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Your data matches 42 different statistics following compositions of up to 3 maps.
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Matching statistic: St000854

Mapping

St000854: Finite Cartan types ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

['A',2]
 => 1
['B',2]
 => 2
['G',2]
 => 2
['A',3]
 => 1
['B',3]
 => 2
['C',3]
 => 2
['A',4]
 => 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: St000860

Mapping

St000860: Finite Cartan types ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

['A',2]
 => 1
['B',2]
 => 2
['G',2]
 => 2
['A',3]
 => 1
['B',3]
 => 2
['C',3]
 => 2
['A',4]
 => 1

Description

The size of the center of the Weyl group of a finite Cartan type.

Matching statistic: St000312
(load all 2 compositions to match this statistic)

Mapping

Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000312: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

['A',2]
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 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)
 => 3 = 2 + 1
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => 2 = 1 + 1
['B',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => ([(0,8),(1,7),(2,6),(3,6),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8)],9)
 => 3 = 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)
 => ([(0,8),(1,7),(2,6),(3,6),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8)],9)
 => 3 = 2 + 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)
 => ([(0,8),(1,7),(2,5),(2,6),(3,7),(3,9),(4,8),(4,9),(5,7),(5,9),(6,8),(6,9)],10)
 => 2 = 1 + 1

Description

The number of leaves in a graph. That is, the number of vertices of a graph that have degree 1.

Matching statistic: St000811

Mapping

Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000811: 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]
 => 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
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => [3,2,1]
 => 0 = 1 - 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
['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]
 => 0 = 1 - 1

Description

The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to Schur symmetric functions. For example,

$p_{22} = s_{1111} - s_{211} + 2s_{22} - s_{31} + s_4$ , so the statistic on the partition

$22$ is 2. This is also the sum of the character values at the given conjugacy class over all irreducible characters of the symmetric group. [2] For a permutation

$\pi$ of given cycle type, this is also the number of permutations whose square equals

$\pi$ . [2]

Matching statistic: St000897
(load all 3 compositions to match this statistic)

Mapping

Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000897: 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)
 => ([(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
['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)
 => [5,1]
 => 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]
 => 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
['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)
 => [9,1]
 => 1

Description

The number of different multiplicities of parts of an integer partition.

Matching statistic: St000955

Mapping

Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000955: 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]
 => [1,0,1,0,1,0]
 => 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
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 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]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => 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
['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]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1

Description

Number of times one has

$Ext^i(D(A),A)>0$ for

$i>0$ for the corresponding LNakayama algebra.

Matching statistic: St001568

Mapping

Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001568: 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)
 => ([(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
['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)
 => [5,1]
 => 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]
 => 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
['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)
 => [9,1]
 => 1

Description

The smallest positive integer that does not appear twice in the partition.

Matching statistic: St000143

Mapping

Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000143: 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)
 => ([(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
['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)
 => [5,1]
 => 0 = 1 - 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
['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)
 => [9,1]
 => 0 = 1 - 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.

Matching statistic: St000257
(load all 2 compositions to match this statistic)

Mapping

Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000257: 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)
 => ([(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
['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)
 => [5,1]
 => 0 = 1 - 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
['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)
 => [9,1]
 => 0 = 1 - 1

Description

The number of distinct parts of a partition that occur at least twice. See Section 3.3.1 of [2].

Matching statistic: St000659

Mapping

Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000659: 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]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 1 = 2 - 1
['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]
 => 1 = 2 - 1
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 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]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => 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]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => 1 = 2 - 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]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1

Description

The number of rises of length at least 2 of a Dyck path.

The following 32 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001092The number of distinct even parts of a partition. 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. 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. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000636The hull number of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001654The monophonic hull number of a graph. St001672The restrained domination number of a graph. St001282The number of graphs with the same chromatic polynomial. St000313The number of degree 2 vertices of a graph. St000477The weight of a partition according to Alladi. St000478Another weight of a partition according to Alladi. St001057The Grundy value of the game of creating an independent set in 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. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001368The number of vertices of maximal degree in a graph. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001578The minimal number of edges to add or remove to make a graph a line graph. St001692The number of vertices with higher degree than the average degree in a graph. 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. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St000481The number of upper covers of a partition in dominance order. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition.