Your data matches 29 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001753
St001753: Finite Cartan types ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
['A',1]
=> 1 = 0 + 1
['A',2]
=> 3 = 2 + 1
['B',2]
=> 6 = 5 + 1
['G',2]
=> 15 = 14 + 1
Description
The number of subsets of the positive roots that form a basis of the associated vector space. For the group $W$ and an associated set of positive roots $\Phi^+ \subseteq V$ this counts the number of subsets $S \subseteq \Phi^+$ that form a basis of $V$. This is also the number of subsets of the reflections $R \subseteq W$ that form a minimal set of generators of a reflection subgroup of full rank. The Coxeter permutahedron can be defined as the Minkowski sum of the line segments $[- \frac{\alpha}{2}, \frac{\alpha}{2}]$ for $\alpha \in \Phi^+$. As a zonotope this polytope can be decomposed into a (disjoint) union of (half-open) parallel epipeds [1]. This also counts the number of full dimensional parallel epipeds among this decomposition.
Matching statistic: St001541
Mp00148: Finite Cartan types to root posetPosets
Mp00110: Posets Greene-Kleitman invariantInteger partitions
St001541: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
['A',1]
=> ([],1)
=> [1]
=> 0
['A',2]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> 2
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> 5
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [5,1]
=> 14
Description
The Gini index of an integer partition. As discussed in [1], this statistic is equal to [[St000567]] applied to the conjugate partition.
Mp00148: Finite Cartan types to root posetPosets
Mp00198: Posets incomparability graphGraphs
St000349: Graphs ⟶ 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)
=> ([(1,2)],3)
=> 3 = 2 + 1
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 6 = 5 + 1
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> 15 = 14 + 1
Description
The number of different adjacency matrices of a graph. This is the number of different labellings of the graph, or $\frac{|G|!}{|\operatorname{Aut}(G)|}$.
Matching statistic: St001833
Mp00148: Finite Cartan types to root posetPosets
Mp00205: Posets maximal antichainsLattices
St001833: Lattices ⟶ 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)
=> ([(0,1)],2)
=> 3 = 2 + 1
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 6 = 5 + 1
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 15 = 14 + 1
Description
The number of linear intervals in a lattice.
Matching statistic: St000009
Mp00148: Finite Cartan types to root posetPosets
Mp00110: Posets Greene-Kleitman invariantInteger partitions
Mp00042: Integer partitions initial tableauStandard tableaux
St000009: Standard tableaux ⟶ 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]
=> [[1,2],[3]]
=> 2
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 5
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> 14
Description
The charge of a standard tableau.
Matching statistic: St000081
Mp00148: Finite Cartan types to root posetPosets
Mp00198: Posets incomparability graphGraphs
Mp00111: Graphs complementGraphs
St000081: Graphs ⟶ 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)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 5
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 14
Description
The number of edges of a graph.
Matching statistic: St000448
Mp00148: Finite Cartan types to root posetPosets
Mp00198: Posets incomparability graphGraphs
Mp00203: Graphs coneGraphs
St000448: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
['A',1]
=> ([],1)
=> ([],1)
=> ([(0,1)],2)
=> 0
['A',2]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
['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)
=> 14
Description
The number of pairs of vertices of a graph with distance 2. This is the coefficient of the quadratic term of the Wiener polynomial.
Matching statistic: St001308
Mp00148: Finite Cartan types to root posetPosets
Mp00198: Posets incomparability graphGraphs
Mp00203: Graphs coneGraphs
St001308: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
['A',1]
=> ([],1)
=> ([],1)
=> ([(0,1)],2)
=> 0
['A',2]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
['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)
=> 14
Description
The number of induced paths on three vertices in a graph.
Matching statistic: St001350
Mp00148: Finite Cartan types to root posetPosets
Mp00198: Posets incomparability graphGraphs
Mp00203: Graphs coneGraphs
St001350: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
['A',1]
=> ([],1)
=> ([],1)
=> ([(0,1)],2)
=> 0
['A',2]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
['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)
=> 14
Description
Half of the Albertson index of a graph. This is $\frac{1}{2}\sum_{\{u,v\}\in E}|d(u)-d(v)|$, where $E$ is the set of edges and $d_v$ is the degree of vertex $v$, see [1]. In particular, this statistic vanishes on graphs whose components are all regular, see [2].
Matching statistic: St001646
Mp00148: Finite Cartan types to root posetPosets
Mp00198: Posets incomparability graphGraphs
Mp00203: Graphs coneGraphs
St001646: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
['A',1]
=> ([],1)
=> ([],1)
=> ([(0,1)],2)
=> 0
['A',2]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
['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)
=> 14
Description
The number of edges that can be added without increasing the maximal degree of a graph. This statistic is (except for the degenerate case of two vertices) maximized by the star-graph on $n$ vertices, which has maximal degree $n-1$ and therefore has statistic $\binom{n-1}{2}$.
The following 19 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000049The number of set partitions whose sorted block sizes correspond to the partition. St000182The number of permutations whose cycle type is the given integer partition. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000517The Kreweras number of an integer partition. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001694The number of maximal dissociation sets in a graph. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001909The number of interval-closed sets of a poset. St000309The number of vertices with even degree. St000567The sum of the products of all pairs of parts. St000456The monochromatic index of a connected graph. St000479The Ramsey number of a graph. St000509The diagonal index (content) of a partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000639The number of relations in a poset. St001243The sum of coefficients in the Schur basis of certain LLT polynomials associated with a Dyck path. St001345The Hamming dimension of a graph.