Your data matches 130 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000139
Mapping
St000139: Finite Cartan types ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',2]
 => 3 = 1 + 2
['B',2]
 => 4 = 2 + 2
['G',2]
 => 6 = 4 + 2
['A',3]
 => 4 = 2 + 2
['B',3]
 => 6 = 4 + 2
['C',3]
 => 6 = 4 + 2
['A',4]
 => 5 = 3 + 2
Description
The Coxeter number of a finite Cartan type. The Coxeter number $h$ for the Weyl group $W$ of the given finite Cartan type is defined as the order of the product of the Coxeter generators of $W$. Equivalently, this is equal to the maximal degree of a fundamental invariant of $W$, see also [[St000138]].
Matching statistic: St001897
Mapping
St001897: Finite Cartan types ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',2]
 => 3 = 1 + 2
['B',2]
 => 4 = 2 + 2
['G',2]
 => 6 = 4 + 2
['A',3]
 => 4 = 2 + 2
['B',3]
 => 6 = 4 + 2
['C',3]
 => 6 = 4 + 2
['A',4]
 => 5 = 3 + 2
Description
The minimal degree of a faithful permutation representation of a Weyl group. Data are from [1, Table 1].
Matching statistic: St000225
(load all 2 compositions to match this statistic)
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000225: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',2]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => 1
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => 2
['G',2]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => 4
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => [3,2,1]
 => 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]
 => 4
['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]
 => 4
['A',4]
 => ([(0,8),(1,7),(2,7),(2,9),(3,8),(3,9),(5,4),(6,4),(7,5),(8,6),(9,5),(9,6)],10)
 => [4,3,2,1]
 => 3
Description
Difference between largest and smallest parts in a partition.
Matching statistic: St000093
(load all 2 compositions to match this statistic)
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',2]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => 2 = 1 + 1
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => 3 = 2 + 1
['G',2]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ([(4,5)],6)
 => 5 = 4 + 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)
 => 3 = 2 + 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)
 => 5 = 4 + 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)
 => 5 = 4 + 1
['A',4]
 => ([(0,8),(1,7),(2,7),(2,9),(3,8),(3,9),(5,4),(6,4),(7,5),(8,6),(9,5),(9,6)],10)
 => ([(1,2),(1,7),(1,9),(2,6),(2,8),(3,4),(3,6),(3,8),(3,9),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,9),(7,8),(8,9)],10)
 => 4 = 3 + 1
Description
The cardinality of a maximal independent set of vertices of a graph. An independent set of a graph is a set of pairwise non-adjacent vertices. A maximum independent set is an independent set of maximum cardinality. This statistic is also called the independence number or stability number $\alpha(G)$ of $G$.
Matching statistic: St000147
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',2]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => 2 = 1 + 1
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => 3 = 2 + 1
['G',2]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => 5 = 4 + 1
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => [3,2,1]
 => 3 = 2 + 1
['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]
 => 5 = 4 + 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]
 => 5 = 4 + 1
['A',4]
 => ([(0,8),(1,7),(2,7),(2,9),(3,8),(3,9),(5,4),(6,4),(7,5),(8,6),(9,5),(9,6)],10)
 => [4,3,2,1]
 => 4 = 3 + 1
Description
The largest part of an integer partition.
Matching statistic: St000384
(load all 4 compositions to match this statistic)
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000384: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',2]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => 2 = 1 + 1
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => 3 = 2 + 1
['G',2]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => 5 = 4 + 1
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => [3,2,1]
 => 3 = 2 + 1
['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]
 => 5 = 4 + 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]
 => 5 = 4 + 1
['A',4]
 => ([(0,8),(1,7),(2,7),(2,9),(3,8),(3,9),(5,4),(6,4),(7,5),(8,6),(9,5),(9,6)],10)
 => [4,3,2,1]
 => 4 = 3 + 1
Description
The maximal part of the shifted composition of an integer partition. A partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ is shifted into a composition by adding $i-1$ to the $i$-th part. The statistic is then $\operatorname{max}_i\{ \lambda_i + i - 1 \}$. See also [[St000380]].
Matching statistic: St000784
(load all 4 compositions to match this statistic)
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000784: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',2]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => 2 = 1 + 1
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => 3 = 2 + 1
['G',2]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => 5 = 4 + 1
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => [3,2,1]
 => 3 = 2 + 1
['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]
 => 5 = 4 + 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]
 => 5 = 4 + 1
['A',4]
 => ([(0,8),(1,7),(2,7),(2,9),(3,8),(3,9),(5,4),(6,4),(7,5),(8,6),(9,5),(9,6)],10)
 => [4,3,2,1]
 => 4 = 3 + 1
Description
The maximum of the length and the largest part of the integer partition. This is the side length of the smallest square the Ferrers diagram of the partition fits into. It is also the minimal number of colours required to colour the cells of the Ferrers diagram such that no two cells in a column or in a row have the same colour, see [1]. See also [[St001214]].
Matching statistic: St000380
(load all 4 compositions to match this statistic)
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000380: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',2]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => 3 = 1 + 2
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => 4 = 2 + 2
['G',2]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => 6 = 4 + 2
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => [3,2,1]
 => 4 = 2 + 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]
 => 6 = 4 + 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]
 => 6 = 4 + 2
['A',4]
 => ([(0,8),(1,7),(2,7),(2,9),(3,8),(3,9),(5,4),(6,4),(7,5),(8,6),(9,5),(9,6)],10)
 => [4,3,2,1]
 => 5 = 3 + 2
Description
Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. Put differently, this is the smallest number $n$ such that the partition fits into the triangular partition $(n-1,n-2,\dots,1)$.
Matching statistic: St000053
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',2]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 2
['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]
 => 4
['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]
 => 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]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => 4
['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]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => 4
['A',4]
 => ([(0,8),(1,7),(2,7),(2,9),(3,8),(3,9),(5,4),(6,4),(7,5),(8,6),(9,5),(9,6)],10)
 => [4,3,2,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
Description
The number of valleys of the Dyck path.
Matching statistic: St000970
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000970: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',2]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 2
['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]
 => 4
['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]
 => 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]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => 4
['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]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => 4
['A',4]
 => ([(0,8),(1,7),(2,7),(2,9),(3,8),(3,9),(5,4),(6,4),(7,5),(8,6),(9,5),(9,6)],10)
 => [4,3,2,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
Description
Number of peaks minus the dominant dimension of the corresponding LNakayama algebra.
The following 120 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000010The length of the partition. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000676The number of odd rises of a Dyck path. St000734The last entry in the first row of a standard tableau. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St000393The number of strictly increasing runs in a binary word. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001622The number of join-irreducible elements of a lattice. St000080The rank of the poset. St000528The height of a poset. St000906The length of the shortest maximal chain in a poset. St000680The Grundy value for Hackendot on posets. St000776The maximal multiplicity of an eigenvalue in a graph. St000778The metric dimension of a graph. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001820The size of the image of the pop stack sorting operator. St000258The burning number of a graph. St000273The domination number of a graph. St000482The (zero)-forcing number of a graph. St000544The cop number of a graph. St000553The number of blocks of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001626The number of maximal proper sublattices of a lattice. St001829The common independence number of a graph. St000636The hull number of a graph. St000918The 2-limited packing number of a graph. St001315The dissociation number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000272The treewidth of a graph. St000310The minimal degree of a vertex of a graph. St000331The number of upper interactions of a Dyck path. St000536The pathwidth of a graph. St000744The length of the path to the largest entry in a standard Young tableau. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001277The degeneracy of a graph. St001323The independence gap of a graph. St001358The largest degree of a regular subgraph of a graph. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001962The proper pathwidth of a graph. St000015The number of peaks of a Dyck path. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000822The Hadwiger number of the graph. St000907The number of maximal antichains of minimal length in a poset. St000916The packing number of a graph. St001029The size of the core of a graph. St001119The length of a shortest maximal path in a graph. St001286The annihilation number of a graph. St001316The domatic number of a graph. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001494The Alon-Tarsi number of a graph. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001580The acyclic chromatic number of a graph. St001720The minimal length of a chain of small intervals in a lattice. St000144The pyramid weight of the Dyck path. St000917The open packing number of a graph. St001002Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001672The restrained domination number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001875The number of simple modules with projective dimension at most 1. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St000145The Dyson rank of a partition. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000992The alternating sum of the parts of an integer partition. St001055The Grundy value for the game of removing cells of a row in an integer partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000474Dyson's crank of a partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000228The size of a partition. St000517The Kreweras number of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000160The multiplicity of the smallest part of a partition. St000475The number of parts equal to 1 in a partition. St001091The number of parts in an integer partition whose next smaller part has the same size. St001644The dimension of a graph. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. 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 $eA$. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001638The book thickness of a graph. St000479The Ramsey number of a graph. St001917The order of toric promotion on the set of labellings of a graph. St000081The number of edges of a graph. St000459The hook length of the base cell of a partition. St001725The harmonious chromatic number of a graph. St000548The number of different non-empty partial sums of an integer partition. St000867The sum of the hook lengths in the first row of an integer partition. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St001127The sum of the squares of the parts of a partition.