Your data matches 32 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
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',1]
 => 1 = 0 + 1
['A',2]
 => 1 = 0 + 1
['B',2]
 => 2 = 1 + 1
['G',2]
 => 2 = 1 + 1
['A',3]
 => 1 = 0 + 1
['B',3]
 => 2 = 1 + 1
['C',3]
 => 2 = 1 + 1
['A',4]
 => 1 = 0 + 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: 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',1]
 => ([],1)
 => ([],1)
 => [1]
 => 0
['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]
 => 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
['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
['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
['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
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',1]
 => ([],1)
 => ([],1)
 => [1]
 => 0
['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]
 => 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
['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
['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
['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
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',1]
 => ([],1)
 => [1]
 => [1,0,1,0]
 => 0
['A',2]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => [1,0,1,0,1,0]
 => 0
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 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
['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
['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
['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
['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
Description
The number of rises of length at least 2 of a Dyck path.
Matching statistic: St001092
(load all 2 compositions to match this statistic)
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St001092: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
 => ([],1)
 => [1]
 => [1]
 => 0
['A',2]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => [3]
 => 0
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [2,1,1]
 => 1
['G',2]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => [2,1,1,1,1]
 => 1
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [5,1]
 => 0
['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]
 => 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]
 => [3,2,2,1,1]
 => 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]
 => [7,3]
 => 0
Description
The number of distinct even parts of a partition. See Section 3.3.1 of [1].
Matching statistic: St001172
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001172: 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]
 => 0
['A',2]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => [1,0,1,0,1,0]
 => 0
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 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
['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
['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
['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
['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
Description
The number of 1-rises at odd height of a Dyck path.
Matching statistic: St001413
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St001413: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
 => ([],1)
 => [1]
 => 1 => 0
['A',2]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => 01 => 0
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => 11 => 1
['G',2]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => 11 => 1
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => [3,2,1]
 => 101 => 0
['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]
 => 111 => 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]
 => 111 => 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]
 => 0101 => 0
Description
Half the length of the longest even length palindromic prefix of a binary word.
Matching statistic: St001587
(load all 2 compositions to match this statistic)
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St001587: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
 => ([],1)
 => [1]
 => [1]
 => 0
['A',2]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => [3]
 => 0
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [2,1,1]
 => 1
['G',2]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => [2,1,1,1,1]
 => 1
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [5,1]
 => 0
['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]
 => 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]
 => [3,2,2,1,1]
 => 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]
 => [7,3]
 => 0
Description
Half of the largest even part of an integer partition. The largest even part is recorded by [[St000995]].
Matching statistic: St001588
(load all 2 compositions to match this statistic)
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St001588: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
 => ([],1)
 => [1]
 => [1]
 => 0
['A',2]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => [3]
 => 0
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [2,1,1]
 => 1
['G',2]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => [2,1,1,1,1]
 => 1
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [5,1]
 => 0
['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]
 => 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]
 => [3,2,2,1,1]
 => 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]
 => [7,3]
 => 0
Description
The number of distinct odd parts smaller than the largest even part in an integer partition.
Matching statistic: St001730
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001730: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
 => ([],1)
 => [1]
 => 10 => 0
['A',2]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => 1010 => 0
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => 10010 => 1
['G',2]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => 1000010 => 1
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => [3,2,1]
 => 101010 => 0
['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]
 => 10010010 => 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]
 => 10010010 => 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]
 => 10101010 => 0
Description
The number of times the path corresponding to a binary word crosses the base line. Interpret each $0$ as a step $(1,-1)$ and $1$ as a step $(1,1)$. Then this statistic counts the number of times the path crosses the $x$-axis.
The following 22 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000897The number of different multiplicities of parts of an integer partition. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St001568The smallest positive integer that does not appear twice in the partition. 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. St001578The minimal number of edges to add or remove to make a graph a line graph. St001282The number of graphs with the same chromatic polynomial. St001366The maximal multiplicity of a degree of a vertex of 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. St001654The monophonic hull number of 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. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St000477The weight of a partition according to Alladi. St000478Another weight of a partition according to Alladi. 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. St001964The interval resolution global dimension of a poset. St001624The breadth of a lattice. St001783The number of odd automorphisms of a graph.