Your data matches 135 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',1]
 => 2 = 0 + 2
['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
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: St001495
Mapping
St001495: Finite Cartan types ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
 => 2 = 0 + 2
['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
Description
The maximal order of an element in the Weyl group of a given Cartan type. For the symmetric group, this is [[oeis:A000793]]
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',1]
 => 2 = 0 + 2
['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
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',1]
 => ([],1)
 => [1]
 => 0
['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
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',1]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
['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
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',1]
 => ([],1)
 => [1]
 => 1 = 0 + 1
['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
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',1]
 => ([],1)
 => [1]
 => 1 = 0 + 1
['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
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',1]
 => ([],1)
 => [1]
 => 1 = 0 + 1
['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
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',1]
 => ([],1)
 => [1]
 => 2 = 0 + 2
['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
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',1]
 => ([],1)
 => [1]
 => [1,0]
 => 0
['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
Description
The number of valleys of the Dyck path.
The following 125 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000970Number of peaks minus the dominant dimension of the corresponding LNakayama 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. 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. 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. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St000706The product of the factorials of the multiplicities of an integer partition. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000080The rank of the poset. St000528The height of a poset. St000912The number of maximal antichains in a poset. St000778The metric dimension of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. 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. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. 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. St001829The common independence number of a graph. St000171The degree of the graph. 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. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. 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. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. 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. St001586The number of odd parts smaller than the largest even part in an integer partition. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001949The rigidity index of a graph. 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. St000189The number of elements in the poset. St000271The chromatic index 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. St001286The annihilation number of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001316The domatic number of a graph. St001343The dimension of the reduced incidence algebra of a poset. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001616The number of neutral elements in a lattice. St001656The monophonic position number of a graph. St001717The largest size of an interval in a poset. St001720The minimal length of a chain of small intervals in a lattice. St001883The mutual visibility number of a graph. St000636The hull number of a graph. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001655The general position number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St000906The length of the shortest maximal chain in a poset. St000656The number of cuts of a poset. St000680The Grundy value for Hackendot on posets. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000474Dyson's crank of a partition. St000744The length of the path to the largest entry in a standard Young tableau. St000940The number of characters of the symmetric group whose value on the partition is zero. St001644The dimension of a graph. St001118The acyclic chromatic index of a graph. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001651The Frankl number of a lattice. St001875The number of simple modules with projective dimension at most 1. St001003The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000422The energy of a graph, if it is integral. St000454The largest eigenvalue of a graph if it is integral. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001621The number of atoms of a lattice. St001638The book thickness of a graph. St001703The villainy of a graph. St001742The difference of the maximal and the minimal degree in a graph. St001812The biclique partition number of a graph. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000741The Colin de Verdière graph invariant. St001330The hat guessing number of a graph. St001391The disjunction number of a graph. St001642The Prague dimension of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000509The diagonal index (content) of a partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001738The minimal order of a graph which is not an induced subgraph of the given graph.