searching the database
Your data matches 8 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: St001176
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,1]
=> 1
[2,1] => [2,1] => [2]
=> 0
[-2,1] => [-2,-1] => [2]
=> 0
[1,2,3] => [1,2,3] => [1,1,1]
=> 2
[1,2,-3] => [1,2,-3] => [1,1]
=> 1
[1,3,2] => [1,3,2] => [2,1]
=> 1
[1,3,-2] => [1,-2,3] => [1,1]
=> 1
[1,-3,2] => [1,-3,-2] => [2,1]
=> 1
[2,1,3] => [2,1,3] => [2,1]
=> 1
[2,1,-3] => [2,1,-3] => [2]
=> 0
[-2,1,3] => [-2,-1,3] => [2,1]
=> 1
[-2,1,-3] => [-2,-1,-3] => [2]
=> 0
[2,3,1] => [3,2,1] => [2,1]
=> 1
[2,3,-1] => [-1,2,3] => [1,1]
=> 1
[2,-3,1] => [-3,2,-1] => [2,1]
=> 1
[-2,3,1] => [-2,-1,3] => [2,1]
=> 1
[-2,-3,1] => [-2,-1,-3] => [2]
=> 0
[3,1,2] => [3,2,1] => [2,1]
=> 1
[3,1,-2] => [3,-2,1] => [2]
=> 0
[-3,1,2] => [-3,2,-1] => [2,1]
=> 1
[-3,1,-2] => [-3,-2,-1] => [2]
=> 0
[3,2,1] => [3,2,1] => [2,1]
=> 1
[3,2,-1] => [-1,3,2] => [2]
=> 0
[3,-2,1] => [-2,-1,3] => [2,1]
=> 1
[-3,2,1] => [-3,2,-1] => [2,1]
=> 1
[-3,2,-1] => [-1,-3,-2] => [2]
=> 0
[-3,-2,1] => [-2,-1,-3] => [2]
=> 0
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 3
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
=> 2
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
=> 1
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> 1
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
=> 1
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
=> 1
[1,2,4,3] => [1,2,4,3] => [2,1,1]
=> 2
[1,2,4,-3] => [1,2,-3,4] => [1,1,1]
=> 2
[1,2,-4,3] => [1,2,-4,-3] => [2,1,1]
=> 2
[1,2,-4,-3] => [1,2,-3,-4] => [1,1]
=> 1
[1,-2,4,3] => [1,-2,-3,4] => [1,1]
=> 1
[-1,2,4,3] => [-1,-2,4,3] => [2]
=> 0
[-1,2,-4,3] => [-1,-2,-4,-3] => [2]
=> 0
[1,3,2,4] => [1,3,2,4] => [2,1,1]
=> 2
[1,3,2,-4] => [1,3,2,-4] => [2,1]
=> 1
[1,3,-2,4] => [1,-2,3,-4] => [1,1]
=> 1
[1,3,-2,-4] => [1,-2,3,-4] => [1,1]
=> 1
[1,-3,2,4] => [1,-3,-2,4] => [2,1,1]
=> 2
[1,-3,2,-4] => [1,-3,-2,-4] => [2,1]
=> 1
[-1,3,2,4] => [-1,-2,3,4] => [1,1]
=> 1
[1,3,4,2] => [1,4,3,2] => [2,1,1]
=> 2
[1,3,4,-2] => [1,-2,3,4] => [1,1,1]
=> 2
[1,3,-4,2] => [1,-4,3,-2] => [2,1,1]
=> 2
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
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ 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%
Mp00166: Signed permutations —even cycle type⟶ 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] => [1,2] => [1,1]
=> [1]
=> 1
[2,1] => [2,1] => [2]
=> []
=> 0
[-2,1] => [-2,-1] => [2]
=> []
=> 0
[1,2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 2
[1,2,-3] => [1,2,-3] => [1,1]
=> [1]
=> 1
[1,3,2] => [1,3,2] => [2,1]
=> [1]
=> 1
[1,3,-2] => [1,-2,3] => [1,1]
=> [1]
=> 1
[1,-3,2] => [1,-3,-2] => [2,1]
=> [1]
=> 1
[2,1,3] => [2,1,3] => [2,1]
=> [1]
=> 1
[2,1,-3] => [2,1,-3] => [2]
=> []
=> 0
[-2,1,3] => [-2,-1,3] => [2,1]
=> [1]
=> 1
[-2,1,-3] => [-2,-1,-3] => [2]
=> []
=> 0
[2,3,1] => [3,2,1] => [2,1]
=> [1]
=> 1
[2,3,-1] => [-1,2,3] => [1,1]
=> [1]
=> 1
[2,-3,1] => [-3,2,-1] => [2,1]
=> [1]
=> 1
[-2,3,1] => [-2,-1,3] => [2,1]
=> [1]
=> 1
[-2,-3,1] => [-2,-1,-3] => [2]
=> []
=> 0
[3,1,2] => [3,2,1] => [2,1]
=> [1]
=> 1
[3,1,-2] => [3,-2,1] => [2]
=> []
=> 0
[-3,1,2] => [-3,2,-1] => [2,1]
=> [1]
=> 1
[-3,1,-2] => [-3,-2,-1] => [2]
=> []
=> 0
[3,2,1] => [3,2,1] => [2,1]
=> [1]
=> 1
[3,2,-1] => [-1,3,2] => [2]
=> []
=> 0
[3,-2,1] => [-2,-1,3] => [2,1]
=> [1]
=> 1
[-3,2,1] => [-3,2,-1] => [2,1]
=> [1]
=> 1
[-3,2,-1] => [-1,-3,-2] => [2]
=> []
=> 0
[-3,-2,1] => [-2,-1,-3] => [2]
=> []
=> 0
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 3
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
=> [1,1]
=> 2
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
=> [1]
=> 1
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> [1]
=> 1
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
=> [1]
=> 1
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
=> [1]
=> 1
[1,2,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 2
[1,2,4,-3] => [1,2,-3,4] => [1,1,1]
=> [1,1]
=> 2
[1,2,-4,3] => [1,2,-4,-3] => [2,1,1]
=> [1,1]
=> 2
[1,2,-4,-3] => [1,2,-3,-4] => [1,1]
=> [1]
=> 1
[1,-2,4,3] => [1,-2,-3,4] => [1,1]
=> [1]
=> 1
[-1,2,4,3] => [-1,-2,4,3] => [2]
=> []
=> 0
[-1,2,-4,3] => [-1,-2,-4,-3] => [2]
=> []
=> 0
[1,3,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 2
[1,3,2,-4] => [1,3,2,-4] => [2,1]
=> [1]
=> 1
[1,3,-2,4] => [1,-2,3,-4] => [1,1]
=> [1]
=> 1
[1,3,-2,-4] => [1,-2,3,-4] => [1,1]
=> [1]
=> 1
[1,-3,2,4] => [1,-3,-2,4] => [2,1,1]
=> [1,1]
=> 2
[1,-3,2,-4] => [1,-3,-2,-4] => [2,1]
=> [1]
=> 1
[-1,3,2,4] => [-1,-2,3,4] => [1,1]
=> [1]
=> 1
[1,3,4,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 2
[1,3,4,-2] => [1,-2,3,4] => [1,1,1]
=> [1,1]
=> 2
[1,3,-4,2] => [1,-4,3,-2] => [2,1,1]
=> [1,1]
=> 2
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St000394
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,1]
=> [1,1,0,0]
=> 1
[2,1] => [2,1] => [2]
=> [1,0,1,0]
=> 0
[-2,1] => [-2,-1] => [2]
=> [1,0,1,0]
=> 0
[1,2,3] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,2,-3] => [1,2,-3] => [1,1]
=> [1,1,0,0]
=> 1
[1,3,2] => [1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,3,-2] => [1,-2,3] => [1,1]
=> [1,1,0,0]
=> 1
[1,-3,2] => [1,-3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,1,3] => [2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,1,-3] => [2,1,-3] => [2]
=> [1,0,1,0]
=> 0
[-2,1,3] => [-2,-1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[-2,1,-3] => [-2,-1,-3] => [2]
=> [1,0,1,0]
=> 0
[2,3,1] => [3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,3,-1] => [-1,2,3] => [1,1]
=> [1,1,0,0]
=> 1
[2,-3,1] => [-3,2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[-2,3,1] => [-2,-1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[-2,-3,1] => [-2,-1,-3] => [2]
=> [1,0,1,0]
=> 0
[3,1,2] => [3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[3,1,-2] => [3,-2,1] => [2]
=> [1,0,1,0]
=> 0
[-3,1,2] => [-3,2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[-3,1,-2] => [-3,-2,-1] => [2]
=> [1,0,1,0]
=> 0
[3,2,1] => [3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[3,2,-1] => [-1,3,2] => [2]
=> [1,0,1,0]
=> 0
[3,-2,1] => [-2,-1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[-3,2,1] => [-3,2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[-3,2,-1] => [-1,-3,-2] => [2]
=> [1,0,1,0]
=> 0
[-3,-2,1] => [-2,-1,-3] => [2]
=> [1,0,1,0]
=> 0
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> 1
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> 1
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> 1
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
=> [1,1,0,0]
=> 1
[1,2,4,3] => [1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,2,4,-3] => [1,2,-3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,2,-4,3] => [1,2,-4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,2,-4,-3] => [1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> 1
[1,-2,4,3] => [1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> 1
[-1,2,4,3] => [-1,-2,4,3] => [2]
=> [1,0,1,0]
=> 0
[-1,2,-4,3] => [-1,-2,-4,-3] => [2]
=> [1,0,1,0]
=> 0
[1,3,2,4] => [1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,3,2,-4] => [1,3,2,-4] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,3,-2,4] => [1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> 1
[1,3,-2,-4] => [1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> 1
[1,-3,2,4] => [1,-3,-2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,-3,2,-4] => [1,-3,-2,-4] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[-1,3,2,4] => [-1,-2,3,4] => [1,1]
=> [1,1,0,0]
=> 1
[1,3,4,2] => [1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,3,4,-2] => [1,-2,3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,3,-4,2] => [1,-4,3,-2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
Description
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
Matching statistic: St000459
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000459: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000459: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,1]
=> [1]
=> 1
[2,1] => [2,1] => [2]
=> []
=> 0
[-2,1] => [-2,-1] => [2]
=> []
=> 0
[1,2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 2
[1,2,-3] => [1,2,-3] => [1,1]
=> [1]
=> 1
[1,3,2] => [1,3,2] => [2,1]
=> [1]
=> 1
[1,3,-2] => [1,-2,3] => [1,1]
=> [1]
=> 1
[1,-3,2] => [1,-3,-2] => [2,1]
=> [1]
=> 1
[2,1,3] => [2,1,3] => [2,1]
=> [1]
=> 1
[2,1,-3] => [2,1,-3] => [2]
=> []
=> 0
[-2,1,3] => [-2,-1,3] => [2,1]
=> [1]
=> 1
[-2,1,-3] => [-2,-1,-3] => [2]
=> []
=> 0
[2,3,1] => [3,2,1] => [2,1]
=> [1]
=> 1
[2,3,-1] => [-1,2,3] => [1,1]
=> [1]
=> 1
[2,-3,1] => [-3,2,-1] => [2,1]
=> [1]
=> 1
[-2,3,1] => [-2,-1,3] => [2,1]
=> [1]
=> 1
[-2,-3,1] => [-2,-1,-3] => [2]
=> []
=> 0
[3,1,2] => [3,2,1] => [2,1]
=> [1]
=> 1
[3,1,-2] => [3,-2,1] => [2]
=> []
=> 0
[-3,1,2] => [-3,2,-1] => [2,1]
=> [1]
=> 1
[-3,1,-2] => [-3,-2,-1] => [2]
=> []
=> 0
[3,2,1] => [3,2,1] => [2,1]
=> [1]
=> 1
[3,2,-1] => [-1,3,2] => [2]
=> []
=> 0
[3,-2,1] => [-2,-1,3] => [2,1]
=> [1]
=> 1
[-3,2,1] => [-3,2,-1] => [2,1]
=> [1]
=> 1
[-3,2,-1] => [-1,-3,-2] => [2]
=> []
=> 0
[-3,-2,1] => [-2,-1,-3] => [2]
=> []
=> 0
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 3
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
=> [1,1]
=> 2
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
=> [1]
=> 1
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> [1]
=> 1
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
=> [1]
=> 1
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
=> [1]
=> 1
[1,2,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 2
[1,2,4,-3] => [1,2,-3,4] => [1,1,1]
=> [1,1]
=> 2
[1,2,-4,3] => [1,2,-4,-3] => [2,1,1]
=> [1,1]
=> 2
[1,2,-4,-3] => [1,2,-3,-4] => [1,1]
=> [1]
=> 1
[1,-2,4,3] => [1,-2,-3,4] => [1,1]
=> [1]
=> 1
[-1,2,4,3] => [-1,-2,4,3] => [2]
=> []
=> 0
[-1,2,-4,3] => [-1,-2,-4,-3] => [2]
=> []
=> 0
[1,3,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 2
[1,3,2,-4] => [1,3,2,-4] => [2,1]
=> [1]
=> 1
[1,3,-2,4] => [1,-2,3,-4] => [1,1]
=> [1]
=> 1
[1,3,-2,-4] => [1,-2,3,-4] => [1,1]
=> [1]
=> 1
[1,-3,2,4] => [1,-3,-2,4] => [2,1,1]
=> [1,1]
=> 2
[1,-3,2,-4] => [1,-3,-2,-4] => [2,1]
=> [1]
=> 1
[-1,3,2,4] => [-1,-2,3,4] => [1,1]
=> [1]
=> 1
[1,3,4,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 2
[1,3,4,-2] => [1,-2,3,4] => [1,1,1]
=> [1,1]
=> 2
[1,3,-4,2] => [1,-4,3,-2] => [2,1,1]
=> [1,1]
=> 2
Description
The hook length of the base cell of a partition.
This is also known as the perimeter of a partition. In particular, the perimeter of the empty partition is zero.
Matching statistic: St001480
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001480: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001480: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,1]
=> [1,1,0,0]
=> 1
[2,1] => [2,1] => [2]
=> [1,0,1,0]
=> 0
[-2,1] => [-2,-1] => [2]
=> [1,0,1,0]
=> 0
[1,2,3] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,2,-3] => [1,2,-3] => [1,1]
=> [1,1,0,0]
=> 1
[1,3,2] => [1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,3,-2] => [1,-2,3] => [1,1]
=> [1,1,0,0]
=> 1
[1,-3,2] => [1,-3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,1,3] => [2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,1,-3] => [2,1,-3] => [2]
=> [1,0,1,0]
=> 0
[-2,1,3] => [-2,-1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[-2,1,-3] => [-2,-1,-3] => [2]
=> [1,0,1,0]
=> 0
[2,3,1] => [3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,3,-1] => [-1,2,3] => [1,1]
=> [1,1,0,0]
=> 1
[2,-3,1] => [-3,2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[-2,3,1] => [-2,-1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[-2,-3,1] => [-2,-1,-3] => [2]
=> [1,0,1,0]
=> 0
[3,1,2] => [3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[3,1,-2] => [3,-2,1] => [2]
=> [1,0,1,0]
=> 0
[-3,1,2] => [-3,2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[-3,1,-2] => [-3,-2,-1] => [2]
=> [1,0,1,0]
=> 0
[3,2,1] => [3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[3,2,-1] => [-1,3,2] => [2]
=> [1,0,1,0]
=> 0
[3,-2,1] => [-2,-1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[-3,2,1] => [-3,2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[-3,2,-1] => [-1,-3,-2] => [2]
=> [1,0,1,0]
=> 0
[-3,-2,1] => [-2,-1,-3] => [2]
=> [1,0,1,0]
=> 0
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> 1
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> 1
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> 1
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
=> [1,1,0,0]
=> 1
[1,2,4,3] => [1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,2,4,-3] => [1,2,-3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,2,-4,3] => [1,2,-4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,2,-4,-3] => [1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> 1
[1,-2,4,3] => [1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> 1
[-1,2,4,3] => [-1,-2,4,3] => [2]
=> [1,0,1,0]
=> 0
[-1,2,-4,3] => [-1,-2,-4,-3] => [2]
=> [1,0,1,0]
=> 0
[1,3,2,4] => [1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,3,2,-4] => [1,3,2,-4] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,3,-2,4] => [1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> 1
[1,3,-2,-4] => [1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> 1
[1,-3,2,4] => [1,-3,-2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,-3,2,-4] => [1,-3,-2,-4] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[-1,3,2,4] => [-1,-2,3,4] => [1,1]
=> [1,1,0,0]
=> 1
[1,3,4,2] => [1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,3,4,-2] => [1,-2,3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,3,-4,2] => [1,-4,3,-2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
Description
The number of simple summands of the module J^2/J^3. Here J is the Jacobson radical of the Nakayama algebra algebra corresponding to the Dyck path.
Matching statistic: St000738
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[2,1] => [2,1] => [2]
=> [[1,2]]
=> 1 = 0 + 1
[-2,1] => [-2,-1] => [2]
=> [[1,2]]
=> 1 = 0 + 1
[1,2,3] => [1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> 3 = 2 + 1
[1,2,-3] => [1,2,-3] => [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[1,3,2] => [1,3,2] => [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[1,3,-2] => [1,-2,3] => [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[1,-3,2] => [1,-3,-2] => [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[2,1,3] => [2,1,3] => [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[2,1,-3] => [2,1,-3] => [2]
=> [[1,2]]
=> 1 = 0 + 1
[-2,1,3] => [-2,-1,3] => [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[-2,1,-3] => [-2,-1,-3] => [2]
=> [[1,2]]
=> 1 = 0 + 1
[2,3,1] => [3,2,1] => [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[2,3,-1] => [-1,2,3] => [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[2,-3,1] => [-3,2,-1] => [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[-2,3,1] => [-2,-1,3] => [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[-2,-3,1] => [-2,-1,-3] => [2]
=> [[1,2]]
=> 1 = 0 + 1
[3,1,2] => [3,2,1] => [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[3,1,-2] => [3,-2,1] => [2]
=> [[1,2]]
=> 1 = 0 + 1
[-3,1,2] => [-3,2,-1] => [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[-3,1,-2] => [-3,-2,-1] => [2]
=> [[1,2]]
=> 1 = 0 + 1
[3,2,1] => [3,2,1] => [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[3,2,-1] => [-1,3,2] => [2]
=> [[1,2]]
=> 1 = 0 + 1
[3,-2,1] => [-2,-1,3] => [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[-3,2,1] => [-3,2,-1] => [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[-3,2,-1] => [-1,-3,-2] => [2]
=> [[1,2]]
=> 1 = 0 + 1
[-3,-2,1] => [-2,-1,-3] => [2]
=> [[1,2]]
=> 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4 = 3 + 1
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
=> [[1],[2],[3]]
=> 3 = 2 + 1
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[1,2,4,3] => [1,2,4,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[1,2,4,-3] => [1,2,-3,4] => [1,1,1]
=> [[1],[2],[3]]
=> 3 = 2 + 1
[1,2,-4,3] => [1,2,-4,-3] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[1,2,-4,-3] => [1,2,-3,-4] => [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[1,-2,4,3] => [1,-2,-3,4] => [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[-1,2,4,3] => [-1,-2,4,3] => [2]
=> [[1,2]]
=> 1 = 0 + 1
[-1,2,-4,3] => [-1,-2,-4,-3] => [2]
=> [[1,2]]
=> 1 = 0 + 1
[1,3,2,4] => [1,3,2,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[1,3,2,-4] => [1,3,2,-4] => [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[1,3,-2,4] => [1,-2,3,-4] => [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[1,3,-2,-4] => [1,-2,3,-4] => [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[1,-3,2,4] => [1,-3,-2,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[1,-3,2,-4] => [1,-3,-2,-4] => [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[-1,3,2,4] => [-1,-2,3,4] => [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[1,3,4,2] => [1,4,3,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[1,3,4,-2] => [1,-2,3,4] => [1,1,1]
=> [[1],[2],[3]]
=> 3 = 2 + 1
[1,3,-4,2] => [1,-4,3,-2] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 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: St000460
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000460: Integer partitions ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 80%
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000460: Integer partitions ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 80%
Values
[1,2] => [1,2] => [1,1]
=> [1]
=> 1
[2,1] => [2,1] => [2]
=> []
=> ? = 0
[-2,1] => [-2,-1] => [2]
=> []
=> ? = 0
[1,2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 2
[1,2,-3] => [1,2,-3] => [1,1]
=> [1]
=> 1
[1,3,2] => [1,3,2] => [2,1]
=> [1]
=> 1
[1,3,-2] => [1,-2,3] => [1,1]
=> [1]
=> 1
[1,-3,2] => [1,-3,-2] => [2,1]
=> [1]
=> 1
[2,1,3] => [2,1,3] => [2,1]
=> [1]
=> 1
[2,1,-3] => [2,1,-3] => [2]
=> []
=> ? = 0
[-2,1,3] => [-2,-1,3] => [2,1]
=> [1]
=> 1
[-2,1,-3] => [-2,-1,-3] => [2]
=> []
=> ? = 0
[2,3,1] => [3,2,1] => [2,1]
=> [1]
=> 1
[2,3,-1] => [-1,2,3] => [1,1]
=> [1]
=> 1
[2,-3,1] => [-3,2,-1] => [2,1]
=> [1]
=> 1
[-2,3,1] => [-2,-1,3] => [2,1]
=> [1]
=> 1
[-2,-3,1] => [-2,-1,-3] => [2]
=> []
=> ? = 0
[3,1,2] => [3,2,1] => [2,1]
=> [1]
=> 1
[3,1,-2] => [3,-2,1] => [2]
=> []
=> ? = 0
[-3,1,2] => [-3,2,-1] => [2,1]
=> [1]
=> 1
[-3,1,-2] => [-3,-2,-1] => [2]
=> []
=> ? = 0
[3,2,1] => [3,2,1] => [2,1]
=> [1]
=> 1
[3,2,-1] => [-1,3,2] => [2]
=> []
=> ? = 0
[3,-2,1] => [-2,-1,3] => [2,1]
=> [1]
=> 1
[-3,2,1] => [-3,2,-1] => [2,1]
=> [1]
=> 1
[-3,2,-1] => [-1,-3,-2] => [2]
=> []
=> ? = 0
[-3,-2,1] => [-2,-1,-3] => [2]
=> []
=> ? = 0
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 3
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
=> [1,1]
=> 2
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
=> [1]
=> 1
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> [1]
=> 1
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
=> [1]
=> 1
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
=> [1]
=> 1
[1,2,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 2
[1,2,4,-3] => [1,2,-3,4] => [1,1,1]
=> [1,1]
=> 2
[1,2,-4,3] => [1,2,-4,-3] => [2,1,1]
=> [1,1]
=> 2
[1,2,-4,-3] => [1,2,-3,-4] => [1,1]
=> [1]
=> 1
[1,-2,4,3] => [1,-2,-3,4] => [1,1]
=> [1]
=> 1
[-1,2,4,3] => [-1,-2,4,3] => [2]
=> []
=> ? = 0
[-1,2,-4,3] => [-1,-2,-4,-3] => [2]
=> []
=> ? = 0
[1,3,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 2
[1,3,2,-4] => [1,3,2,-4] => [2,1]
=> [1]
=> 1
[1,3,-2,4] => [1,-2,3,-4] => [1,1]
=> [1]
=> 1
[1,3,-2,-4] => [1,-2,3,-4] => [1,1]
=> [1]
=> 1
[1,-3,2,4] => [1,-3,-2,4] => [2,1,1]
=> [1,1]
=> 2
[1,-3,2,-4] => [1,-3,-2,-4] => [2,1]
=> [1]
=> 1
[-1,3,2,4] => [-1,-2,3,4] => [1,1]
=> [1]
=> 1
[1,3,4,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 2
[1,3,4,-2] => [1,-2,3,4] => [1,1,1]
=> [1,1]
=> 2
[1,3,-4,2] => [1,-4,3,-2] => [2,1,1]
=> [1,1]
=> 2
[1,3,-4,-2] => [1,-2,3,-4] => [1,1]
=> [1]
=> 1
[1,-3,4,2] => [1,-3,-2,4] => [2,1,1]
=> [1,1]
=> 2
[1,-3,-4,2] => [1,-3,-2,-4] => [2,1]
=> [1]
=> 1
[-1,3,4,2] => [-1,-2,4,3] => [2]
=> []
=> ? = 0
[-1,3,-4,2] => [-1,-2,-4,-3] => [2]
=> []
=> ? = 0
[1,4,2,3] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 2
[1,4,2,-3] => [1,4,-3,2] => [2,1]
=> [1]
=> 1
[1,4,-2,3] => [1,-2,-3,4] => [1,1]
=> [1]
=> 1
[1,4,-2,-3] => [1,-2,-3,4] => [1,1]
=> [1]
=> 1
[1,-4,2,3] => [1,-4,3,-2] => [2,1,1]
=> [1,1]
=> 2
[1,-4,2,-3] => [1,-4,-3,-2] => [2,1]
=> [1]
=> 1
[-1,4,2,3] => [-1,-2,4,3] => [2]
=> []
=> ? = 0
[-1,-4,2,3] => [-1,-2,-4,-3] => [2]
=> []
=> ? = 0
[1,4,3,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 2
[1,4,3,-2] => [1,-2,4,3] => [2,1]
=> [1]
=> 1
[1,4,-3,2] => [1,-3,-2,4] => [2,1,1]
=> [1,1]
=> 2
[-1,4,3,2] => [-1,-2,4,3] => [2]
=> []
=> ? = 0
[-1,-4,3,2] => [-1,-2,-4,-3] => [2]
=> []
=> ? = 0
[2,1,-3,4] => [2,1,-3,-4] => [2]
=> []
=> ? = 0
[2,1,-3,-4] => [2,1,-3,-4] => [2]
=> []
=> ? = 0
[-2,1,-3,4] => [-2,-1,-3,-4] => [2]
=> []
=> ? = 0
[-2,1,-3,-4] => [-2,-1,-3,-4] => [2]
=> []
=> ? = 0
[2,1,-4,-3] => [2,1,-3,-4] => [2]
=> []
=> ? = 0
[-2,1,-4,-3] => [-2,-1,-3,-4] => [2]
=> []
=> ? = 0
[-2,-3,1,4] => [-2,-1,-3,-4] => [2]
=> []
=> ? = 0
[-2,-3,1,-4] => [-2,-1,-3,-4] => [2]
=> []
=> ? = 0
[-2,-3,4,1] => [-2,-1,-3,-4] => [2]
=> []
=> ? = 0
[-2,-3,-4,1] => [-2,-1,-3,-4] => [2]
=> []
=> ? = 0
[-2,-4,1,-3] => [-2,-1,-3,-4] => [2]
=> []
=> ? = 0
[-2,4,-3,1] => [-2,-1,-3,-4] => [2]
=> []
=> ? = 0
[-2,-4,-3,1] => [-2,-1,-3,-4] => [2]
=> []
=> ? = 0
[3,1,-2,4] => [3,-2,1,-4] => [2]
=> []
=> ? = 0
[3,1,-2,-4] => [3,-2,1,-4] => [2]
=> []
=> ? = 0
[-3,1,-2,4] => [-3,-2,-1,-4] => [2]
=> []
=> ? = 0
[-3,1,-2,-4] => [-3,-2,-1,-4] => [2]
=> []
=> ? = 0
[3,1,-4,-2] => [3,-2,1,-4] => [2]
=> []
=> ? = 0
[-3,1,-4,-2] => [-3,-2,-1,-4] => [2]
=> []
=> ? = 0
[3,2,-1,4] => [-1,3,2,-4] => [2]
=> []
=> ? = 0
[3,2,-1,-4] => [-1,3,2,-4] => [2]
=> []
=> ? = 0
[-3,2,-1,4] => [-1,-3,-2,-4] => [2]
=> []
=> ? = 0
[-3,2,-1,-4] => [-1,-3,-2,-4] => [2]
=> []
=> ? = 0
[-3,-2,1,4] => [-2,-1,-3,-4] => [2]
=> []
=> ? = 0
[-3,-2,1,-4] => [-2,-1,-3,-4] => [2]
=> []
=> ? = 0
[3,2,-4,-1] => [-1,3,2,-4] => [2]
=> []
=> ? = 0
[-3,2,-4,-1] => [-1,-3,-2,-4] => [2]
=> []
=> ? = 0
[-3,-2,4,1] => [-2,-1,-3,-4] => [2]
=> []
=> ? = 0
[-3,-2,-4,1] => [-2,-1,-3,-4] => [2]
=> []
=> ? = 0
[-3,-4,1,-2] => [-3,-2,-1,-4] => [2]
=> []
=> ? = 0
[-3,4,-2,1] => [-2,-1,-3,-4] => [2]
=> []
=> ? = 0
[-3,-4,2,-1] => [-1,-3,-2,-4] => [2]
=> []
=> ? = 0
Description
The hook length of the last cell along the main diagonal of an integer partition.
Matching statistic: St000870
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000870: Integer partitions ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 80%
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000870: Integer partitions ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 80%
Values
[1,2] => [1,2] => [1,1]
=> [1]
=> 1
[2,1] => [2,1] => [2]
=> []
=> ? = 0
[-2,1] => [-2,-1] => [2]
=> []
=> ? = 0
[1,2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 2
[1,2,-3] => [1,2,-3] => [1,1]
=> [1]
=> 1
[1,3,2] => [1,3,2] => [2,1]
=> [1]
=> 1
[1,3,-2] => [1,-2,3] => [1,1]
=> [1]
=> 1
[1,-3,2] => [1,-3,-2] => [2,1]
=> [1]
=> 1
[2,1,3] => [2,1,3] => [2,1]
=> [1]
=> 1
[2,1,-3] => [2,1,-3] => [2]
=> []
=> ? = 0
[-2,1,3] => [-2,-1,3] => [2,1]
=> [1]
=> 1
[-2,1,-3] => [-2,-1,-3] => [2]
=> []
=> ? = 0
[2,3,1] => [3,2,1] => [2,1]
=> [1]
=> 1
[2,3,-1] => [-1,2,3] => [1,1]
=> [1]
=> 1
[2,-3,1] => [-3,2,-1] => [2,1]
=> [1]
=> 1
[-2,3,1] => [-2,-1,3] => [2,1]
=> [1]
=> 1
[-2,-3,1] => [-2,-1,-3] => [2]
=> []
=> ? = 0
[3,1,2] => [3,2,1] => [2,1]
=> [1]
=> 1
[3,1,-2] => [3,-2,1] => [2]
=> []
=> ? = 0
[-3,1,2] => [-3,2,-1] => [2,1]
=> [1]
=> 1
[-3,1,-2] => [-3,-2,-1] => [2]
=> []
=> ? = 0
[3,2,1] => [3,2,1] => [2,1]
=> [1]
=> 1
[3,2,-1] => [-1,3,2] => [2]
=> []
=> ? = 0
[3,-2,1] => [-2,-1,3] => [2,1]
=> [1]
=> 1
[-3,2,1] => [-3,2,-1] => [2,1]
=> [1]
=> 1
[-3,2,-1] => [-1,-3,-2] => [2]
=> []
=> ? = 0
[-3,-2,1] => [-2,-1,-3] => [2]
=> []
=> ? = 0
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 3
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
=> [1,1]
=> 2
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
=> [1]
=> 1
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> [1]
=> 1
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
=> [1]
=> 1
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
=> [1]
=> 1
[1,2,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 2
[1,2,4,-3] => [1,2,-3,4] => [1,1,1]
=> [1,1]
=> 2
[1,2,-4,3] => [1,2,-4,-3] => [2,1,1]
=> [1,1]
=> 2
[1,2,-4,-3] => [1,2,-3,-4] => [1,1]
=> [1]
=> 1
[1,-2,4,3] => [1,-2,-3,4] => [1,1]
=> [1]
=> 1
[-1,2,4,3] => [-1,-2,4,3] => [2]
=> []
=> ? = 0
[-1,2,-4,3] => [-1,-2,-4,-3] => [2]
=> []
=> ? = 0
[1,3,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 2
[1,3,2,-4] => [1,3,2,-4] => [2,1]
=> [1]
=> 1
[1,3,-2,4] => [1,-2,3,-4] => [1,1]
=> [1]
=> 1
[1,3,-2,-4] => [1,-2,3,-4] => [1,1]
=> [1]
=> 1
[1,-3,2,4] => [1,-3,-2,4] => [2,1,1]
=> [1,1]
=> 2
[1,-3,2,-4] => [1,-3,-2,-4] => [2,1]
=> [1]
=> 1
[-1,3,2,4] => [-1,-2,3,4] => [1,1]
=> [1]
=> 1
[1,3,4,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 2
[1,3,4,-2] => [1,-2,3,4] => [1,1,1]
=> [1,1]
=> 2
[1,3,-4,2] => [1,-4,3,-2] => [2,1,1]
=> [1,1]
=> 2
[1,3,-4,-2] => [1,-2,3,-4] => [1,1]
=> [1]
=> 1
[1,-3,4,2] => [1,-3,-2,4] => [2,1,1]
=> [1,1]
=> 2
[1,-3,-4,2] => [1,-3,-2,-4] => [2,1]
=> [1]
=> 1
[-1,3,4,2] => [-1,-2,4,3] => [2]
=> []
=> ? = 0
[-1,3,-4,2] => [-1,-2,-4,-3] => [2]
=> []
=> ? = 0
[1,4,2,3] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 2
[1,4,2,-3] => [1,4,-3,2] => [2,1]
=> [1]
=> 1
[1,4,-2,3] => [1,-2,-3,4] => [1,1]
=> [1]
=> 1
[1,4,-2,-3] => [1,-2,-3,4] => [1,1]
=> [1]
=> 1
[1,-4,2,3] => [1,-4,3,-2] => [2,1,1]
=> [1,1]
=> 2
[1,-4,2,-3] => [1,-4,-3,-2] => [2,1]
=> [1]
=> 1
[-1,4,2,3] => [-1,-2,4,3] => [2]
=> []
=> ? = 0
[-1,-4,2,3] => [-1,-2,-4,-3] => [2]
=> []
=> ? = 0
[1,4,3,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 2
[1,4,3,-2] => [1,-2,4,3] => [2,1]
=> [1]
=> 1
[1,4,-3,2] => [1,-3,-2,4] => [2,1,1]
=> [1,1]
=> 2
[-1,4,3,2] => [-1,-2,4,3] => [2]
=> []
=> ? = 0
[-1,-4,3,2] => [-1,-2,-4,-3] => [2]
=> []
=> ? = 0
[2,1,-3,4] => [2,1,-3,-4] => [2]
=> []
=> ? = 0
[2,1,-3,-4] => [2,1,-3,-4] => [2]
=> []
=> ? = 0
[-2,1,-3,4] => [-2,-1,-3,-4] => [2]
=> []
=> ? = 0
[-2,1,-3,-4] => [-2,-1,-3,-4] => [2]
=> []
=> ? = 0
[2,1,-4,-3] => [2,1,-3,-4] => [2]
=> []
=> ? = 0
[-2,1,-4,-3] => [-2,-1,-3,-4] => [2]
=> []
=> ? = 0
[-2,-3,1,4] => [-2,-1,-3,-4] => [2]
=> []
=> ? = 0
[-2,-3,1,-4] => [-2,-1,-3,-4] => [2]
=> []
=> ? = 0
[-2,-3,4,1] => [-2,-1,-3,-4] => [2]
=> []
=> ? = 0
[-2,-3,-4,1] => [-2,-1,-3,-4] => [2]
=> []
=> ? = 0
[-2,-4,1,-3] => [-2,-1,-3,-4] => [2]
=> []
=> ? = 0
[-2,4,-3,1] => [-2,-1,-3,-4] => [2]
=> []
=> ? = 0
[-2,-4,-3,1] => [-2,-1,-3,-4] => [2]
=> []
=> ? = 0
[3,1,-2,4] => [3,-2,1,-4] => [2]
=> []
=> ? = 0
[3,1,-2,-4] => [3,-2,1,-4] => [2]
=> []
=> ? = 0
[-3,1,-2,4] => [-3,-2,-1,-4] => [2]
=> []
=> ? = 0
[-3,1,-2,-4] => [-3,-2,-1,-4] => [2]
=> []
=> ? = 0
[3,1,-4,-2] => [3,-2,1,-4] => [2]
=> []
=> ? = 0
[-3,1,-4,-2] => [-3,-2,-1,-4] => [2]
=> []
=> ? = 0
[3,2,-1,4] => [-1,3,2,-4] => [2]
=> []
=> ? = 0
[3,2,-1,-4] => [-1,3,2,-4] => [2]
=> []
=> ? = 0
[-3,2,-1,4] => [-1,-3,-2,-4] => [2]
=> []
=> ? = 0
[-3,2,-1,-4] => [-1,-3,-2,-4] => [2]
=> []
=> ? = 0
[-3,-2,1,4] => [-2,-1,-3,-4] => [2]
=> []
=> ? = 0
[-3,-2,1,-4] => [-2,-1,-3,-4] => [2]
=> []
=> ? = 0
[3,2,-4,-1] => [-1,3,2,-4] => [2]
=> []
=> ? = 0
[-3,2,-4,-1] => [-1,-3,-2,-4] => [2]
=> []
=> ? = 0
[-3,-2,4,1] => [-2,-1,-3,-4] => [2]
=> []
=> ? = 0
[-3,-2,-4,1] => [-2,-1,-3,-4] => [2]
=> []
=> ? = 0
[-3,-4,1,-2] => [-3,-2,-1,-4] => [2]
=> []
=> ? = 0
[-3,4,-2,1] => [-2,-1,-3,-4] => [2]
=> []
=> ? = 0
[-3,-4,2,-1] => [-1,-3,-2,-4] => [2]
=> []
=> ? = 0
Description
The product of the hook lengths of the diagonal cells in an integer partition.
For a cell in the Ferrers diagram of a partition, the hook length is given by the number of boxes to its right plus the number of boxes below + 1. This statistic is the product of the hook lengths of the diagonal cells $(i,i)$ of a partition.
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!