Your data matches 11 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001384
Mapping
Mp00057: Parking functions —to touch composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001384: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,1] => [1,1]
 => [1]
 => 0
[2,1] => [1,1] => [1,1]
 => [1]
 => 0
[1,1,3] => [2,1] => [2,1]
 => [1]
 => 0
[1,3,1] => [2,1] => [2,1]
 => [1]
 => 0
[3,1,1] => [2,1] => [2,1]
 => [1]
 => 0
[1,2,2] => [1,2] => [2,1]
 => [1]
 => 0
[2,1,2] => [1,2] => [2,1]
 => [1]
 => 0
[2,2,1] => [1,2] => [2,1]
 => [1]
 => 0
[1,2,3] => [1,1,1] => [1,1,1]
 => [1,1]
 => 1
[1,3,2] => [1,1,1] => [1,1,1]
 => [1,1]
 => 1
[2,1,3] => [1,1,1] => [1,1,1]
 => [1,1]
 => 1
[2,3,1] => [1,1,1] => [1,1,1]
 => [1,1]
 => 1
[3,1,2] => [1,1,1] => [1,1,1]
 => [1,1]
 => 1
[3,2,1] => [1,1,1] => [1,1,1]
 => [1,1]
 => 1
[1,1,1,4] => [3,1] => [3,1]
 => [1]
 => 0
[1,1,4,1] => [3,1] => [3,1]
 => [1]
 => 0
[1,4,1,1] => [3,1] => [3,1]
 => [1]
 => 0
[4,1,1,1] => [3,1] => [3,1]
 => [1]
 => 0
[1,1,2,4] => [3,1] => [3,1]
 => [1]
 => 0
[1,1,4,2] => [3,1] => [3,1]
 => [1]
 => 0
[1,2,1,4] => [3,1] => [3,1]
 => [1]
 => 0
[1,2,4,1] => [3,1] => [3,1]
 => [1]
 => 0
[1,4,1,2] => [3,1] => [3,1]
 => [1]
 => 0
[1,4,2,1] => [3,1] => [3,1]
 => [1]
 => 0
[2,1,1,4] => [3,1] => [3,1]
 => [1]
 => 0
[2,1,4,1] => [3,1] => [3,1]
 => [1]
 => 0
[2,4,1,1] => [3,1] => [3,1]
 => [1]
 => 0
[4,1,1,2] => [3,1] => [3,1]
 => [1]
 => 0
[4,1,2,1] => [3,1] => [3,1]
 => [1]
 => 0
[4,2,1,1] => [3,1] => [3,1]
 => [1]
 => 0
[1,1,3,3] => [2,2] => [2,2]
 => [2]
 => 1
[1,3,1,3] => [2,2] => [2,2]
 => [2]
 => 1
[1,3,3,1] => [2,2] => [2,2]
 => [2]
 => 1
[3,1,1,3] => [2,2] => [2,2]
 => [2]
 => 1
[3,1,3,1] => [2,2] => [2,2]
 => [2]
 => 1
[3,3,1,1] => [2,2] => [2,2]
 => [2]
 => 1
[1,1,3,4] => [2,1,1] => [2,1,1]
 => [1,1]
 => 1
[1,1,4,3] => [2,1,1] => [2,1,1]
 => [1,1]
 => 1
[1,3,1,4] => [2,1,1] => [2,1,1]
 => [1,1]
 => 1
[1,3,4,1] => [2,1,1] => [2,1,1]
 => [1,1]
 => 1
[1,4,1,3] => [2,1,1] => [2,1,1]
 => [1,1]
 => 1
[1,4,3,1] => [2,1,1] => [2,1,1]
 => [1,1]
 => 1
[3,1,1,4] => [2,1,1] => [2,1,1]
 => [1,1]
 => 1
[3,1,4,1] => [2,1,1] => [2,1,1]
 => [1,1]
 => 1
[3,4,1,1] => [2,1,1] => [2,1,1]
 => [1,1]
 => 1
[4,1,1,3] => [2,1,1] => [2,1,1]
 => [1,1]
 => 1
[4,1,3,1] => [2,1,1] => [2,1,1]
 => [1,1]
 => 1
[4,3,1,1] => [2,1,1] => [2,1,1]
 => [1,1]
 => 1
[1,2,2,2] => [1,3] => [3,1]
 => [1]
 => 0
[2,1,2,2] => [1,3] => [3,1]
 => [1]
 => 0
Description
The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains.
Matching statistic: St000531
Mapping
Mp00057: Parking functions —to touch composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000531: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,1] => [1,1]
 => [1]
 => 1 = 0 + 1
[2,1] => [1,1] => [1,1]
 => [1]
 => 1 = 0 + 1
[1,1,3] => [2,1] => [2,1]
 => [1]
 => 1 = 0 + 1
[1,3,1] => [2,1] => [2,1]
 => [1]
 => 1 = 0 + 1
[3,1,1] => [2,1] => [2,1]
 => [1]
 => 1 = 0 + 1
[1,2,2] => [1,2] => [2,1]
 => [1]
 => 1 = 0 + 1
[2,1,2] => [1,2] => [2,1]
 => [1]
 => 1 = 0 + 1
[2,2,1] => [1,2] => [2,1]
 => [1]
 => 1 = 0 + 1
[1,2,3] => [1,1,1] => [1,1,1]
 => [1,1]
 => 2 = 1 + 1
[1,3,2] => [1,1,1] => [1,1,1]
 => [1,1]
 => 2 = 1 + 1
[2,1,3] => [1,1,1] => [1,1,1]
 => [1,1]
 => 2 = 1 + 1
[2,3,1] => [1,1,1] => [1,1,1]
 => [1,1]
 => 2 = 1 + 1
[3,1,2] => [1,1,1] => [1,1,1]
 => [1,1]
 => 2 = 1 + 1
[3,2,1] => [1,1,1] => [1,1,1]
 => [1,1]
 => 2 = 1 + 1
[1,1,1,4] => [3,1] => [3,1]
 => [1]
 => 1 = 0 + 1
[1,1,4,1] => [3,1] => [3,1]
 => [1]
 => 1 = 0 + 1
[1,4,1,1] => [3,1] => [3,1]
 => [1]
 => 1 = 0 + 1
[4,1,1,1] => [3,1] => [3,1]
 => [1]
 => 1 = 0 + 1
[1,1,2,4] => [3,1] => [3,1]
 => [1]
 => 1 = 0 + 1
[1,1,4,2] => [3,1] => [3,1]
 => [1]
 => 1 = 0 + 1
[1,2,1,4] => [3,1] => [3,1]
 => [1]
 => 1 = 0 + 1
[1,2,4,1] => [3,1] => [3,1]
 => [1]
 => 1 = 0 + 1
[1,4,1,2] => [3,1] => [3,1]
 => [1]
 => 1 = 0 + 1
[1,4,2,1] => [3,1] => [3,1]
 => [1]
 => 1 = 0 + 1
[2,1,1,4] => [3,1] => [3,1]
 => [1]
 => 1 = 0 + 1
[2,1,4,1] => [3,1] => [3,1]
 => [1]
 => 1 = 0 + 1
[2,4,1,1] => [3,1] => [3,1]
 => [1]
 => 1 = 0 + 1
[4,1,1,2] => [3,1] => [3,1]
 => [1]
 => 1 = 0 + 1
[4,1,2,1] => [3,1] => [3,1]
 => [1]
 => 1 = 0 + 1
[4,2,1,1] => [3,1] => [3,1]
 => [1]
 => 1 = 0 + 1
[1,1,3,3] => [2,2] => [2,2]
 => [2]
 => 2 = 1 + 1
[1,3,1,3] => [2,2] => [2,2]
 => [2]
 => 2 = 1 + 1
[1,3,3,1] => [2,2] => [2,2]
 => [2]
 => 2 = 1 + 1
[3,1,1,3] => [2,2] => [2,2]
 => [2]
 => 2 = 1 + 1
[3,1,3,1] => [2,2] => [2,2]
 => [2]
 => 2 = 1 + 1
[3,3,1,1] => [2,2] => [2,2]
 => [2]
 => 2 = 1 + 1
[1,1,3,4] => [2,1,1] => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[1,1,4,3] => [2,1,1] => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[1,3,1,4] => [2,1,1] => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[1,3,4,1] => [2,1,1] => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[1,4,1,3] => [2,1,1] => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[1,4,3,1] => [2,1,1] => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[3,1,1,4] => [2,1,1] => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[3,1,4,1] => [2,1,1] => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[3,4,1,1] => [2,1,1] => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[4,1,1,3] => [2,1,1] => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[4,1,3,1] => [2,1,1] => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[4,3,1,1] => [2,1,1] => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[1,2,2,2] => [1,3] => [3,1]
 => [1]
 => 1 = 0 + 1
[2,1,2,2] => [1,3] => [3,1]
 => [1]
 => 1 = 0 + 1
Description
The leading coefficient of the rook polynomial of an integer partition. Let $m$ be the minimum of the number of parts and the size of the first part of an integer partition $\lambda$. Then this statistic yields the number of ways to place $m$ non-attacking rooks on the Ferrers board of $\lambda$.
Matching statistic: St001659
Mapping
Mp00057: Parking functions —to touch composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001659: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,1] => [1,1]
 => [1]
 => 1 = 0 + 1
[2,1] => [1,1] => [1,1]
 => [1]
 => 1 = 0 + 1
[1,1,3] => [2,1] => [2,1]
 => [1]
 => 1 = 0 + 1
[1,3,1] => [2,1] => [2,1]
 => [1]
 => 1 = 0 + 1
[3,1,1] => [2,1] => [2,1]
 => [1]
 => 1 = 0 + 1
[1,2,2] => [1,2] => [2,1]
 => [1]
 => 1 = 0 + 1
[2,1,2] => [1,2] => [2,1]
 => [1]
 => 1 = 0 + 1
[2,2,1] => [1,2] => [2,1]
 => [1]
 => 1 = 0 + 1
[1,2,3] => [1,1,1] => [1,1,1]
 => [1,1]
 => 2 = 1 + 1
[1,3,2] => [1,1,1] => [1,1,1]
 => [1,1]
 => 2 = 1 + 1
[2,1,3] => [1,1,1] => [1,1,1]
 => [1,1]
 => 2 = 1 + 1
[2,3,1] => [1,1,1] => [1,1,1]
 => [1,1]
 => 2 = 1 + 1
[3,1,2] => [1,1,1] => [1,1,1]
 => [1,1]
 => 2 = 1 + 1
[3,2,1] => [1,1,1] => [1,1,1]
 => [1,1]
 => 2 = 1 + 1
[1,1,1,4] => [3,1] => [3,1]
 => [1]
 => 1 = 0 + 1
[1,1,4,1] => [3,1] => [3,1]
 => [1]
 => 1 = 0 + 1
[1,4,1,1] => [3,1] => [3,1]
 => [1]
 => 1 = 0 + 1
[4,1,1,1] => [3,1] => [3,1]
 => [1]
 => 1 = 0 + 1
[1,1,2,4] => [3,1] => [3,1]
 => [1]
 => 1 = 0 + 1
[1,1,4,2] => [3,1] => [3,1]
 => [1]
 => 1 = 0 + 1
[1,2,1,4] => [3,1] => [3,1]
 => [1]
 => 1 = 0 + 1
[1,2,4,1] => [3,1] => [3,1]
 => [1]
 => 1 = 0 + 1
[1,4,1,2] => [3,1] => [3,1]
 => [1]
 => 1 = 0 + 1
[1,4,2,1] => [3,1] => [3,1]
 => [1]
 => 1 = 0 + 1
[2,1,1,4] => [3,1] => [3,1]
 => [1]
 => 1 = 0 + 1
[2,1,4,1] => [3,1] => [3,1]
 => [1]
 => 1 = 0 + 1
[2,4,1,1] => [3,1] => [3,1]
 => [1]
 => 1 = 0 + 1
[4,1,1,2] => [3,1] => [3,1]
 => [1]
 => 1 = 0 + 1
[4,1,2,1] => [3,1] => [3,1]
 => [1]
 => 1 = 0 + 1
[4,2,1,1] => [3,1] => [3,1]
 => [1]
 => 1 = 0 + 1
[1,1,3,3] => [2,2] => [2,2]
 => [2]
 => 2 = 1 + 1
[1,3,1,3] => [2,2] => [2,2]
 => [2]
 => 2 = 1 + 1
[1,3,3,1] => [2,2] => [2,2]
 => [2]
 => 2 = 1 + 1
[3,1,1,3] => [2,2] => [2,2]
 => [2]
 => 2 = 1 + 1
[3,1,3,1] => [2,2] => [2,2]
 => [2]
 => 2 = 1 + 1
[3,3,1,1] => [2,2] => [2,2]
 => [2]
 => 2 = 1 + 1
[1,1,3,4] => [2,1,1] => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[1,1,4,3] => [2,1,1] => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[1,3,1,4] => [2,1,1] => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[1,3,4,1] => [2,1,1] => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[1,4,1,3] => [2,1,1] => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[1,4,3,1] => [2,1,1] => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[3,1,1,4] => [2,1,1] => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[3,1,4,1] => [2,1,1] => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[3,4,1,1] => [2,1,1] => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[4,1,1,3] => [2,1,1] => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[4,1,3,1] => [2,1,1] => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[4,3,1,1] => [2,1,1] => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[1,2,2,2] => [1,3] => [3,1]
 => [1]
 => 1 = 0 + 1
[2,1,2,2] => [1,3] => [3,1]
 => [1]
 => 1 = 0 + 1
Description
The number of ways to place as many non-attacking rooks as possible on a Ferrers board.
Matching statistic: St001330
(load all 2 compositions to match this statistic)
Mapping
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,0,1,0]
 => [2,1] => ([(0,1)],2)
 => 2 = 0 + 2
[2,1] => [1,0,1,0]
 => [2,1] => ([(0,1)],2)
 => 2 = 0 + 2
[1,1,3] => [1,1,0,0,1,0]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 2 = 0 + 2
[1,3,1] => [1,1,0,0,1,0]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 2 = 0 + 2
[3,1,1] => [1,1,0,0,1,0]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 2 = 0 + 2
[1,2,2] => [1,0,1,1,0,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 2 = 0 + 2
[2,1,2] => [1,0,1,1,0,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 2 = 0 + 2
[2,2,1] => [1,0,1,1,0,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 2 = 0 + 2
[1,2,3] => [1,0,1,0,1,0]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 1 + 2
[1,3,2] => [1,0,1,0,1,0]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 1 + 2
[2,1,3] => [1,0,1,0,1,0]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 1 + 2
[2,3,1] => [1,0,1,0,1,0]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 1 + 2
[3,1,2] => [1,0,1,0,1,0]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 1 + 2
[3,2,1] => [1,0,1,0,1,0]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 1 + 2
[1,1,1,4] => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 0 + 2
[1,1,4,1] => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 0 + 2
[1,4,1,1] => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 0 + 2
[4,1,1,1] => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 0 + 2
[1,1,2,4] => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 2
[1,1,4,2] => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 2
[1,2,1,4] => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 2
[1,2,4,1] => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 2
[1,4,1,2] => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 2
[1,4,2,1] => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 2
[2,1,1,4] => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 2
[2,1,4,1] => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 2
[2,4,1,1] => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 2
[4,1,1,2] => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 2
[4,1,2,1] => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 2
[4,2,1,1] => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 2
[1,1,3,3] => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 3 = 1 + 2
[1,3,1,3] => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 3 = 1 + 2
[1,3,3,1] => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 3 = 1 + 2
[3,1,1,3] => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 3 = 1 + 2
[3,1,3,1] => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 3 = 1 + 2
[3,3,1,1] => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 3 = 1 + 2
[1,1,3,4] => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
[1,1,4,3] => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
[1,3,1,4] => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
[1,3,4,1] => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
[1,4,1,3] => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
[1,4,3,1] => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
[3,1,1,4] => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
[3,1,4,1] => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
[3,4,1,1] => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
[4,1,1,3] => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
[4,1,3,1] => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
[4,3,1,1] => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
[1,2,2,2] => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 0 + 2
[2,1,2,2] => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 0 + 2
[2,2,1,2] => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 0 + 2
[2,2,2,1] => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 0 + 2
[1,2,2,3] => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 2
[1,2,3,2] => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 2
[1,3,2,2] => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 2
[2,1,2,3] => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 2
[2,1,3,2] => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 2
[2,2,1,3] => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 2
[2,2,3,1] => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 2
[2,3,1,2] => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 2
[2,3,2,1] => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 2
[3,1,2,2] => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 2
[3,2,1,2] => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 2
[3,2,2,1] => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 2
[1,2,2,4] => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
[1,2,4,2] => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
[1,4,2,2] => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
[2,1,2,4] => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
[2,1,4,2] => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
[2,2,1,4] => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
[2,2,4,1] => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
[2,4,1,2] => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
[2,4,2,1] => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
[4,1,2,2] => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
[4,2,1,2] => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
[4,2,2,1] => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
[1,2,3,3] => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
[1,3,2,3] => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 2 + 2
[1,2,4,3] => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 2 + 2
[1,3,2,4] => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 2 + 2
[1,3,4,2] => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 2 + 2
[1,4,2,3] => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 2 + 2
[1,4,3,2] => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 2 + 2
[2,1,3,4] => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 2 + 2
[2,1,4,3] => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 2 + 2
[2,3,1,4] => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 2 + 2
[2,3,4,1] => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 2 + 2
[2,4,1,3] => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 2 + 2
[2,4,3,1] => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 2 + 2
[3,1,2,4] => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 2 + 2
[3,1,4,2] => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 2 + 2
[3,2,1,4] => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 2 + 2
[3,2,4,1] => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 2 + 2
[3,4,1,2] => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 2 + 2
[3,4,2,1] => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 2 + 2
[4,1,2,3] => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 2 + 2
[4,1,3,2] => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 2 + 2
[4,2,1,3] => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 2 + 2
[4,2,3,1] => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 2 + 2
Description
The hat guessing number of a graph. Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors. Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St001232
Mapping
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 2 = 0 + 2
[2,1] => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 2 = 0 + 2
[1,1,3] => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => ? = 0 + 2
[1,3,1] => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => ? = 0 + 2
[3,1,1] => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => ? = 0 + 2
[1,2,2] => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? = 0 + 2
[2,1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? = 0 + 2
[2,2,1] => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? = 0 + 2
[1,2,3] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3 = 1 + 2
[1,3,2] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3 = 1 + 2
[2,1,3] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3 = 1 + 2
[2,3,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3 = 1 + 2
[3,1,2] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3 = 1 + 2
[3,2,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3 = 1 + 2
[1,1,1,4] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => ? = 0 + 2
[1,1,4,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => ? = 0 + 2
[1,4,1,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => ? = 0 + 2
[4,1,1,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => ? = 0 + 2
[1,1,2,4] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => ? = 0 + 2
[1,1,4,2] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => ? = 0 + 2
[1,2,1,4] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => ? = 0 + 2
[1,2,4,1] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => ? = 0 + 2
[1,4,1,2] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => ? = 0 + 2
[1,4,2,1] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => ? = 0 + 2
[2,1,1,4] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => ? = 0 + 2
[2,1,4,1] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => ? = 0 + 2
[2,4,1,1] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => ? = 0 + 2
[4,1,1,2] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => ? = 0 + 2
[4,1,2,1] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => ? = 0 + 2
[4,2,1,1] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => ? = 0 + 2
[1,1,3,3] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => ? = 1 + 2
[1,3,1,3] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => ? = 1 + 2
[1,3,3,1] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => ? = 1 + 2
[3,1,1,3] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => ? = 1 + 2
[3,1,3,1] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => ? = 1 + 2
[3,3,1,1] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => ? = 1 + 2
[1,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => ? = 1 + 2
[1,1,4,3] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => ? = 1 + 2
[1,3,1,4] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => ? = 1 + 2
[1,3,4,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => ? = 1 + 2
[1,4,1,3] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => ? = 1 + 2
[1,4,3,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => ? = 1 + 2
[3,1,1,4] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => ? = 1 + 2
[3,1,4,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => ? = 1 + 2
[3,4,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => ? = 1 + 2
[4,1,1,3] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => ? = 1 + 2
[4,1,3,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => ? = 1 + 2
[4,3,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => ? = 1 + 2
[1,2,2,2] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => ? = 0 + 2
[2,1,2,2] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => ? = 0 + 2
[2,2,1,2] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => ? = 0 + 2
[2,2,2,1] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => ? = 0 + 2
[1,2,2,3] => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ? = 0 + 2
[1,2,3,2] => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ? = 0 + 2
[1,3,2,2] => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ? = 0 + 2
[2,1,2,3] => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ? = 0 + 2
[2,1,3,2] => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ? = 0 + 2
[2,2,1,3] => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ? = 0 + 2
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 2 + 2
[1,2,4,3] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 2 + 2
[1,3,2,4] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 2 + 2
[1,3,4,2] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 2 + 2
[1,4,2,3] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 2 + 2
[1,4,3,2] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 2 + 2
[2,1,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 2 + 2
[2,1,4,3] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 2 + 2
[2,3,1,4] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 2 + 2
[2,3,4,1] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 2 + 2
[2,4,1,3] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 2 + 2
[2,4,3,1] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 2 + 2
[3,1,2,4] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 2 + 2
[3,1,4,2] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 2 + 2
[3,2,1,4] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 2 + 2
[3,2,4,1] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 2 + 2
[3,4,1,2] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 2 + 2
[3,4,2,1] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 2 + 2
[4,1,2,3] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 2 + 2
[4,1,3,2] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 2 + 2
[4,2,1,3] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 2 + 2
[4,2,3,1] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 2 + 2
[4,3,1,2] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 2 + 2
[4,3,2,1] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 2 + 2
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 3 + 2
[1,2,3,5,4] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 3 + 2
[1,2,4,3,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 3 + 2
[1,2,4,5,3] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 3 + 2
[1,2,5,3,4] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 3 + 2
[1,2,5,4,3] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 3 + 2
[1,3,2,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 3 + 2
[1,3,2,5,4] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 3 + 2
[1,3,4,2,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 3 + 2
[1,3,4,5,2] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 3 + 2
[1,3,5,2,4] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 3 + 2
[1,3,5,4,2] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 3 + 2
[1,4,2,3,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 3 + 2
[1,4,2,5,3] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 3 + 2
[1,4,3,2,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 3 + 2
[1,4,3,5,2] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 3 + 2
[1,4,5,2,3] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 3 + 2
[1,4,5,3,2] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 3 + 2
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000718
(load all 2 compositions to match this statistic)
Mapping
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St000718: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,0,1,0]
 => [[],[]]
 => ([(0,2),(1,2)],3)
 => 3 = 0 + 3
[2,1] => [1,0,1,0]
 => [[],[]]
 => ([(0,2),(1,2)],3)
 => 3 = 0 + 3
[1,1,3] => [1,1,0,0,1,0]
 => [[[]],[]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 3
[1,3,1] => [1,1,0,0,1,0]
 => [[[]],[]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 3
[3,1,1] => [1,1,0,0,1,0]
 => [[[]],[]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 3
[1,2,2] => [1,0,1,1,0,0]
 => [[],[[]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 3
[2,1,2] => [1,0,1,1,0,0]
 => [[],[[]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 3
[2,2,1] => [1,0,1,1,0,0]
 => [[],[[]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 3
[1,2,3] => [1,0,1,0,1,0]
 => [[],[],[]]
 => ([(0,3),(1,3),(2,3)],4)
 => 4 = 1 + 3
[1,3,2] => [1,0,1,0,1,0]
 => [[],[],[]]
 => ([(0,3),(1,3),(2,3)],4)
 => 4 = 1 + 3
[2,1,3] => [1,0,1,0,1,0]
 => [[],[],[]]
 => ([(0,3),(1,3),(2,3)],4)
 => 4 = 1 + 3
[2,3,1] => [1,0,1,0,1,0]
 => [[],[],[]]
 => ([(0,3),(1,3),(2,3)],4)
 => 4 = 1 + 3
[3,1,2] => [1,0,1,0,1,0]
 => [[],[],[]]
 => ([(0,3),(1,3),(2,3)],4)
 => 4 = 1 + 3
[3,2,1] => [1,0,1,0,1,0]
 => [[],[],[]]
 => ([(0,3),(1,3),(2,3)],4)
 => 4 = 1 + 3
[1,1,1,4] => [1,1,1,0,0,0,1,0]
 => [[[[]]],[]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 0 + 3
[1,1,4,1] => [1,1,1,0,0,0,1,0]
 => [[[[]]],[]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 0 + 3
[1,4,1,1] => [1,1,1,0,0,0,1,0]
 => [[[[]]],[]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 0 + 3
[4,1,1,1] => [1,1,1,0,0,0,1,0]
 => [[[[]]],[]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 0 + 3
[1,1,2,4] => [1,1,0,1,0,0,1,0]
 => [[[],[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 0 + 3
[1,1,4,2] => [1,1,0,1,0,0,1,0]
 => [[[],[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 0 + 3
[1,2,1,4] => [1,1,0,1,0,0,1,0]
 => [[[],[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 0 + 3
[1,2,4,1] => [1,1,0,1,0,0,1,0]
 => [[[],[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 0 + 3
[1,4,1,2] => [1,1,0,1,0,0,1,0]
 => [[[],[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 0 + 3
[1,4,2,1] => [1,1,0,1,0,0,1,0]
 => [[[],[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 0 + 3
[2,1,1,4] => [1,1,0,1,0,0,1,0]
 => [[[],[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 0 + 3
[2,1,4,1] => [1,1,0,1,0,0,1,0]
 => [[[],[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 0 + 3
[2,4,1,1] => [1,1,0,1,0,0,1,0]
 => [[[],[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 0 + 3
[4,1,1,2] => [1,1,0,1,0,0,1,0]
 => [[[],[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 0 + 3
[4,1,2,1] => [1,1,0,1,0,0,1,0]
 => [[[],[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 0 + 3
[4,2,1,1] => [1,1,0,1,0,0,1,0]
 => [[[],[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 0 + 3
[1,1,3,3] => [1,1,0,0,1,1,0,0]
 => [[[]],[[]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 + 3
[1,3,1,3] => [1,1,0,0,1,1,0,0]
 => [[[]],[[]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 + 3
[1,3,3,1] => [1,1,0,0,1,1,0,0]
 => [[[]],[[]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 + 3
[3,1,1,3] => [1,1,0,0,1,1,0,0]
 => [[[]],[[]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 + 3
[3,1,3,1] => [1,1,0,0,1,1,0,0]
 => [[[]],[[]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 + 3
[3,3,1,1] => [1,1,0,0,1,1,0,0]
 => [[[]],[[]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 + 3
[1,1,3,4] => [1,1,0,0,1,0,1,0]
 => [[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 1 + 3
[1,1,4,3] => [1,1,0,0,1,0,1,0]
 => [[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 1 + 3
[1,3,1,4] => [1,1,0,0,1,0,1,0]
 => [[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 1 + 3
[1,3,4,1] => [1,1,0,0,1,0,1,0]
 => [[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 1 + 3
[1,4,1,3] => [1,1,0,0,1,0,1,0]
 => [[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 1 + 3
[1,4,3,1] => [1,1,0,0,1,0,1,0]
 => [[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 1 + 3
[3,1,1,4] => [1,1,0,0,1,0,1,0]
 => [[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 1 + 3
[3,1,4,1] => [1,1,0,0,1,0,1,0]
 => [[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 1 + 3
[3,4,1,1] => [1,1,0,0,1,0,1,0]
 => [[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 1 + 3
[4,1,1,3] => [1,1,0,0,1,0,1,0]
 => [[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 1 + 3
[4,1,3,1] => [1,1,0,0,1,0,1,0]
 => [[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 1 + 3
[4,3,1,1] => [1,1,0,0,1,0,1,0]
 => [[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 1 + 3
[1,2,2,2] => [1,0,1,1,1,0,0,0]
 => [[],[[[]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 0 + 3
[2,1,2,2] => [1,0,1,1,1,0,0,0]
 => [[],[[[]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 0 + 3
[2,2,1,2] => [1,0,1,1,1,0,0,0]
 => [[],[[[]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 0 + 3
[2,2,2,1] => [1,0,1,1,1,0,0,0]
 => [[],[[[]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 0 + 3
[1,2,2,3] => [1,0,1,1,0,1,0,0]
 => [[],[[],[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 0 + 3
[1,2,3,2] => [1,0,1,1,0,1,0,0]
 => [[],[[],[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 0 + 3
[1,3,2,2] => [1,0,1,1,0,1,0,0]
 => [[],[[],[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 0 + 3
[2,1,2,3] => [1,0,1,1,0,1,0,0]
 => [[],[[],[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 0 + 3
[2,1,3,2] => [1,0,1,1,0,1,0,0]
 => [[],[[],[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 0 + 3
[2,2,1,3] => [1,0,1,1,0,1,0,0]
 => [[],[[],[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 0 + 3
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 5 = 2 + 3
[1,2,4,3] => [1,0,1,0,1,0,1,0]
 => [[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 5 = 2 + 3
[1,3,2,4] => [1,0,1,0,1,0,1,0]
 => [[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 5 = 2 + 3
[1,3,4,2] => [1,0,1,0,1,0,1,0]
 => [[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 5 = 2 + 3
[1,4,2,3] => [1,0,1,0,1,0,1,0]
 => [[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 5 = 2 + 3
[1,4,3,2] => [1,0,1,0,1,0,1,0]
 => [[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 5 = 2 + 3
[2,1,3,4] => [1,0,1,0,1,0,1,0]
 => [[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 5 = 2 + 3
[2,1,4,3] => [1,0,1,0,1,0,1,0]
 => [[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 5 = 2 + 3
[2,3,1,4] => [1,0,1,0,1,0,1,0]
 => [[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 5 = 2 + 3
[2,3,4,1] => [1,0,1,0,1,0,1,0]
 => [[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 5 = 2 + 3
[2,4,1,3] => [1,0,1,0,1,0,1,0]
 => [[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 5 = 2 + 3
[2,4,3,1] => [1,0,1,0,1,0,1,0]
 => [[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 5 = 2 + 3
[3,1,2,4] => [1,0,1,0,1,0,1,0]
 => [[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 5 = 2 + 3
[3,1,4,2] => [1,0,1,0,1,0,1,0]
 => [[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 5 = 2 + 3
[3,2,1,4] => [1,0,1,0,1,0,1,0]
 => [[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 5 = 2 + 3
[3,2,4,1] => [1,0,1,0,1,0,1,0]
 => [[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 5 = 2 + 3
[3,4,1,2] => [1,0,1,0,1,0,1,0]
 => [[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 5 = 2 + 3
[3,4,2,1] => [1,0,1,0,1,0,1,0]
 => [[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 5 = 2 + 3
[4,1,2,3] => [1,0,1,0,1,0,1,0]
 => [[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 5 = 2 + 3
[4,1,3,2] => [1,0,1,0,1,0,1,0]
 => [[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 5 = 2 + 3
[4,2,1,3] => [1,0,1,0,1,0,1,0]
 => [[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 5 = 2 + 3
[4,2,3,1] => [1,0,1,0,1,0,1,0]
 => [[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 5 = 2 + 3
[4,3,1,2] => [1,0,1,0,1,0,1,0]
 => [[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 5 = 2 + 3
[4,3,2,1] => [1,0,1,0,1,0,1,0]
 => [[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 5 = 2 + 3
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 6 = 3 + 3
[1,2,3,5,4] => [1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 6 = 3 + 3
[1,2,4,3,5] => [1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 6 = 3 + 3
[1,2,4,5,3] => [1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 6 = 3 + 3
[1,2,5,3,4] => [1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 6 = 3 + 3
[1,2,5,4,3] => [1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 6 = 3 + 3
[1,3,2,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 6 = 3 + 3
[1,3,2,5,4] => [1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 6 = 3 + 3
[1,3,4,2,5] => [1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 6 = 3 + 3
[1,3,4,5,2] => [1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 6 = 3 + 3
[1,3,5,2,4] => [1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 6 = 3 + 3
[1,3,5,4,2] => [1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 6 = 3 + 3
[1,4,2,3,5] => [1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 6 = 3 + 3
[1,4,2,5,3] => [1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 6 = 3 + 3
[1,4,3,2,5] => [1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 6 = 3 + 3
[1,4,3,5,2] => [1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 6 = 3 + 3
[1,4,5,2,3] => [1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 6 = 3 + 3
[1,4,5,3,2] => [1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 6 = 3 + 3
Description
The largest Laplacian eigenvalue of a graph if it is integral. This statistic is undefined if the largest Laplacian eigenvalue of the graph is not integral. Various results are collected in Section 3.9 of [1]
Matching statistic: St001879
(load all 2 compositions to match this statistic)
Mapping
Mp00057: Parking functions —to touch composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 80%
Values
[1,2] => [1,1] => [[1,1],[]]
 => ([(0,1)],2)
 => ? = 0 + 1
[2,1] => [1,1] => [[1,1],[]]
 => ([(0,1)],2)
 => ? = 0 + 1
[1,1,3] => [2,1] => [[2,2],[1]]
 => ([(0,2),(1,2)],3)
 => ? = 0 + 1
[1,3,1] => [2,1] => [[2,2],[1]]
 => ([(0,2),(1,2)],3)
 => ? = 0 + 1
[3,1,1] => [2,1] => [[2,2],[1]]
 => ([(0,2),(1,2)],3)
 => ? = 0 + 1
[1,2,2] => [1,2] => [[2,1],[]]
 => ([(0,1),(0,2)],3)
 => ? = 0 + 1
[2,1,2] => [1,2] => [[2,1],[]]
 => ([(0,1),(0,2)],3)
 => ? = 0 + 1
[2,2,1] => [1,2] => [[2,1],[]]
 => ([(0,1),(0,2)],3)
 => ? = 0 + 1
[1,2,3] => [1,1,1] => [[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => 2 = 1 + 1
[1,3,2] => [1,1,1] => [[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => 2 = 1 + 1
[2,1,3] => [1,1,1] => [[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => 2 = 1 + 1
[2,3,1] => [1,1,1] => [[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => 2 = 1 + 1
[3,1,2] => [1,1,1] => [[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => 2 = 1 + 1
[3,2,1] => [1,1,1] => [[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => 2 = 1 + 1
[1,1,1,4] => [3,1] => [[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 1
[1,1,4,1] => [3,1] => [[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 1
[1,4,1,1] => [3,1] => [[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 1
[4,1,1,1] => [3,1] => [[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 1
[1,1,2,4] => [3,1] => [[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 1
[1,1,4,2] => [3,1] => [[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 1
[1,2,1,4] => [3,1] => [[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 1
[1,2,4,1] => [3,1] => [[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 1
[1,4,1,2] => [3,1] => [[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 1
[1,4,2,1] => [3,1] => [[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 1
[2,1,1,4] => [3,1] => [[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 1
[2,1,4,1] => [3,1] => [[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 1
[2,4,1,1] => [3,1] => [[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 1
[4,1,1,2] => [3,1] => [[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 1
[4,1,2,1] => [3,1] => [[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 1
[4,2,1,1] => [3,1] => [[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 1
[1,1,3,3] => [2,2] => [[3,2],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => ? = 1 + 1
[1,3,1,3] => [2,2] => [[3,2],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => ? = 1 + 1
[1,3,3,1] => [2,2] => [[3,2],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => ? = 1 + 1
[3,1,1,3] => [2,2] => [[3,2],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => ? = 1 + 1
[3,1,3,1] => [2,2] => [[3,2],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => ? = 1 + 1
[3,3,1,1] => [2,2] => [[3,2],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => ? = 1 + 1
[1,1,3,4] => [2,1,1] => [[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 1
[1,1,4,3] => [2,1,1] => [[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 1
[1,3,1,4] => [2,1,1] => [[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 1
[1,3,4,1] => [2,1,1] => [[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 1
[1,4,1,3] => [2,1,1] => [[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 1
[1,4,3,1] => [2,1,1] => [[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 1
[3,1,1,4] => [2,1,1] => [[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 1
[3,1,4,1] => [2,1,1] => [[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 1
[3,4,1,1] => [2,1,1] => [[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 1
[4,1,1,3] => [2,1,1] => [[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 1
[4,1,3,1] => [2,1,1] => [[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 1
[4,3,1,1] => [2,1,1] => [[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 1
[1,2,2,2] => [1,3] => [[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ? = 0 + 1
[2,1,2,2] => [1,3] => [[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ? = 0 + 1
[2,2,1,2] => [1,3] => [[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ? = 0 + 1
[2,2,2,1] => [1,3] => [[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ? = 0 + 1
[1,2,2,3] => [1,3] => [[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ? = 0 + 1
[1,2,3,2] => [1,3] => [[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ? = 0 + 1
[1,3,2,2] => [1,3] => [[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ? = 0 + 1
[2,1,2,3] => [1,3] => [[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ? = 0 + 1
[1,2,3,4] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 2 + 1
[1,2,4,3] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 2 + 1
[1,3,2,4] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 2 + 1
[1,3,4,2] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 2 + 1
[1,4,2,3] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 2 + 1
[1,4,3,2] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 2 + 1
[2,1,3,4] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 2 + 1
[2,1,4,3] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 2 + 1
[2,3,1,4] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 2 + 1
[2,3,4,1] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 2 + 1
[2,4,1,3] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 2 + 1
[2,4,3,1] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 2 + 1
[3,1,2,4] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 2 + 1
[3,1,4,2] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 2 + 1
[3,2,1,4] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 2 + 1
[3,2,4,1] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 2 + 1
[3,4,1,2] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 2 + 1
[3,4,2,1] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 2 + 1
[4,1,2,3] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 2 + 1
[4,1,3,2] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 2 + 1
[4,2,1,3] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 2 + 1
[4,2,3,1] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 2 + 1
[4,3,1,2] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 2 + 1
[4,3,2,1] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 2 + 1
[1,2,3,4,5] => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[1,2,3,5,4] => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[1,2,4,3,5] => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[1,2,4,5,3] => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[1,2,5,3,4] => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[1,2,5,4,3] => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[1,3,2,4,5] => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[1,3,2,5,4] => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[1,3,4,2,5] => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[1,3,4,5,2] => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[1,3,5,2,4] => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[1,3,5,4,2] => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[1,4,2,3,5] => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[1,4,2,5,3] => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[1,4,3,2,5] => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[1,4,3,5,2] => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[1,4,5,2,3] => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[1,4,5,3,2] => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[1,5,2,3,4] => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[1,5,2,4,3] => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
Description
The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.
Matching statistic: St001880
(load all 2 compositions to match this statistic)
Mapping
Mp00057: Parking functions —to touch composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 80%
Values
[1,2] => [1,1] => [[1,1],[]]
 => ([(0,1)],2)
 => ? = 0 + 2
[2,1] => [1,1] => [[1,1],[]]
 => ([(0,1)],2)
 => ? = 0 + 2
[1,1,3] => [2,1] => [[2,2],[1]]
 => ([(0,2),(1,2)],3)
 => ? = 0 + 2
[1,3,1] => [2,1] => [[2,2],[1]]
 => ([(0,2),(1,2)],3)
 => ? = 0 + 2
[3,1,1] => [2,1] => [[2,2],[1]]
 => ([(0,2),(1,2)],3)
 => ? = 0 + 2
[1,2,2] => [1,2] => [[2,1],[]]
 => ([(0,1),(0,2)],3)
 => ? = 0 + 2
[2,1,2] => [1,2] => [[2,1],[]]
 => ([(0,1),(0,2)],3)
 => ? = 0 + 2
[2,2,1] => [1,2] => [[2,1],[]]
 => ([(0,1),(0,2)],3)
 => ? = 0 + 2
[1,2,3] => [1,1,1] => [[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => 3 = 1 + 2
[1,3,2] => [1,1,1] => [[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => 3 = 1 + 2
[2,1,3] => [1,1,1] => [[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => 3 = 1 + 2
[2,3,1] => [1,1,1] => [[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => 3 = 1 + 2
[3,1,2] => [1,1,1] => [[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => 3 = 1 + 2
[3,2,1] => [1,1,1] => [[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => 3 = 1 + 2
[1,1,1,4] => [3,1] => [[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 2
[1,1,4,1] => [3,1] => [[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 2
[1,4,1,1] => [3,1] => [[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 2
[4,1,1,1] => [3,1] => [[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 2
[1,1,2,4] => [3,1] => [[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 2
[1,1,4,2] => [3,1] => [[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 2
[1,2,1,4] => [3,1] => [[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 2
[1,2,4,1] => [3,1] => [[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 2
[1,4,1,2] => [3,1] => [[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 2
[1,4,2,1] => [3,1] => [[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 2
[2,1,1,4] => [3,1] => [[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 2
[2,1,4,1] => [3,1] => [[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 2
[2,4,1,1] => [3,1] => [[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 2
[4,1,1,2] => [3,1] => [[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 2
[4,1,2,1] => [3,1] => [[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 2
[4,2,1,1] => [3,1] => [[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 2
[1,1,3,3] => [2,2] => [[3,2],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => ? = 1 + 2
[1,3,1,3] => [2,2] => [[3,2],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => ? = 1 + 2
[1,3,3,1] => [2,2] => [[3,2],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => ? = 1 + 2
[3,1,1,3] => [2,2] => [[3,2],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => ? = 1 + 2
[3,1,3,1] => [2,2] => [[3,2],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => ? = 1 + 2
[3,3,1,1] => [2,2] => [[3,2],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => ? = 1 + 2
[1,1,3,4] => [2,1,1] => [[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 2
[1,1,4,3] => [2,1,1] => [[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 2
[1,3,1,4] => [2,1,1] => [[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 2
[1,3,4,1] => [2,1,1] => [[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 2
[1,4,1,3] => [2,1,1] => [[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 2
[1,4,3,1] => [2,1,1] => [[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 2
[3,1,1,4] => [2,1,1] => [[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 2
[3,1,4,1] => [2,1,1] => [[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 2
[3,4,1,1] => [2,1,1] => [[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 2
[4,1,1,3] => [2,1,1] => [[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 2
[4,1,3,1] => [2,1,1] => [[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 2
[4,3,1,1] => [2,1,1] => [[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 2
[1,2,2,2] => [1,3] => [[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ? = 0 + 2
[2,1,2,2] => [1,3] => [[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ? = 0 + 2
[2,2,1,2] => [1,3] => [[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ? = 0 + 2
[2,2,2,1] => [1,3] => [[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ? = 0 + 2
[1,2,2,3] => [1,3] => [[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ? = 0 + 2
[1,2,3,2] => [1,3] => [[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ? = 0 + 2
[1,3,2,2] => [1,3] => [[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ? = 0 + 2
[2,1,2,3] => [1,3] => [[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ? = 0 + 2
[1,2,3,4] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[1,2,4,3] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[1,3,2,4] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[1,3,4,2] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[1,4,2,3] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[1,4,3,2] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[2,1,3,4] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[2,1,4,3] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[2,3,1,4] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[2,3,4,1] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[2,4,1,3] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[2,4,3,1] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[3,1,2,4] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[3,1,4,2] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[3,2,1,4] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[3,2,4,1] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[3,4,1,2] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[3,4,2,1] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[4,1,2,3] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[4,1,3,2] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[4,2,1,3] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[4,2,3,1] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[4,3,1,2] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[4,3,2,1] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[1,2,3,4,5] => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[1,2,3,5,4] => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[1,2,4,3,5] => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[1,2,4,5,3] => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[1,2,5,3,4] => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[1,2,5,4,3] => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[1,3,2,4,5] => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[1,3,2,5,4] => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[1,3,4,2,5] => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[1,3,4,5,2] => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[1,3,5,2,4] => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[1,3,5,4,2] => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[1,4,2,3,5] => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[1,4,2,5,3] => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[1,4,3,2,5] => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[1,4,3,5,2] => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[1,4,5,2,3] => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[1,4,5,3,2] => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[1,5,2,3,4] => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[1,5,2,4,3] => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St000454
(load all 2 compositions to match this statistic)
Mapping
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 40%
Values
[1,2] => [1,0,1,0]
 => [[.,.],.]
 => ([(0,1)],2)
 => 1 = 0 + 1
[2,1] => [1,0,1,0]
 => [[.,.],.]
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,1,3] => [1,1,0,0,1,0]
 => [[.,[.,.]],.]
 => ([(0,2),(1,2)],3)
 => ? = 0 + 1
[1,3,1] => [1,1,0,0,1,0]
 => [[.,[.,.]],.]
 => ([(0,2),(1,2)],3)
 => ? = 0 + 1
[3,1,1] => [1,1,0,0,1,0]
 => [[.,[.,.]],.]
 => ([(0,2),(1,2)],3)
 => ? = 0 + 1
[1,2,2] => [1,0,1,1,0,0]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ? = 0 + 1
[2,1,2] => [1,0,1,1,0,0]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ? = 0 + 1
[2,2,1] => [1,0,1,1,0,0]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ? = 0 + 1
[1,2,3] => [1,0,1,0,1,0]
 => [[[.,.],.],.]
 => ([(0,2),(1,2)],3)
 => ? = 1 + 1
[1,3,2] => [1,0,1,0,1,0]
 => [[[.,.],.],.]
 => ([(0,2),(1,2)],3)
 => ? = 1 + 1
[2,1,3] => [1,0,1,0,1,0]
 => [[[.,.],.],.]
 => ([(0,2),(1,2)],3)
 => ? = 1 + 1
[2,3,1] => [1,0,1,0,1,0]
 => [[[.,.],.],.]
 => ([(0,2),(1,2)],3)
 => ? = 1 + 1
[3,1,2] => [1,0,1,0,1,0]
 => [[[.,.],.],.]
 => ([(0,2),(1,2)],3)
 => ? = 1 + 1
[3,2,1] => [1,0,1,0,1,0]
 => [[[.,.],.],.]
 => ([(0,2),(1,2)],3)
 => ? = 1 + 1
[1,1,1,4] => [1,1,1,0,0,0,1,0]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 1
[1,1,4,1] => [1,1,1,0,0,0,1,0]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 1
[1,4,1,1] => [1,1,1,0,0,0,1,0]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 1
[4,1,1,1] => [1,1,1,0,0,0,1,0]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 1
[1,1,2,4] => [1,1,0,1,0,0,1,0]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 1
[1,1,4,2] => [1,1,0,1,0,0,1,0]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 1
[1,2,1,4] => [1,1,0,1,0,0,1,0]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 1
[1,2,4,1] => [1,1,0,1,0,0,1,0]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 1
[1,4,1,2] => [1,1,0,1,0,0,1,0]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 1
[1,4,2,1] => [1,1,0,1,0,0,1,0]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 1
[2,1,1,4] => [1,1,0,1,0,0,1,0]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 1
[2,1,4,1] => [1,1,0,1,0,0,1,0]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 1
[2,4,1,1] => [1,1,0,1,0,0,1,0]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 1
[4,1,1,2] => [1,1,0,1,0,0,1,0]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 1
[4,1,2,1] => [1,1,0,1,0,0,1,0]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 1
[4,2,1,1] => [1,1,0,1,0,0,1,0]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 1
[1,1,3,3] => [1,1,0,0,1,1,0,0]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 1
[1,3,1,3] => [1,1,0,0,1,1,0,0]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 1
[1,3,3,1] => [1,1,0,0,1,1,0,0]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 1
[3,1,1,3] => [1,1,0,0,1,1,0,0]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 1
[3,1,3,1] => [1,1,0,0,1,1,0,0]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 1
[3,3,1,1] => [1,1,0,0,1,1,0,0]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 1
[1,1,3,4] => [1,1,0,0,1,0,1,0]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 1
[1,1,4,3] => [1,1,0,0,1,0,1,0]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 1
[1,3,1,4] => [1,1,0,0,1,0,1,0]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 1
[1,3,4,1] => [1,1,0,0,1,0,1,0]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 1
[1,4,1,3] => [1,1,0,0,1,0,1,0]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 1
[1,4,3,1] => [1,1,0,0,1,0,1,0]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 1
[3,1,1,4] => [1,1,0,0,1,0,1,0]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 1
[3,1,4,1] => [1,1,0,0,1,0,1,0]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 1
[3,4,1,1] => [1,1,0,0,1,0,1,0]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 1
[4,1,1,3] => [1,1,0,0,1,0,1,0]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 1
[4,1,3,1] => [1,1,0,0,1,0,1,0]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 1
[4,3,1,1] => [1,1,0,0,1,0,1,0]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 1
[1,2,2,2] => [1,0,1,1,1,0,0,0]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 1
[2,1,2,2] => [1,0,1,1,1,0,0,0]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 1
[2,2,1,2] => [1,0,1,1,1,0,0,0]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 1
[2,2,2,1] => [1,0,1,1,1,0,0,0]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 1
[1,2,2,3,3,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,2,2,3,6,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,2,2,6,3,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,2,3,2,3,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,2,3,2,6,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,2,3,3,2,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,2,3,3,6,2] => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,2,3,6,2,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,2,3,6,3,2] => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,2,6,2,3,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,2,6,3,2,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,2,6,3,3,2] => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,3,2,2,3,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,3,2,2,6,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,3,2,3,2,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,3,2,3,6,2] => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,3,2,6,2,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,3,2,6,3,2] => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,3,3,2,2,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,3,3,2,6,2] => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,3,3,6,2,2] => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,3,6,2,2,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,3,6,2,3,2] => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,3,6,3,2,2] => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,6,2,2,3,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,6,2,3,2,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,6,2,3,3,2] => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,6,3,2,2,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,6,3,2,3,2] => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,6,3,3,2,2] => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,1,2,3,3,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,1,2,3,6,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,1,2,6,3,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,1,3,2,3,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,1,3,2,6,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,1,3,3,2,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,1,3,3,6,2] => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,1,3,6,2,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,1,3,6,3,2] => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,1,6,2,3,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,1,6,3,2,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,1,6,3,3,2] => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,2,1,3,3,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,2,1,3,6,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,2,1,6,3,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,2,3,1,3,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,2,3,1,6,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,2,3,3,1,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
Description
The largest eigenvalue of a graph if it is integral. If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St001514
Mapping
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00228: Dyck paths —reflect parallelogram polyomino⟶ Dyck paths
St001514: Dyck paths ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 60%
Values
[1,2] => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2 = 0 + 2
[2,1] => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2 = 0 + 2
[1,1,3] => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2 = 0 + 2
[1,3,1] => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2 = 0 + 2
[3,1,1] => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2 = 0 + 2
[1,2,2] => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[2,1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[2,2,1] => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[1,2,3] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3 = 1 + 2
[1,3,2] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3 = 1 + 2
[2,1,3] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3 = 1 + 2
[2,3,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3 = 1 + 2
[3,1,2] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3 = 1 + 2
[3,2,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3 = 1 + 2
[1,1,1,4] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2 = 0 + 2
[1,1,4,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2 = 0 + 2
[1,4,1,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2 = 0 + 2
[4,1,1,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2 = 0 + 2
[1,1,2,4] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 0 + 2
[1,1,4,2] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 0 + 2
[1,2,1,4] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 0 + 2
[1,2,4,1] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 0 + 2
[1,4,1,2] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 0 + 2
[1,4,2,1] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 0 + 2
[2,1,1,4] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 0 + 2
[2,1,4,1] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 0 + 2
[2,4,1,1] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 0 + 2
[4,1,1,2] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 0 + 2
[4,1,2,1] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 0 + 2
[4,2,1,1] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 0 + 2
[1,1,3,3] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 3 = 1 + 2
[1,3,1,3] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 3 = 1 + 2
[1,3,3,1] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 3 = 1 + 2
[3,1,1,3] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 3 = 1 + 2
[3,1,3,1] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 3 = 1 + 2
[3,3,1,1] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 3 = 1 + 2
[1,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 1 + 2
[1,1,4,3] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 1 + 2
[1,3,1,4] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 1 + 2
[1,3,4,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 1 + 2
[1,4,1,3] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 1 + 2
[1,4,3,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 1 + 2
[3,1,1,4] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 1 + 2
[3,1,4,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 1 + 2
[3,4,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 1 + 2
[4,1,1,3] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 1 + 2
[4,1,3,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 1 + 2
[4,3,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 1 + 2
[1,2,2,2] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 0 + 2
[2,1,2,2] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 0 + 2
[1,1,1,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 0 + 2
[1,1,1,5,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 0 + 2
[1,1,5,1,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 0 + 2
[1,5,1,1,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 0 + 2
[5,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 0 + 2
[1,1,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => ? = 0 + 2
[1,1,1,5,2] => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => ? = 0 + 2
[1,1,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => ? = 0 + 2
[1,1,2,5,1] => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => ? = 0 + 2
[1,1,5,1,2] => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => ? = 0 + 2
[1,1,5,2,1] => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => ? = 0 + 2
[1,2,1,1,5] => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => ? = 0 + 2
[1,2,1,5,1] => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => ? = 0 + 2
[1,2,5,1,1] => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => ? = 0 + 2
[1,5,1,1,2] => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => ? = 0 + 2
[1,5,1,2,1] => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => ? = 0 + 2
[1,5,2,1,1] => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => ? = 0 + 2
[2,1,1,1,5] => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => ? = 0 + 2
[2,1,1,5,1] => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => ? = 0 + 2
[2,1,5,1,1] => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => ? = 0 + 2
[2,5,1,1,1] => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => ? = 0 + 2
[5,1,1,1,2] => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => ? = 0 + 2
[5,1,1,2,1] => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => ? = 0 + 2
[5,1,2,1,1] => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => ? = 0 + 2
[5,2,1,1,1] => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => ? = 0 + 2
[1,1,1,3,5] => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => ? = 0 + 2
[1,1,1,5,3] => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => ? = 0 + 2
[1,1,3,1,5] => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => ? = 0 + 2
[1,1,3,5,1] => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => ? = 0 + 2
[1,1,5,1,3] => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => ? = 0 + 2
[1,1,5,3,1] => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => ? = 0 + 2
[1,3,1,1,5] => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => ? = 0 + 2
[1,3,1,5,1] => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => ? = 0 + 2
[1,3,5,1,1] => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => ? = 0 + 2
[1,5,1,1,3] => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => ? = 0 + 2
[1,5,1,3,1] => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => ? = 0 + 2
[1,5,3,1,1] => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => ? = 0 + 2
[3,1,1,1,5] => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => ? = 0 + 2
[3,1,1,5,1] => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => ? = 0 + 2
[3,1,5,1,1] => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => ? = 0 + 2
[3,5,1,1,1] => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => ? = 0 + 2
[5,1,1,1,3] => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => ? = 0 + 2
[5,1,1,3,1] => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => ? = 0 + 2
[5,1,3,1,1] => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => ? = 0 + 2
[5,3,1,1,1] => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => ? = 0 + 2
[1,1,1,4,4] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => ? = 1 + 2
[1,1,4,1,4] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => ? = 1 + 2
[1,1,4,4,1] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => ? = 1 + 2
[1,4,1,1,4] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => ? = 1 + 2
[1,4,1,4,1] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => ? = 1 + 2
Description
The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule.
The following 1 statistic also match your data. Click on any of them to see the details.
St001875The number of simple modules with projective dimension at most 1.