.,[.,.]],[.,[.,.]]]
=> 1
[1,3,5,4,2] => [[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [[[.,.],.],[.,[.,.]]]
=> 0
[1,4,2,3,5] => [[1,2,3,5],[4]]
=> [4,1,2,3,5] => [[.,.],[.,[.,[.,.]]]]
=> 0
[1,4,2,5,3] => [[1,2,3],[4,5]]
=> [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
=> 1
[1,4,3,2,5] => [[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [[[.,.],.],[.,[.,.]]]
=> 0
[1,4,3,5,2] => [[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [[[.,.],.],[.,[.,.]]]
=> 0
[1,4,5,2,3] => [[1,2,3],[4,5]]
=> [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
=> 1
Description
The number of internal nodes of a binary tree.
That is, the total number of nodes of the tree minus [[St000203]]. A counting formula for the total number of internal nodes across all binary trees of size $n$ is given in [1]. This is equivalent to the number of internal triangles in all triangulations of an $(n+1)$-gon.
Matching statistic: St000371
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000371: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000371: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1,0]
=> [1] => 0
[1,2] => [2]
=> [1,0,1,0]
=> [1,2] => 0
[2,1] => [1,1]
=> [1,1,0,0]
=> [2,1] => 0
[1,2,3] => [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[2,3,1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[3,1,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[3,2,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0
[1,2,3,4] => [4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[1,2,4,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[1,3,2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[1,3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[2,1,3,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[2,1,4,3] => [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 1
[2,3,1,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[2,3,4,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[2,4,1,3] => [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 1
[2,4,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[3,1,2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[3,1,4,2] => [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 1
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[3,2,4,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[3,4,1,2] => [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 1
[3,4,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[4,1,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[4,1,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[4,2,1,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[4,3,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[4,3,2,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 0
[1,2,3,4,5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[1,2,3,5,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
[1,2,4,3,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
[1,2,4,5,3] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
[1,2,5,3,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
[1,2,5,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0
[1,3,2,4,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
[1,3,2,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[1,3,4,2,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
[1,3,4,5,2] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
[1,3,5,2,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0
[1,4,2,3,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
[1,4,2,5,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[1,4,3,2,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0
[1,4,5,2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
Description
The number of mid points of decreasing subsequences of length 3 in a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the number of indices $j$ such that there exist indices $i,k$ with $i < j < k$ and $\pi(i) > \pi(j) > \pi(k)$. 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: St000377
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000377: 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
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000377: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> []
=> []
=> 0
[1,2] => [2]
=> []
=> []
=> 0
[2,1] => [1,1]
=> [1]
=> [1]
=> 0
[1,2,3] => [3]
=> []
=> []
=> 0
[1,3,2] => [2,1]
=> [1]
=> [1]
=> 0
[2,1,3] => [2,1]
=> [1]
=> [1]
=> 0
[2,3,1] => [2,1]
=> [1]
=> [1]
=> 0
[3,1,2] => [2,1]
=> [1]
=> [1]
=> 0
[3,2,1] => [1,1,1]
=> [1,1]
=> [2]
=> 0
[1,2,3,4] => [4]
=> []
=> []
=> 0
[1,2,4,3] => [3,1]
=> [1]
=> [1]
=> 0
[1,3,2,4] => [3,1]
=> [1]
=> [1]
=> 0
[1,3,4,2] => [3,1]
=> [1]
=> [1]
=> 0
[1,4,2,3] => [3,1]
=> [1]
=> [1]
=> 0
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [2]
=> 0
[2,1,3,4] => [3,1]
=> [1]
=> [1]
=> 0
[2,1,4,3] => [2,2]
=> [2]
=> [1,1]
=> 1
[2,3,1,4] => [3,1]
=> [1]
=> [1]
=> 0
[2,3,4,1] => [3,1]
=> [1]
=> [1]
=> 0
[2,4,1,3] => [2,2]
=> [2]
=> [1,1]
=> 1
[2,4,3,1] => [2,1,1]
=> [1,1]
=> [2]
=> 0
[3,1,2,4] => [3,1]
=> [1]
=> [1]
=> 0
[3,1,4,2] => [2,2]
=> [2]
=> [1,1]
=> 1
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [2]
=> 0
[3,2,4,1] => [2,1,1]
=> [1,1]
=> [2]
=> 0
[3,4,1,2] => [2,2]
=> [2]
=> [1,1]
=> 1
[3,4,2,1] => [2,1,1]
=> [1,1]
=> [2]
=> 0
[4,1,2,3] => [3,1]
=> [1]
=> [1]
=> 0
[4,1,3,2] => [2,1,1]
=> [1,1]
=> [2]
=> 0
[4,2,1,3] => [2,1,1]
=> [1,1]
=> [2]
=> 0
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [2]
=> 0
[4,3,1,2] => [2,1,1]
=> [1,1]
=> [2]
=> 0
[4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[1,2,3,4,5] => [5]
=> []
=> []
=> 0
[1,2,3,5,4] => [4,1]
=> [1]
=> [1]
=> 0
[1,2,4,3,5] => [4,1]
=> [1]
=> [1]
=> 0
[1,2,4,5,3] => [4,1]
=> [1]
=> [1]
=> 0
[1,2,5,3,4] => [4,1]
=> [1]
=> [1]
=> 0
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> [2]
=> 0
[1,3,2,4,5] => [4,1]
=> [1]
=> [1]
=> 0
[1,3,2,5,4] => [3,2]
=> [2]
=> [1,1]
=> 1
[1,3,4,2,5] => [4,1]
=> [1]
=> [1]
=> 0
[1,3,4,5,2] => [4,1]
=> [1]
=> [1]
=> 0
[1,3,5,2,4] => [3,2]
=> [2]
=> [1,1]
=> 1
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [2]
=> 0
[1,4,2,3,5] => [4,1]
=> [1]
=> [1]
=> 0
[1,4,2,5,3] => [3,2]
=> [2]
=> [1,1]
=> 1
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> [2]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [2]
=> 0
[1,4,5,2,3] => [3,2]
=> [2]
=> [1,1]
=> 1
Description
The dinv defect of an integer partition.
This is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \not\in \{0,1\}$.
Matching statistic: St001727
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St001727: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St001727: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1,0]
=> [1] => 0
[1,2] => [2]
=> [1,0,1,0]
=> [1,2] => 0
[2,1] => [1,1]
=> [1,1,0,0]
=> [2,1] => 0
[1,2,3] => [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[2,3,1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[3,1,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[3,2,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0
[1,2,3,4] => [4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[1,2,4,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[1,3,2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[1,3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[2,1,3,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[2,1,4,3] => [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 1
[2,3,1,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[2,3,4,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[2,4,1,3] => [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 1
[2,4,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[3,1,2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[3,1,4,2] => [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 1
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[3,2,4,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[3,4,1,2] => [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 1
[3,4,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[4,1,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[4,1,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[4,2,1,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[4,3,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[4,3,2,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 0
[1,2,3,4,5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[1,2,3,5,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
[1,2,4,3,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
[1,2,4,5,3] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
[1,2,5,3,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
[1,2,5,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0
[1,3,2,4,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
[1,3,2,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[1,3,4,2,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
[1,3,4,5,2] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
[1,3,5,2,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0
[1,4,2,3,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
[1,4,2,5,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[1,4,3,2,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0
[1,4,5,2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
Description
The number of invisible inversions of a permutation.
A visible inversion of a permutation $\pi$ is a pair $i < j$ such that $\pi(j) \leq \min(i, \pi(i))$. Thus, an invisible inversion satisfies $\pi(i) > \pi(j) > i$.
Matching statistic: St001176
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ 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
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> []
=> []
=> ? = 0
[1,2] => [2]
=> []
=> []
=> ? = 0
[2,1] => [1,1]
=> [1]
=> [1]
=> 0
[1,2,3] => [3]
=> []
=> []
=> ? = 0
[1,3,2] => [2,1]
=> [1]
=> [1]
=> 0
[2,1,3] => [2,1]
=> [1]
=> [1]
=> 0
[2,3,1] => [2,1]
=> [1]
=> [1]
=> 0
[3,1,2] => [2,1]
=> [1]
=> [1]
=> 0
[3,2,1] => [1,1,1]
=> [1,1]
=> [2]
=> 0
[1,2,3,4] => [4]
=> []
=> []
=> ? = 0
[1,2,4,3] => [3,1]
=> [1]
=> [1]
=> 0
[1,3,2,4] => [3,1]
=> [1]
=> [1]
=> 0
[1,3,4,2] => [3,1]
=> [1]
=> [1]
=> 0
[1,4,2,3] => [3,1]
=> [1]
=> [1]
=> 0
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [2]
=> 0
[2,1,3,4] => [3,1]
=> [1]
=> [1]
=> 0
[2,1,4,3] => [2,2]
=> [2]
=> [1,1]
=> 1
[2,3,1,4] => [3,1]
=> [1]
=> [1]
=> 0
[2,3,4,1] => [3,1]
=> [1]
=> [1]
=> 0
[2,4,1,3] => [2,2]
=> [2]
=> [1,1]
=> 1
[2,4,3,1] => [2,1,1]
=> [1,1]
=> [2]
=> 0
[3,1,2,4] => [3,1]
=> [1]
=> [1]
=> 0
[3,1,4,2] => [2,2]
=> [2]
=> [1,1]
=> 1
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [2]
=> 0
[3,2,4,1] => [2,1,1]
=> [1,1]
=> [2]
=> 0
[3,4,1,2] => [2,2]
=> [2]
=> [1,1]
=> 1
[3,4,2,1] => [2,1,1]
=> [1,1]
=> [2]
=> 0
[4,1,2,3] => [3,1]
=> [1]
=> [1]
=> 0
[4,1,3,2] => [2,1,1]
=> [1,1]
=> [2]
=> 0
[4,2,1,3] => [2,1,1]
=> [1,1]
=> [2]
=> 0
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [2]
=> 0
[4,3,1,2] => [2,1,1]
=> [1,1]
=> [2]
=> 0
[4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> [3]
=> 0
[1,2,3,4,5] => [5]
=> []
=> []
=> ? = 0
[1,2,3,5,4] => [4,1]
=> [1]
=> [1]
=> 0
[1,2,4,3,5] => [4,1]
=> [1]
=> [1]
=> 0
[1,2,4,5,3] => [4,1]
=> [1]
=> [1]
=> 0
[1,2,5,3,4] => [4,1]
=> [1]
=> [1]
=> 0
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> [2]
=> 0
[1,3,2,4,5] => [4,1]
=> [1]
=> [1]
=> 0
[1,3,2,5,4] => [3,2]
=> [2]
=> [1,1]
=> 1
[1,3,4,2,5] => [4,1]
=> [1]
=> [1]
=> 0
[1,3,4,5,2] => [4,1]
=> [1]
=> [1]
=> 0
[1,3,5,2,4] => [3,2]
=> [2]
=> [1,1]
=> 1
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [2]
=> 0
[1,4,2,3,5] => [4,1]
=> [1]
=> [1]
=> 0
[1,4,2,5,3] => [3,2]
=> [2]
=> [1,1]
=> 1
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> [2]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [2]
=> 0
[1,4,5,2,3] => [3,2]
=> [2]
=> [1,1]
=> 1
[1,4,5,3,2] => [3,1,1]
=> [1,1]
=> [2]
=> 0
[1,5,2,3,4] => [4,1]
=> [1]
=> [1]
=> 0
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [2]
=> 0
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [2]
=> 0
[1,5,3,4,2] => [3,1,1]
=> [1,1]
=> [2]
=> 0
[1,2,3,4,5,6] => [6]
=> []
=> []
=> ? = 0
[1,2,3,4,5,6,7] => [7]
=> []
=> []
=> ? = 0
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.
The following 142 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000223The number of nestings in the permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001082The number of boxed occurrences of 123 in a permutation. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St001394The genus of a permutation. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001498The normalised height of a Nakayama algebra with magnitude 1. St000668The least common multiple of the parts of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001128The exponens consonantiae of a partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000260The radius of a connected graph. St001871The number of triconnected components of a graph. St001877Number of indecomposable injective modules with projective dimension 2. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St000624The normalized sum of the minimal distances to a greater element. St000779The tier of a permutation. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000159The number of distinct parts of the integer partition. St001432The order dimension of the partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000781The number of proper colouring schemes of a Ferrers diagram. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000225Difference between largest and smallest parts in a partition. St000455The second largest eigenvalue of a graph if it is integral. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001964The interval resolution global dimension of a poset. St001875The number of simple modules with projective dimension at most 1. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St001803The maximal overlap of the cylindrical tableau associated with a tableau. 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 $K[x]/(x^n)$. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St000908The length of the shortest maximal antichain in a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000914The sum of the values of the Möbius function of a poset. St000068The number of minimal elements in a poset. St001846The number of elements which do not have a complement in the lattice. St001820The size of the image of the pop stack sorting operator. St001568The smallest positive integer that does not appear twice in the partition. St000259The diameter of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001330The hat guessing number of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001561The value of the elementary symmetric function evaluated at 1. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001487The number of inner corners of a skew partition. St000454The largest eigenvalue of a graph if it is integral. St001862The number of crossings of a signed permutation. St001866The nesting alignments of a signed permutation. St001621The number of atoms of a lattice. St001845The number of join irreducibles minus the rank of a lattice. St000022The number of fixed points of a permutation. St000731The number of double exceedences of a permutation. 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. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St001490The number of connected components of a skew partition. St000632The jump number of the poset. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001510The number of self-evacuating linear extensions of a finite poset. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001779The order of promotion on the set of linear extensions of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St000143The largest repeated part of a partition. St000150The floored half-sum of the multiplicities of a partition. St000257The number of distinct parts of a partition that occur at least twice. St001091The number of parts in an integer partition whose next smaller part has the same size. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000160The multiplicity of the smallest part of a partition. St000897The number of different multiplicities of parts of an integer partition. St001933The largest multiplicity of a part in an integer partition. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St000706The product of the factorials of the multiplicities of an integer partition. St000664The number of right ropes of a permutation. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001856The number of edges in the reduced word graph of a permutation. St001867The number of alignments of type EN of a signed permutation. St001768The number of reduced words of a signed permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001722The number of minimal chains with small intervals between a binary word and the top element. St000322The skewness of a graph. St000095The number of triangles of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000310The minimal degree of a vertex of a graph. St000449The number of pairs of vertices of a graph with distance 4. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001578The minimal number of edges to add or remove to make a graph a line graph. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St001765The number of connected components of the friends and strangers graph.
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