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Your data matches 57 different statistics following compositions of up to 3 maps.
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Matching statistic: St001173
St001173: Finite Cartan types ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
=> 1
['A',2]
=> 3
['B',2]
=> 3
['G',2]
=> 3
['A',3]
=> 6
Description
The number of commutative positive roots in the root system of the given finite Cartan type.
An upper ideal $I$ in the root poset $\Phi^+$ is called '''abelian''' if $\alpha,\beta \in I$ implies that $\alpha+\beta \notin \Phi^+$. A positive root is called '''commutative''' if the upper ideal it generates is abelian.
The numbers are then given in [1, Theorem 4.4].
Matching statistic: St000548
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000548: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000548: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
=> ([],1)
=> [1]
=> 1
['A',2]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> 3
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> 3
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [5,1]
=> 3
['A',3]
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,2,1]
=> 6
Description
The number of different non-empty partial sums of an integer partition.
Matching statistic: St000722
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Values
['A',1]
=> ([],1)
=> ([],1)
=> 1
['A',2]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 3
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 3
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> 3
['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)
=> 6
Description
The number of different neighbourhoods in a graph.
Matching statistic: St000299
Values
['A',1]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
['A',2]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 3
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
['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)
=> 3
['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)
=> 6
Description
The number of nonisomorphic vertex-induced subtrees.
Matching statistic: St000453
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
['A',1]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
['A',2]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 3
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
['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)
=> 3
['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)
=> 6
Description
The number of distinct Laplacian eigenvalues of a graph.
Matching statistic: St000763
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
St000763: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
St000763: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
=> ([],1)
=> ([],1)
=> [1] => 1
['A',2]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> [1,2] => 3
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> [1,3] => 3
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> [1,5] => 3
['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)
=> [1,1,1,1,2] => 6
Description
The sum of the positions of the strong records of an integer composition.
A strong record is an element $a_i$ such that $a_i > a_j$ for all $j < i$. This statistic is the sum of the positions of the strong records.
Matching statistic: St000777
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
['A',1]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
['A',2]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 3
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
['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)
=> 3
['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)
=> 6
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Matching statistic: St001120
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)
=> 3
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
['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)
=> 3
['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)
=> 6
Description
The length of a longest path in a graph.
Matching statistic: St001342
Values
['A',1]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
['A',2]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 3
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(1,2)],3)
=> 3
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> ([(1,2)],3)
=> 3
['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),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 6
Description
The number of vertices in the center of a graph.
The center of a graph is the set of vertices whose maximal distance to any other vertex is minimal. In particular, if the graph is disconnected, all vertices are in the certer.
Matching statistic: St001746
Values
['A',1]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
['A',2]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 3
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 3
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> ([(4,5)],6)
=> 3
['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)
=> 6
Description
The coalition number of a graph.
This is the maximal cardinality of a set partition such that each block is either a dominating set of cardinality one, or is not a dominating set but can be joined with a second block to form a dominating set.
The following 47 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000718The largest Laplacian eigenvalue of a graph if it is integral. St001279The sum of the parts of an integer partition that are at least two. St001458The rank of the adjacency matrix of a graph. St001459The number of zero columns in the nullspace of a graph. St001692The number of vertices with higher degree than the average degree in a graph. St000849The number of 1/3-balanced pairs in a poset. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St000185The weighted size of a partition. St000422The energy of a graph, if it is integral. St000511The number of invariant subsets when acting with a permutation of given cycle type. St001214The aft of an integer partition. St001592The maximal number of simple paths between any two different vertices of a graph. St001877Number of indecomposable injective modules with projective dimension 2. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000935The number of ordered refinements of an integer partition. St000142The number of even parts of a partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000995The largest even part of an integer partition. St001128The exponens consonantiae of a partition. St001252Half the sum of the even parts of a partition. St001703The villainy of a graph. St001814The number of partitions interlacing the given partition. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000635The number of strictly order preserving maps of a poset into itself. St000693The modular (standard) major index of a standard tableau. St000984The number of boxes below precisely one peak. St001378The product of the cohook lengths of the integer partition. St001563The value of the power-sum symmetric function evaluated at 1. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St000477The weight of a partition according to Alladi. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St000741The Colin de Verdière graph invariant. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000455The second largest eigenvalue of a graph if it is integral. St000096The number of spanning trees of a graph. St001488The number of corners of a skew partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001626The number of maximal proper sublattices of a lattice. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path.
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