Your data matches 63 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001369
St001369: 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]
=> 1 = 0 + 1
['B',2]
=> 2 = 1 + 1
['G',2]
=> 3 = 2 + 1
['A',3]
=> 1 = 0 + 1
['B',3]
=> 2 = 1 + 1
['C',3]
=> 2 = 1 + 1
Description
The largest coefficient in the highest root in the root system of a Cartan type.
Matching statistic: St000150
Mp00148: Finite Cartan types to root posetPosets
Mp00198: Posets incomparability graphGraphs
Mp00037: Graphs to partition of connected componentsInteger partitions
St000150: Integer partitions ⟶ 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)
=> [2,1]
=> 0
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> 1
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> [2,1,1,1,1]
=> 2
['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)
=> [5,1]
=> 0
['B',3]
=> ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
=> ([(2,7),(3,5),(3,8),(4,6),(4,8),(5,6),(5,7),(6,8),(7,8)],9)
=> [7,1,1]
=> 1
['C',3]
=> ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
=> ([(2,7),(3,5),(3,8),(4,6),(4,8),(5,6),(5,7),(6,8),(7,8)],9)
=> [7,1,1]
=> 1
Description
The floored half-sum of the multiplicities of a partition. This statistic is equidistributed with [[St000143]] and [[St000149]], see [1].
Matching statistic: St001031
Mp00148: Finite Cartan types to root posetPosets
Mp00110: Posets Greene-Kleitman invariantInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
St001031: 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]
=> 0
['A',2]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 1
['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]
=> 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]
=> 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]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 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]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 1
Description
The height of the bicoloured Motzkin path associated with the Dyck path.
Mp00148: Finite Cartan types to root posetPosets
Mp00110: Posets Greene-Kleitman invariantInteger partitions
St000478: Integer partitions ⟶ ℤResult quality: 86% values known / values provided: 86%distinct values known / distinct values provided: 100%
Values
['A',1]
=> ([],1)
=> [1]
=> ? = 0
['A',2]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> 0
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> 1
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [5,1]
=> 2
['A',3]
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,2,1]
=> 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
['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
Description
Another weight of a partition according to Alladi. According to Theorem 3.4 (Alladi 2012) in [1] $$ \sum_{\pi\in GG_1(r)} w_1(\pi) $$ equals the number of partitions of $r$ whose odd parts are all distinct. $GG_1(r)$ is the set of partitions of $r$ where consecutive entries differ by at least $2$, and consecutive even entries differ by at least $4$.
Matching statistic: St000997
Mp00148: Finite Cartan types to root posetPosets
Mp00110: Posets Greene-Kleitman invariantInteger partitions
Mp00322: Integer partitions Loehr-WarringtonInteger partitions
St000997: Integer partitions ⟶ ℤResult quality: 86% values known / values provided: 86%distinct values known / distinct values provided: 100%
Values
['A',1]
=> ([],1)
=> [1]
=> [1]
=> ? = 0 - 2
['A',2]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> [3]
=> -1 = 1 - 2
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [2,1,1]
=> 0 = 2 - 2
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [5,1]
=> [2,1,1,1,1]
=> -2 = 0 - 2
['A',3]
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [5,1]
=> -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)
=> [5,3,1]
=> [3,2,2,1,1]
=> -1 = 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)
=> [5,3,1]
=> [3,2,2,1,1]
=> -1 = 1 - 2
Description
The even-odd crank of an integer partition. This is the largest even part minus the number of odd parts.
Mp00148: Finite Cartan types to root posetPosets
Mp00074: Posets to graphGraphs
St001056: Graphs ⟶ ℤResult quality: 71% values known / values provided: 71%distinct values known / distinct values provided: 100%
Values
['A',1]
=> ([],1)
=> ([],1)
=> 0
['A',2]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
['B',2]
=> ([(0,3),(1,3),(3,2)],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
['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
['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)
=> ? ∊ {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)
=> ? ∊ {1,1}
Description
The Grundy value for the game of deleting vertices of a graph until it has no edges.
Mp00148: Finite Cartan types to root posetPosets
Mp00198: Posets incomparability graphGraphs
St000785: Graphs ⟶ ℤResult quality: 71% values known / values provided: 71%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)
=> 1 = 0 + 1
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 2 = 1 + 1
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> 3 = 2 + 1
['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)
=> 2 = 1 + 1
['B',3]
=> ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
=> ([(2,7),(3,5),(3,8),(4,6),(4,8),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {0,1} + 1
['C',3]
=> ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
=> ([(2,7),(3,5),(3,8),(4,6),(4,8),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {0,1} + 1
Description
The number of distinct colouring schemes of a graph. To any proper colouring with the minimal number of colours possible we associate the integer partition recording how often each colour is used. This statistic records the number of distinct partitions that occur. For example, the graph on six vertices consisting of a square together with two attached triangles - ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) in the list of values - is three-colourable and admits two colouring schemes, $[2,2,2]$ and $[3,2,1]$.
Matching statistic: St000671
Mp00148: Finite Cartan types to root posetPosets
Mp00198: Posets incomparability graphGraphs
Mp00111: Graphs complementGraphs
St000671: Graphs ⟶ ℤResult quality: 71% values known / values provided: 71%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)
=> 0
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
['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)
=> 2
['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)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 1
['B',3]
=> ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
=> ([(2,7),(3,5),(3,8),(4,6),(4,8),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,3),(0,6),(0,7),(0,8),(1,2),(1,4),(1,6),(1,7),(1,8),(2,4),(2,5),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {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)
=> ([(2,7),(3,5),(3,8),(4,6),(4,8),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,3),(0,6),(0,7),(0,8),(1,2),(1,4),(1,6),(1,7),(1,8),(2,4),(2,5),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,1}
Description
The maximin edge-connectivity for choosing a subgraph. This is $\max_X \min(\lambda(G[X]), \lambda(G[V\setminus X]))$, where $X$ ranges over all subsets of the vertex set $V$ and $\lambda$ is the edge-connectivity of a graph.
Matching statistic: St001323
Mp00148: Finite Cartan types to root posetPosets
Mp00074: Posets to graphGraphs
Mp00157: Graphs connected complementGraphs
St001323: Graphs ⟶ ℤResult quality: 71% values known / values provided: 71%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)
=> 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
['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)
=> 0
['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)
=> ? ∊ {1,1}
['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}
Description
The independence gap of a graph. This is the difference between the independence number [[St000093]] and the minimal size of a maximally independent set of a graph. In particular, this statistic is $0$ for well covered graphs
Matching statistic: St001646
Mp00148: Finite Cartan types to root posetPosets
Mp00198: Posets incomparability graphGraphs
Mp00117: Graphs Ore closureGraphs
St001646: Graphs ⟶ ℤResult quality: 71% values known / values provided: 71%distinct values known / distinct values provided: 100%
Values
['A',1]
=> ([],1)
=> ([],1)
=> ([],1)
=> 0
['A',2]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 0
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 1
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> ([(4,5)],6)
=> 2
['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)
=> ([(1,2),(1,3),(1,4),(1,5),(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)
=> ([(2,7),(3,5),(3,8),(4,6),(4,8),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(2,6),(3,7),(3,8),(4,5),(4,8),(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)
=> ([(2,7),(3,5),(3,8),(4,6),(4,8),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(2,6),(3,7),(3,8),(4,5),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,1}
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 53 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001689The number of celebrities in a graph. St000553The number of blocks of a graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St001111The weak 2-dynamic chromatic number of a graph. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001739The number of graphs with the same edge polytope as the given graph. St001949The rigidity index of a graph. St000636The hull number of a graph. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001570The minimal number of edges to add to make a graph Hamiltonian. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001638The book thickness of a graph. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001002Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001964The interval resolution global dimension of a poset. St000454The largest eigenvalue of a graph if it is integral. St000477The weight of a partition according to Alladi. St001578The minimal number of edges to add or remove to make a graph a line graph. St001621The number of atoms of a lattice. St001623The number of doubly irreducible elements of a lattice. St001644The dimension of a graph. St001742The difference of the maximal and the minimal degree in a graph. St001812The biclique partition number of a graph. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000741The Colin de Verdière graph invariant. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001330The hat guessing number of a graph. St001386The number of prime labellings of a graph. St001391The disjunction number of a graph. St001624The breadth of a lattice. St001642The Prague dimension of a graph. St001783The number of odd automorphisms of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000567The sum of the products of all pairs of parts. St000668The least common multiple of the parts of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000708The product of the parts of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000933The number of multipartitions of sizes given by an integer partition. St001128The exponens consonantiae of a partition. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000770The major index of an integer partition when read from bottom to top. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition.