Your data matches 3 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000438
(load all 2 compositions to match this statistic)
Mapping
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000438: 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]
 => [1,0,1,0]
 => 3
[2,-1] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[-2,1] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[1,-2,-3] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 3
[-1,2,-3] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 3
[-1,-2,3] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 3
[-1,-2,-3] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 4
[1,3,-2] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[1,-3,2] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[-1,3,-2] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 4
[-1,-3,2] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 4
[2,-1,3] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[2,-1,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 4
[-2,1,3] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[-2,1,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 4
[2,3,-1] => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
[2,-3,1] => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
[-2,3,1] => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
[-2,-3,-1] => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
[3,1,-2] => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
[3,-1,2] => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
[-3,1,2] => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
[-3,-1,-2] => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
[3,2,-1] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[3,-2,-1] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 4
[-3,2,1] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[-3,-2,1] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 4
[1,2,-3,-4] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 3
[1,-2,3,-4] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 3
[1,-2,-3,4] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 3
[1,-2,-3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 4
[-1,2,3,-4] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 3
[-1,2,-3,4] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 3
[-1,2,-3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 4
[-1,-2,3,4] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 3
[-1,-2,3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 4
[-1,-2,-3,4] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 4
[-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 5
[1,2,4,-3] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[1,2,-4,3] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[1,-2,4,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 4
[1,-2,-4,3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 4
[-1,2,4,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 4
[-1,2,-4,3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 4
[-1,-2,4,3] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 3
[-1,-2,4,-3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 5
[-1,-2,-4,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 5
[-1,-2,-4,-3] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 3
[1,3,-2,4] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[1,3,-2,-4] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 4
Description
The position of the last up step in a Dyck path.
Matching statistic: St001213
Mapping
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001213: Dyck paths ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[-1,-2] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 4 = 3 + 1
[2,-1] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 3 = 2 + 1
[-2,1] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 3 = 2 + 1
[1,-2,-3] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 4 = 3 + 1
[-1,2,-3] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 4 = 3 + 1
[-1,-2,3] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 4 = 3 + 1
[-1,-2,-3] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 5 = 4 + 1
[1,3,-2] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 3 = 2 + 1
[1,-3,2] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 3 = 2 + 1
[-1,3,-2] => [2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 5 = 4 + 1
[-1,-3,2] => [2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 5 = 4 + 1
[2,-1,3] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 3 = 2 + 1
[2,-1,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 5 = 4 + 1
[-2,1,3] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 3 = 2 + 1
[-2,1,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 5 = 4 + 1
[2,3,-1] => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 4 = 3 + 1
[2,-3,1] => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 4 = 3 + 1
[-2,3,1] => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 4 = 3 + 1
[-2,-3,-1] => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 4 = 3 + 1
[3,1,-2] => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 4 = 3 + 1
[3,-1,2] => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 4 = 3 + 1
[-3,1,2] => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 4 = 3 + 1
[-3,-1,-2] => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 4 = 3 + 1
[3,2,-1] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 3 = 2 + 1
[3,-2,-1] => [2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 5 = 4 + 1
[-3,2,1] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 3 = 2 + 1
[-3,-2,1] => [2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 5 = 4 + 1
[1,2,-3,-4] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 4 = 3 + 1
[1,-2,3,-4] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 4 = 3 + 1
[1,-2,-3,4] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 4 = 3 + 1
[1,-2,-3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 5 = 4 + 1
[-1,2,3,-4] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 4 = 3 + 1
[-1,2,-3,4] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 4 = 3 + 1
[-1,2,-3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 5 = 4 + 1
[-1,-2,3,4] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 4 = 3 + 1
[-1,-2,3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 5 = 4 + 1
[-1,-2,-3,4] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 5 = 4 + 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]
 => 6 = 5 + 1
[1,2,4,-3] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 3 = 2 + 1
[1,2,-4,3] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 3 = 2 + 1
[1,-2,4,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 5 = 4 + 1
[1,-2,-4,3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 5 = 4 + 1
[-1,2,4,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 5 = 4 + 1
[-1,2,-4,3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 5 = 4 + 1
[-1,-2,4,3] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 4 = 3 + 1
[-1,-2,4,-3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 6 = 5 + 1
[-1,-2,-4,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 6 = 5 + 1
[-1,-2,-4,-3] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 4 = 3 + 1
[1,3,-2,4] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 3 = 2 + 1
[1,3,-2,-4] => [2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 5 = 4 + 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]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 8 + 1
[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]
 => ? = 9 + 1
[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]
 => ? = 9 + 1
[-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]
 => ? = 9 + 1
[-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]
 => ? = 9 + 1
[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]
 => ? = 9 + 1
[-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]
 => ? = 9 + 1
[-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]
 => ? = 9 + 1
[-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]
 => ? = 8 + 1
[-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]
 => ? = 9 + 1
[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]
 => ? = 8 + 1
[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]
 => ? = 8 + 1
[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]
 => ? = 7 + 1
[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]
 => ? = 7 + 1
[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]
 => ? = 7 + 1
[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]
 => ? = 7 + 1
[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]
 => ? = 7 + 1
[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]
 => ? = 7 + 1
[-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]
 => ? = 8 + 1
[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]
 => ? = 7 + 1
[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]
 => ? = 7 + 1
[-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]
 => ? = 7 + 1
[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]
 => ? = 7 + 1
[-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]
 => ? = 7 + 1
[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]
 => ? = 7 + 1
[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]
 => ? = 7 + 1
[-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]
 => ? = 7 + 1
[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]
 => ? = 7 + 1
[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]
 => ? = 7 + 1
[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]
 => ? = 7 + 1
[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]
 => ? = 7 + 1
[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]
 => ? = 7 + 1
[1,7,6,8,3,4,5,-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]
 => ? = 7 + 1
[1,5,8,3,7,4,6,-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]
 => ? = 7 + 1
[1,6,8,7,3,4,5,-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]
 => ? = 7 + 1
[-8,-7,2,4,-5,-3,1,6] => [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]
 => ? = 8 + 1
[-8,6,3,-7,-5,-4,1,2] => [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]
 => ? = 8 + 1
[-7,-3,2,8,-5,1,4,6] => [5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 10 + 1
[-8,-3,2,6,-7,-5,1,4] => [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]
 => ? = 9 + 1
[7,8,-6,-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]
 => ? = 8 + 1
[-8,-7,-5,4,-6,-2,1,3] => [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]
 => ? = 7 + 1
[6,7,8,3,5,-4,1,2] => [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]
 => ? = 7 + 1
[5,6,2,3,-4,1,-8,7] => [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]
 => ? = 9 + 1
[5,6,1,-3,2,4,-8,7] => [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]
 => ? = 9 + 1
[7,8,-3,-6,2,4,-5,1] => [5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 10 + 1
[7,8,2,3,-6,5,-4,1] => [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]
 => ? = 9 + 1
[2,8,-7,-6,-5,4,1,3] => [5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 10 + 1
[7,8,1,-3,-6,5,2,4] => [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]
 => ? = 9 + 1
[4,5,2,3,-6,1,-8,7] => [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]
 => ? = 9 + 1
[-5,4,1,-3,-6,2,-8,7] => [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]
 => ? = 9 + 1
Description
The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module.
Matching statistic: St001491
Mapping
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 22%
Values
[-1,-2] => [1,1]
 => [1,1,0,0]
 => 1100 => 1 = 3 - 2
[2,-1] => [2]
 => [1,0,1,0]
 => 1010 => 0 = 2 - 2
[-2,1] => [2]
 => [1,0,1,0]
 => 1010 => 0 = 2 - 2
[1,-2,-3] => [1,1]
 => [1,1,0,0]
 => 1100 => 1 = 3 - 2
[-1,2,-3] => [1,1]
 => [1,1,0,0]
 => 1100 => 1 = 3 - 2
[-1,-2,3] => [1,1]
 => [1,1,0,0]
 => 1100 => 1 = 3 - 2
[-1,-2,-3] => [1,1,1]
 => [1,1,0,1,0,0]
 => 110100 => ? = 4 - 2
[1,3,-2] => [2]
 => [1,0,1,0]
 => 1010 => 0 = 2 - 2
[1,-3,2] => [2]
 => [1,0,1,0]
 => 1010 => 0 = 2 - 2
[-1,3,-2] => [2,1]
 => [1,0,1,1,0,0]
 => 101100 => ? = 4 - 2
[-1,-3,2] => [2,1]
 => [1,0,1,1,0,0]
 => 101100 => ? = 4 - 2
[2,-1,3] => [2]
 => [1,0,1,0]
 => 1010 => 0 = 2 - 2
[2,-1,-3] => [2,1]
 => [1,0,1,1,0,0]
 => 101100 => ? = 4 - 2
[-2,1,3] => [2]
 => [1,0,1,0]
 => 1010 => 0 = 2 - 2
[-2,1,-3] => [2,1]
 => [1,0,1,1,0,0]
 => 101100 => ? = 4 - 2
[2,3,-1] => [3]
 => [1,0,1,0,1,0]
 => 101010 => ? = 3 - 2
[2,-3,1] => [3]
 => [1,0,1,0,1,0]
 => 101010 => ? = 3 - 2
[-2,3,1] => [3]
 => [1,0,1,0,1,0]
 => 101010 => ? = 3 - 2
[-2,-3,-1] => [3]
 => [1,0,1,0,1,0]
 => 101010 => ? = 3 - 2
[3,1,-2] => [3]
 => [1,0,1,0,1,0]
 => 101010 => ? = 3 - 2
[3,-1,2] => [3]
 => [1,0,1,0,1,0]
 => 101010 => ? = 3 - 2
[-3,1,2] => [3]
 => [1,0,1,0,1,0]
 => 101010 => ? = 3 - 2
[-3,-1,-2] => [3]
 => [1,0,1,0,1,0]
 => 101010 => ? = 3 - 2
[3,2,-1] => [2]
 => [1,0,1,0]
 => 1010 => 0 = 2 - 2
[3,-2,-1] => [2,1]
 => [1,0,1,1,0,0]
 => 101100 => ? = 4 - 2
[-3,2,1] => [2]
 => [1,0,1,0]
 => 1010 => 0 = 2 - 2
[-3,-2,1] => [2,1]
 => [1,0,1,1,0,0]
 => 101100 => ? = 4 - 2
[1,2,-3,-4] => [1,1]
 => [1,1,0,0]
 => 1100 => 1 = 3 - 2
[1,-2,3,-4] => [1,1]
 => [1,1,0,0]
 => 1100 => 1 = 3 - 2
[1,-2,-3,4] => [1,1]
 => [1,1,0,0]
 => 1100 => 1 = 3 - 2
[1,-2,-3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => 110100 => ? = 4 - 2
[-1,2,3,-4] => [1,1]
 => [1,1,0,0]
 => 1100 => 1 = 3 - 2
[-1,2,-3,4] => [1,1]
 => [1,1,0,0]
 => 1100 => 1 = 3 - 2
[-1,2,-3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => 110100 => ? = 4 - 2
[-1,-2,3,4] => [1,1]
 => [1,1,0,0]
 => 1100 => 1 = 3 - 2
[-1,-2,3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => 110100 => ? = 4 - 2
[-1,-2,-3,4] => [1,1,1]
 => [1,1,0,1,0,0]
 => 110100 => ? = 4 - 2
[-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 11010100 => ? = 5 - 2
[1,2,4,-3] => [2]
 => [1,0,1,0]
 => 1010 => 0 = 2 - 2
[1,2,-4,3] => [2]
 => [1,0,1,0]
 => 1010 => 0 = 2 - 2
[1,-2,4,-3] => [2,1]
 => [1,0,1,1,0,0]
 => 101100 => ? = 4 - 2
[1,-2,-4,3] => [2,1]
 => [1,0,1,1,0,0]
 => 101100 => ? = 4 - 2
[-1,2,4,-3] => [2,1]
 => [1,0,1,1,0,0]
 => 101100 => ? = 4 - 2
[-1,2,-4,3] => [2,1]
 => [1,0,1,1,0,0]
 => 101100 => ? = 4 - 2
[-1,-2,4,3] => [1,1]
 => [1,1,0,0]
 => 1100 => 1 = 3 - 2
[-1,-2,4,-3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => ? = 5 - 2
[-1,-2,-4,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => ? = 5 - 2
[-1,-2,-4,-3] => [1,1]
 => [1,1,0,0]
 => 1100 => 1 = 3 - 2
[1,3,-2,4] => [2]
 => [1,0,1,0]
 => 1010 => 0 = 2 - 2
[1,3,-2,-4] => [2,1]
 => [1,0,1,1,0,0]
 => 101100 => ? = 4 - 2
[1,-3,2,4] => [2]
 => [1,0,1,0]
 => 1010 => 0 = 2 - 2
[1,-3,2,-4] => [2,1]
 => [1,0,1,1,0,0]
 => 101100 => ? = 4 - 2
[-1,3,2,-4] => [1,1]
 => [1,1,0,0]
 => 1100 => 1 = 3 - 2
[-1,3,-2,4] => [2,1]
 => [1,0,1,1,0,0]
 => 101100 => ? = 4 - 2
[-1,3,-2,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => ? = 5 - 2
[-1,-3,2,4] => [2,1]
 => [1,0,1,1,0,0]
 => 101100 => ? = 4 - 2
[-1,-3,2,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => ? = 5 - 2
[-1,-3,-2,-4] => [1,1]
 => [1,1,0,0]
 => 1100 => 1 = 3 - 2
[1,3,4,-2] => [3]
 => [1,0,1,0,1,0]
 => 101010 => ? = 3 - 2
[1,3,-4,2] => [3]
 => [1,0,1,0,1,0]
 => 101010 => ? = 3 - 2
[1,-3,4,2] => [3]
 => [1,0,1,0,1,0]
 => 101010 => ? = 3 - 2
[1,-3,-4,-2] => [3]
 => [1,0,1,0,1,0]
 => 101010 => ? = 3 - 2
[-1,3,4,-2] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => ? = 5 - 2
[-1,3,-4,2] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => ? = 5 - 2
[-1,-3,4,2] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => ? = 5 - 2
[-1,-3,-4,-2] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => ? = 5 - 2
[1,4,2,-3] => [3]
 => [1,0,1,0,1,0]
 => 101010 => ? = 3 - 2
[1,4,-2,3] => [3]
 => [1,0,1,0,1,0]
 => 101010 => ? = 3 - 2
[1,-4,2,3] => [3]
 => [1,0,1,0,1,0]
 => 101010 => ? = 3 - 2
[1,-4,-2,-3] => [3]
 => [1,0,1,0,1,0]
 => 101010 => ? = 3 - 2
[-1,4,2,-3] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => ? = 5 - 2
[-1,4,-2,3] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => ? = 5 - 2
[-1,-4,2,3] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => ? = 5 - 2
[-1,-4,-2,-3] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => ? = 5 - 2
[1,4,3,-2] => [2]
 => [1,0,1,0]
 => 1010 => 0 = 2 - 2
[1,4,-3,-2] => [2,1]
 => [1,0,1,1,0,0]
 => 101100 => ? = 4 - 2
[1,-4,3,2] => [2]
 => [1,0,1,0]
 => 1010 => 0 = 2 - 2
[1,-4,-3,2] => [2,1]
 => [1,0,1,1,0,0]
 => 101100 => ? = 4 - 2
[-1,4,-3,2] => [1,1]
 => [1,1,0,0]
 => 1100 => 1 = 3 - 2
[-1,-4,-3,-2] => [1,1]
 => [1,1,0,0]
 => 1100 => 1 = 3 - 2
[2,1,-3,-4] => [1,1]
 => [1,1,0,0]
 => 1100 => 1 = 3 - 2
[2,-1,3,4] => [2]
 => [1,0,1,0]
 => 1010 => 0 = 2 - 2
[-2,1,3,4] => [2]
 => [1,0,1,0]
 => 1010 => 0 = 2 - 2
[-2,-1,-3,-4] => [1,1]
 => [1,1,0,0]
 => 1100 => 1 = 3 - 2
[2,1,4,-3] => [2]
 => [1,0,1,0]
 => 1010 => 0 = 2 - 2
[2,1,-4,3] => [2]
 => [1,0,1,0]
 => 1010 => 0 = 2 - 2
[2,-1,4,3] => [2]
 => [1,0,1,0]
 => 1010 => 0 = 2 - 2
[2,-1,-4,-3] => [2]
 => [1,0,1,0]
 => 1010 => 0 = 2 - 2
[-2,1,4,3] => [2]
 => [1,0,1,0]
 => 1010 => 0 = 2 - 2
[-2,1,-4,-3] => [2]
 => [1,0,1,0]
 => 1010 => 0 = 2 - 2
[-2,-1,4,-3] => [2]
 => [1,0,1,0]
 => 1010 => 0 = 2 - 2
[-2,-1,-4,3] => [2]
 => [1,0,1,0]
 => 1010 => 0 = 2 - 2
[3,2,-1,4] => [2]
 => [1,0,1,0]
 => 1010 => 0 = 2 - 2
[3,-2,1,-4] => [1,1]
 => [1,1,0,0]
 => 1100 => 1 = 3 - 2
[-3,2,1,4] => [2]
 => [1,0,1,0]
 => 1010 => 0 = 2 - 2
[-3,-2,-1,-4] => [1,1]
 => [1,1,0,0]
 => 1100 => 1 = 3 - 2
[3,4,1,-2] => [2]
 => [1,0,1,0]
 => 1010 => 0 = 2 - 2
[3,4,-1,2] => [2]
 => [1,0,1,0]
 => 1010 => 0 = 2 - 2
[3,-4,1,2] => [2]
 => [1,0,1,0]
 => 1010 => 0 = 2 - 2
[3,-4,-1,-2] => [2]
 => [1,0,1,0]
 => 1010 => 0 = 2 - 2
Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset. Let $A_n=K[x]/(x^n)$. We associate to a nonempty subset S of an (n-1)-set the module $M_S$, which is the direct sum of $A_n$-modules with indecomposable non-projective direct summands of dimension $i$ when $i$ is in $S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of $M_S$. We decode the subset as a binary word so that for example the subset $S=\{1,3 \} $ of $\{1,2,3 \}$ is decoded as 101.