Your data matches 3 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000480
(load all 4 compositions to match this statistic)
Mapping
St000480: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 0
[2]
 => 1
[1,1]
 => 0
[3]
 => 1
[2,1]
 => 1
[1,1,1]
 => 0
[4]
 => 1
[3,1]
 => 1
[2,2]
 => 1
[2,1,1]
 => 1
[1,1,1,1]
 => 0
[5]
 => 1
[4,1]
 => 1
[3,2]
 => 1
[3,1,1]
 => 1
[2,2,1]
 => 1
[2,1,1,1]
 => 1
[1,1,1,1,1]
 => 0
[6]
 => 1
[5,1]
 => 1
[4,2]
 => 2
[4,1,1]
 => 1
[3,3]
 => 1
[3,2,1]
 => 2
[3,1,1,1]
 => 1
[2,2,2]
 => 1
[2,2,1,1]
 => 1
[2,1,1,1,1]
 => 1
[1,1,1,1,1,1]
 => 0
[7]
 => 1
[6,1]
 => 1
[5,2]
 => 2
[5,1,1]
 => 1
[4,3]
 => 1
[4,2,1]
 => 2
[4,1,1,1]
 => 1
[3,3,1]
 => 1
[3,2,2]
 => 1
[3,2,1,1]
 => 2
[3,1,1,1,1]
 => 1
[2,2,2,1]
 => 1
[2,2,1,1,1]
 => 1
[2,1,1,1,1,1]
 => 1
[1,1,1,1,1,1,1]
 => 0
[8]
 => 1
[7,1]
 => 1
[6,2]
 => 2
[6,1,1]
 => 1
[5,3]
 => 2
[5,2,1]
 => 2
Description
The number of lower covers of a partition in dominance order. According to [1], Corollary 2.4, the maximum number of elements one element (apparently for $n\neq 2$) can cover is $$ \frac{1}{2}(\sqrt{1+8n}-3) $$ and an element which covers this number of elements is given by $(c+i,c,c-1,\dots,3,2,1)$, where $1\leq i\leq c+2$.
Matching statistic: St000481
Mapping
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000481: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1]
 => 0
[2]
 => [1,1]
 => 1
[1,1]
 => [2]
 => 0
[3]
 => [1,1,1]
 => 1
[2,1]
 => [2,1]
 => 1
[1,1,1]
 => [3]
 => 0
[4]
 => [1,1,1,1]
 => 1
[3,1]
 => [2,1,1]
 => 1
[2,2]
 => [2,2]
 => 1
[2,1,1]
 => [3,1]
 => 1
[1,1,1,1]
 => [4]
 => 0
[5]
 => [1,1,1,1,1]
 => 1
[4,1]
 => [2,1,1,1]
 => 1
[3,2]
 => [2,2,1]
 => 1
[3,1,1]
 => [3,1,1]
 => 1
[2,2,1]
 => [3,2]
 => 1
[2,1,1,1]
 => [4,1]
 => 1
[1,1,1,1,1]
 => [5]
 => 0
[6]
 => [1,1,1,1,1,1]
 => 1
[5,1]
 => [2,1,1,1,1]
 => 1
[4,2]
 => [2,2,1,1]
 => 2
[4,1,1]
 => [3,1,1,1]
 => 1
[3,3]
 => [2,2,2]
 => 1
[3,2,1]
 => [3,2,1]
 => 2
[3,1,1,1]
 => [4,1,1]
 => 1
[2,2,2]
 => [3,3]
 => 1
[2,2,1,1]
 => [4,2]
 => 1
[2,1,1,1,1]
 => [5,1]
 => 1
[1,1,1,1,1,1]
 => [6]
 => 0
[7]
 => [1,1,1,1,1,1,1]
 => 1
[6,1]
 => [2,1,1,1,1,1]
 => 1
[5,2]
 => [2,2,1,1,1]
 => 2
[5,1,1]
 => [3,1,1,1,1]
 => 1
[4,3]
 => [2,2,2,1]
 => 1
[4,2,1]
 => [3,2,1,1]
 => 2
[4,1,1,1]
 => [4,1,1,1]
 => 1
[3,3,1]
 => [3,2,2]
 => 1
[3,2,2]
 => [3,3,1]
 => 1
[3,2,1,1]
 => [4,2,1]
 => 2
[3,1,1,1,1]
 => [5,1,1]
 => 1
[2,2,2,1]
 => [4,3]
 => 1
[2,2,1,1,1]
 => [5,2]
 => 1
[2,1,1,1,1,1]
 => [6,1]
 => 1
[1,1,1,1,1,1,1]
 => [7]
 => 0
[8]
 => [1,1,1,1,1,1,1,1]
 => 1
[7,1]
 => [2,1,1,1,1,1,1]
 => 1
[6,2]
 => [2,2,1,1,1,1]
 => 2
[6,1,1]
 => [3,1,1,1,1,1]
 => 1
[5,3]
 => [2,2,2,1,1]
 => 2
[5,2,1]
 => [3,2,1,1,1]
 => 2
Description
The number of upper covers of a partition in dominance order.
Matching statistic: St001195
(load all 3 compositions to match this statistic)
Mapping
Mp00323: Integer partitions —Loehr-Warrington inverse⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Values
[1]
 => [1]
 => [1,0]
 => ? = 0
[2]
 => [1,1]
 => [1,1,0,0]
 => ? = 1
[1,1]
 => [2]
 => [1,0,1,0]
 => ? = 0
[3]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[2,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => 0
[4]
 => [2,2]
 => [1,1,1,0,0,0]
 => 1
[3,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 1
[2,2]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
[2,1,1]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 1
[1,1,1,1]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0
[5]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1
[4,1]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
[3,2]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
[3,1,1]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1
[2,2,1]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 1
[2,1,1,1]
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,1,1,1,1]
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[6]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 1
[5,1]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1
[4,2]
 => [2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => ? = 2
[4,1,1]
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1
[3,3]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1
[3,2,1]
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2
[3,1,1,1]
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 1
[2,2,2]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 1
[2,2,1,1]
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
[2,1,1,1,1]
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[1,1,1,1,1,1]
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[7]
 => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1
[6,1]
 => [3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1
[5,2]
 => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => ? = 2
[5,1,1]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => ? = 1
[4,3]
 => [2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => ? = 1
[4,2,1]
 => [1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2
[4,1,1,1]
 => [2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[3,3,1]
 => [2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
[3,2,2]
 => [3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => ? = 1
[3,2,1,1]
 => [4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 2
[3,1,1,1,1]
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 1
[2,2,2,1]
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[2,2,1,1,1]
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 1
[2,1,1,1,1,1]
 => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[1,1,1,1,1,1,1]
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[8]
 => [4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[7,1]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => ? = 1
[6,2]
 => [3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => ? = 2
[6,1,1]
 => [4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 1
[5,3]
 => [2,2,2,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => ? = 2
[5,2,1]
 => [4,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => ? = 2
[5,1,1,1]
 => [2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1
[4,4]
 => [4,2,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => ? = 1
[4,3,1]
 => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
[4,2,2]
 => [2,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2
[4,2,1,1]
 => [3,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2
[4,1,1,1,1]
 => [5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => ? = 1
[3,3,2]
 => [2,2,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
[3,3,1,1]
 => [3,2,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => ? = 1
[3,2,2,1]
 => [3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => ? = 2
[3,2,1,1,1]
 => [5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 2
[3,1,1,1,1,1]
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 1
[2,2,2,2]
 => [3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[2,2,2,1,1]
 => [5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => ? = 1
[2,2,1,1,1,1]
 => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 1
[2,1,1,1,1,1,1]
 => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[1,1,1,1,1,1,1,1]
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[9]
 => [5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => ? = 1
[8,1]
 => [4,4,1]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => ? = 1
[7,2]
 => [4,3,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => ? = 2
[7,1,1]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => ? = 1
[6,3]
 => [4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => ? = 2
[6,2,1]
 => [4,2,1,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => ? = 2
[6,1,1,1]
 => [5,2,2]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 1
[5,4]
 => [2,2,2,2,1]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 1
[5,3,1]
 => [3,1,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2
[5,2,2]
 => [3,2,1,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 2
[5,2,1,1]
 => [3,3,1,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => ? = 2
[2,2,2,2,1]
 => [3,3,3]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
Description
The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$.