Your data matches 8 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000932
(load all 5 compositions to match this statistic)
Mapping
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000932: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1,-2] => [1,1]
 => [1,1,0,0]
 => 0
[2,-1] => [2]
 => [1,0,1,0]
 => 1
[-2,1] => [2]
 => [1,0,1,0]
 => 1
[1,-2,-3] => [1,1]
 => [1,1,0,0]
 => 0
[-1,2,-3] => [1,1]
 => [1,1,0,0]
 => 0
[-1,-2,3] => [1,1]
 => [1,1,0,0]
 => 0
[-1,-2,-3] => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[1,3,-2] => [2]
 => [1,0,1,0]
 => 1
[1,-3,2] => [2]
 => [1,0,1,0]
 => 1
[-1,3,-2] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[-1,-3,2] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[2,-1,3] => [2]
 => [1,0,1,0]
 => 1
[2,-1,-3] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[-2,1,3] => [2]
 => [1,0,1,0]
 => 1
[-2,1,-3] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[2,3,-1] => [3]
 => [1,0,1,0,1,0]
 => 2
[2,-3,1] => [3]
 => [1,0,1,0,1,0]
 => 2
[-2,3,1] => [3]
 => [1,0,1,0,1,0]
 => 2
[-2,-3,-1] => [3]
 => [1,0,1,0,1,0]
 => 2
[3,1,-2] => [3]
 => [1,0,1,0,1,0]
 => 2
[3,-1,2] => [3]
 => [1,0,1,0,1,0]
 => 2
[-3,1,2] => [3]
 => [1,0,1,0,1,0]
 => 2
[-3,-1,-2] => [3]
 => [1,0,1,0,1,0]
 => 2
[3,2,-1] => [2]
 => [1,0,1,0]
 => 1
[3,-2,-1] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[-3,2,1] => [2]
 => [1,0,1,0]
 => 1
[-3,-2,1] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[1,2,-3,-4] => [1,1]
 => [1,1,0,0]
 => 0
[1,-2,3,-4] => [1,1]
 => [1,1,0,0]
 => 0
[1,-2,-3,4] => [1,1]
 => [1,1,0,0]
 => 0
[1,-2,-3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[-1,2,3,-4] => [1,1]
 => [1,1,0,0]
 => 0
[-1,2,-3,4] => [1,1]
 => [1,1,0,0]
 => 0
[-1,2,-3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[-1,-2,3,4] => [1,1]
 => [1,1,0,0]
 => 0
[-1,-2,3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[-1,-2,-3,4] => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 2
[1,2,4,-3] => [2]
 => [1,0,1,0]
 => 1
[1,2,-4,3] => [2]
 => [1,0,1,0]
 => 1
[1,-2,4,-3] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[1,-2,-4,3] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[-1,2,4,-3] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[-1,2,-4,3] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[-1,-2,4,3] => [1,1]
 => [1,1,0,0]
 => 0
[-1,-2,4,-3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
[-1,-2,-4,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
[-1,-2,-4,-3] => [1,1]
 => [1,1,0,0]
 => 0
[1,3,-2,4] => [2]
 => [1,0,1,0]
 => 1
[1,3,-2,-4] => [2,1]
 => [1,0,1,1,0,0]
 => 1
Description
The number of occurrences of the pattern UDU in a Dyck path. The number of Dyck paths with statistic value 0 are counted by the Motzkin numbers [1].
Matching statistic: St001067
(load all 5 compositions to match this statistic)
Mapping
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001067: Dyck paths ⟶ ℤResult quality: 88% ●values known / values provided: 97%●distinct values known / distinct values provided: 88%
Values
[-1,-2] => [1,1]
 => [1,1,0,0]
 => 0
[2,-1] => [2]
 => [1,0,1,0]
 => 1
[-2,1] => [2]
 => [1,0,1,0]
 => 1
[1,-2,-3] => [1,1]
 => [1,1,0,0]
 => 0
[-1,2,-3] => [1,1]
 => [1,1,0,0]
 => 0
[-1,-2,3] => [1,1]
 => [1,1,0,0]
 => 0
[-1,-2,-3] => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[1,3,-2] => [2]
 => [1,0,1,0]
 => 1
[1,-3,2] => [2]
 => [1,0,1,0]
 => 1
[-1,3,-2] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[-1,-3,2] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[2,-1,3] => [2]
 => [1,0,1,0]
 => 1
[2,-1,-3] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[-2,1,3] => [2]
 => [1,0,1,0]
 => 1
[-2,1,-3] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[2,3,-1] => [3]
 => [1,0,1,0,1,0]
 => 2
[2,-3,1] => [3]
 => [1,0,1,0,1,0]
 => 2
[-2,3,1] => [3]
 => [1,0,1,0,1,0]
 => 2
[-2,-3,-1] => [3]
 => [1,0,1,0,1,0]
 => 2
[3,1,-2] => [3]
 => [1,0,1,0,1,0]
 => 2
[3,-1,2] => [3]
 => [1,0,1,0,1,0]
 => 2
[-3,1,2] => [3]
 => [1,0,1,0,1,0]
 => 2
[-3,-1,-2] => [3]
 => [1,0,1,0,1,0]
 => 2
[3,2,-1] => [2]
 => [1,0,1,0]
 => 1
[3,-2,-1] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[-3,2,1] => [2]
 => [1,0,1,0]
 => 1
[-3,-2,1] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[1,2,-3,-4] => [1,1]
 => [1,1,0,0]
 => 0
[1,-2,3,-4] => [1,1]
 => [1,1,0,0]
 => 0
[1,-2,-3,4] => [1,1]
 => [1,1,0,0]
 => 0
[1,-2,-3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[-1,2,3,-4] => [1,1]
 => [1,1,0,0]
 => 0
[-1,2,-3,4] => [1,1]
 => [1,1,0,0]
 => 0
[-1,2,-3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[-1,-2,3,4] => [1,1]
 => [1,1,0,0]
 => 0
[-1,-2,3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[-1,-2,-3,4] => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 2
[1,2,4,-3] => [2]
 => [1,0,1,0]
 => 1
[1,2,-4,3] => [2]
 => [1,0,1,0]
 => 1
[1,-2,4,-3] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[1,-2,-4,3] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[-1,2,4,-3] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[-1,2,-4,3] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[-1,-2,4,3] => [1,1]
 => [1,1,0,0]
 => 0
[-1,-2,4,-3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
[-1,-2,-4,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
[-1,-2,-4,-3] => [1,1]
 => [1,1,0,0]
 => 0
[1,3,-2,4] => [2]
 => [1,0,1,0]
 => 1
[1,3,-2,-4] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[4,5,2,3,-8,-6,1,7] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 6
[2,-8,-4,-6,-5,1,3,7] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 6
[-4,3,-5,2,-8,-6,1,7] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 6
[-4,3,5,8,7,-6,1,2] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 6
[3,-7,2,8,-5,1,4,6] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 6
[-7,6,-8,-4,-2,1,3,5] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 6
[5,-6,-7,-4,2,3,-8,1] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 6
[-8,-6,5,-4,2,-7,1,3] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 6
[-7,6,1,-8,-5,-4,2,3] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 6
[4,-8,-5,-7,1,-6,2,3] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 6
[3,1,5,2,8,4,6,-7] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[4,5,1,2,8,3,6,-7] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[7,4,1,8,2,3,5,-6] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[8,1,5,2,7,4,6,-3] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[8,5,1,2,7,3,6,-4] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[8,4,1,6,2,3,5,-7] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[8,6,7,1,2,3,4,-5] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[8,6,1,7,2,3,5,-4] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[8,1,6,7,2,4,5,-3] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[5,6,8,1,2,3,4,-7] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[6,7,8,1,2,3,4,-5] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[6,1,7,8,2,4,5,-3] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[5,1,7,2,8,4,6,-3] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[5,7,1,2,8,3,6,-4] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[4,6,1,8,2,3,5,-7] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[3,1,6,8,2,4,5,-7] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[4,1,2,7,8,5,6,-3] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[3,1,8,2,7,4,6,-5] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[3,1,8,7,2,4,5,-6] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[4,8,1,7,2,3,5,-6] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[4,8,1,2,7,3,6,-5] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[5,8,7,1,2,3,4,-6] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[6,8,5,1,2,3,4,-7] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[7,8,1,6,2,3,5,-4] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[8,7,5,1,2,3,4,-6] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[4,5,2,3,8,1,6,-7] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[7,5,1,3,8,2,6,-4] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[8,5,6,3,4,1,2,-7] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[8,5,7,3,1,4,2,-6] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[8,6,7,1,3,4,2,-5] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[8,6,1,7,3,5,2,-4] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[8,1,6,7,4,5,2,-3] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[7,1,6,8,4,2,5,-3] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[7,5,8,3,1,2,4,-6] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[5,6,8,2,3,1,4,-7] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[4,7,2,8,1,3,5,-6] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[4,6,2,8,3,1,5,-7] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[4,8,2,7,1,5,3,-6] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[6,8,1,7,2,5,3,-4] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[5,8,6,2,4,1,3,-7] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
Description
The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra.
Matching statistic: St001189
(load all 2 compositions to match this statistic)
Mapping
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001189: Dyck paths ⟶ ℤResult quality: 88% ●values known / values provided: 97%●distinct values known / distinct values provided: 88%
Values
[-1,-2] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
[2,-1] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
[-2,1] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
[1,-2,-3] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
[-1,2,-3] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
[-1,-2,3] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
[-1,-2,-3] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[1,3,-2] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
[1,-3,2] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
[-1,3,-2] => [2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 1
[-1,-3,2] => [2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 1
[2,-1,3] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
[2,-1,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 1
[-2,1,3] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
[-2,1,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 1
[2,3,-1] => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 2
[2,-3,1] => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 2
[-2,3,1] => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 2
[-2,-3,-1] => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 2
[3,1,-2] => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 2
[3,-1,2] => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 2
[-3,1,2] => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 2
[-3,-1,-2] => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 2
[3,2,-1] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
[3,-2,-1] => [2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 1
[-3,2,1] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
[-3,-2,1] => [2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 1
[1,2,-3,-4] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
[1,-2,3,-4] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
[1,-2,-3,4] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
[1,-2,-3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[-1,2,3,-4] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
[-1,2,-3,4] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
[-1,2,-3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[-1,-2,3,4] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
[-1,-2,3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[-1,-2,-3,4] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2
[1,2,4,-3] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
[1,2,-4,3] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
[1,-2,4,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 1
[1,-2,-4,3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 1
[-1,2,4,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 1
[-1,2,-4,3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 1
[-1,-2,4,3] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
[-1,-2,4,-3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[-1,-2,-4,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[-1,-2,-4,-3] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
[1,3,-2,4] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
[1,3,-2,-4] => [2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 1
[4,5,2,3,-8,-6,1,7] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6
[2,-8,-4,-6,-5,1,3,7] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6
[-4,3,-5,2,-8,-6,1,7] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6
[-4,3,5,8,7,-6,1,2] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6
[3,-7,2,8,-5,1,4,6] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6
[-7,6,-8,-4,-2,1,3,5] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6
[5,-6,-7,-4,2,3,-8,1] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6
[-8,-6,5,-4,2,-7,1,3] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6
[-7,6,1,-8,-5,-4,2,3] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6
[4,-8,-5,-7,1,-6,2,3] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6
[3,1,5,2,8,4,6,-7] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[4,5,1,2,8,3,6,-7] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[7,4,1,8,2,3,5,-6] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[8,1,5,2,7,4,6,-3] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[8,5,1,2,7,3,6,-4] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[8,4,1,6,2,3,5,-7] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[8,6,7,1,2,3,4,-5] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[8,6,1,7,2,3,5,-4] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[8,1,6,7,2,4,5,-3] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[5,6,8,1,2,3,4,-7] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[6,7,8,1,2,3,4,-5] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[6,1,7,8,2,4,5,-3] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[5,1,7,2,8,4,6,-3] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[5,7,1,2,8,3,6,-4] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[4,6,1,8,2,3,5,-7] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[3,1,6,8,2,4,5,-7] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[4,1,2,7,8,5,6,-3] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[3,1,8,2,7,4,6,-5] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[3,1,8,7,2,4,5,-6] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[4,8,1,7,2,3,5,-6] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[4,8,1,2,7,3,6,-5] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[5,8,7,1,2,3,4,-6] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[6,8,5,1,2,3,4,-7] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[7,8,1,6,2,3,5,-4] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[8,7,5,1,2,3,4,-6] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[4,5,2,3,8,1,6,-7] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[7,5,1,3,8,2,6,-4] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[8,5,6,3,4,1,2,-7] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[8,5,7,3,1,4,2,-6] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[8,6,7,1,3,4,2,-5] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[8,6,1,7,3,5,2,-4] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[8,1,6,7,4,5,2,-3] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[7,1,6,8,4,2,5,-3] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[7,5,8,3,1,2,4,-6] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[5,6,8,2,3,1,4,-7] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[4,7,2,8,1,3,5,-6] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[4,6,2,8,3,1,5,-7] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[4,8,2,7,1,5,3,-6] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[6,8,1,7,2,5,3,-4] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[5,8,6,2,4,1,3,-7] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
Description
The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001088
Mapping
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001088: Dyck paths ⟶ ℤResult quality: 88% ●values known / values provided: 97%●distinct values known / distinct values provided: 88%
Values
[-1,-2] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 0 + 1
[2,-1] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[-2,1] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[1,-2,-3] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 0 + 1
[-1,2,-3] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 0 + 1
[-1,-2,3] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 0 + 1
[-1,-2,-3] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 2 = 1 + 1
[1,3,-2] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[1,-3,2] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[-1,3,-2] => [2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[-1,-3,2] => [2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[2,-1,3] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[2,-1,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[-2,1,3] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[-2,1,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[2,3,-1] => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3 = 2 + 1
[2,-3,1] => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3 = 2 + 1
[-2,3,1] => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3 = 2 + 1
[-2,-3,-1] => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3 = 2 + 1
[3,1,-2] => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3 = 2 + 1
[3,-1,2] => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3 = 2 + 1
[-3,1,2] => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3 = 2 + 1
[-3,-1,-2] => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3 = 2 + 1
[3,2,-1] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[3,-2,-1] => [2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[-3,2,1] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[-3,-2,1] => [2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[1,2,-3,-4] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 0 + 1
[1,-2,3,-4] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 0 + 1
[1,-2,-3,4] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 0 + 1
[1,-2,-3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 2 = 1 + 1
[-1,2,3,-4] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 0 + 1
[-1,2,-3,4] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 0 + 1
[-1,2,-3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 2 = 1 + 1
[-1,-2,3,4] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 0 + 1
[-1,-2,3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 2 = 1 + 1
[-1,-2,-3,4] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 2 = 1 + 1
[-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[1,2,4,-3] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[1,2,-4,3] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[1,-2,4,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[1,-2,-4,3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[-1,2,4,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[-1,2,-4,3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[-1,-2,4,3] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 0 + 1
[-1,-2,4,-3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[-1,-2,-4,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[-1,-2,-4,-3] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 0 + 1
[1,3,-2,4] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[1,3,-2,-4] => [2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[4,5,2,3,-8,-6,1,7] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[2,-8,-4,-6,-5,1,3,7] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[-4,3,-5,2,-8,-6,1,7] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[-4,3,5,8,7,-6,1,2] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[3,-7,2,8,-5,1,4,6] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[-7,6,-8,-4,-2,1,3,5] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[5,-6,-7,-4,2,3,-8,1] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[-8,-6,5,-4,2,-7,1,3] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[-7,6,1,-8,-5,-4,2,3] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[4,-8,-5,-7,1,-6,2,3] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[3,1,5,2,8,4,6,-7] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7 + 1
[4,5,1,2,8,3,6,-7] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7 + 1
[7,4,1,8,2,3,5,-6] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7 + 1
[8,1,5,2,7,4,6,-3] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7 + 1
[8,5,1,2,7,3,6,-4] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7 + 1
[8,4,1,6,2,3,5,-7] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7 + 1
[8,6,7,1,2,3,4,-5] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7 + 1
[8,6,1,7,2,3,5,-4] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7 + 1
[8,1,6,7,2,4,5,-3] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7 + 1
[5,6,8,1,2,3,4,-7] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7 + 1
[6,7,8,1,2,3,4,-5] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7 + 1
[6,1,7,8,2,4,5,-3] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7 + 1
[5,1,7,2,8,4,6,-3] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7 + 1
[5,7,1,2,8,3,6,-4] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7 + 1
[4,6,1,8,2,3,5,-7] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7 + 1
[3,1,6,8,2,4,5,-7] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7 + 1
[4,1,2,7,8,5,6,-3] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7 + 1
[3,1,8,2,7,4,6,-5] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7 + 1
[3,1,8,7,2,4,5,-6] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7 + 1
[4,8,1,7,2,3,5,-6] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7 + 1
[4,8,1,2,7,3,6,-5] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7 + 1
[5,8,7,1,2,3,4,-6] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7 + 1
[6,8,5,1,2,3,4,-7] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7 + 1
[7,8,1,6,2,3,5,-4] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7 + 1
[8,7,5,1,2,3,4,-6] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7 + 1
[4,5,2,3,8,1,6,-7] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7 + 1
[7,5,1,3,8,2,6,-4] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7 + 1
[8,5,6,3,4,1,2,-7] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7 + 1
[8,5,7,3,1,4,2,-6] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7 + 1
[8,6,7,1,3,4,2,-5] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7 + 1
[8,6,1,7,3,5,2,-4] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7 + 1
[8,1,6,7,4,5,2,-3] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7 + 1
[7,1,6,8,4,2,5,-3] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7 + 1
[7,5,8,3,1,2,4,-6] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7 + 1
[5,6,8,2,3,1,4,-7] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7 + 1
[4,7,2,8,1,3,5,-6] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7 + 1
[4,6,2,8,3,1,5,-7] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7 + 1
[4,8,2,7,1,5,3,-6] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7 + 1
[6,8,1,7,2,5,3,-4] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7 + 1
[5,8,6,2,4,1,3,-7] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7 + 1
Description
Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra.
Matching statistic: St001223
(load all 5 compositions to match this statistic)
Mapping
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001223: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 96%●distinct values known / distinct values provided: 75%
Values
[-1,-2] => [1,1]
 => [1,1,0,0]
 => 0
[2,-1] => [2]
 => [1,0,1,0]
 => 1
[-2,1] => [2]
 => [1,0,1,0]
 => 1
[1,-2,-3] => [1,1]
 => [1,1,0,0]
 => 0
[-1,2,-3] => [1,1]
 => [1,1,0,0]
 => 0
[-1,-2,3] => [1,1]
 => [1,1,0,0]
 => 0
[-1,-2,-3] => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[1,3,-2] => [2]
 => [1,0,1,0]
 => 1
[1,-3,2] => [2]
 => [1,0,1,0]
 => 1
[-1,3,-2] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[-1,-3,2] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[2,-1,3] => [2]
 => [1,0,1,0]
 => 1
[2,-1,-3] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[-2,1,3] => [2]
 => [1,0,1,0]
 => 1
[-2,1,-3] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[2,3,-1] => [3]
 => [1,0,1,0,1,0]
 => 2
[2,-3,1] => [3]
 => [1,0,1,0,1,0]
 => 2
[-2,3,1] => [3]
 => [1,0,1,0,1,0]
 => 2
[-2,-3,-1] => [3]
 => [1,0,1,0,1,0]
 => 2
[3,1,-2] => [3]
 => [1,0,1,0,1,0]
 => 2
[3,-1,2] => [3]
 => [1,0,1,0,1,0]
 => 2
[-3,1,2] => [3]
 => [1,0,1,0,1,0]
 => 2
[-3,-1,-2] => [3]
 => [1,0,1,0,1,0]
 => 2
[3,2,-1] => [2]
 => [1,0,1,0]
 => 1
[3,-2,-1] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[-3,2,1] => [2]
 => [1,0,1,0]
 => 1
[-3,-2,1] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[1,2,-3,-4] => [1,1]
 => [1,1,0,0]
 => 0
[1,-2,3,-4] => [1,1]
 => [1,1,0,0]
 => 0
[1,-2,-3,4] => [1,1]
 => [1,1,0,0]
 => 0
[1,-2,-3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[-1,2,3,-4] => [1,1]
 => [1,1,0,0]
 => 0
[-1,2,-3,4] => [1,1]
 => [1,1,0,0]
 => 0
[-1,2,-3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[-1,-2,3,4] => [1,1]
 => [1,1,0,0]
 => 0
[-1,-2,3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[-1,-2,-3,4] => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 2
[1,2,4,-3] => [2]
 => [1,0,1,0]
 => 1
[1,2,-4,3] => [2]
 => [1,0,1,0]
 => 1
[1,-2,4,-3] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[1,-2,-4,3] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[-1,2,4,-3] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[-1,2,-4,3] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[-1,-2,4,3] => [1,1]
 => [1,1,0,0]
 => 0
[-1,-2,4,-3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
[-1,-2,-4,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
[-1,-2,-4,-3] => [1,1]
 => [1,1,0,0]
 => 0
[1,3,-2,4] => [2]
 => [1,0,1,0]
 => 1
[1,3,-2,-4] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[3,2,8,-6,-5,1,4,7] => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 5
[4,5,2,3,-8,-6,1,7] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 6
[2,-8,-4,-6,-5,1,3,7] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 6
[2,8,-6,-5,1,3,4,7] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 4
[-4,3,-5,2,-8,-6,1,7] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 6
[5,8,-4,3,-7,1,2,6] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 4
[-7,-5,4,6,-8,-2,1,3] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 4
[-4,3,5,8,7,-6,1,2] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 6
[-7,-6,5,3,-8,-4,1,2] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 4
[3,-7,-5,-6,2,4,-8,1] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 4
[-8,-4,-6,2,3,-7,1,5] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 4
[3,-7,2,8,-5,1,4,6] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 6
[-7,6,-8,-4,-2,1,3,5] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 6
[5,-6,-7,-4,2,3,-8,1] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 6
[-8,-6,5,-4,2,-7,1,3] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 6
[-7,6,1,-8,-5,-4,2,3] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 6
[-8,6,7,5,-4,1,2,3] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 4
[-8,2,5,7,4,-6,1,3] => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 5
[4,-8,-5,-7,1,-6,2,3] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 6
[-8,-7,5,6,4,1,2,3] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 4
[8,6,-7,-4,5,1,2,3] => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 5
[5,7,-8,4,3,-6,1,2] => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 5
[3,4,5,6,7,1,-2] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6
[3,4,6,1,7,5,-2] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6
[4,1,5,6,7,3,-2] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6
[4,1,6,3,7,5,-2] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6
[5,6,2,7,4,1,-3] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6
[6,1,2,-7,3,4,5] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6
[-7,4,-3,1,2,5,6] => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 5
[6,1,7,-5,2,3,4] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6
[7,-5,-4,-6,1,2,3] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6
[-2,-5,-7,1,3,4,6] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6
[7,-6,2,1,3,4,5] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6
[-3,-5,7,-6,1,2,4] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6
[5,4,1,-7,2,3,6] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6
[7,4,1,-6,2,3,5] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6
[-5,4,1,-7,-6,2,3] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6
[3,1,6,2,4,8,7,-5] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6
[3,1,5,2,8,4,6,-7] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[4,5,1,2,8,3,6,-7] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[4,6,1,2,3,8,7,-5] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6
[6,1,5,2,4,8,7,-3] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6
[6,5,1,2,3,8,7,-4] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6
[7,4,1,8,2,3,5,-6] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[1,7,5,3,8,4,6,-2] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6
[8,1,5,2,7,4,6,-3] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[8,5,1,2,7,3,6,-4] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[8,4,1,6,2,3,5,-7] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[8,6,7,1,2,3,4,-5] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[8,6,1,7,2,3,5,-4] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
Description
Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless.
Matching statistic: St001233
Mapping
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001233: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 96%●distinct values known / distinct values provided: 75%
Values
[-1,-2] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
[2,-1] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
[-2,1] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
[1,-2,-3] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
[-1,2,-3] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
[-1,-2,3] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
[-1,-2,-3] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 1
[1,3,-2] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
[1,-3,2] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
[-1,3,-2] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[-1,-3,2] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[2,-1,3] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
[2,-1,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[-2,1,3] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
[-2,1,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[2,3,-1] => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 2
[2,-3,1] => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 2
[-2,3,1] => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 2
[-2,-3,-1] => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 2
[3,1,-2] => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 2
[3,-1,2] => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 2
[-3,1,2] => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 2
[-3,-1,-2] => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 2
[3,2,-1] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
[3,-2,-1] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[-3,2,1] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
[-3,-2,1] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[1,2,-3,-4] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
[1,-2,3,-4] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
[1,-2,-3,4] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
[1,-2,-3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 1
[-1,2,3,-4] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
[-1,2,-3,4] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
[-1,2,-3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 1
[-1,-2,3,4] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
[-1,-2,3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 1
[-1,-2,-3,4] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 1
[-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2
[1,2,4,-3] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
[1,2,-4,3] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
[1,-2,4,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[1,-2,-4,3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[-1,2,4,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[-1,2,-4,3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[-1,-2,4,3] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
[-1,-2,4,-3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[-1,-2,-4,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[-1,-2,-4,-3] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
[1,3,-2,4] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
[1,3,-2,-4] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[3,2,8,-6,-5,1,4,7] => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 5
[4,5,2,3,-8,-6,1,7] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 6
[2,-8,-4,-6,-5,1,3,7] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 6
[2,8,-6,-5,1,3,4,7] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? = 4
[-4,3,-5,2,-8,-6,1,7] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 6
[5,8,-4,3,-7,1,2,6] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? = 4
[-7,-5,4,6,-8,-2,1,3] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? = 4
[-4,3,5,8,7,-6,1,2] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 6
[-7,-6,5,3,-8,-4,1,2] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? = 4
[3,-7,-5,-6,2,4,-8,1] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? = 4
[-8,-4,-6,2,3,-7,1,5] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? = 4
[3,-7,2,8,-5,1,4,6] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 6
[-7,6,-8,-4,-2,1,3,5] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 6
[5,-6,-7,-4,2,3,-8,1] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 6
[-8,-6,5,-4,2,-7,1,3] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 6
[-7,6,1,-8,-5,-4,2,3] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 6
[-8,6,7,5,-4,1,2,3] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? = 4
[-8,2,5,7,4,-6,1,3] => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 5
[4,-8,-5,-7,1,-6,2,3] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 6
[-8,-7,5,6,4,1,2,3] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? = 4
[8,6,-7,-4,5,1,2,3] => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 5
[5,7,-8,4,3,-6,1,2] => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 5
[3,4,5,6,7,1,-2] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6
[3,4,6,1,7,5,-2] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6
[4,1,5,6,7,3,-2] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6
[4,1,6,3,7,5,-2] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6
[5,6,2,7,4,1,-3] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6
[6,1,2,-7,3,4,5] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6
[-7,4,-3,1,2,5,6] => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 5
[6,1,7,-5,2,3,4] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6
[7,-5,-4,-6,1,2,3] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6
[-2,-5,-7,1,3,4,6] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6
[7,-6,2,1,3,4,5] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6
[-3,-5,7,-6,1,2,4] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6
[5,4,1,-7,2,3,6] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6
[7,4,1,-6,2,3,5] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6
[-5,4,1,-7,-6,2,3] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6
[3,1,6,2,4,8,7,-5] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6
[3,1,5,2,8,4,6,-7] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[4,5,1,2,8,3,6,-7] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[4,6,1,2,3,8,7,-5] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6
[6,1,5,2,4,8,7,-3] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6
[6,5,1,2,3,8,7,-4] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6
[7,4,1,8,2,3,5,-6] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[1,7,5,3,8,4,6,-2] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6
[8,1,5,2,7,4,6,-3] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[8,5,1,2,7,3,6,-4] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[8,4,1,6,2,3,5,-7] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[8,6,7,1,2,3,4,-5] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[8,6,1,7,2,3,5,-4] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
Description
The number of indecomposable 2-dimensional modules with projective dimension one.
Matching statistic: St001226
Mapping
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001226: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 96%●distinct values known / distinct values provided: 75%
Values
[-1,-2] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2 = 0 + 2
[2,-1] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 3 = 1 + 2
[-2,1] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 3 = 1 + 2
[1,-2,-3] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2 = 0 + 2
[-1,2,-3] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2 = 0 + 2
[-1,-2,3] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2 = 0 + 2
[-1,-2,-3] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 3 = 1 + 2
[1,3,-2] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 3 = 1 + 2
[1,-3,2] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 3 = 1 + 2
[-1,3,-2] => [2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 3 = 1 + 2
[-1,-3,2] => [2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 3 = 1 + 2
[2,-1,3] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 3 = 1 + 2
[2,-1,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 3 = 1 + 2
[-2,1,3] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 3 = 1 + 2
[-2,1,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 3 = 1 + 2
[2,3,-1] => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 4 = 2 + 2
[2,-3,1] => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 4 = 2 + 2
[-2,3,1] => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 4 = 2 + 2
[-2,-3,-1] => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 4 = 2 + 2
[3,1,-2] => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 4 = 2 + 2
[3,-1,2] => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 4 = 2 + 2
[-3,1,2] => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 4 = 2 + 2
[-3,-1,-2] => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 4 = 2 + 2
[3,2,-1] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 3 = 1 + 2
[3,-2,-1] => [2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 3 = 1 + 2
[-3,2,1] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 3 = 1 + 2
[-3,-2,1] => [2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 3 = 1 + 2
[1,2,-3,-4] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2 = 0 + 2
[1,-2,3,-4] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2 = 0 + 2
[1,-2,-3,4] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2 = 0 + 2
[1,-2,-3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 3 = 1 + 2
[-1,2,3,-4] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2 = 0 + 2
[-1,2,-3,4] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2 = 0 + 2
[-1,2,-3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 3 = 1 + 2
[-1,-2,3,4] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2 = 0 + 2
[-1,-2,3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 3 = 1 + 2
[-1,-2,-3,4] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 3 = 1 + 2
[-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 4 = 2 + 2
[1,2,4,-3] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 3 = 1 + 2
[1,2,-4,3] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 3 = 1 + 2
[1,-2,4,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 3 = 1 + 2
[1,-2,-4,3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 3 = 1 + 2
[-1,2,4,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 3 = 1 + 2
[-1,2,-4,3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 3 = 1 + 2
[-1,-2,4,3] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2 = 0 + 2
[-1,-2,4,-3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 4 = 2 + 2
[-1,-2,-4,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 4 = 2 + 2
[-1,-2,-4,-3] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2 = 0 + 2
[1,3,-2,4] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 3 = 1 + 2
[1,3,-2,-4] => [2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 3 = 1 + 2
[3,2,8,-6,-5,1,4,7] => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 5 + 2
[4,5,2,3,-8,-6,1,7] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 2
[2,-8,-4,-6,-5,1,3,7] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 2
[2,8,-6,-5,1,3,4,7] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 4 + 2
[-4,3,-5,2,-8,-6,1,7] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 2
[5,8,-4,3,-7,1,2,6] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 4 + 2
[-7,-5,4,6,-8,-2,1,3] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 4 + 2
[-4,3,5,8,7,-6,1,2] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 2
[-7,-6,5,3,-8,-4,1,2] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 4 + 2
[3,-7,-5,-6,2,4,-8,1] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 4 + 2
[-8,-4,-6,2,3,-7,1,5] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 4 + 2
[3,-7,2,8,-5,1,4,6] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 2
[-7,6,-8,-4,-2,1,3,5] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 2
[5,-6,-7,-4,2,3,-8,1] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 2
[-8,-6,5,-4,2,-7,1,3] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 2
[-7,6,1,-8,-5,-4,2,3] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 2
[-8,6,7,5,-4,1,2,3] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 4 + 2
[-8,2,5,7,4,-6,1,3] => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 5 + 2
[4,-8,-5,-7,1,-6,2,3] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 2
[-8,-7,5,6,4,1,2,3] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 4 + 2
[8,6,-7,-4,5,1,2,3] => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 5 + 2
[5,7,-8,4,3,-6,1,2] => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 5 + 2
[3,4,5,6,7,1,-2] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 2
[3,4,6,1,7,5,-2] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 2
[4,1,5,6,7,3,-2] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 2
[4,1,6,3,7,5,-2] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 2
[5,6,2,7,4,1,-3] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 2
[6,1,2,-7,3,4,5] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 2
[-7,4,-3,1,2,5,6] => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 5 + 2
[6,1,7,-5,2,3,4] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 2
[7,-5,-4,-6,1,2,3] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 2
[-2,-5,-7,1,3,4,6] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 2
[7,-6,2,1,3,4,5] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 2
[-3,-5,7,-6,1,2,4] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 2
[5,4,1,-7,2,3,6] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 2
[7,4,1,-6,2,3,5] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 2
[-5,4,1,-7,-6,2,3] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 2
[3,1,6,2,4,8,7,-5] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 2
[3,1,5,2,8,4,6,-7] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7 + 2
[4,5,1,2,8,3,6,-7] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7 + 2
[4,6,1,2,3,8,7,-5] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 2
[6,1,5,2,4,8,7,-3] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 2
[6,5,1,2,3,8,7,-4] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 2
[7,4,1,8,2,3,5,-6] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7 + 2
[1,7,5,3,8,4,6,-2] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 2
[8,1,5,2,7,4,6,-3] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7 + 2
[8,5,1,2,7,3,6,-4] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7 + 2
[8,4,1,6,2,3,5,-7] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7 + 2
[8,6,7,1,2,3,4,-5] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7 + 2
[8,6,1,7,2,3,5,-4] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7 + 2
Description
The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. That is the number of i such that $Ext_A^1(J,e_i J)=0$.
Matching statistic: St001948
Mapping
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St001948: Permutations ⟶ ℤResult quality: 62% ●values known / values provided: 88%●distinct values known / distinct values provided: 62%
Values
[-1,-2] => [1,1]
 => [1,1,0,0]
 => [2,1] => 0
[2,-1] => [2]
 => [1,0,1,0]
 => [1,2] => 1
[-2,1] => [2]
 => [1,0,1,0]
 => [1,2] => 1
[1,-2,-3] => [1,1]
 => [1,1,0,0]
 => [2,1] => 0
[-1,2,-3] => [1,1]
 => [1,1,0,0]
 => [2,1] => 0
[-1,-2,3] => [1,1]
 => [1,1,0,0]
 => [2,1] => 0
[-1,-2,-3] => [1,1,1]
 => [1,1,0,1,0,0]
 => [2,3,1] => 1
[1,3,-2] => [2]
 => [1,0,1,0]
 => [1,2] => 1
[1,-3,2] => [2]
 => [1,0,1,0]
 => [1,2] => 1
[-1,3,-2] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1
[-1,-3,2] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1
[2,-1,3] => [2]
 => [1,0,1,0]
 => [1,2] => 1
[2,-1,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1
[-2,1,3] => [2]
 => [1,0,1,0]
 => [1,2] => 1
[-2,1,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1
[2,3,-1] => [3]
 => [1,0,1,0,1,0]
 => [1,2,3] => 2
[2,-3,1] => [3]
 => [1,0,1,0,1,0]
 => [1,2,3] => 2
[-2,3,1] => [3]
 => [1,0,1,0,1,0]
 => [1,2,3] => 2
[-2,-3,-1] => [3]
 => [1,0,1,0,1,0]
 => [1,2,3] => 2
[3,1,-2] => [3]
 => [1,0,1,0,1,0]
 => [1,2,3] => 2
[3,-1,2] => [3]
 => [1,0,1,0,1,0]
 => [1,2,3] => 2
[-3,1,2] => [3]
 => [1,0,1,0,1,0]
 => [1,2,3] => 2
[-3,-1,-2] => [3]
 => [1,0,1,0,1,0]
 => [1,2,3] => 2
[3,2,-1] => [2]
 => [1,0,1,0]
 => [1,2] => 1
[3,-2,-1] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1
[-3,2,1] => [2]
 => [1,0,1,0]
 => [1,2] => 1
[-3,-2,1] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1
[1,2,-3,-4] => [1,1]
 => [1,1,0,0]
 => [2,1] => 0
[1,-2,3,-4] => [1,1]
 => [1,1,0,0]
 => [2,1] => 0
[1,-2,-3,4] => [1,1]
 => [1,1,0,0]
 => [2,1] => 0
[1,-2,-3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => [2,3,1] => 1
[-1,2,3,-4] => [1,1]
 => [1,1,0,0]
 => [2,1] => 0
[-1,2,-3,4] => [1,1]
 => [1,1,0,0]
 => [2,1] => 0
[-1,2,-3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => [2,3,1] => 1
[-1,-2,3,4] => [1,1]
 => [1,1,0,0]
 => [2,1] => 0
[-1,-2,3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => [2,3,1] => 1
[-1,-2,-3,4] => [1,1,1]
 => [1,1,0,1,0,0]
 => [2,3,1] => 1
[-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 2
[1,2,4,-3] => [2]
 => [1,0,1,0]
 => [1,2] => 1
[1,2,-4,3] => [2]
 => [1,0,1,0]
 => [1,2] => 1
[1,-2,4,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1
[1,-2,-4,3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1
[-1,2,4,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1
[-1,2,-4,3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1
[-1,-2,4,3] => [1,1]
 => [1,1,0,0]
 => [2,1] => 0
[-1,-2,4,-3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 2
[-1,-2,-4,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 2
[-1,-2,-4,-3] => [1,1]
 => [1,1,0,0]
 => [2,1] => 0
[1,3,-2,4] => [2]
 => [1,0,1,0]
 => [1,2] => 1
[1,3,-2,-4] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1
[-6,-4,-2,1,3,5] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => ? = 5
[4,5,2,3,-6,1] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => ? = 5
[2,-6,-4,-5,1,3] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => ? = 5
[-5,3,6,1,2,4] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => ? = 5
[4,-5,-6,-3,1,2] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => ? = 5
[-4,3,-5,2,-6,1] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => ? = 5
[3,2,8,-6,-5,1,4,7] => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,7,6] => ? = 5
[5,-3,2,4,8,-7,1,6] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,6,5,4] => ? = 3
[4,5,2,3,-8,-6,1,7] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,8,7] => ? = 6
[2,-8,-4,-6,-5,1,3,7] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,8,7] => ? = 6
[2,8,-6,-5,1,3,4,7] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,4,7,6,5] => ? = 4
[-5,4,-8,-6,-2,1,3,7] => [5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,2,5,6,4,3] => ? = 3
[4,-5,-8,-6,-3,1,2,7] => [5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,2,5,6,4,3] => ? = 3
[3,-8,-5,-6,-4,1,2,7] => [5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,2,5,6,4,3] => ? = 3
[-4,3,-5,2,-8,-6,1,7] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,8,7] => ? = 6
[5,8,-4,3,-7,1,2,6] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,4,7,6,5] => ? = 4
[-7,-5,4,6,-8,-2,1,3] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,4,7,6,5] => ? = 4
[-4,3,5,8,7,-6,1,2] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,8,7] => ? = 6
[-7,-6,3,-8,4,-5,1,2] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,6,5,4] => ? = 3
[-7,-6,5,3,-8,-4,1,2] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,4,7,6,5] => ? = 4
[3,-7,-5,-6,2,4,-8,1] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,4,7,6,5] => ? = 4
[4,2,-7,8,-6,1,3,5] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,6,5,4] => ? = 3
[-8,-4,-6,2,3,-7,1,5] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,4,7,6,5] => ? = 4
[3,-7,2,8,-5,1,4,6] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,8,7] => ? = 6
[-7,6,-8,-4,-2,1,3,5] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,8,7] => ? = 6
[-7,-8,-3,2,5,-6,1,4] => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,4,3,5,6,2] => ? = 2
[5,-6,-7,-4,2,3,-8,1] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,8,7] => ? = 6
[-8,-6,5,-4,2,-7,1,3] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,8,7] => ? = 6
[-7,2,-4,6,-8,-5,1,3] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,6,5,4] => ? = 3
[-7,6,1,-8,-5,-4,2,3] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,8,7] => ? = 6
[-8,6,7,5,-4,1,2,3] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,4,7,6,5] => ? = 4
[-8,2,5,7,4,-6,1,3] => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,7,6] => ? = 5
[5,7,-6,4,2,3,-8,1] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,6,5,4] => ? = 3
[4,-8,-5,-7,1,-6,2,3] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,8,7] => ? = 6
[-8,-7,5,6,4,1,2,3] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,4,7,6,5] => ? = 4
[8,6,-7,-4,5,1,2,3] => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,7,6] => ? = 5
[5,7,-8,4,3,-6,1,2] => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,7,6] => ? = 5
[6,1,2,3,4,-5] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => ? = 5
[5,1,2,3,6,-4] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => ? = 5
[5,1,2,6,4,-3] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => ? = 5
[4,1,2,5,6,-3] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => ? = 5
[5,1,6,3,4,-2] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => ? = 5
[4,1,5,3,6,-2] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => ? = 5
[5,1,6,2,3,-4] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => ? = 5
[3,1,5,6,4,-2] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => ? = 5
[4,1,5,6,2,-3] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => ? = 5
[3,1,4,5,6,-2] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => ? = 5
[5,6,2,3,4,-1] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => ? = 5
[4,5,2,3,6,-1] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => ? = 5
[3,5,2,6,4,-1] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => ? = 5
Description
The number of augmented double ascents of a permutation. An augmented double ascent of a permutation $\pi$ is a double ascent of the augmented permutation $\tilde\pi$ obtained from $\pi$ by adding an initial $0$. A double ascent of $\tilde\pi$ then is a position $i$ such that $\tilde\pi(i) < \tilde\pi(i+1) < \tilde\pi(i+2)$.