Your data matches 8 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001213
Mapping
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001213: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1,-2] => [1,1]
 => [1]
 => [1,0,1,0]
 => 4
[1,-2,-3] => [1,1]
 => [1]
 => [1,0,1,0]
 => 4
[-1,2,-3] => [1,1]
 => [1]
 => [1,0,1,0]
 => 4
[-1,-2,3] => [1,1]
 => [1]
 => [1,0,1,0]
 => 4
[-1,-2,-3] => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 5
[-1,3,-2] => [2,1]
 => [1]
 => [1,0,1,0]
 => 4
[-1,-3,2] => [2,1]
 => [1]
 => [1,0,1,0]
 => 4
[2,-1,-3] => [2,1]
 => [1]
 => [1,0,1,0]
 => 4
[-2,1,-3] => [2,1]
 => [1]
 => [1,0,1,0]
 => 4
[3,-2,-1] => [2,1]
 => [1]
 => [1,0,1,0]
 => 4
[-3,-2,1] => [2,1]
 => [1]
 => [1,0,1,0]
 => 4
[1,2,-3,-4] => [1,1]
 => [1]
 => [1,0,1,0]
 => 4
[1,-2,3,-4] => [1,1]
 => [1]
 => [1,0,1,0]
 => 4
[1,-2,-3,4] => [1,1]
 => [1]
 => [1,0,1,0]
 => 4
[1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 5
[-1,2,3,-4] => [1,1]
 => [1]
 => [1,0,1,0]
 => 4
[-1,2,-3,4] => [1,1]
 => [1]
 => [1,0,1,0]
 => 4
[-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 5
[-1,-2,3,4] => [1,1]
 => [1]
 => [1,0,1,0]
 => 4
[-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 5
[-1,-2,-3,4] => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 5
[-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 6
[1,-2,4,-3] => [2,1]
 => [1]
 => [1,0,1,0]
 => 4
[1,-2,-4,3] => [2,1]
 => [1]
 => [1,0,1,0]
 => 4
[-1,2,4,-3] => [2,1]
 => [1]
 => [1,0,1,0]
 => 4
[-1,2,-4,3] => [2,1]
 => [1]
 => [1,0,1,0]
 => 4
[-1,-2,4,3] => [1,1]
 => [1]
 => [1,0,1,0]
 => 4
[-1,-2,4,-3] => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 5
[-1,-2,-4,3] => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 5
[-1,-2,-4,-3] => [1,1]
 => [1]
 => [1,0,1,0]
 => 4
[1,3,-2,-4] => [2,1]
 => [1]
 => [1,0,1,0]
 => 4
[1,-3,2,-4] => [2,1]
 => [1]
 => [1,0,1,0]
 => 4
[-1,3,2,-4] => [1,1]
 => [1]
 => [1,0,1,0]
 => 4
[-1,3,-2,4] => [2,1]
 => [1]
 => [1,0,1,0]
 => 4
[-1,3,-2,-4] => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 5
[-1,-3,2,4] => [2,1]
 => [1]
 => [1,0,1,0]
 => 4
[-1,-3,2,-4] => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 5
[-1,-3,-2,-4] => [1,1]
 => [1]
 => [1,0,1,0]
 => 4
[-1,3,4,-2] => [3,1]
 => [1]
 => [1,0,1,0]
 => 4
[-1,3,-4,2] => [3,1]
 => [1]
 => [1,0,1,0]
 => 4
[-1,-3,4,2] => [3,1]
 => [1]
 => [1,0,1,0]
 => 4
[-1,-3,-4,-2] => [3,1]
 => [1]
 => [1,0,1,0]
 => 4
[-1,4,2,-3] => [3,1]
 => [1]
 => [1,0,1,0]
 => 4
[-1,4,-2,3] => [3,1]
 => [1]
 => [1,0,1,0]
 => 4
[-1,-4,2,3] => [3,1]
 => [1]
 => [1,0,1,0]
 => 4
[-1,-4,-2,-3] => [3,1]
 => [1]
 => [1,0,1,0]
 => 4
[1,4,-3,-2] => [2,1]
 => [1]
 => [1,0,1,0]
 => 4
[1,-4,-3,2] => [2,1]
 => [1]
 => [1,0,1,0]
 => 4
[-1,4,3,-2] => [2,1]
 => [1]
 => [1,0,1,0]
 => 4
[-1,4,-3,2] => [1,1]
 => [1]
 => [1,0,1,0]
 => 4
Description
The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module.
Matching statistic: St001232
(load all 4 compositions to match this statistic)
Mapping
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 80%
Values
[-1,-2] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 4 - 3
[1,-2,-3] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 4 - 3
[-1,2,-3] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 4 - 3
[-1,-2,3] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 4 - 3
[-1,-2,-3] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => ? = 5 - 3
[-1,3,-2] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 4 - 3
[-1,-3,2] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 4 - 3
[2,-1,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 4 - 3
[-2,1,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 4 - 3
[3,-2,-1] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 4 - 3
[-3,-2,1] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 4 - 3
[1,2,-3,-4] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 4 - 3
[1,-2,3,-4] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 4 - 3
[1,-2,-3,4] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 4 - 3
[1,-2,-3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => ? = 5 - 3
[-1,2,3,-4] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 4 - 3
[-1,2,-3,4] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 4 - 3
[-1,2,-3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => ? = 5 - 3
[-1,-2,3,4] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 4 - 3
[-1,-2,3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => ? = 5 - 3
[-1,-2,-3,4] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => ? = 5 - 3
[-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? = 6 - 3
[1,-2,4,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 4 - 3
[1,-2,-4,3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 4 - 3
[-1,2,4,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 4 - 3
[-1,2,-4,3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 4 - 3
[-1,-2,4,3] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 4 - 3
[-1,-2,4,-3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? = 5 - 3
[-1,-2,-4,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? = 5 - 3
[-1,-2,-4,-3] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 4 - 3
[1,3,-2,-4] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 4 - 3
[1,-3,2,-4] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 4 - 3
[-1,3,2,-4] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 4 - 3
[-1,3,-2,4] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 4 - 3
[-1,3,-2,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? = 5 - 3
[-1,-3,2,4] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 4 - 3
[-1,-3,2,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? = 5 - 3
[-1,-3,-2,-4] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 4 - 3
[-1,3,4,-2] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1 = 4 - 3
[-1,3,-4,2] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1 = 4 - 3
[-1,-3,4,2] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1 = 4 - 3
[-1,-3,-4,-2] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1 = 4 - 3
[-1,4,2,-3] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1 = 4 - 3
[-1,4,-2,3] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1 = 4 - 3
[-1,-4,2,3] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1 = 4 - 3
[-1,-4,-2,-3] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1 = 4 - 3
[1,4,-3,-2] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 4 - 3
[1,-4,-3,2] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 4 - 3
[-1,4,3,-2] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 4 - 3
[-1,4,-3,2] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 4 - 3
[-1,4,-3,-2] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? = 5 - 3
[-1,-4,3,2] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 4 - 3
[-1,-4,-3,2] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? = 5 - 3
[-1,-4,-3,-2] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 4 - 3
[2,1,-3,-4] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 4 - 3
[2,-1,3,-4] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 4 - 3
[2,-1,-3,4] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 4 - 3
[2,-1,-3,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? = 5 - 3
[-2,1,3,-4] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 4 - 3
[-2,1,-3,4] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 4 - 3
[-2,1,-3,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? = 5 - 3
[-2,-1,-3,-4] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 4 - 3
[2,-1,4,-3] => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2 = 5 - 3
[2,-1,-4,3] => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2 = 5 - 3
[3,-2,-1,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? = 5 - 3
[-3,-2,1,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? = 5 - 3
[4,-2,-3,-1] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? = 5 - 3
[-4,-2,-3,1] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? = 5 - 3
[1,2,-3,-4,-5] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => ? = 5 - 3
[1,-2,3,-4,-5] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => ? = 5 - 3
[1,-2,-3,4,-5] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => ? = 5 - 3
[1,-2,-3,-4,5] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => ? = 5 - 3
[1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? = 6 - 3
[-1,2,3,-4,-5] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => ? = 5 - 3
[-1,2,-3,4,-5] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => ? = 5 - 3
[-1,2,-3,-4,5] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => ? = 5 - 3
[-1,2,-3,-4,-5] => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? = 6 - 3
[-1,-2,3,4,-5] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => ? = 5 - 3
[-1,-2,3,-4,5] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => ? = 5 - 3
[-1,-2,3,-4,-5] => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? = 6 - 3
[-1,-2,-3,4,5] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => ? = 5 - 3
[-1,-2,-3,4,-5] => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? = 6 - 3
[-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? = 6 - 3
[-1,-2,-3,-4,-5] => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => ? = 7 - 3
[1,-2,-3,5,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? = 5 - 3
[1,-2,-3,-5,4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? = 5 - 3
[-1,2,-3,5,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? = 5 - 3
[-1,2,-3,-5,4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? = 5 - 3
[-1,-2,3,5,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? = 5 - 3
[-1,-2,3,-5,4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? = 5 - 3
[-1,-2,-3,5,4] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => ? = 5 - 3
[-1,-2,-3,5,-4] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? = 6 - 3
[-1,-2,-3,-5,4] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? = 6 - 3
[-1,-2,-3,-5,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => ? = 5 - 3
[1,-2,4,-3,-5] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? = 5 - 3
[1,-2,-4,3,-5] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? = 5 - 3
[-1,2,4,-3,-5] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? = 5 - 3
[-1,2,-4,3,-5] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? = 5 - 3
[-1,-2,4,3,-5] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => ? = 5 - 3
[-1,-2,4,-3,5] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? = 5 - 3
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001207
(load all 3 compositions to match this statistic)
Mapping
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 60% ●values known / values provided: 65%●distinct values known / distinct values provided: 60%
Values
[-1,-2] => [1,1]
 => [1,1,0,0]
 => [2,1] => 1 = 4 - 3
[1,-2,-3] => [1,1]
 => [1,1,0,0]
 => [2,1] => 1 = 4 - 3
[-1,2,-3] => [1,1]
 => [1,1,0,0]
 => [2,1] => 1 = 4 - 3
[-1,-2,3] => [1,1]
 => [1,1,0,0]
 => [2,1] => 1 = 4 - 3
[-1,-2,-3] => [1,1,1]
 => [1,1,0,1,0,0]
 => [2,3,1] => 2 = 5 - 3
[-1,3,-2] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1 = 4 - 3
[-1,-3,2] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1 = 4 - 3
[2,-1,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1 = 4 - 3
[-2,1,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1 = 4 - 3
[3,-2,-1] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1 = 4 - 3
[-3,-2,1] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1 = 4 - 3
[1,2,-3,-4] => [1,1]
 => [1,1,0,0]
 => [2,1] => 1 = 4 - 3
[1,-2,3,-4] => [1,1]
 => [1,1,0,0]
 => [2,1] => 1 = 4 - 3
[1,-2,-3,4] => [1,1]
 => [1,1,0,0]
 => [2,1] => 1 = 4 - 3
[1,-2,-3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => [2,3,1] => 2 = 5 - 3
[-1,2,3,-4] => [1,1]
 => [1,1,0,0]
 => [2,1] => 1 = 4 - 3
[-1,2,-3,4] => [1,1]
 => [1,1,0,0]
 => [2,1] => 1 = 4 - 3
[-1,2,-3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => [2,3,1] => 2 = 5 - 3
[-1,-2,3,4] => [1,1]
 => [1,1,0,0]
 => [2,1] => 1 = 4 - 3
[-1,-2,3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => [2,3,1] => 2 = 5 - 3
[-1,-2,-3,4] => [1,1,1]
 => [1,1,0,1,0,0]
 => [2,3,1] => 2 = 5 - 3
[-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 3 = 6 - 3
[1,-2,4,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1 = 4 - 3
[1,-2,-4,3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1 = 4 - 3
[-1,2,4,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1 = 4 - 3
[-1,2,-4,3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1 = 4 - 3
[-1,-2,4,3] => [1,1]
 => [1,1,0,0]
 => [2,1] => 1 = 4 - 3
[-1,-2,4,-3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 2 = 5 - 3
[-1,-2,-4,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 2 = 5 - 3
[-1,-2,-4,-3] => [1,1]
 => [1,1,0,0]
 => [2,1] => 1 = 4 - 3
[1,3,-2,-4] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1 = 4 - 3
[1,-3,2,-4] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1 = 4 - 3
[-1,3,2,-4] => [1,1]
 => [1,1,0,0]
 => [2,1] => 1 = 4 - 3
[-1,3,-2,4] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1 = 4 - 3
[-1,3,-2,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 2 = 5 - 3
[-1,-3,2,4] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1 = 4 - 3
[-1,-3,2,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 2 = 5 - 3
[-1,-3,-2,-4] => [1,1]
 => [1,1,0,0]
 => [2,1] => 1 = 4 - 3
[-1,3,4,-2] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 1 = 4 - 3
[-1,3,-4,2] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 1 = 4 - 3
[-1,-3,4,2] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 1 = 4 - 3
[-1,-3,-4,-2] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 1 = 4 - 3
[-1,4,2,-3] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 1 = 4 - 3
[-1,4,-2,3] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 1 = 4 - 3
[-1,-4,2,3] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 1 = 4 - 3
[-1,-4,-2,-3] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 1 = 4 - 3
[1,4,-3,-2] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1 = 4 - 3
[1,-4,-3,2] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1 = 4 - 3
[-1,4,3,-2] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1 = 4 - 3
[-1,4,-3,2] => [1,1]
 => [1,1,0,0]
 => [2,1] => 1 = 4 - 3
[-1,-2,-3,-4,-5] => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => ? = 7 - 3
[-1,-2,-3,5,-4] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => ? = 6 - 3
[-1,-2,-3,-5,4] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => ? = 6 - 3
[-1,-2,4,-3,-5] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => ? = 6 - 3
[-1,-2,-4,3,-5] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => ? = 6 - 3
[-1,-2,4,5,-3] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => ? = 5 - 3
[-1,-2,4,-5,3] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => ? = 5 - 3
[-1,-2,-4,5,3] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => ? = 5 - 3
[-1,-2,-4,-5,-3] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => ? = 5 - 3
[-1,-2,5,3,-4] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => ? = 5 - 3
[-1,-2,5,-3,4] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => ? = 5 - 3
[-1,-2,-5,3,4] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => ? = 5 - 3
[-1,-2,-5,-3,-4] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => ? = 5 - 3
[-1,-2,5,-4,-3] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => ? = 6 - 3
[-1,-2,-5,-4,3] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => ? = 6 - 3
[-1,3,-2,-4,-5] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => ? = 6 - 3
[-1,-3,2,-4,-5] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => ? = 6 - 3
[-1,3,4,-2,-5] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => ? = 5 - 3
[-1,3,-4,2,-5] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => ? = 5 - 3
[-1,-3,4,2,-5] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => ? = 5 - 3
[-1,-3,-4,-2,-5] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => ? = 5 - 3
[-1,3,4,5,-2] => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => ? = 4 - 3
[-1,3,4,-5,2] => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => ? = 4 - 3
[-1,3,-4,5,2] => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => ? = 4 - 3
[-1,3,-4,-5,-2] => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => ? = 4 - 3
[-1,-3,4,5,2] => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => ? = 4 - 3
[-1,-3,4,-5,-2] => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => ? = 4 - 3
[-1,-3,-4,5,-2] => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => ? = 4 - 3
[-1,-3,-4,-5,2] => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => ? = 4 - 3
[-1,3,5,2,-4] => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => ? = 4 - 3
[-1,3,5,-2,4] => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => ? = 4 - 3
[-1,3,-5,2,4] => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => ? = 4 - 3
[-1,3,-5,-2,-4] => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => ? = 4 - 3
[-1,-3,5,2,4] => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => ? = 4 - 3
[-1,-3,5,-2,-4] => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => ? = 4 - 3
[-1,-3,-5,2,-4] => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => ? = 4 - 3
[-1,-3,-5,-2,4] => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => ? = 4 - 3
[-1,3,5,-4,-2] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => ? = 5 - 3
[-1,3,-5,-4,2] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => ? = 5 - 3
[-1,-3,5,-4,2] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => ? = 5 - 3
[-1,-3,-5,-4,-2] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => ? = 5 - 3
[-1,4,2,-3,-5] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => ? = 5 - 3
[-1,4,-2,3,-5] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => ? = 5 - 3
[-1,-4,2,3,-5] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => ? = 5 - 3
[-1,-4,-2,-3,-5] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => ? = 5 - 3
[-1,4,2,5,-3] => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => ? = 4 - 3
[-1,4,2,-5,3] => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => ? = 4 - 3
[-1,4,-2,5,3] => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => ? = 4 - 3
[-1,4,-2,-5,-3] => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => ? = 4 - 3
[-1,-4,2,5,3] => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => ? = 4 - 3
Description
The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
Matching statistic: St001491
(load all 3 compositions to match this statistic)
Mapping
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 40%
Values
[-1,-2] => [1,1]
 => 110 => 110 => 1 = 4 - 3
[1,-2,-3] => [1,1]
 => 110 => 110 => 1 = 4 - 3
[-1,2,-3] => [1,1]
 => 110 => 110 => 1 = 4 - 3
[-1,-2,3] => [1,1]
 => 110 => 110 => 1 = 4 - 3
[-1,-2,-3] => [1,1,1]
 => 1110 => 1110 => 2 = 5 - 3
[-1,3,-2] => [2,1]
 => 1010 => 1100 => 1 = 4 - 3
[-1,-3,2] => [2,1]
 => 1010 => 1100 => 1 = 4 - 3
[2,-1,-3] => [2,1]
 => 1010 => 1100 => 1 = 4 - 3
[-2,1,-3] => [2,1]
 => 1010 => 1100 => 1 = 4 - 3
[3,-2,-1] => [2,1]
 => 1010 => 1100 => 1 = 4 - 3
[-3,-2,1] => [2,1]
 => 1010 => 1100 => 1 = 4 - 3
[1,2,-3,-4] => [1,1]
 => 110 => 110 => 1 = 4 - 3
[1,-2,3,-4] => [1,1]
 => 110 => 110 => 1 = 4 - 3
[1,-2,-3,4] => [1,1]
 => 110 => 110 => 1 = 4 - 3
[1,-2,-3,-4] => [1,1,1]
 => 1110 => 1110 => 2 = 5 - 3
[-1,2,3,-4] => [1,1]
 => 110 => 110 => 1 = 4 - 3
[-1,2,-3,4] => [1,1]
 => 110 => 110 => 1 = 4 - 3
[-1,2,-3,-4] => [1,1,1]
 => 1110 => 1110 => 2 = 5 - 3
[-1,-2,3,4] => [1,1]
 => 110 => 110 => 1 = 4 - 3
[-1,-2,3,-4] => [1,1,1]
 => 1110 => 1110 => 2 = 5 - 3
[-1,-2,-3,4] => [1,1,1]
 => 1110 => 1110 => 2 = 5 - 3
[-1,-2,-3,-4] => [1,1,1,1]
 => 11110 => 11110 => ? = 6 - 3
[1,-2,4,-3] => [2,1]
 => 1010 => 1100 => 1 = 4 - 3
[1,-2,-4,3] => [2,1]
 => 1010 => 1100 => 1 = 4 - 3
[-1,2,4,-3] => [2,1]
 => 1010 => 1100 => 1 = 4 - 3
[-1,2,-4,3] => [2,1]
 => 1010 => 1100 => 1 = 4 - 3
[-1,-2,4,3] => [1,1]
 => 110 => 110 => 1 = 4 - 3
[-1,-2,4,-3] => [2,1,1]
 => 10110 => 11010 => ? = 5 - 3
[-1,-2,-4,3] => [2,1,1]
 => 10110 => 11010 => ? = 5 - 3
[-1,-2,-4,-3] => [1,1]
 => 110 => 110 => 1 = 4 - 3
[1,3,-2,-4] => [2,1]
 => 1010 => 1100 => 1 = 4 - 3
[1,-3,2,-4] => [2,1]
 => 1010 => 1100 => 1 = 4 - 3
[-1,3,2,-4] => [1,1]
 => 110 => 110 => 1 = 4 - 3
[-1,3,-2,4] => [2,1]
 => 1010 => 1100 => 1 = 4 - 3
[-1,3,-2,-4] => [2,1,1]
 => 10110 => 11010 => ? = 5 - 3
[-1,-3,2,4] => [2,1]
 => 1010 => 1100 => 1 = 4 - 3
[-1,-3,2,-4] => [2,1,1]
 => 10110 => 11010 => ? = 5 - 3
[-1,-3,-2,-4] => [1,1]
 => 110 => 110 => 1 = 4 - 3
[-1,3,4,-2] => [3,1]
 => 10010 => 10100 => ? = 4 - 3
[-1,3,-4,2] => [3,1]
 => 10010 => 10100 => ? = 4 - 3
[-1,-3,4,2] => [3,1]
 => 10010 => 10100 => ? = 4 - 3
[-1,-3,-4,-2] => [3,1]
 => 10010 => 10100 => ? = 4 - 3
[-1,4,2,-3] => [3,1]
 => 10010 => 10100 => ? = 4 - 3
[-1,4,-2,3] => [3,1]
 => 10010 => 10100 => ? = 4 - 3
[-1,-4,2,3] => [3,1]
 => 10010 => 10100 => ? = 4 - 3
[-1,-4,-2,-3] => [3,1]
 => 10010 => 10100 => ? = 4 - 3
[1,4,-3,-2] => [2,1]
 => 1010 => 1100 => 1 = 4 - 3
[1,-4,-3,2] => [2,1]
 => 1010 => 1100 => 1 = 4 - 3
[-1,4,3,-2] => [2,1]
 => 1010 => 1100 => 1 = 4 - 3
[-1,4,-3,2] => [1,1]
 => 110 => 110 => 1 = 4 - 3
[-1,4,-3,-2] => [2,1,1]
 => 10110 => 11010 => ? = 5 - 3
[-1,-4,3,2] => [2,1]
 => 1010 => 1100 => 1 = 4 - 3
[-1,-4,-3,2] => [2,1,1]
 => 10110 => 11010 => ? = 5 - 3
[-1,-4,-3,-2] => [1,1]
 => 110 => 110 => 1 = 4 - 3
[2,1,-3,-4] => [1,1]
 => 110 => 110 => 1 = 4 - 3
[2,-1,3,-4] => [2,1]
 => 1010 => 1100 => 1 = 4 - 3
[2,-1,-3,4] => [2,1]
 => 1010 => 1100 => 1 = 4 - 3
[2,-1,-3,-4] => [2,1,1]
 => 10110 => 11010 => ? = 5 - 3
[-2,1,3,-4] => [2,1]
 => 1010 => 1100 => 1 = 4 - 3
[-2,1,-3,4] => [2,1]
 => 1010 => 1100 => 1 = 4 - 3
[-2,1,-3,-4] => [2,1,1]
 => 10110 => 11010 => ? = 5 - 3
[-2,-1,-3,-4] => [1,1]
 => 110 => 110 => 1 = 4 - 3
[2,-1,4,-3] => [2,2]
 => 1100 => 0110 => 2 = 5 - 3
[2,-1,-4,3] => [2,2]
 => 1100 => 0110 => 2 = 5 - 3
[-2,1,4,-3] => [2,2]
 => 1100 => 0110 => 2 = 5 - 3
[-2,1,-4,3] => [2,2]
 => 1100 => 0110 => 2 = 5 - 3
[2,3,-1,-4] => [3,1]
 => 10010 => 10100 => ? = 4 - 3
[2,-3,1,-4] => [3,1]
 => 10010 => 10100 => ? = 4 - 3
[-2,3,1,-4] => [3,1]
 => 10010 => 10100 => ? = 4 - 3
[-2,-3,-1,-4] => [3,1]
 => 10010 => 10100 => ? = 4 - 3
[2,4,-3,-1] => [3,1]
 => 10010 => 10100 => ? = 4 - 3
[2,-4,-3,1] => [3,1]
 => 10010 => 10100 => ? = 4 - 3
[-2,4,-3,1] => [3,1]
 => 10010 => 10100 => ? = 4 - 3
[-2,-4,-3,-1] => [3,1]
 => 10010 => 10100 => ? = 4 - 3
[3,1,-2,-4] => [3,1]
 => 10010 => 10100 => ? = 4 - 3
[3,-1,2,-4] => [3,1]
 => 10010 => 10100 => ? = 4 - 3
[-3,1,2,-4] => [3,1]
 => 10010 => 10100 => ? = 4 - 3
[-3,-1,-2,-4] => [3,1]
 => 10010 => 10100 => ? = 4 - 3
[3,2,-1,-4] => [2,1]
 => 1010 => 1100 => 1 = 4 - 3
[3,-2,-1,-4] => [2,1,1]
 => 10110 => 11010 => ? = 5 - 3
[-3,-2,1,-4] => [2,1,1]
 => 10110 => 11010 => ? = 5 - 3
[3,-2,4,-1] => [3,1]
 => 10010 => 10100 => ? = 4 - 3
[3,-2,-4,1] => [3,1]
 => 10010 => 10100 => ? = 4 - 3
[-3,-2,4,1] => [3,1]
 => 10010 => 10100 => ? = 4 - 3
[-3,-2,-4,-1] => [3,1]
 => 10010 => 10100 => ? = 4 - 3
[4,1,-3,-2] => [3,1]
 => 10010 => 10100 => ? = 4 - 3
[4,-1,-3,2] => [3,1]
 => 10010 => 10100 => ? = 4 - 3
[-4,1,-3,2] => [3,1]
 => 10010 => 10100 => ? = 4 - 3
[-4,-1,-3,-2] => [3,1]
 => 10010 => 10100 => ? = 4 - 3
[4,-2,1,-3] => [3,1]
 => 10010 => 10100 => ? = 4 - 3
[4,-2,-1,3] => [3,1]
 => 10010 => 10100 => ? = 4 - 3
[-4,-2,1,3] => [3,1]
 => 10010 => 10100 => ? = 4 - 3
[-4,-2,-1,-3] => [3,1]
 => 10010 => 10100 => ? = 4 - 3
[4,-2,-3,-1] => [2,1,1]
 => 10110 => 11010 => ? = 5 - 3
[-4,-2,-3,1] => [2,1,1]
 => 10110 => 11010 => ? = 5 - 3
[1,-2,-3,-4,-5] => [1,1,1,1]
 => 11110 => 11110 => ? = 6 - 3
[-1,2,-3,-4,-5] => [1,1,1,1]
 => 11110 => 11110 => ? = 6 - 3
[-1,-2,3,-4,-5] => [1,1,1,1]
 => 11110 => 11110 => ? = 6 - 3
[-1,-2,-3,4,-5] => [1,1,1,1]
 => 11110 => 11110 => ? = 6 - 3
[-1,-2,-3,-4,5] => [1,1,1,1]
 => 11110 => 11110 => ? = 6 - 3
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.
Matching statistic: St001583
Mapping
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001583: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 25%●distinct values known / distinct values provided: 20%
Values
[-1,-2] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3 = 4 - 1
[1,-2,-3] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3 = 4 - 1
[-1,2,-3] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3 = 4 - 1
[-1,-2,3] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3 = 4 - 1
[-1,-2,-3] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ? = 5 - 1
[-1,3,-2] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 4 - 1
[-1,-3,2] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 4 - 1
[2,-1,-3] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 4 - 1
[-2,1,-3] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 4 - 1
[3,-2,-1] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 4 - 1
[-3,-2,1] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 4 - 1
[1,2,-3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3 = 4 - 1
[1,-2,3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3 = 4 - 1
[1,-2,-3,4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3 = 4 - 1
[1,-2,-3,-4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ? = 5 - 1
[-1,2,3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3 = 4 - 1
[-1,2,-3,4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3 = 4 - 1
[-1,2,-3,-4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ? = 5 - 1
[-1,-2,3,4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3 = 4 - 1
[-1,-2,3,-4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ? = 5 - 1
[-1,-2,-3,4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ? = 5 - 1
[-1,-2,-3,-4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => ? = 6 - 1
[1,-2,4,-3] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 4 - 1
[1,-2,-4,3] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 4 - 1
[-1,2,4,-3] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 4 - 1
[-1,2,-4,3] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 4 - 1
[-1,-2,4,3] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3 = 4 - 1
[-1,-2,4,-3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 5 - 1
[-1,-2,-4,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 5 - 1
[-1,-2,-4,-3] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3 = 4 - 1
[1,3,-2,-4] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 4 - 1
[1,-3,2,-4] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 4 - 1
[-1,3,2,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3 = 4 - 1
[-1,3,-2,4] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 4 - 1
[-1,3,-2,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 5 - 1
[-1,-3,2,4] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 4 - 1
[-1,-3,2,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 5 - 1
[-1,-3,-2,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3 = 4 - 1
[-1,3,4,-2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4 - 1
[-1,3,-4,2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4 - 1
[-1,-3,4,2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4 - 1
[-1,-3,-4,-2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4 - 1
[-1,4,2,-3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4 - 1
[-1,4,-2,3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4 - 1
[-1,-4,2,3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4 - 1
[-1,-4,-2,-3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4 - 1
[1,4,-3,-2] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 4 - 1
[1,-4,-3,2] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 4 - 1
[-1,4,3,-2] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 4 - 1
[-1,4,-3,2] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3 = 4 - 1
[-1,4,-3,-2] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 5 - 1
[-1,-4,3,2] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 4 - 1
[-1,-4,-3,2] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 5 - 1
[-1,-4,-3,-2] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3 = 4 - 1
[2,1,-3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3 = 4 - 1
[2,-1,3,-4] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 4 - 1
[2,-1,-3,4] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 4 - 1
[2,-1,-3,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 5 - 1
[-2,1,3,-4] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 4 - 1
[-2,1,-3,4] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 4 - 1
[-2,1,-3,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 5 - 1
[-2,-1,-3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3 = 4 - 1
[2,-1,4,-3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ? = 5 - 1
[2,-1,-4,3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ? = 5 - 1
[-2,1,4,-3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ? = 5 - 1
[-2,1,-4,3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ? = 5 - 1
[2,3,-1,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4 - 1
[2,-3,1,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4 - 1
[-2,3,1,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4 - 1
[-2,-3,-1,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4 - 1
[2,4,-3,-1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4 - 1
[2,-4,-3,1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4 - 1
[-2,4,-3,1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4 - 1
[-2,-4,-3,-1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4 - 1
[3,1,-2,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4 - 1
[3,-1,2,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4 - 1
[-3,1,2,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4 - 1
[-3,-1,-2,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4 - 1
[3,2,-1,-4] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 4 - 1
[3,-2,1,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3 = 4 - 1
[3,-2,-1,4] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 4 - 1
[3,-2,-1,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 5 - 1
[-3,2,1,-4] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 4 - 1
[-3,-2,1,4] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 4 - 1
[-3,-2,1,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 5 - 1
[-3,-2,-1,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3 = 4 - 1
[3,-2,4,-1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4 - 1
[3,-2,-4,1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4 - 1
[-3,-2,4,1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4 - 1
[-3,-2,-4,-1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4 - 1
[3,4,-1,-2] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ? = 5 - 1
[3,-4,-1,2] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ? = 5 - 1
[-3,4,1,-2] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ? = 5 - 1
[-3,-4,1,2] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ? = 5 - 1
[4,1,-3,-2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4 - 1
[4,-1,-3,2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4 - 1
[4,2,-3,-1] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 4 - 1
[4,-2,3,-1] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 4 - 1
[4,-2,-3,1] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3 = 4 - 1
[-4,2,-3,1] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 4 - 1
Description
The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order.
Matching statistic: St000782
Mapping
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000782: Perfect matchings ⟶ ℤResult quality: 20% ●values known / values provided: 25%●distinct values known / distinct values provided: 20%
Values
[-1,-2] => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 4 - 3
[1,-2,-3] => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 4 - 3
[-1,2,-3] => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 4 - 3
[-1,-2,3] => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 4 - 3
[-1,-2,-3] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => ? = 5 - 3
[-1,3,-2] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 4 - 3
[-1,-3,2] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 4 - 3
[2,-1,-3] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 4 - 3
[-2,1,-3] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 4 - 3
[3,-2,-1] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 4 - 3
[-3,-2,1] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 4 - 3
[1,2,-3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 4 - 3
[1,-2,3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 4 - 3
[1,-2,-3,4] => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 4 - 3
[1,-2,-3,-4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => ? = 5 - 3
[-1,2,3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 4 - 3
[-1,2,-3,4] => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 4 - 3
[-1,2,-3,-4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => ? = 5 - 3
[-1,-2,3,4] => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 4 - 3
[-1,-2,3,-4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => ? = 5 - 3
[-1,-2,-3,4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => ? = 5 - 3
[-1,-2,-3,-4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,10),(4,9),(5,8),(6,7)]
 => ? = 6 - 3
[1,-2,4,-3] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 4 - 3
[1,-2,-4,3] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 4 - 3
[-1,2,4,-3] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 4 - 3
[-1,2,-4,3] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 4 - 3
[-1,-2,4,3] => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 4 - 3
[-1,-2,4,-3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => ? = 5 - 3
[-1,-2,-4,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => ? = 5 - 3
[-1,-2,-4,-3] => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 4 - 3
[1,3,-2,-4] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 4 - 3
[1,-3,2,-4] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 4 - 3
[-1,3,2,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 4 - 3
[-1,3,-2,4] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 4 - 3
[-1,3,-2,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => ? = 5 - 3
[-1,-3,2,4] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 4 - 3
[-1,-3,2,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => ? = 5 - 3
[-1,-3,-2,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 4 - 3
[-1,3,4,-2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 4 - 3
[-1,3,-4,2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 4 - 3
[-1,-3,4,2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 4 - 3
[-1,-3,-4,-2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 4 - 3
[-1,4,2,-3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 4 - 3
[-1,4,-2,3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 4 - 3
[-1,-4,2,3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 4 - 3
[-1,-4,-2,-3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 4 - 3
[1,4,-3,-2] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 4 - 3
[1,-4,-3,2] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 4 - 3
[-1,4,3,-2] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 4 - 3
[-1,4,-3,2] => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 4 - 3
[-1,4,-3,-2] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => ? = 5 - 3
[-1,-4,3,2] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 4 - 3
[-1,-4,-3,2] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => ? = 5 - 3
[-1,-4,-3,-2] => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 4 - 3
[2,1,-3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 4 - 3
[2,-1,3,-4] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 4 - 3
[2,-1,-3,4] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 4 - 3
[2,-1,-3,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => ? = 5 - 3
[-2,1,3,-4] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 4 - 3
[-2,1,-3,4] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 4 - 3
[-2,1,-3,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => ? = 5 - 3
[-2,-1,-3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 4 - 3
[2,-1,4,-3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => ? = 5 - 3
[2,-1,-4,3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => ? = 5 - 3
[-2,1,4,-3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => ? = 5 - 3
[-2,1,-4,3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => ? = 5 - 3
[2,3,-1,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 4 - 3
[2,-3,1,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 4 - 3
[-2,3,1,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 4 - 3
[-2,-3,-1,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 4 - 3
[2,4,-3,-1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 4 - 3
[2,-4,-3,1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 4 - 3
[-2,4,-3,1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 4 - 3
[-2,-4,-3,-1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 4 - 3
[3,1,-2,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 4 - 3
[3,-1,2,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 4 - 3
[-3,1,2,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 4 - 3
[-3,-1,-2,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 4 - 3
[3,2,-1,-4] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 4 - 3
[3,-2,1,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 4 - 3
[3,-2,-1,4] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 4 - 3
[3,-2,-1,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => ? = 5 - 3
[-3,2,1,-4] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 4 - 3
[-3,-2,1,4] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 4 - 3
[-3,-2,1,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => ? = 5 - 3
[-3,-2,-1,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 4 - 3
[3,-2,4,-1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 4 - 3
[3,-2,-4,1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 4 - 3
[-3,-2,4,1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 4 - 3
[-3,-2,-4,-1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 4 - 3
[3,4,-1,-2] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => ? = 5 - 3
[3,-4,-1,2] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => ? = 5 - 3
[-3,4,1,-2] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => ? = 5 - 3
[-3,-4,1,2] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => ? = 5 - 3
[4,1,-3,-2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 4 - 3
[4,-1,-3,2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 4 - 3
[4,2,-3,-1] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 4 - 3
[4,-2,3,-1] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 4 - 3
[4,-2,-3,1] => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 4 - 3
[-4,2,-3,1] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 4 - 3
Description
The indicator function of whether a given perfect matching is an L & P matching. An L&P matching is built inductively as follows: starting with either a single edge, or a hairpin $([1,3],[2,4])$, insert a noncrossing matching or inflate an edge by a ladder, that is, a number of nested edges. The number of L&P matchings is (see [thm. 1, 2]) $$\frac{1}{2} \cdot 4^{n} + \frac{1}{n + 1}{2 \, n \choose n} - {2 \, n + 1 \choose n} + {2 \, n - 1 \choose n - 1}$$
Matching statistic: St001722
Mapping
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001722: Binary words ⟶ ℤResult quality: 20% ●values known / values provided: 25%●distinct values known / distinct values provided: 20%
Values
[-1,-2] => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 4 - 3
[1,-2,-3] => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 4 - 3
[-1,2,-3] => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 4 - 3
[-1,-2,3] => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 4 - 3
[-1,-2,-3] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => ? = 5 - 3
[-1,3,-2] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 4 - 3
[-1,-3,2] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 4 - 3
[2,-1,-3] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 4 - 3
[-2,1,-3] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 4 - 3
[3,-2,-1] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 4 - 3
[-3,-2,1] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 4 - 3
[1,2,-3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 4 - 3
[1,-2,3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 4 - 3
[1,-2,-3,4] => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 4 - 3
[1,-2,-3,-4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => ? = 5 - 3
[-1,2,3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 4 - 3
[-1,2,-3,4] => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 4 - 3
[-1,2,-3,-4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => ? = 5 - 3
[-1,-2,3,4] => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 4 - 3
[-1,-2,3,-4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => ? = 5 - 3
[-1,-2,-3,4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => ? = 5 - 3
[-1,-2,-3,-4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => ? = 6 - 3
[1,-2,4,-3] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 4 - 3
[1,-2,-4,3] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 4 - 3
[-1,2,4,-3] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 4 - 3
[-1,2,-4,3] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 4 - 3
[-1,-2,4,3] => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 4 - 3
[-1,-2,4,-3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => ? = 5 - 3
[-1,-2,-4,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => ? = 5 - 3
[-1,-2,-4,-3] => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 4 - 3
[1,3,-2,-4] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 4 - 3
[1,-3,2,-4] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 4 - 3
[-1,3,2,-4] => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 4 - 3
[-1,3,-2,4] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 4 - 3
[-1,3,-2,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => ? = 5 - 3
[-1,-3,2,4] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 4 - 3
[-1,-3,2,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => ? = 5 - 3
[-1,-3,-2,-4] => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 4 - 3
[-1,3,4,-2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 4 - 3
[-1,3,-4,2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 4 - 3
[-1,-3,4,2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 4 - 3
[-1,-3,-4,-2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 4 - 3
[-1,4,2,-3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 4 - 3
[-1,4,-2,3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 4 - 3
[-1,-4,2,3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 4 - 3
[-1,-4,-2,-3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 4 - 3
[1,4,-3,-2] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 4 - 3
[1,-4,-3,2] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 4 - 3
[-1,4,3,-2] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 4 - 3
[-1,4,-3,2] => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 4 - 3
[-1,4,-3,-2] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => ? = 5 - 3
[-1,-4,3,2] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 4 - 3
[-1,-4,-3,2] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => ? = 5 - 3
[-1,-4,-3,-2] => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 4 - 3
[2,1,-3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 4 - 3
[2,-1,3,-4] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 4 - 3
[2,-1,-3,4] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 4 - 3
[2,-1,-3,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => ? = 5 - 3
[-2,1,3,-4] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 4 - 3
[-2,1,-3,4] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 4 - 3
[-2,1,-3,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => ? = 5 - 3
[-2,-1,-3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 4 - 3
[2,-1,4,-3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => ? = 5 - 3
[2,-1,-4,3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => ? = 5 - 3
[-2,1,4,-3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => ? = 5 - 3
[-2,1,-4,3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => ? = 5 - 3
[2,3,-1,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 4 - 3
[2,-3,1,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 4 - 3
[-2,3,1,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 4 - 3
[-2,-3,-1,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 4 - 3
[2,4,-3,-1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 4 - 3
[2,-4,-3,1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 4 - 3
[-2,4,-3,1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 4 - 3
[-2,-4,-3,-1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 4 - 3
[3,1,-2,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 4 - 3
[3,-1,2,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 4 - 3
[-3,1,2,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 4 - 3
[-3,-1,-2,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 4 - 3
[3,2,-1,-4] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 4 - 3
[3,-2,1,-4] => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 4 - 3
[3,-2,-1,4] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 4 - 3
[3,-2,-1,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => ? = 5 - 3
[-3,2,1,-4] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 4 - 3
[-3,-2,1,4] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 4 - 3
[-3,-2,1,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => ? = 5 - 3
[-3,-2,-1,-4] => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 4 - 3
[3,-2,4,-1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 4 - 3
[3,-2,-4,1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 4 - 3
[-3,-2,4,1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 4 - 3
[-3,-2,-4,-1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 4 - 3
[3,4,-1,-2] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => ? = 5 - 3
[3,-4,-1,2] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => ? = 5 - 3
[-3,4,1,-2] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => ? = 5 - 3
[-3,-4,1,2] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => ? = 5 - 3
[4,1,-3,-2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 4 - 3
[4,-1,-3,2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 4 - 3
[4,2,-3,-1] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 4 - 3
[4,-2,3,-1] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 4 - 3
[4,-2,-3,1] => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 4 - 3
[-4,2,-3,1] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 4 - 3
Description
The number of minimal chains with small intervals between a binary word and the top element. A valley in a binary word is a subsequence $01$, or a trailing $0$. A peak is a subsequence $10$ or a trailing $1$. Let $P$ be the lattice on binary words of length $n$, where the covering elements of a word are obtained by replacing a valley with a peak. An interval $[w_1, w_2]$ in $P$ is small if $w_2$ is obtained from $w_1$ by replacing some valleys with peaks. This statistic counts the number of chains $w = w_1 < \dots < w_d = 1\dots 1$ to the top element of minimal length. For example, there are two such chains for the word $0110$: $$ 0110 < 1011 < 1101 < 1110 < 1111 $$ and $$ 0110 < 1010 < 1101 < 1110 < 1111. $$
Matching statistic: St001816
Mapping
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001816: Standard tableaux ⟶ ℤResult quality: 20% ●values known / values provided: 25%●distinct values known / distinct values provided: 20%
Values
[-1,-2] => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 0 = 4 - 4
[1,-2,-3] => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 0 = 4 - 4
[-1,2,-3] => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 0 = 4 - 4
[-1,-2,3] => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 0 = 4 - 4
[-1,-2,-3] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => ? = 5 - 4
[-1,3,-2] => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0 = 4 - 4
[-1,-3,2] => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0 = 4 - 4
[2,-1,-3] => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0 = 4 - 4
[-2,1,-3] => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0 = 4 - 4
[3,-2,-1] => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0 = 4 - 4
[-3,-2,1] => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0 = 4 - 4
[1,2,-3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 0 = 4 - 4
[1,-2,3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 0 = 4 - 4
[1,-2,-3,4] => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 0 = 4 - 4
[1,-2,-3,-4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => ? = 5 - 4
[-1,2,3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 0 = 4 - 4
[-1,2,-3,4] => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 0 = 4 - 4
[-1,2,-3,-4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => ? = 5 - 4
[-1,-2,3,4] => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 0 = 4 - 4
[-1,-2,3,-4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => ? = 5 - 4
[-1,-2,-3,4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => ? = 5 - 4
[-1,-2,-3,-4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[1,3,4,5,6],[2,7,8,9,10]]
 => ? = 6 - 4
[1,-2,4,-3] => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0 = 4 - 4
[1,-2,-4,3] => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0 = 4 - 4
[-1,2,4,-3] => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0 = 4 - 4
[-1,2,-4,3] => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0 = 4 - 4
[-1,-2,4,3] => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 0 = 4 - 4
[-1,-2,4,-3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => ? = 5 - 4
[-1,-2,-4,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => ? = 5 - 4
[-1,-2,-4,-3] => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 0 = 4 - 4
[1,3,-2,-4] => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0 = 4 - 4
[1,-3,2,-4] => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0 = 4 - 4
[-1,3,2,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 0 = 4 - 4
[-1,3,-2,4] => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0 = 4 - 4
[-1,3,-2,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => ? = 5 - 4
[-1,-3,2,4] => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0 = 4 - 4
[-1,-3,2,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => ? = 5 - 4
[-1,-3,-2,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 0 = 4 - 4
[-1,3,4,-2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => ? = 4 - 4
[-1,3,-4,2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => ? = 4 - 4
[-1,-3,4,2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => ? = 4 - 4
[-1,-3,-4,-2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => ? = 4 - 4
[-1,4,2,-3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => ? = 4 - 4
[-1,4,-2,3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => ? = 4 - 4
[-1,-4,2,3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => ? = 4 - 4
[-1,-4,-2,-3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => ? = 4 - 4
[1,4,-3,-2] => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0 = 4 - 4
[1,-4,-3,2] => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0 = 4 - 4
[-1,4,3,-2] => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0 = 4 - 4
[-1,4,-3,2] => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 0 = 4 - 4
[-1,4,-3,-2] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => ? = 5 - 4
[-1,-4,3,2] => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0 = 4 - 4
[-1,-4,-3,2] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => ? = 5 - 4
[-1,-4,-3,-2] => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 0 = 4 - 4
[2,1,-3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 0 = 4 - 4
[2,-1,3,-4] => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0 = 4 - 4
[2,-1,-3,4] => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0 = 4 - 4
[2,-1,-3,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => ? = 5 - 4
[-2,1,3,-4] => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0 = 4 - 4
[-2,1,-3,4] => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0 = 4 - 4
[-2,1,-3,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => ? = 5 - 4
[-2,-1,-3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 0 = 4 - 4
[2,-1,4,-3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [[1,2,5,6],[3,4,7,8]]
 => ? = 5 - 4
[2,-1,-4,3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [[1,2,5,6],[3,4,7,8]]
 => ? = 5 - 4
[-2,1,4,-3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [[1,2,5,6],[3,4,7,8]]
 => ? = 5 - 4
[-2,1,-4,3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [[1,2,5,6],[3,4,7,8]]
 => ? = 5 - 4
[2,3,-1,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => ? = 4 - 4
[2,-3,1,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => ? = 4 - 4
[-2,3,1,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => ? = 4 - 4
[-2,-3,-1,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => ? = 4 - 4
[2,4,-3,-1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => ? = 4 - 4
[2,-4,-3,1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => ? = 4 - 4
[-2,4,-3,1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => ? = 4 - 4
[-2,-4,-3,-1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => ? = 4 - 4
[3,1,-2,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => ? = 4 - 4
[3,-1,2,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => ? = 4 - 4
[-3,1,2,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => ? = 4 - 4
[-3,-1,-2,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => ? = 4 - 4
[3,2,-1,-4] => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0 = 4 - 4
[3,-2,1,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 0 = 4 - 4
[3,-2,-1,4] => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0 = 4 - 4
[3,-2,-1,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => ? = 5 - 4
[-3,2,1,-4] => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0 = 4 - 4
[-3,-2,1,4] => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0 = 4 - 4
[-3,-2,1,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => ? = 5 - 4
[-3,-2,-1,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 0 = 4 - 4
[3,-2,4,-1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => ? = 4 - 4
[3,-2,-4,1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => ? = 4 - 4
[-3,-2,4,1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => ? = 4 - 4
[-3,-2,-4,-1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => ? = 4 - 4
[3,4,-1,-2] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [[1,2,5,6],[3,4,7,8]]
 => ? = 5 - 4
[3,-4,-1,2] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [[1,2,5,6],[3,4,7,8]]
 => ? = 5 - 4
[-3,4,1,-2] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [[1,2,5,6],[3,4,7,8]]
 => ? = 5 - 4
[-3,-4,1,2] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [[1,2,5,6],[3,4,7,8]]
 => ? = 5 - 4
[4,1,-3,-2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => ? = 4 - 4
[4,-1,-3,2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => ? = 4 - 4
[4,2,-3,-1] => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0 = 4 - 4
[4,-2,3,-1] => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0 = 4 - 4
[4,-2,-3,1] => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 0 = 4 - 4
[-4,2,-3,1] => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0 = 4 - 4
Description
Eigenvalues of the top-to-random operator acting on a simple module. These eigenvalues are given in [1] and [3]. The simple module of the symmetric group indexed by a partition $\lambda$ has dimension equal to the number of standard tableaux of shape $\lambda$. Hence, the eigenvalues of any linear operator defined on this module can be indexed by standard tableaux of shape $\lambda$; this statistic gives all the eigenvalues of the operator acting on the module. This statistic bears different names, such as the type in [2] or eig in [3]. Similarly, the eigenvalues of the random-to-random operator acting on a simple module is [[St000508]].