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Your data matches 47 different statistics following compositions of up to 3 maps.
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Matching statistic: St001147
St001147: Finite Cartan types ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
=> 1
['A',2]
=> 2
['B',2]
=> 1
['G',2]
=> 0
Description
The number of minuscule dominant weights in the weight lattice of a finite Cartan type.
In short, this is the number of simple roots that appear with multiplicity one in the hightest root of the root system.
By definition, a weight $\lambda \neq 0$ in the weight lattice is '''dominant''' if $\langle \lambda, \alpha\rangle \geq 0$ for all simple roots $\alpha$ and a dominant weight is '''minuscule''' if $\langle \lambda, \beta\rangle \in \{0,\pm 1\}$ for all roots $\beta$. Since $\langle \lambda, \alpha\rangle \in \{0,1\}$ for simple roots $\alpha$, we have that $\lambda$ is minuscule if and only if it is fundamental and $\langle \lambda, \rho\rangle = 1$ for the unique highest root $\rho$.
The number of minuscule dominant weights is one less than the determinant of the Cartan matrix [[St000821]]. They index the nontrivial minuscule representations, see [1].
Matching statistic: St000821
St000821: Finite Cartan types ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
=> 2 = 1 + 1
['A',2]
=> 3 = 2 + 1
['B',2]
=> 2 = 1 + 1
['G',2]
=> 1 = 0 + 1
Description
The determinant of the Cartan matrix.
This is also the order of the center of the corresponding simply connected group.
Matching statistic: St000309
Values
['A',1]
=> ([],1)
=> ([],1)
=> 1
['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)
=> 0
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2
Description
The number of vertices with even degree.
Matching statistic: St001057
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
['A',1]
=> ([],1)
=> ([],1)
=> 1
['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
Description
The Grundy value of the game of creating an independent set in a graph.
Two players alternately add a vertex to an initially empty set, which is not adjacent to any of the vertices it already contains.
Alternatively, the game can be described as starting with a graph, the players remove vertices together with their neighbors, until the graph is empty.
Matching statistic: St000986
Values
['A',1]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
['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
Description
The multiplicity of the eigenvalue zero of the adjacency matrix of the graph.
Matching statistic: St001056
Values
['A',1]
=> ([],1)
=> ([],1)
=> ([(0,1)],2)
=> 1
['A',2]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
Description
The Grundy value for the game of deleting vertices of a graph until it has no edges.
Matching statistic: St001283
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001283: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001283: Integer partitions ⟶ 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)
=> [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
Description
The number of finite solvable groups that are realised by the given partition over the complex numbers.
A finite group $G$ is ''realised'' by the partition $(a_1,\dots,a_m)$ if its group algebra over the complex numbers is isomorphic to the direct product of $a_i\times a_i$ matrix rings over the complex numbers.
The smallest partition which does not realise a solvable group, but does realise a finite group, is $(5,4,3,3,1)$.
Matching statistic: St001284
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001284: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001284: Integer partitions ⟶ 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)
=> [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
Description
The number of finite groups that are realised by the given partition over the complex numbers.
A finite group $G$ is 'realised' by the partition $(a_1,...,a_m)$ if its group algebra over the complex numbers is isomorphic to the direct product of $a_i\times a_i$ matrix rings over the complex numbers.
Matching statistic: St001367
Values
['A',1]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
['A',2]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(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)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 0
Description
The smallest number which does not occur as degree of a vertex in a graph.
Matching statistic: St001913
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001913: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001913: Integer partitions ⟶ 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)
=> [2,1]
=> 2
['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]
=> 0
Description
The number of preimages of an integer partition in Bulgarian solitaire.
A move in Bulgarian solitaire consists of removing the first column of the Ferrers diagram and inserting it as a new row.
Partitions without preimages are called garden of eden partitions [1].
The following 37 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St000478Another weight of a partition according to Alladi. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000936The number of even 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. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001964The interval resolution global dimension of a poset. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000274The number of perfect matchings of a graph. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000928The sum of the coefficients of the character polynomial of an integer partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St001618The cardinality of the Frattini sublattice of a lattice. St001624The breadth of a lattice. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001783The number of odd automorphisms of a graph. St001881The number of factors of a lattice as a Cartesian product of lattices. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. 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. St001578The minimal number of edges to add or remove to make a graph a line graph. St001638The book thickness of a graph. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000708The product of the parts of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000997The even-odd crank of an integer partition. 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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