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Your data matches 27 different statistics following compositions of up to 3 maps.
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Matching statistic: St000013
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000013: 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
St000013: 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]
=> 2
[2,1] => [2,1] => [2]
=> [1,0,1,0]
=> 1
[-2,1] => [-2,-1] => [2]
=> [1,0,1,0]
=> 1
[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]
=> 2
[1,3,2] => [1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,3,-2] => [1,-2,3] => [1,1]
=> [1,1,0,0]
=> 2
[1,-3,2] => [1,-3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,1,3] => [2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,1,-3] => [2,1,-3] => [2]
=> [1,0,1,0]
=> 1
[-2,1,3] => [-2,-1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[-2,1,-3] => [-2,-1,-3] => [2]
=> [1,0,1,0]
=> 1
[2,3,1] => [3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,3,-1] => [-1,2,3] => [1,1]
=> [1,1,0,0]
=> 2
[2,-3,1] => [-3,2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[-2,3,1] => [-2,-1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[-2,-3,1] => [-2,-1,-3] => [2]
=> [1,0,1,0]
=> 1
[3,1,2] => [3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[3,1,-2] => [3,-2,1] => [2]
=> [1,0,1,0]
=> 1
[-3,1,2] => [-3,2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[-3,1,-2] => [-3,-2,-1] => [2]
=> [1,0,1,0]
=> 1
[3,2,1] => [3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[3,2,-1] => [-1,3,2] => [2]
=> [1,0,1,0]
=> 1
[3,-2,1] => [-2,-1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[-3,2,1] => [-3,2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[-3,2,-1] => [-1,-3,-2] => [2]
=> [1,0,1,0]
=> 1
[-3,-2,1] => [-2,-1,-3] => [2]
=> [1,0,1,0]
=> 1
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2
[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]
=> 2
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> 2
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> 2
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
=> [1,1,0,0]
=> 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]
=> [1,1,0,1,0,0]
=> 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]
=> 2
[1,-2,4,3] => [1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> 2
[-1,2,4,3] => [-1,-2,4,3] => [2]
=> [1,0,1,0]
=> 1
[-1,2,-4,3] => [-1,-2,-4,-3] => [2]
=> [1,0,1,0]
=> 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]
=> 2
[1,3,-2,4] => [1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> 2
[1,3,-2,-4] => [1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> 2
[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]
=> 2
[-1,3,2,4] => [-1,-2,3,4] => [1,1]
=> [1,1,0,0]
=> 2
[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 height of a Dyck path.
The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.
Matching statistic: St000443
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000443: 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
St000443: 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]
=> 2
[2,1] => [2,1] => [2]
=> [1,0,1,0]
=> 1
[-2,1] => [-2,-1] => [2]
=> [1,0,1,0]
=> 1
[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]
=> 2
[1,3,2] => [1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,3,-2] => [1,-2,3] => [1,1]
=> [1,1,0,0]
=> 2
[1,-3,2] => [1,-3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,1,3] => [2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,1,-3] => [2,1,-3] => [2]
=> [1,0,1,0]
=> 1
[-2,1,3] => [-2,-1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[-2,1,-3] => [-2,-1,-3] => [2]
=> [1,0,1,0]
=> 1
[2,3,1] => [3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,3,-1] => [-1,2,3] => [1,1]
=> [1,1,0,0]
=> 2
[2,-3,1] => [-3,2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[-2,3,1] => [-2,-1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[-2,-3,1] => [-2,-1,-3] => [2]
=> [1,0,1,0]
=> 1
[3,1,2] => [3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[3,1,-2] => [3,-2,1] => [2]
=> [1,0,1,0]
=> 1
[-3,1,2] => [-3,2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[-3,1,-2] => [-3,-2,-1] => [2]
=> [1,0,1,0]
=> 1
[3,2,1] => [3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[3,2,-1] => [-1,3,2] => [2]
=> [1,0,1,0]
=> 1
[3,-2,1] => [-2,-1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[-3,2,1] => [-3,2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[-3,2,-1] => [-1,-3,-2] => [2]
=> [1,0,1,0]
=> 1
[-3,-2,1] => [-2,-1,-3] => [2]
=> [1,0,1,0]
=> 1
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2
[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]
=> 2
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> 2
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> 2
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
=> [1,1,0,0]
=> 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]
=> [1,1,0,1,0,0]
=> 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]
=> 2
[1,-2,4,3] => [1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> 2
[-1,2,4,3] => [-1,-2,4,3] => [2]
=> [1,0,1,0]
=> 1
[-1,2,-4,3] => [-1,-2,-4,-3] => [2]
=> [1,0,1,0]
=> 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]
=> 2
[1,3,-2,4] => [1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> 2
[1,3,-2,-4] => [1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> 2
[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]
=> 2
[-1,3,2,4] => [-1,-2,3,4] => [1,1]
=> [1,1,0,0]
=> 2
[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 long tunnels of a Dyck path.
A long tunnel of a Dyck path is a longest sequence of consecutive usual tunnels, i.e., a longest sequence of tunnels where the end point of one is the starting point of the next. See [1] for the definition of tunnels.
Matching statistic: St000444
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000444: 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
St000444: 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]
=> 2
[2,1] => [2,1] => [2]
=> [1,0,1,0]
=> 1
[-2,1] => [-2,-1] => [2]
=> [1,0,1,0]
=> 1
[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]
=> 2
[1,3,2] => [1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,3,-2] => [1,-2,3] => [1,1]
=> [1,1,0,0]
=> 2
[1,-3,2] => [1,-3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,1,3] => [2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,1,-3] => [2,1,-3] => [2]
=> [1,0,1,0]
=> 1
[-2,1,3] => [-2,-1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[-2,1,-3] => [-2,-1,-3] => [2]
=> [1,0,1,0]
=> 1
[2,3,1] => [3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,3,-1] => [-1,2,3] => [1,1]
=> [1,1,0,0]
=> 2
[2,-3,1] => [-3,2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[-2,3,1] => [-2,-1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[-2,-3,1] => [-2,-1,-3] => [2]
=> [1,0,1,0]
=> 1
[3,1,2] => [3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[3,1,-2] => [3,-2,1] => [2]
=> [1,0,1,0]
=> 1
[-3,1,2] => [-3,2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[-3,1,-2] => [-3,-2,-1] => [2]
=> [1,0,1,0]
=> 1
[3,2,1] => [3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[3,2,-1] => [-1,3,2] => [2]
=> [1,0,1,0]
=> 1
[3,-2,1] => [-2,-1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[-3,2,1] => [-3,2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[-3,2,-1] => [-1,-3,-2] => [2]
=> [1,0,1,0]
=> 1
[-3,-2,1] => [-2,-1,-3] => [2]
=> [1,0,1,0]
=> 1
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2
[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]
=> 2
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> 2
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> 2
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
=> [1,1,0,0]
=> 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]
=> [1,1,0,1,0,0]
=> 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]
=> 2
[1,-2,4,3] => [1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> 2
[-1,2,4,3] => [-1,-2,4,3] => [2]
=> [1,0,1,0]
=> 1
[-1,2,-4,3] => [-1,-2,-4,-3] => [2]
=> [1,0,1,0]
=> 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]
=> 2
[1,3,-2,4] => [1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> 2
[1,3,-2,-4] => [1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> 2
[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]
=> 2
[-1,3,2,4] => [-1,-2,3,4] => [1,1]
=> [1,1,0,0]
=> 2
[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 length of the maximal rise of a Dyck path.
Matching statistic: St001007
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001007: 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
St001007: 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]
=> 2
[2,1] => [2,1] => [2]
=> [1,0,1,0]
=> 1
[-2,1] => [-2,-1] => [2]
=> [1,0,1,0]
=> 1
[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]
=> 2
[1,3,2] => [1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,3,-2] => [1,-2,3] => [1,1]
=> [1,1,0,0]
=> 2
[1,-3,2] => [1,-3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,1,3] => [2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,1,-3] => [2,1,-3] => [2]
=> [1,0,1,0]
=> 1
[-2,1,3] => [-2,-1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[-2,1,-3] => [-2,-1,-3] => [2]
=> [1,0,1,0]
=> 1
[2,3,1] => [3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,3,-1] => [-1,2,3] => [1,1]
=> [1,1,0,0]
=> 2
[2,-3,1] => [-3,2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[-2,3,1] => [-2,-1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[-2,-3,1] => [-2,-1,-3] => [2]
=> [1,0,1,0]
=> 1
[3,1,2] => [3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[3,1,-2] => [3,-2,1] => [2]
=> [1,0,1,0]
=> 1
[-3,1,2] => [-3,2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[-3,1,-2] => [-3,-2,-1] => [2]
=> [1,0,1,0]
=> 1
[3,2,1] => [3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[3,2,-1] => [-1,3,2] => [2]
=> [1,0,1,0]
=> 1
[3,-2,1] => [-2,-1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[-3,2,1] => [-3,2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[-3,2,-1] => [-1,-3,-2] => [2]
=> [1,0,1,0]
=> 1
[-3,-2,1] => [-2,-1,-3] => [2]
=> [1,0,1,0]
=> 1
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2
[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]
=> 2
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> 2
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> 2
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
=> [1,1,0,0]
=> 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]
=> [1,1,0,1,0,0]
=> 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]
=> 2
[1,-2,4,3] => [1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> 2
[-1,2,4,3] => [-1,-2,4,3] => [2]
=> [1,0,1,0]
=> 1
[-1,2,-4,3] => [-1,-2,-4,-3] => [2]
=> [1,0,1,0]
=> 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]
=> 2
[1,3,-2,4] => [1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> 2
[1,3,-2,-4] => [1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> 2
[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]
=> 2
[-1,3,2,4] => [-1,-2,3,4] => [1,1]
=> [1,1,0,0]
=> 2
[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
Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001187
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001187: 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
St001187: 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]
=> 2
[2,1] => [2,1] => [2]
=> [1,0,1,0]
=> 1
[-2,1] => [-2,-1] => [2]
=> [1,0,1,0]
=> 1
[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]
=> 2
[1,3,2] => [1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,3,-2] => [1,-2,3] => [1,1]
=> [1,1,0,0]
=> 2
[1,-3,2] => [1,-3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,1,3] => [2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,1,-3] => [2,1,-3] => [2]
=> [1,0,1,0]
=> 1
[-2,1,3] => [-2,-1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[-2,1,-3] => [-2,-1,-3] => [2]
=> [1,0,1,0]
=> 1
[2,3,1] => [3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,3,-1] => [-1,2,3] => [1,1]
=> [1,1,0,0]
=> 2
[2,-3,1] => [-3,2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[-2,3,1] => [-2,-1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[-2,-3,1] => [-2,-1,-3] => [2]
=> [1,0,1,0]
=> 1
[3,1,2] => [3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[3,1,-2] => [3,-2,1] => [2]
=> [1,0,1,0]
=> 1
[-3,1,2] => [-3,2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[-3,1,-2] => [-3,-2,-1] => [2]
=> [1,0,1,0]
=> 1
[3,2,1] => [3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[3,2,-1] => [-1,3,2] => [2]
=> [1,0,1,0]
=> 1
[3,-2,1] => [-2,-1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[-3,2,1] => [-3,2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[-3,2,-1] => [-1,-3,-2] => [2]
=> [1,0,1,0]
=> 1
[-3,-2,1] => [-2,-1,-3] => [2]
=> [1,0,1,0]
=> 1
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2
[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]
=> 2
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> 2
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> 2
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
=> [1,1,0,0]
=> 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]
=> [1,1,0,1,0,0]
=> 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]
=> 2
[1,-2,4,3] => [1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> 2
[-1,2,4,3] => [-1,-2,4,3] => [2]
=> [1,0,1,0]
=> 1
[-1,2,-4,3] => [-1,-2,-4,-3] => [2]
=> [1,0,1,0]
=> 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]
=> 2
[1,3,-2,4] => [1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> 2
[1,3,-2,-4] => [1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> 2
[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]
=> 2
[-1,3,2,4] => [-1,-2,3,4] => [1,1]
=> [1,1,0,0]
=> 2
[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 modules with grade at least one in the corresponding Nakayama algebra.
Matching statistic: St001224
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001224: 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
St001224: 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]
=> 2
[2,1] => [2,1] => [2]
=> [1,0,1,0]
=> 1
[-2,1] => [-2,-1] => [2]
=> [1,0,1,0]
=> 1
[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]
=> 2
[1,3,2] => [1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,3,-2] => [1,-2,3] => [1,1]
=> [1,1,0,0]
=> 2
[1,-3,2] => [1,-3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,1,3] => [2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,1,-3] => [2,1,-3] => [2]
=> [1,0,1,0]
=> 1
[-2,1,3] => [-2,-1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[-2,1,-3] => [-2,-1,-3] => [2]
=> [1,0,1,0]
=> 1
[2,3,1] => [3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,3,-1] => [-1,2,3] => [1,1]
=> [1,1,0,0]
=> 2
[2,-3,1] => [-3,2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[-2,3,1] => [-2,-1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[-2,-3,1] => [-2,-1,-3] => [2]
=> [1,0,1,0]
=> 1
[3,1,2] => [3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[3,1,-2] => [3,-2,1] => [2]
=> [1,0,1,0]
=> 1
[-3,1,2] => [-3,2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[-3,1,-2] => [-3,-2,-1] => [2]
=> [1,0,1,0]
=> 1
[3,2,1] => [3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[3,2,-1] => [-1,3,2] => [2]
=> [1,0,1,0]
=> 1
[3,-2,1] => [-2,-1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[-3,2,1] => [-3,2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[-3,2,-1] => [-1,-3,-2] => [2]
=> [1,0,1,0]
=> 1
[-3,-2,1] => [-2,-1,-3] => [2]
=> [1,0,1,0]
=> 1
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2
[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]
=> 2
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> 2
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> 2
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
=> [1,1,0,0]
=> 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]
=> [1,1,0,1,0,0]
=> 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]
=> 2
[1,-2,4,3] => [1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> 2
[-1,2,4,3] => [-1,-2,4,3] => [2]
=> [1,0,1,0]
=> 1
[-1,2,-4,3] => [-1,-2,-4,-3] => [2]
=> [1,0,1,0]
=> 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]
=> 2
[1,3,-2,4] => [1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> 2
[1,3,-2,-4] => [1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> 2
[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]
=> 2
[-1,3,2,4] => [-1,-2,3,4] => [1,1]
=> [1,1,0,0]
=> 2
[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
Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. Then the statistic gives the vector space dimension of the first Ext-group between X and the regular module.
Matching statistic: St000024
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000024: 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
St000024: 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] => [2]
=> [1,0,1,0]
=> 0 = 1 - 1
[-2,1] => [-2,-1] => [2]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,2,3] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,2,-3] => [1,2,-3] => [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,3,2] => [1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,3,-2] => [1,-2,3] => [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,-3,2] => [1,-3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[2,1,3] => [2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[2,1,-3] => [2,1,-3] => [2]
=> [1,0,1,0]
=> 0 = 1 - 1
[-2,1,3] => [-2,-1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[-2,1,-3] => [-2,-1,-3] => [2]
=> [1,0,1,0]
=> 0 = 1 - 1
[2,3,1] => [3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[2,3,-1] => [-1,2,3] => [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
[2,-3,1] => [-3,2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[-2,3,1] => [-2,-1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[-2,-3,1] => [-2,-1,-3] => [2]
=> [1,0,1,0]
=> 0 = 1 - 1
[3,1,2] => [3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[3,1,-2] => [3,-2,1] => [2]
=> [1,0,1,0]
=> 0 = 1 - 1
[-3,1,2] => [-3,2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[-3,1,-2] => [-3,-2,-1] => [2]
=> [1,0,1,0]
=> 0 = 1 - 1
[3,2,1] => [3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[3,2,-1] => [-1,3,2] => [2]
=> [1,0,1,0]
=> 0 = 1 - 1
[3,-2,1] => [-2,-1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[-3,2,1] => [-3,2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[-3,2,-1] => [-1,-3,-2] => [2]
=> [1,0,1,0]
=> 0 = 1 - 1
[-3,-2,1] => [-2,-1,-3] => [2]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,2,4,3] => [1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,2,4,-3] => [1,2,-3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,2,-4,3] => [1,2,-4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,2,-4,-3] => [1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,-2,4,3] => [1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
[-1,2,4,3] => [-1,-2,4,3] => [2]
=> [1,0,1,0]
=> 0 = 1 - 1
[-1,2,-4,3] => [-1,-2,-4,-3] => [2]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,3,2,4] => [1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,3,2,-4] => [1,3,2,-4] => [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,3,-2,4] => [1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,3,-2,-4] => [1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,-3,2,4] => [1,-3,-2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,-3,2,-4] => [1,-3,-2,-4] => [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[-1,3,2,4] => [-1,-2,3,4] => [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,3,4,2] => [1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,3,4,-2] => [1,-2,3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,3,-4,2] => [1,-4,3,-2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
Description
The number of double up and double down steps of a Dyck path.
In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.
Matching statistic: St000147
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
St000147: 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
St000147: 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] => [2,1] => [2]
=> []
=> 0 = 1 - 1
[-2,1] => [-2,-1] => [2]
=> []
=> 0 = 1 - 1
[1,2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[1,2,-3] => [1,2,-3] => [1,1]
=> [1]
=> 1 = 2 - 1
[1,3,2] => [1,3,2] => [2,1]
=> [1]
=> 1 = 2 - 1
[1,3,-2] => [1,-2,3] => [1,1]
=> [1]
=> 1 = 2 - 1
[1,-3,2] => [1,-3,-2] => [2,1]
=> [1]
=> 1 = 2 - 1
[2,1,3] => [2,1,3] => [2,1]
=> [1]
=> 1 = 2 - 1
[2,1,-3] => [2,1,-3] => [2]
=> []
=> 0 = 1 - 1
[-2,1,3] => [-2,-1,3] => [2,1]
=> [1]
=> 1 = 2 - 1
[-2,1,-3] => [-2,-1,-3] => [2]
=> []
=> 0 = 1 - 1
[2,3,1] => [3,2,1] => [2,1]
=> [1]
=> 1 = 2 - 1
[2,3,-1] => [-1,2,3] => [1,1]
=> [1]
=> 1 = 2 - 1
[2,-3,1] => [-3,2,-1] => [2,1]
=> [1]
=> 1 = 2 - 1
[-2,3,1] => [-2,-1,3] => [2,1]
=> [1]
=> 1 = 2 - 1
[-2,-3,1] => [-2,-1,-3] => [2]
=> []
=> 0 = 1 - 1
[3,1,2] => [3,2,1] => [2,1]
=> [1]
=> 1 = 2 - 1
[3,1,-2] => [3,-2,1] => [2]
=> []
=> 0 = 1 - 1
[-3,1,2] => [-3,2,-1] => [2,1]
=> [1]
=> 1 = 2 - 1
[-3,1,-2] => [-3,-2,-1] => [2]
=> []
=> 0 = 1 - 1
[3,2,1] => [3,2,1] => [2,1]
=> [1]
=> 1 = 2 - 1
[3,2,-1] => [-1,3,2] => [2]
=> []
=> 0 = 1 - 1
[3,-2,1] => [-2,-1,3] => [2,1]
=> [1]
=> 1 = 2 - 1
[-3,2,1] => [-3,2,-1] => [2,1]
=> [1]
=> 1 = 2 - 1
[-3,2,-1] => [-1,-3,-2] => [2]
=> []
=> 0 = 1 - 1
[-3,-2,1] => [-2,-1,-3] => [2]
=> []
=> 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
=> [1]
=> 1 = 2 - 1
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> [1]
=> 1 = 2 - 1
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
=> [1]
=> 1 = 2 - 1
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
=> [1]
=> 1 = 2 - 1
[1,2,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 1 = 2 - 1
[1,2,4,-3] => [1,2,-3,4] => [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[1,2,-4,3] => [1,2,-4,-3] => [2,1,1]
=> [1,1]
=> 1 = 2 - 1
[1,2,-4,-3] => [1,2,-3,-4] => [1,1]
=> [1]
=> 1 = 2 - 1
[1,-2,4,3] => [1,-2,-3,4] => [1,1]
=> [1]
=> 1 = 2 - 1
[-1,2,4,3] => [-1,-2,4,3] => [2]
=> []
=> 0 = 1 - 1
[-1,2,-4,3] => [-1,-2,-4,-3] => [2]
=> []
=> 0 = 1 - 1
[1,3,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 1 = 2 - 1
[1,3,2,-4] => [1,3,2,-4] => [2,1]
=> [1]
=> 1 = 2 - 1
[1,3,-2,4] => [1,-2,3,-4] => [1,1]
=> [1]
=> 1 = 2 - 1
[1,3,-2,-4] => [1,-2,3,-4] => [1,1]
=> [1]
=> 1 = 2 - 1
[1,-3,2,4] => [1,-3,-2,4] => [2,1,1]
=> [1,1]
=> 1 = 2 - 1
[1,-3,2,-4] => [1,-3,-2,-4] => [2,1]
=> [1]
=> 1 = 2 - 1
[-1,3,2,4] => [-1,-2,3,4] => [1,1]
=> [1]
=> 1 = 2 - 1
[1,3,4,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1 = 2 - 1
[1,3,4,-2] => [1,-2,3,4] => [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[1,3,-4,2] => [1,-4,3,-2] => [2,1,1]
=> [1,1]
=> 1 = 2 - 1
Description
The largest part of an integer partition.
Matching statistic: St000442
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000442: 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
St000442: 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] => [2]
=> [1,0,1,0]
=> 0 = 1 - 1
[-2,1] => [-2,-1] => [2]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,2,3] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,2,-3] => [1,2,-3] => [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,3,2] => [1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,3,-2] => [1,-2,3] => [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,-3,2] => [1,-3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[2,1,3] => [2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[2,1,-3] => [2,1,-3] => [2]
=> [1,0,1,0]
=> 0 = 1 - 1
[-2,1,3] => [-2,-1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[-2,1,-3] => [-2,-1,-3] => [2]
=> [1,0,1,0]
=> 0 = 1 - 1
[2,3,1] => [3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[2,3,-1] => [-1,2,3] => [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
[2,-3,1] => [-3,2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[-2,3,1] => [-2,-1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[-2,-3,1] => [-2,-1,-3] => [2]
=> [1,0,1,0]
=> 0 = 1 - 1
[3,1,2] => [3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[3,1,-2] => [3,-2,1] => [2]
=> [1,0,1,0]
=> 0 = 1 - 1
[-3,1,2] => [-3,2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[-3,1,-2] => [-3,-2,-1] => [2]
=> [1,0,1,0]
=> 0 = 1 - 1
[3,2,1] => [3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[3,2,-1] => [-1,3,2] => [2]
=> [1,0,1,0]
=> 0 = 1 - 1
[3,-2,1] => [-2,-1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[-3,2,1] => [-3,2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[-3,2,-1] => [-1,-3,-2] => [2]
=> [1,0,1,0]
=> 0 = 1 - 1
[-3,-2,1] => [-2,-1,-3] => [2]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,2,4,3] => [1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,2,4,-3] => [1,2,-3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,2,-4,3] => [1,2,-4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,2,-4,-3] => [1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,-2,4,3] => [1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
[-1,2,4,3] => [-1,-2,4,3] => [2]
=> [1,0,1,0]
=> 0 = 1 - 1
[-1,2,-4,3] => [-1,-2,-4,-3] => [2]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,3,2,4] => [1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,3,2,-4] => [1,3,2,-4] => [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,3,-2,4] => [1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,3,-2,-4] => [1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,-3,2,4] => [1,-3,-2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,-3,2,-4] => [1,-3,-2,-4] => [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[-1,3,2,4] => [-1,-2,3,4] => [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,3,4,2] => [1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,3,4,-2] => [1,-2,3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,3,-4,2] => [1,-4,3,-2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
Description
The maximal area to the right of an up step of a Dyck path.
Matching statistic: St000473
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000473: 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
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000473: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,1]
=> [2]
=> 1 = 2 - 1
[2,1] => [2,1] => [2]
=> [1,1]
=> 0 = 1 - 1
[-2,1] => [-2,-1] => [2]
=> [1,1]
=> 0 = 1 - 1
[1,2,3] => [1,2,3] => [1,1,1]
=> [3]
=> 1 = 2 - 1
[1,2,-3] => [1,2,-3] => [1,1]
=> [2]
=> 1 = 2 - 1
[1,3,2] => [1,3,2] => [2,1]
=> [2,1]
=> 1 = 2 - 1
[1,3,-2] => [1,-2,3] => [1,1]
=> [2]
=> 1 = 2 - 1
[1,-3,2] => [1,-3,-2] => [2,1]
=> [2,1]
=> 1 = 2 - 1
[2,1,3] => [2,1,3] => [2,1]
=> [2,1]
=> 1 = 2 - 1
[2,1,-3] => [2,1,-3] => [2]
=> [1,1]
=> 0 = 1 - 1
[-2,1,3] => [-2,-1,3] => [2,1]
=> [2,1]
=> 1 = 2 - 1
[-2,1,-3] => [-2,-1,-3] => [2]
=> [1,1]
=> 0 = 1 - 1
[2,3,1] => [3,2,1] => [2,1]
=> [2,1]
=> 1 = 2 - 1
[2,3,-1] => [-1,2,3] => [1,1]
=> [2]
=> 1 = 2 - 1
[2,-3,1] => [-3,2,-1] => [2,1]
=> [2,1]
=> 1 = 2 - 1
[-2,3,1] => [-2,-1,3] => [2,1]
=> [2,1]
=> 1 = 2 - 1
[-2,-3,1] => [-2,-1,-3] => [2]
=> [1,1]
=> 0 = 1 - 1
[3,1,2] => [3,2,1] => [2,1]
=> [2,1]
=> 1 = 2 - 1
[3,1,-2] => [3,-2,1] => [2]
=> [1,1]
=> 0 = 1 - 1
[-3,1,2] => [-3,2,-1] => [2,1]
=> [2,1]
=> 1 = 2 - 1
[-3,1,-2] => [-3,-2,-1] => [2]
=> [1,1]
=> 0 = 1 - 1
[3,2,1] => [3,2,1] => [2,1]
=> [2,1]
=> 1 = 2 - 1
[3,2,-1] => [-1,3,2] => [2]
=> [1,1]
=> 0 = 1 - 1
[3,-2,1] => [-2,-1,3] => [2,1]
=> [2,1]
=> 1 = 2 - 1
[-3,2,1] => [-3,2,-1] => [2,1]
=> [2,1]
=> 1 = 2 - 1
[-3,2,-1] => [-1,-3,-2] => [2]
=> [1,1]
=> 0 = 1 - 1
[-3,-2,1] => [-2,-1,-3] => [2]
=> [1,1]
=> 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [4]
=> 1 = 2 - 1
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
=> [3]
=> 1 = 2 - 1
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
=> [2]
=> 1 = 2 - 1
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> [2]
=> 1 = 2 - 1
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
=> [2]
=> 1 = 2 - 1
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
=> [2]
=> 1 = 2 - 1
[1,2,4,3] => [1,2,4,3] => [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[1,2,4,-3] => [1,2,-3,4] => [1,1,1]
=> [3]
=> 1 = 2 - 1
[1,2,-4,3] => [1,2,-4,-3] => [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[1,2,-4,-3] => [1,2,-3,-4] => [1,1]
=> [2]
=> 1 = 2 - 1
[1,-2,4,3] => [1,-2,-3,4] => [1,1]
=> [2]
=> 1 = 2 - 1
[-1,2,4,3] => [-1,-2,4,3] => [2]
=> [1,1]
=> 0 = 1 - 1
[-1,2,-4,3] => [-1,-2,-4,-3] => [2]
=> [1,1]
=> 0 = 1 - 1
[1,3,2,4] => [1,3,2,4] => [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[1,3,2,-4] => [1,3,2,-4] => [2,1]
=> [2,1]
=> 1 = 2 - 1
[1,3,-2,4] => [1,-2,3,-4] => [1,1]
=> [2]
=> 1 = 2 - 1
[1,3,-2,-4] => [1,-2,3,-4] => [1,1]
=> [2]
=> 1 = 2 - 1
[1,-3,2,4] => [1,-3,-2,4] => [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[1,-3,2,-4] => [1,-3,-2,-4] => [2,1]
=> [2,1]
=> 1 = 2 - 1
[-1,3,2,4] => [-1,-2,3,4] => [1,1]
=> [2]
=> 1 = 2 - 1
[1,3,4,2] => [1,4,3,2] => [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[1,3,4,-2] => [1,-2,3,4] => [1,1,1]
=> [3]
=> 1 = 2 - 1
[1,3,-4,2] => [1,-4,3,-2] => [2,1,1]
=> [3,1]
=> 1 = 2 - 1
Description
The number of parts of a partition that are strictly bigger than the number of ones.
This is part of the definition of Dyson's crank of a partition, see [[St000474]].
The following 17 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001280The number of parts of an integer partition that are at least two. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001389The number of partitions of the same length below the given integer partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000933The number of multipartitions of sizes given by an integer 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.
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