Your data matches 58 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
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
['A',2]
=> 1
['B',2]
=> 2
['G',2]
=> 2
['B',3]
=> 2
['C',3]
=> 2
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: St001950
St001950: Finite Cartan types ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
['A',1]
=> 1
['A',2]
=> 1
['B',2]
=> 2
['G',2]
=> 2
['B',3]
=> 2
['C',3]
=> 2
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.
Mp00148: Finite Cartan types to root posetPosets
Mp00306: Posets rowmotion cycle typeInteger partitions
St001092: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
['A',1]
=> ([],1)
=> [2]
=> 1
['A',2]
=> ([(0,2),(1,2)],3)
=> [3,2]
=> 1
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> 2
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [6,2]
=> 2
['B',3]
=> ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
=> [6,6,6,2]
=> 2
['C',3]
=> ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
=> [6,6,6,2]
=> 2
Description
The number of distinct even parts of a partition. See Section 3.3.1 of [1].
Mp00148: Finite Cartan types to root posetPosets
Mp00110: Posets Greene-Kleitman invariantInteger partitions
St001767: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
['A',1]
=> ([],1)
=> [1]
=> 0 = 1 - 1
['A',2]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> 0 = 1 - 1
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> 1 = 2 - 1
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [5,1]
=> 1 = 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]
=> 1 = 2 - 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]
=> 1 = 2 - 1
Description
The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. Assign to each cell of the Ferrers diagram an arrow pointing north, east, south or west. Then compute for each cell the number of arrows pointing towards it, and take the minimum of those. This statistic is the maximal minimum that can be obtained by assigning arrows in any way.
Matching statistic: St000160
Mp00148: Finite Cartan types to root posetPosets
Mp00110: Posets Greene-Kleitman invariantInteger partitions
Mp00321: Integer partitions 2-conjugateInteger partitions
St000160: 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)
=> [2,1]
=> [3]
=> 1
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [2,1,1]
=> 2
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [5,1]
=> [2,2,1,1]
=> 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]
=> [3,2,2,1,1]
=> 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]
=> [3,2,2,1,1]
=> 2
Description
The multiplicity of the smallest part of a partition. This counts the number of occurrences of the smallest part $spt(\lambda)$ of a partition $\lambda$. The sum $spt(n) = \sum_{\lambda \vdash n} spt(\lambda)$ satisfies the congruences \begin{align*} spt(5n+4) &\equiv 0\quad \pmod{5}\\\ spt(7n+5) &\equiv 0\quad \pmod{7}\\\ spt(13n+6) &\equiv 0\quad \pmod{13}, \end{align*} analogous to those of the counting function of partitions, see [1] and [2].
Mp00148: Finite Cartan types to root posetPosets
Mp00306: Posets rowmotion cycle typeInteger partitions
Mp00044: Integer partitions conjugateInteger partitions
St000257: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
['A',1]
=> ([],1)
=> [2]
=> [1,1]
=> 1
['A',2]
=> ([(0,2),(1,2)],3)
=> [3,2]
=> [2,2,1]
=> 1
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [2,2,1,1]
=> 2
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [6,2]
=> [2,2,1,1,1,1]
=> 2
['B',3]
=> ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
=> [6,6,6,2]
=> [4,4,3,3,3,3]
=> 2
['C',3]
=> ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
=> [6,6,6,2]
=> [4,4,3,3,3,3]
=> 2
Description
The number of distinct parts of a partition that occur at least twice. See Section 3.3.1 of [2].
Mp00148: Finite Cartan types to root posetPosets
Mp00198: Posets incomparability graphGraphs
Mp00037: Graphs to partition of connected componentsInteger partitions
St000897: 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]
=> 1
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> 2
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> [2,1,1,1,1]
=> 2
['B',3]
=> ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
=> ([(2,7),(3,5),(3,8),(4,6),(4,8),(5,6),(5,7),(6,8),(7,8)],9)
=> [7,1,1]
=> 2
['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)
=> [7,1,1]
=> 2
Description
The number of different multiplicities of parts of an integer partition.
Matching statistic: St000955
Mp00148: Finite Cartan types to root posetPosets
Mp00110: Posets Greene-Kleitman invariantInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
St000955: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
['A',1]
=> ([],1)
=> [1]
=> [1,0,1,0]
=> 1
['A',2]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> [1,0,1,0,1,0]
=> 1
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 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]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 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]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 2
Description
Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra.
Matching statistic: St001933
Mp00148: Finite Cartan types to root posetPosets
Mp00110: Posets Greene-Kleitman invariantInteger partitions
Mp00321: Integer partitions 2-conjugateInteger partitions
St001933: 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)
=> [2,1]
=> [3]
=> 1
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [2,1,1]
=> 2
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [5,1]
=> [2,2,1,1]
=> 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]
=> [3,2,2,1,1]
=> 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]
=> [3,2,2,1,1]
=> 2
Description
The largest multiplicity of a part in an integer partition.
Matching statistic: St000143
Mp00148: Finite Cartan types to root posetPosets
Mp00198: Posets incomparability graphGraphs
Mp00037: Graphs to partition of connected componentsInteger partitions
St000143: 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 - 1
['A',2]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> 0 = 1 - 1
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> 1 = 2 - 1
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> [2,1,1,1,1]
=> 1 = 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)
=> [7,1,1]
=> 1 = 2 - 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)
=> [7,1,1]
=> 1 = 2 - 1
Description
The largest repeated part of a partition. If the parts of the partition are all distinct, the value of the statistic is defined to be zero.
The following 48 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000659The number of rises of length at least 2 of a Dyck path. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001172The number of 1-rises at odd height of a Dyck path. St001413Half the length of the longest even length palindromic prefix of a binary word. St001424The number of distinct squares in a binary word. St001524The degree of symmetry of a binary word. 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. St001730The number of times the path corresponding to a binary word crosses the base line. St001568The smallest positive integer that does not appear twice in the partition. St001964The interval resolution global dimension of a poset. St000256The number of parts from which one can substract 2 and still get an integer partition. 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. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. 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. St001307The number of induced stars on four vertices in a graph. St001320The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph. St001578The minimal number of edges to add or remove to make a graph a line graph. St001638The book thickness of a graph. St000309The number of vertices with even degree. St000481The number of upper covers of a partition in dominance order. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001282The number of graphs with the same chromatic polynomial. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001716The 1-improper chromatic number of a graph. St001743The discrepancy of a graph. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. 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. St001366The maximal multiplicity of a degree of a vertex of a graph. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001654The monophonic hull number of a graph. 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. St001624The breadth of a lattice. 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. 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. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001002Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path.