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Your data matches 166 different statistics following compositions of up to 3 maps.
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Matching statistic: St000854
St000854: 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]
=> 2 = 1 + 1
Description
The number of orbits of reflections of a finite Cartan type.
Let $W$ be the Weyl group of a Cartan type. The reflections in $W$ are closed under conjugation, and this statistic counts the number of conjugacy classes of $W$ that are reflections.
It is well-known that there are either one or two such conjugacy classes.
Matching statistic: St001158
St001158: 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]
=> 2 = 1 + 1
Description
The size of the mutation class of quivers of given type.
Matching statistic: St001950
St001950: 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]
=> 2 = 1 + 1
Description
The minimal size of a base for the Weyl group of the Cartan type.
A base of a permutation group is a set $B$ such that the pointwise stabilizer of $B$ is trivial. For example, a base of the symmetric group on $n$ letters must contain all but one letter.
Any base has at least $\log |G|/n$ elements, where $n$ is the degree of the group, i.e., the size of its domain.
Matching statistic: St001964
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Mp00148: Finite Cartan types —to root poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001964: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
=> ([],1)
=> 0
['A',2]
=> ([(0,2),(1,2)],3)
=> 0
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> 1
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> 1
Description
The interval resolution global dimension of a poset.
This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.
Matching statistic: St000256
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(load all 3 compositions to match this statistic)
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000256: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000256: 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]
=> 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]
=> 1
Description
The number of parts from which one can substract 2 and still get an integer partition.
Matching statistic: St000928
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(load all 3 compositions to match this statistic)
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St000928: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St000928: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
=> ([],1)
=> [2]
=> 0
['A',2]
=> ([(0,2),(1,2)],3)
=> [3,2]
=> 1
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> 1
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [6,2]
=> 0
Description
The sum of the coefficients of the character polynomial of an integer partition.
The definition of the character polynomial can be found in [1].
Matching statistic: St001307
Values
['A',1]
=> ([],1)
=> ([],1)
=> 0
['A',2]
=> ([(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)
=> 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)
=> 1
Description
The number of induced stars on four vertices in a graph.
Matching statistic: St001320
Values
['A',1]
=> ([],1)
=> ([],1)
=> 0
['A',2]
=> ([(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)
=> 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)
=> 1
Description
The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph.
A graph is a disjoint union of paths if and only if in any linear ordering of its vertices, there are no three vertices $a < b < c$ such that $(a,c)$ is an edge. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.
Matching statistic: St001354
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(load all 2 compositions to match this statistic)
Values
['A',1]
=> ([],1)
=> ([],1)
=> 0
['A',2]
=> ([(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)
=> 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
Description
The number of series nodes in the modular decomposition of a graph.
Matching statistic: St001578
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Values
['A',1]
=> ([],1)
=> ([],1)
=> 0
['A',2]
=> ([(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)
=> 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)
=> 1
Description
The minimal number of edges to add or remove to make a graph a line graph.
The following 156 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001638The book thickness of a graph. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St000142The number of even parts of a partition. St000148The number of odd parts of a partition. St000286The number of connected components of the complement of a graph. St000480The number of lower covers of a partition in dominance order. St000482The (zero)-forcing number of a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001092The number of distinct even parts of a partition. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St000052The number of valleys of a Dyck path not on the x-axis. St000143The largest repeated part of a partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St000257The number of distinct parts of a partition that occur at least twice. St000274The number of perfect matchings of a graph. St000295The length of the border of a binary word. St000313The number of degree 2 vertices of a graph. St000386The number of factors DDU in a Dyck path. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000481The number of upper covers of a partition in dominance order. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000547The number of even non-empty partial sums of an integer partition. St000552The number of cut vertices of a graph. St000659The number of rises of length at least 2 of a Dyck path. St000944The 3-degree of an integer partition. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001071The beta invariant of the graph. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001214The aft of an integer partition. St001252Half the sum of the even parts of a partition. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001280The number of parts of an integer partition that are at least two. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St001341The number of edges in the center of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001413Half the length of the longest even length palindromic prefix of a binary word. St001424The number of distinct squares in a binary word. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001524The degree of symmetry of a binary word. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001587Half of the largest even part of an integer partition. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001657The number of twos in an integer partition. St001689The number of celebrities in a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001730The number of times the path corresponding to a binary word crosses the base line. St001783The number of odd automorphisms of a graph. St001930The weak major index of a binary word. St000159The number of distinct parts of the integer partition. St000160The multiplicity of the smallest part of a partition. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000287The number of connected components of a graph. St000288The number of ones in a binary word. St000309The number of vertices with even degree. St000363The number of minimal vertex covers of a graph. St000392The length of the longest run of ones in a binary word. St000393The number of strictly increasing runs in a binary word. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000553The number of blocks of a graph. St000627The exponent of a binary word. St000631The number of distinct palindromic decompositions of a binary word. St000753The Grundy value for the game of Kayles on a binary word. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000783The side length of the largest staircase partition fitting into a partition. St000897The number of different multiplicities of parts of an integer partition. St000916The packing number of a graph. St000922The minimal number such that all substrings of this length are unique. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000982The length of the longest constant subword. St000992The alternating sum of the parts of an integer partition. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001267The length of the Lyndon factorization of the binary word. St001282The number of graphs with the same chromatic polynomial. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001342The number of vertices in the center of a graph. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001372The length of a longest cyclic run of ones of a binary word. St001415The length of the longest palindromic prefix of a binary word. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001432The order dimension of the partition. St001437The flex of a binary word. St001463The number of distinct columns in the nullspace of a graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001716The 1-improper chromatic number of a graph. St001743The discrepancy of a graph. St001884The number of borders of a binary word. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000676The number of odd rises of a Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001366The maximal multiplicity of a degree of a vertex of a graph. St001654The monophonic hull number of a graph. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001281The normalized isoperimetric number of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001118The acyclic chromatic index of a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000095The number of triangles of a graph. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000454The largest eigenvalue of a graph if it is integral. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000450The number of edges minus the number of vertices plus 2 of a graph. St000455The second largest eigenvalue of a graph if it is integral. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000741The Colin de Verdière graph invariant. St000997The even-odd crank of an integer partition. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001128The exponens consonantiae of a partition. St001330The hat guessing number of a graph. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001568The smallest positive integer that does not appear twice in the partition. St001624The breadth of a lattice. St001642The Prague dimension of a graph. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001720The minimal length of a chain of small intervals in a lattice. St001734The lettericity of a graph. St001812The biclique partition number of a graph. St001881The number of factors of a lattice as a Cartesian product of lattices. St000822The Hadwiger number of the graph. St000939The number of characters of the symmetric group whose value on the partition is positive. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001002Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path.
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