searching the database
Your data matches 164 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000966
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000966: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000966: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 0
[1,1] => [1,0,1,0]
=> 0
[2] => [1,1,0,0]
=> 0
[1,1,1] => [1,0,1,0,1,0]
=> 0
[1,2] => [1,0,1,1,0,0]
=> 0
[2,1] => [1,1,0,0,1,0]
=> 0
[3] => [1,1,1,0,0,0]
=> 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 0
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,3] => [1,0,1,1,1,0,0,0]
=> 0
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
[2,2] => [1,1,0,0,1,1,0,0]
=> 0
[3,1] => [1,1,1,0,0,0,1,0]
=> 0
[4] => [1,1,1,1,0,0,0,0]
=> 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 0
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 0
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 0
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 0
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 0
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 2
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 0
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 2
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 0
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 0
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 0
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 1
Description
Number of peaks minus the global dimension of the corresponding LNakayama algebra.
Matching statistic: St001175
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ 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
[1,1] => [1,1]
=> 0
[2] => [2]
=> 0
[1,1,1] => [1,1,1]
=> 0
[1,2] => [2,1]
=> 0
[2,1] => [2,1]
=> 0
[3] => [3]
=> 0
[1,1,1,1] => [1,1,1,1]
=> 0
[1,1,2] => [2,1,1]
=> 0
[1,2,1] => [2,1,1]
=> 0
[1,3] => [3,1]
=> 0
[2,1,1] => [2,1,1]
=> 0
[2,2] => [2,2]
=> 1
[3,1] => [3,1]
=> 0
[4] => [4]
=> 0
[1,1,1,1,1] => [1,1,1,1,1]
=> 0
[1,1,1,2] => [2,1,1,1]
=> 0
[1,1,2,1] => [2,1,1,1]
=> 0
[1,1,3] => [3,1,1]
=> 0
[1,2,1,1] => [2,1,1,1]
=> 0
[1,2,2] => [2,2,1]
=> 1
[1,3,1] => [3,1,1]
=> 0
[1,4] => [4,1]
=> 0
[2,1,1,1] => [2,1,1,1]
=> 0
[2,1,2] => [2,2,1]
=> 1
[2,2,1] => [2,2,1]
=> 1
[2,3] => [3,2]
=> 1
[3,1,1] => [3,1,1]
=> 0
[3,2] => [3,2]
=> 1
[4,1] => [4,1]
=> 0
[5] => [5]
=> 0
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> 0
[1,1,1,1,2] => [2,1,1,1,1]
=> 0
[1,1,1,2,1] => [2,1,1,1,1]
=> 0
[1,1,1,3] => [3,1,1,1]
=> 0
[1,1,2,1,1] => [2,1,1,1,1]
=> 0
[1,1,2,2] => [2,2,1,1]
=> 1
[1,1,3,1] => [3,1,1,1]
=> 0
[1,1,4] => [4,1,1]
=> 0
[1,2,1,1,1] => [2,1,1,1,1]
=> 0
[1,2,1,2] => [2,2,1,1]
=> 1
[1,2,2,1] => [2,2,1,1]
=> 1
[1,2,3] => [3,2,1]
=> 1
[1,3,1,1] => [3,1,1,1]
=> 0
[1,3,2] => [3,2,1]
=> 1
[1,4,1] => [4,1,1]
=> 0
[1,5] => [5,1]
=> 0
[2,1,1,1,1] => [2,1,1,1,1]
=> 0
[2,1,1,2] => [2,2,1,1]
=> 1
[2,1,2,1] => [2,2,1,1]
=> 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
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ 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
[1,1] => [1,1]
=> [1,1,0,0]
=> 0
[2] => [2]
=> [1,0,1,0]
=> 0
[1,1,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,2] => [2,1]
=> [1,0,1,1,0,0]
=> 0
[2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 0
[3] => [3]
=> [1,0,1,0,1,0]
=> 0
[1,1,1,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[2,2] => [2,2]
=> [1,1,1,0,0,0]
=> 1
[3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,1,1,1,1] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[1,1,1,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
[1,1,2,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
[1,1,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
[1,2,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,3,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
[2,1,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[2,2,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,1,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[3,2] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[4,1] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 0
[1,1,1,1,2] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 0
[1,1,1,2,1] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 0
[1,1,1,3] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 0
[1,1,2,1,1] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 0
[1,1,2,2] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,1,3,1] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 0
[1,1,4] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 0
[1,2,1,1,1] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 0
[1,2,1,2] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,2,2,1] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,2,3] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,3,1,1] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 0
[1,3,2] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,4,1] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 0
[1,5] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 0
[2,1,1,1,1] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 0
[2,1,1,2] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[2,1,2,1] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 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
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ 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
[1,1] => [1,1]
=> [[1,1],[]]
=> 0
[2] => [2]
=> [[2],[]]
=> 0
[1,1,1] => [1,1,1]
=> [[1,1,1],[]]
=> 0
[1,2] => [2,1]
=> [[2,1],[]]
=> 0
[2,1] => [2,1]
=> [[2,1],[]]
=> 0
[3] => [3]
=> [[3],[]]
=> 0
[1,1,1,1] => [1,1,1,1]
=> [[1,1,1,1],[]]
=> 0
[1,1,2] => [2,1,1]
=> [[2,1,1],[]]
=> 0
[1,2,1] => [2,1,1]
=> [[2,1,1],[]]
=> 0
[1,3] => [3,1]
=> [[3,1],[]]
=> 0
[2,1,1] => [2,1,1]
=> [[2,1,1],[]]
=> 0
[2,2] => [2,2]
=> [[2,2],[]]
=> 1
[3,1] => [3,1]
=> [[3,1],[]]
=> 0
[4] => [4]
=> [[4],[]]
=> 0
[1,1,1,1,1] => [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 0
[1,1,1,2] => [2,1,1,1]
=> [[2,1,1,1],[]]
=> 0
[1,1,2,1] => [2,1,1,1]
=> [[2,1,1,1],[]]
=> 0
[1,1,3] => [3,1,1]
=> [[3,1,1],[]]
=> 0
[1,2,1,1] => [2,1,1,1]
=> [[2,1,1,1],[]]
=> 0
[1,2,2] => [2,2,1]
=> [[2,2,1],[]]
=> 1
[1,3,1] => [3,1,1]
=> [[3,1,1],[]]
=> 0
[1,4] => [4,1]
=> [[4,1],[]]
=> 0
[2,1,1,1] => [2,1,1,1]
=> [[2,1,1,1],[]]
=> 0
[2,1,2] => [2,2,1]
=> [[2,2,1],[]]
=> 1
[2,2,1] => [2,2,1]
=> [[2,2,1],[]]
=> 1
[2,3] => [3,2]
=> [[3,2],[]]
=> 1
[3,1,1] => [3,1,1]
=> [[3,1,1],[]]
=> 0
[3,2] => [3,2]
=> [[3,2],[]]
=> 1
[4,1] => [4,1]
=> [[4,1],[]]
=> 0
[5] => [5]
=> [[5],[]]
=> 0
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> 0
[1,1,1,1,2] => [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> 0
[1,1,1,2,1] => [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> 0
[1,1,1,3] => [3,1,1,1]
=> [[3,1,1,1],[]]
=> 0
[1,1,2,1,1] => [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> 0
[1,1,2,2] => [2,2,1,1]
=> [[2,2,1,1],[]]
=> 1
[1,1,3,1] => [3,1,1,1]
=> [[3,1,1,1],[]]
=> 0
[1,1,4] => [4,1,1]
=> [[4,1,1],[]]
=> 0
[1,2,1,1,1] => [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> 0
[1,2,1,2] => [2,2,1,1]
=> [[2,2,1,1],[]]
=> 1
[1,2,2,1] => [2,2,1,1]
=> [[2,2,1,1],[]]
=> 1
[1,2,3] => [3,2,1]
=> [[3,2,1],[]]
=> 1
[1,3,1,1] => [3,1,1,1]
=> [[3,1,1,1],[]]
=> 0
[1,3,2] => [3,2,1]
=> [[3,2,1],[]]
=> 1
[1,4,1] => [4,1,1]
=> [[4,1,1],[]]
=> 0
[1,5] => [5,1]
=> [[5,1],[]]
=> 0
[2,1,1,1,1] => [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> 0
[2,1,1,2] => [2,2,1,1]
=> [[2,2,1,1],[]]
=> 1
[2,1,2,1] => [2,2,1,1]
=> [[2,2,1,1],[]]
=> 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
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ 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
[1,1] => [1,1]
=> [1]
=> 1 = 0 + 1
[2] => [2]
=> []
=> 1 = 0 + 1
[1,1,1] => [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,2] => [2,1]
=> [1]
=> 1 = 0 + 1
[2,1] => [2,1]
=> [1]
=> 1 = 0 + 1
[3] => [3]
=> []
=> 1 = 0 + 1
[1,1,1,1] => [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,1,2] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,2,1] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,3] => [3,1]
=> [1]
=> 1 = 0 + 1
[2,1,1] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[2,2] => [2,2]
=> [2]
=> 2 = 1 + 1
[3,1] => [3,1]
=> [1]
=> 1 = 0 + 1
[4] => [4]
=> []
=> 1 = 0 + 1
[1,1,1,1,1] => [1,1,1,1,1]
=> [1,1,1,1]
=> 1 = 0 + 1
[1,1,1,2] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,1,2,1] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,1,3] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,2,1,1] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,2,2] => [2,2,1]
=> [2,1]
=> 2 = 1 + 1
[1,3,1] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,4] => [4,1]
=> [1]
=> 1 = 0 + 1
[2,1,1,1] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[2,1,2] => [2,2,1]
=> [2,1]
=> 2 = 1 + 1
[2,2,1] => [2,2,1]
=> [2,1]
=> 2 = 1 + 1
[2,3] => [3,2]
=> [2]
=> 2 = 1 + 1
[3,1,1] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[3,2] => [3,2]
=> [2]
=> 2 = 1 + 1
[4,1] => [4,1]
=> [1]
=> 1 = 0 + 1
[5] => [5]
=> []
=> 1 = 0 + 1
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1 = 0 + 1
[1,1,1,1,2] => [2,1,1,1,1]
=> [1,1,1,1]
=> 1 = 0 + 1
[1,1,1,2,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> 1 = 0 + 1
[1,1,1,3] => [3,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,1,2,1,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> 1 = 0 + 1
[1,1,2,2] => [2,2,1,1]
=> [2,1,1]
=> 2 = 1 + 1
[1,1,3,1] => [3,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,1,4] => [4,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,2,1,1,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> 1 = 0 + 1
[1,2,1,2] => [2,2,1,1]
=> [2,1,1]
=> 2 = 1 + 1
[1,2,2,1] => [2,2,1,1]
=> [2,1,1]
=> 2 = 1 + 1
[1,2,3] => [3,2,1]
=> [2,1]
=> 2 = 1 + 1
[1,3,1,1] => [3,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,3,2] => [3,2,1]
=> [2,1]
=> 2 = 1 + 1
[1,4,1] => [4,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,5] => [5,1]
=> [1]
=> 1 = 0 + 1
[2,1,1,1,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> 1 = 0 + 1
[2,1,1,2] => [2,2,1,1]
=> [2,1,1]
=> 2 = 1 + 1
[2,1,2,1] => [2,2,1,1]
=> [2,1,1]
=> 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
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ 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
[1,1] => [1,1]
=> [1]
=> 1 = 0 + 1
[2] => [2]
=> []
=> 1 = 0 + 1
[1,1,1] => [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,2] => [2,1]
=> [1]
=> 1 = 0 + 1
[2,1] => [2,1]
=> [1]
=> 1 = 0 + 1
[3] => [3]
=> []
=> 1 = 0 + 1
[1,1,1,1] => [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,1,2] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,2,1] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,3] => [3,1]
=> [1]
=> 1 = 0 + 1
[2,1,1] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[2,2] => [2,2]
=> [2]
=> 2 = 1 + 1
[3,1] => [3,1]
=> [1]
=> 1 = 0 + 1
[4] => [4]
=> []
=> 1 = 0 + 1
[1,1,1,1,1] => [1,1,1,1,1]
=> [1,1,1,1]
=> 1 = 0 + 1
[1,1,1,2] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,1,2,1] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,1,3] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,2,1,1] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,2,2] => [2,2,1]
=> [2,1]
=> 2 = 1 + 1
[1,3,1] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,4] => [4,1]
=> [1]
=> 1 = 0 + 1
[2,1,1,1] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[2,1,2] => [2,2,1]
=> [2,1]
=> 2 = 1 + 1
[2,2,1] => [2,2,1]
=> [2,1]
=> 2 = 1 + 1
[2,3] => [3,2]
=> [2]
=> 2 = 1 + 1
[3,1,1] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[3,2] => [3,2]
=> [2]
=> 2 = 1 + 1
[4,1] => [4,1]
=> [1]
=> 1 = 0 + 1
[5] => [5]
=> []
=> 1 = 0 + 1
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1 = 0 + 1
[1,1,1,1,2] => [2,1,1,1,1]
=> [1,1,1,1]
=> 1 = 0 + 1
[1,1,1,2,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> 1 = 0 + 1
[1,1,1,3] => [3,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,1,2,1,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> 1 = 0 + 1
[1,1,2,2] => [2,2,1,1]
=> [2,1,1]
=> 2 = 1 + 1
[1,1,3,1] => [3,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,1,4] => [4,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,2,1,1,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> 1 = 0 + 1
[1,2,1,2] => [2,2,1,1]
=> [2,1,1]
=> 2 = 1 + 1
[1,2,2,1] => [2,2,1,1]
=> [2,1,1]
=> 2 = 1 + 1
[1,2,3] => [3,2,1]
=> [2,1]
=> 2 = 1 + 1
[1,3,1,1] => [3,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,3,2] => [3,2,1]
=> [2,1]
=> 2 = 1 + 1
[1,4,1] => [4,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,5] => [5,1]
=> [1]
=> 1 = 0 + 1
[2,1,1,1,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> 1 = 0 + 1
[2,1,1,2] => [2,2,1,1]
=> [2,1,1]
=> 2 = 1 + 1
[2,1,2,1] => [2,2,1,1]
=> [2,1,1]
=> 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: St000052
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> [1,0]
=> 0
[1,1] => [1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[2] => [1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[1,1,1] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 0
[2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 0
[3] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 0
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 0
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 0
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0
[4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 0
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 0
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 0
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 0
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 0
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 0
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 0
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 0
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 0
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 0
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 0
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> 1
Description
The number of valleys of a Dyck path not on the x-axis.
That is, the number of valleys of nonminimal height. This corresponds to the number of -1's in an inclusion of Dyck paths into alternating sign matrices.
Matching statistic: St000371
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ 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
[1,1] => [1,1]
=> [1,1,0,0]
=> [2,1] => 0
[2] => [2]
=> [1,0,1,0]
=> [1,2] => 0
[1,1,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0
[1,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[3] => [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[1,1,1,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 0
[1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[2,2] => [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 1
[3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[4] => [4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[1,1,1,1,1] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0
[1,1,1,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0
[1,1,2,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0
[1,1,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0
[1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0
[1,2,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[1,3,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0
[1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
[2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0
[2,1,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[2,2,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[3,1,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0
[3,2] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[4,1] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
[5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => 0
[1,1,1,1,2] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => 0
[1,1,1,2,1] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => 0
[1,1,1,3] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => 0
[1,1,2,1,1] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => 0
[1,1,2,2] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 1
[1,1,3,1] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => 0
[1,1,4] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => 0
[1,2,1,1,1] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => 0
[1,2,1,2] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 1
[1,2,2,1] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 1
[1,2,3] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 1
[1,3,1,1] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => 0
[1,3,2] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 1
[1,4,1] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => 0
[1,5] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 0
[2,1,1,1,1] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => 0
[2,1,1,2] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 1
[2,1,2,1] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 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
Mp00040: Integer compositions —to partition⟶ 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
[1,1] => [1,1]
=> [1]
=> [1]
=> 0
[2] => [2]
=> []
=> []
=> 0
[1,1,1] => [1,1,1]
=> [1,1]
=> [2]
=> 0
[1,2] => [2,1]
=> [1]
=> [1]
=> 0
[2,1] => [2,1]
=> [1]
=> [1]
=> 0
[3] => [3]
=> []
=> []
=> 0
[1,1,1,1] => [1,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[1,1,2] => [2,1,1]
=> [1,1]
=> [2]
=> 0
[1,2,1] => [2,1,1]
=> [1,1]
=> [2]
=> 0
[1,3] => [3,1]
=> [1]
=> [1]
=> 0
[2,1,1] => [2,1,1]
=> [1,1]
=> [2]
=> 0
[2,2] => [2,2]
=> [2]
=> [1,1]
=> 1
[3,1] => [3,1]
=> [1]
=> [1]
=> 0
[4] => [4]
=> []
=> []
=> 0
[1,1,1,1,1] => [1,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> 0
[1,1,1,2] => [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[1,1,2,1] => [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[1,1,3] => [3,1,1]
=> [1,1]
=> [2]
=> 0
[1,2,1,1] => [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[1,2,2] => [2,2,1]
=> [2,1]
=> [3]
=> 1
[1,3,1] => [3,1,1]
=> [1,1]
=> [2]
=> 0
[1,4] => [4,1]
=> [1]
=> [1]
=> 0
[2,1,1,1] => [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[2,1,2] => [2,2,1]
=> [2,1]
=> [3]
=> 1
[2,2,1] => [2,2,1]
=> [2,1]
=> [3]
=> 1
[2,3] => [3,2]
=> [2]
=> [1,1]
=> 1
[3,1,1] => [3,1,1]
=> [1,1]
=> [2]
=> 0
[3,2] => [3,2]
=> [2]
=> [1,1]
=> 1
[4,1] => [4,1]
=> [1]
=> [1]
=> 0
[5] => [5]
=> []
=> []
=> 0
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [3,2]
=> 0
[1,1,1,1,2] => [2,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> 0
[1,1,1,2,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> 0
[1,1,1,3] => [3,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[1,1,2,1,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> 0
[1,1,2,2] => [2,2,1,1]
=> [2,1,1]
=> [2,2]
=> 1
[1,1,3,1] => [3,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[1,1,4] => [4,1,1]
=> [1,1]
=> [2]
=> 0
[1,2,1,1,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> 0
[1,2,1,2] => [2,2,1,1]
=> [2,1,1]
=> [2,2]
=> 1
[1,2,2,1] => [2,2,1,1]
=> [2,1,1]
=> [2,2]
=> 1
[1,2,3] => [3,2,1]
=> [2,1]
=> [3]
=> 1
[1,3,1,1] => [3,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[1,3,2] => [3,2,1]
=> [2,1]
=> [3]
=> 1
[1,4,1] => [4,1,1]
=> [1,1]
=> [2]
=> 0
[1,5] => [5,1]
=> [1]
=> [1]
=> 0
[2,1,1,1,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> 0
[2,1,1,2] => [2,2,1,1]
=> [2,1,1]
=> [2,2]
=> 1
[2,1,2,1] => [2,2,1,1]
=> [2,1,1]
=> [2,2]
=> 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
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St001727: Permutations ⟶ ℤResult quality: 99% ●values known / values provided: 99%●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: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1,0]
=> [1] => 0
[1,1] => [1,1]
=> [1,1,0,0]
=> [2,1] => 0
[2] => [2]
=> [1,0,1,0]
=> [1,2] => 0
[1,1,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0
[1,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[3] => [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[1,1,1,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 0
[1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[2,2] => [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 1
[3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[4] => [4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[1,1,1,1,1] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0
[1,1,1,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0
[1,1,2,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0
[1,1,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0
[1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0
[1,2,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[1,3,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0
[1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
[2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0
[2,1,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[2,2,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[3,1,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0
[3,2] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[4,1] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
[5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => 0
[1,1,1,1,2] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => 0
[1,1,1,2,1] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => 0
[1,1,1,3] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => 0
[1,1,2,1,1] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => 0
[1,1,2,2] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 1
[1,1,3,1] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => 0
[1,1,4] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => 0
[1,2,1,1,1] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => 0
[1,2,1,2] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 1
[1,2,2,1] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 1
[1,2,3] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 1
[1,3,1,1] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => 0
[1,3,2] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 1
[1,4,1] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => 0
[1,5] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 0
[2,1,1,1,1] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => 0
[2,1,1,2] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 1
[2,1,2,1] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 1
[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,1] => ? = 0
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 $\pi(i) > \pi(j) > i$.
The following 154 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001176The size of a partition minus its first part. 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. St001394The genus of a permutation. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000783The side length of the largest staircase partition fitting into a partition. St001091The number of parts in an integer partition whose next smaller part has the same size. St000358The number of occurrences of the pattern 31-2. St000356The number of occurrences of the pattern 13-2. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000031The number of cycles in the cycle decomposition of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000137The Grundy value of an integer partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001593This is the number of standard Young tableaux of the given shifted shape. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St000624The normalized sum of the minimal distances to a greater element. St000779The tier of a permutation. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001722The number of minimal chains with small intervals between a binary word and the top element. St001220The width of a permutation. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St000370The genus of a graph. St000671The maximin edge-connectivity for choosing a subgraph. St001071The beta invariant of the graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001357The maximal degree of a regular spanning subgraph of a graph. St001638The book thickness of a graph. St001689The number of celebrities in a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001736The total number of cycles in a graph. St001877Number of indecomposable injective modules with projective dimension 2. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001307The number of induced stars on four vertices in a graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001271The competition number of a graph. St000035The number of left outer peaks of a permutation. St000322The skewness of a graph. St000379The number of Hamiltonian cycles in a graph. St001577The minimal number of edges to add or remove to make a graph a cograph. St001578The minimal number of edges to add or remove to make a graph a line 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. 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. St001575The minimal number of edges to add or remove to make a graph edge transitive. 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. St001871The number of triconnected components of a graph. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000929The constant term of the character polynomial of an integer partition. 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. St000360The number of occurrences of the pattern 32-1. St000153The number of adjacent cycles of a permutation. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001280The number of parts of an integer partition that are at least two. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000455The second largest eigenvalue of a graph if it is integral. St001570The minimal number of edges to add to make a graph Hamiltonian. St001728The number of invisible descents of a permutation. St000260The radius of a connected graph. 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 $K[x]/(x^n)$. St000252The number of nodes of degree 3 of a binary tree. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000664The number of right ropes of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000570The Edelman-Greene number of a permutation. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001729The number of visible descents of a permutation. St001928The number of non-overlapping descents in a permutation. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001960The number of descents of a permutation minus one if its first entry is not one. St001846The number of elements which do not have a complement in the lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. 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. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. 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. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001128The exponens consonantiae of a partition. St000408The number of occurrences of the pattern 4231 in a permutation. St000842The breadth of a permutation. St000181The number of connected components of the Hasse diagram for the poset. St001490The number of connected components of a skew partition. St001890The maximum magnitude of the Möbius function of a poset. St000068The number of minimal elements in a poset. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001487The number of inner corners of a skew partition. St000629The defect of a binary word. St001651The Frankl number of a lattice. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001549The number of restricted non-inversions between exceedances. St001715The number of non-records in a permutation. St000486The number of cycles of length at least 3 of a permutation. St000902 The minimal number of repetitions of an integer composition. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St001768The number of reduced words of a signed permutation.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!