Your data matches 19 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001022
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001022: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
 => ([],1)
 => [1]
 => [1,0]
 => 0
['A',2]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => [1,0,1,1,0,0]
 => 0
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 1
['G',2]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 0
['B',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => [5,3,1]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => 1
['C',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => [5,3,1]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => 1
['A',4]
 => ([(0,8),(1,7),(2,7),(2,9),(3,8),(3,9),(5,4),(6,4),(7,5),(8,6),(9,5),(9,6)],10)
 => [4,3,2,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => 1
Description
Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001424
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St001424: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
 => ([],1)
 => [1]
 => 1 => 0
['A',2]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => 01 => 0
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => 11 => 1
['G',2]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => 11 => 1
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => [3,2,1]
 => 101 => 0
['B',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => [5,3,1]
 => 111 => 1
['C',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => [5,3,1]
 => 111 => 1
['A',4]
 => ([(0,8),(1,7),(2,7),(2,9),(3,8),(3,9),(5,4),(6,4),(7,5),(8,6),(9,5),(9,6)],10)
 => [4,3,2,1]
 => 0101 => 1
Description
The number of distinct squares in a binary word. A factor of a word is a sequence of consecutive letters. This statistic records the number of distinct non-empty words $u$ such that $uu$ is a factor of the word. Note that every word of length at least four contains a square.
Matching statistic: St000257
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000257: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 100%
Values
['A',1]
 => ([],1)
 => [2]
 => [1,1]
 => 1 = 0 + 1
['A',2]
 => ([(0,2),(1,2)],3)
 => [3,2]
 => [2,2,1]
 => 1 = 0 + 1
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [2,2,1,1]
 => 2 = 1 + 1
['G',2]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [6,2]
 => [2,2,1,1,1,1]
 => 2 = 1 + 1
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => [8,4,2]
 => [3,3,2,2,1,1,1,1]
 => ? = 0 + 1
['B',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => [6,6,6,2]
 => [4,4,3,3,3,3]
 => 2 = 1 + 1
['C',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => [6,6,6,2]
 => [4,4,3,3,3,3]
 => 2 = 1 + 1
['A',4]
 => ([(0,8),(1,7),(2,7),(2,9),(3,8),(3,9),(5,4),(6,4),(7,5),(8,6),(9,5),(9,6)],10)
 => [10,10,10,5,5,2]
 => [6,6,5,5,5,3,3,3,3,3]
 => ? = 1 + 1
Description
The number of distinct parts of a partition that occur at least twice. See Section 3.3.1 of [2].
Matching statistic: St001113
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001113: Dyck paths ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 100%
Values
['A',1]
 => ([],1)
 => [1]
 => [1,0]
 => 0
['A',2]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => [1,0,1,1,0,0]
 => 0
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 1
['G',2]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 0
['B',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => [5,3,1]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => ? = 1
['C',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => [5,3,1]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => ? = 1
['A',4]
 => ([(0,8),(1,7),(2,7),(2,9),(3,8),(3,9),(5,4),(6,4),(7,5),(8,6),(9,5),(9,6)],10)
 => [4,3,2,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 1
Description
Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra.
Matching statistic: St001186
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001186: Dyck paths ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 100%
Values
['A',1]
 => ([],1)
 => [1]
 => [1,0]
 => 0
['A',2]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => [1,0,1,1,0,0]
 => 0
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 1
['G',2]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 0
['B',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => [5,3,1]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => ? = 1
['C',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => [5,3,1]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => ? = 1
['A',4]
 => ([(0,8),(1,7),(2,7),(2,9),(3,8),(3,9),(5,4),(6,4),(7,5),(8,6),(9,5),(9,6)],10)
 => [4,3,2,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 1
Description
Number of simple modules with grade at least 3 in the corresponding Nakayama algebra.
Matching statistic: St001266
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001266: Dyck paths ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 100%
Values
['A',1]
 => ([],1)
 => [1]
 => [1,0]
 => 0
['A',2]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => [1,0,1,1,0,0]
 => 0
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 1
['G',2]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 0
['B',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => [5,3,1]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => ? = 1
['C',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => [5,3,1]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => ? = 1
['A',4]
 => ([(0,8),(1,7),(2,7),(2,9),(3,8),(3,9),(5,4),(6,4),(7,5),(8,6),(9,5),(9,6)],10)
 => [4,3,2,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 1
Description
The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra.
Matching statistic: St001578
(load all 3 compositions to match this statistic)
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
Mp00157: Graphs —connected complement⟶ Graphs
St001578: Graphs ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 100%
Values
['A',1]
 => ([],1)
 => ([],1)
 => ([],1)
 => 0
['A',2]
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 1
['G',2]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 1
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
['B',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => ([(0,8),(1,7),(2,6),(3,6),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8)],9)
 => ([(0,2),(0,3),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 1
['C',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => ([(0,8),(1,7),(2,6),(3,6),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8)],9)
 => ([(0,2),(0,3),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 1
['A',4]
 => ([(0,8),(1,7),(2,7),(2,9),(3,8),(3,9),(5,4),(6,4),(7,5),(8,6),(9,5),(9,6)],10)
 => ([(0,8),(1,7),(2,5),(2,6),(3,7),(3,9),(4,8),(4,9),(5,7),(5,9),(6,8),(6,9)],10)
 => ([(0,3),(0,4),(0,7),(0,8),(0,9),(1,2),(1,4),(1,5),(1,6),(1,8),(1,9),(2,3),(2,5),(2,6),(2,8),(2,9),(3,4),(3,5),(3,7),(3,8),(4,6),(4,7),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 1
Description
The minimal number of edges to add or remove to make a graph a line graph.
Matching statistic: St001282
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
Mp00157: Graphs —connected complement⟶ Graphs
St001282: Graphs ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 100%
Values
['A',1]
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
['A',2]
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
['G',2]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
['B',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => ([(0,8),(1,7),(2,6),(3,6),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8)],9)
 => ([(0,2),(0,3),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 1 + 1
['C',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => ([(0,8),(1,7),(2,6),(3,6),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8)],9)
 => ([(0,2),(0,3),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 1 + 1
['A',4]
 => ([(0,8),(1,7),(2,7),(2,9),(3,8),(3,9),(5,4),(6,4),(7,5),(8,6),(9,5),(9,6)],10)
 => ([(0,8),(1,7),(2,5),(2,6),(3,7),(3,9),(4,8),(4,9),(5,7),(5,9),(6,8),(6,9)],10)
 => ([(0,3),(0,4),(0,7),(0,8),(0,9),(1,2),(1,4),(1,5),(1,6),(1,8),(1,9),(2,3),(2,5),(2,6),(2,8),(2,9),(3,4),(3,5),(3,7),(3,8),(4,6),(4,7),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 1 + 1
Description
The number of graphs with the same chromatic polynomial.
Matching statistic: St001366
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St001366: Graphs ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 100%
Values
['A',1]
 => ([],1)
 => ([],1)
 => ([(0,1)],2)
 => 2 = 0 + 2
['A',2]
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 0 + 2
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 1 + 2
['G',2]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 1 + 2
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
['B',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => ([(0,8),(1,7),(2,6),(3,6),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8)],9)
 => ([(0,8),(0,9),(1,7),(1,9),(2,6),(2,9),(3,6),(3,8),(3,9),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 1 + 2
['C',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => ([(0,8),(1,7),(2,6),(3,6),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8)],9)
 => ([(0,8),(0,9),(1,7),(1,9),(2,6),(2,9),(3,6),(3,8),(3,9),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 1 + 2
['A',4]
 => ([(0,8),(1,7),(2,7),(2,9),(3,8),(3,9),(5,4),(6,4),(7,5),(8,6),(9,5),(9,6)],10)
 => ([(0,8),(1,7),(2,5),(2,6),(3,7),(3,9),(4,8),(4,9),(5,7),(5,9),(6,8),(6,9)],10)
 => ([(0,8),(0,10),(1,7),(1,10),(2,5),(2,6),(2,10),(3,7),(3,9),(3,10),(4,8),(4,9),(4,10),(5,7),(5,9),(5,10),(6,8),(6,9),(6,10),(7,10),(8,10),(9,10)],11)
 => ? = 1 + 2
Description
The maximal multiplicity of a degree of a vertex of a graph.
Matching statistic: St001654
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St001654: Graphs ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 100%
Values
['A',1]
 => ([],1)
 => ([],1)
 => ([(0,1)],2)
 => 2 = 0 + 2
['A',2]
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 0 + 2
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 1 + 2
['G',2]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 1 + 2
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
['B',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => ([(0,8),(1,7),(2,6),(3,6),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8)],9)
 => ([(0,8),(0,9),(1,7),(1,9),(2,6),(2,9),(3,6),(3,8),(3,9),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 1 + 2
['C',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => ([(0,8),(1,7),(2,6),(3,6),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8)],9)
 => ([(0,8),(0,9),(1,7),(1,9),(2,6),(2,9),(3,6),(3,8),(3,9),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 1 + 2
['A',4]
 => ([(0,8),(1,7),(2,7),(2,9),(3,8),(3,9),(5,4),(6,4),(7,5),(8,6),(9,5),(9,6)],10)
 => ([(0,8),(1,7),(2,5),(2,6),(3,7),(3,9),(4,8),(4,9),(5,7),(5,9),(6,8),(6,9)],10)
 => ([(0,8),(0,10),(1,7),(1,10),(2,5),(2,6),(2,10),(3,7),(3,9),(3,10),(4,8),(4,9),(4,10),(5,7),(5,9),(5,10),(6,8),(6,9),(6,10),(7,10),(8,10),(9,10)],11)
 => ? = 1 + 2
Description
The monophonic hull number of a graph. The monophonic hull of a set of vertices $M$ of a graph $G$ is the set of vertices that lie on at least one induced path between vertices in $M$. The monophonic hull number is the size of the smallest set $M$ such that the monophonic hull of $M$ is all of $G$. For example, the monophonic hull number of a graph $G$ with $n$ vertices is $n$ if and only if $G$ is a disjoint union of complete graphs.
The following 9 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000256The number of parts from which one can substract 2 and still get an integer partition. St000480The number of lower covers of a partition in dominance order. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St000481The number of upper covers of a partition in dominance order. St000937The number of positive values of the symmetric group character corresponding to the partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001964The interval resolution global dimension of a poset. St001624The breadth of a lattice. St001783The number of odd automorphisms of a graph.