Your data matches 17 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000853
Mapping
St000853: Finite Cartan types ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
 => 2 = 1 + 1
['A',2]
 => 5 = 4 + 1
['B',2]
 => 6 = 5 + 1
['G',2]
 => 8 = 7 + 1
['A',3]
 => 9 = 8 + 1
Description
The number of almost positive roots of a finite Cartan type. A root in the root system of a Cartan type is almost positive if it is either positive or simple negative. These are known to be in bijection with cluster variables in the cluster algebra of the given Cartan type, see [1]. This is also equal to the sum of the degrees of the fundamental invariants of the group.
Matching statistic: St000456
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
 => ([],1)
 => ([],1)
 => ([(0,1)],2)
 => 1
['A',2]
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
['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)
 => 5
['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)
 => 7
['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)
 => 8
Description
The monochromatic index of a connected graph. This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path. For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.
Matching statistic: St001254
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001254: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
 => ([],1)
 => [1]
 => [1,0]
 => 1
['A',2]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => [1,0,1,1,0,0]
 => 4
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 5
['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]
 => 7
['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]
 => 8
Description
The vector space dimension of the first extension-group between A/soc(A) and J when A is the corresponding Nakayama algebra with Jacobson radical J.
Matching statistic: St001019
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001019: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
 => ([],1)
 => [1]
 => [1,0,1,0]
 => 3 = 1 + 2
['A',2]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => [1,0,1,0,1,0]
 => 6 = 4 + 2
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 7 = 5 + 2
['G',2]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 9 = 7 + 2
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10 = 8 + 2
Description
Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000524
(load all 2 compositions to match this statistic)
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
St000524: Posets ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 80%
Values
['A',1]
 => ([],1)
 => ? = 1 - 2
['A',2]
 => ([(0,2),(1,2)],3)
 => 2 = 4 - 2
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => 3 = 5 - 2
['G',2]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => 5 = 7 - 2
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => 6 = 8 - 2
Description
The number of posets with the same order polynomial. The order polynomial of a poset $P$ is the polynomial $S$ such that $S(m)$ is the number of order-preserving maps from $P$ to $\{1,\dots,m\}$. See sections 3.12 and 3.15 of [1].
Matching statistic: St000228
(load all 2 compositions to match this statistic)
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 80%
Values
['A',1]
 => ([],1)
 => [2]
 => 2 = 1 + 1
['A',2]
 => ([(0,2),(1,2)],3)
 => [3,2]
 => 5 = 4 + 1
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => 6 = 5 + 1
['G',2]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [6,2]
 => 8 = 7 + 1
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => [8,4,2]
 => ? = 8 + 1
Description
The size of a partition. This statistic is the constant statistic of the level sets.
Matching statistic: St000668
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 80%
Values
['A',1]
 => ([],1)
 => [1]
 => ? = 1 - 2
['A',2]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => 2 = 4 - 2
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => 3 = 5 - 2
['G',2]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => 5 = 7 - 2
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => [3,2,1]
 => 6 = 8 - 2
Description
The least common multiple of the parts of the partition.
Matching statistic: St000708
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 80%
Values
['A',1]
 => ([],1)
 => [1]
 => ? = 1 - 2
['A',2]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => 2 = 4 - 2
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => 3 = 5 - 2
['G',2]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => 5 = 7 - 2
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => [3,2,1]
 => 6 = 8 - 2
Description
The product of the parts of an integer partition.
Matching statistic: St001128
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St001128: Integer partitions ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 80%
Values
['A',1]
 => ([],1)
 => [1]
 => ? = 1 - 2
['A',2]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => 2 = 4 - 2
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => 3 = 5 - 2
['G',2]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => 5 = 7 - 2
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => [3,2,1]
 => 6 = 8 - 2
Description
The exponens consonantiae of a partition. This is the quotient of the least common multiple and the greatest common divior of the parts of the partiton. See [1, Caput sextum, §19-§22].
Matching statistic: St000081
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St000081: Graphs ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 80%
Values
['A',1]
 => ([],1)
 => ([],1)
 => ([(0,1)],2)
 => 1
['A',2]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
['G',2]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ([(4,5)],6)
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => 7
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,6),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 8
Description
The number of edges of a graph.
The following 7 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000770The major index of an integer partition when read from bottom to top. St000548The number of different non-empty partial sums of an integer partition. St001003The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St000327The number of cover relations in a poset. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001649The length of a longest trail in a graph. St001879The 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.