searching the database
Your data matches 9 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000106
St000106: Finite Cartan types ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
=> 2
['A',2]
=> 6
['B',2]
=> 8
['G',2]
=> 12
['A',3]
=> 24
['B',3]
=> 48
['C',3]
=> 48
Description
The size of the associated Weyl group.
Matching statistic: St000063
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000063: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000063: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
=> ([],1)
=> [1]
=> 2
['A',2]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> 6
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> 8
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [5,1]
=> 12
['A',3]
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,2,1]
=> 24
['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]
=> 48
['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]
=> 48
Description
The number of linear extensions of a certain poset defined for an integer partition.
The poset is constructed in David Speyer's answer to Matt Fayers' question [3].
The value at the partition $\lambda$ also counts cover-inclusive Dyck tilings of $\lambda\setminus\mu$, summed over all $\mu$, as noticed by Philippe Nadeau in a comment.
This statistic arises in the homogeneous Garnir relations for the universal graded Specht modules for cyclotomic quiver Hecke algebras.
Matching statistic: St001808
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001808: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001808: 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]
=> 2
['A',2]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> [1,0,1,0,1,0]
=> 6
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 8
['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]
=> 12
['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]
=> 24
['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]
=> 48
['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]
=> 48
Description
The box weight or horizontal decoration of a Dyck path.
Let a Dyck path $D = (d_1,d_2,\dots,d_n)$ with steps $d_i \in \{N=(0,1),E=(1,0)\}$ be given.
For the $i$th step $d_i \in D$ we define the weight
$$
\beta(d_i) = 1, \quad \text{ if } d_i=N,
$$
and
$$
\beta(d_i) = \sum_{k = 1}^{i} [\![ d_k = N]\!], \quad \text{ if } d_i=E,
$$
where we use the Iverson bracket $[\![ A ]\!]$ that is equal to $1$ if $A$ is true, and $0$ otherwise.
The '''box weight''' or '''horizontal deocration''' of $D$ is defined as
$$
\prod_{i=1}^{n} \beta(d_i).
$$
The name describes the fact that between each $E$ step and the line $y=-1$ exactly one unit box is marked.
Matching statistic: St000184
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St000184: Integer partitions ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 67%
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St000184: Integer partitions ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 67%
Values
['A',1]
=> ([],1)
=> [2]
=> 2
['A',2]
=> ([(0,2),(1,2)],3)
=> [3,2]
=> 6
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> 8
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [6,2]
=> 12
['A',3]
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> [8,4,2]
=> ? ∊ {24,48,48}
['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]
=> ? ∊ {24,48,48}
['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]
=> ? ∊ {24,48,48}
Description
The size of the centralizer of any permutation of given cycle type.
The centralizer (or commutant, equivalently normalizer) of an element $g$ of a group $G$ is the set of elements of $G$ that commute with $g$:
$$C_g = \{h \in G : hgh^{-1} = g\}.$$
Its size thus depends only on the conjugacy class of $g$.
The conjugacy classes of a permutation is determined by its cycle type, and the size of the centralizer of a permutation with cycle type $\lambda = (1^{a_1},2^{a_2},\dots)$ is
$$|C| = \Pi j^{a_j} a_j!$$
For example, for any permutation with cycle type $\lambda = (3,2,2,1)$,
$$|C| = (3^1 \cdot 1!)(2^2 \cdot 2!)(1^1 \cdot 1!) = 24.$$
There is exactly one permutation of the empty set, the identity, so the statistic on the empty partition is $1$.
Matching statistic: St000708
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 67%
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 67%
Values
['A',1]
=> ([],1)
=> [2]
=> 2
['A',2]
=> ([(0,2),(1,2)],3)
=> [3,2]
=> 6
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> 8
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [6,2]
=> 12
['A',3]
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> [8,4,2]
=> ? ∊ {24,48,48}
['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]
=> ? ∊ {24,48,48}
['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]
=> ? ∊ {24,48,48}
Description
The product of the parts of an integer partition.
Matching statistic: St001834
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Values
['A',1]
=> ([],1)
=> ([],1)
=> 2
['A',2]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 6
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 8
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> ? = 12
['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)
=> ? ∊ {24,48,48}
['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)
=> ? ∊ {24,48,48}
['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)
=> ? ∊ {24,48,48}
Description
The number of non-isomorphic minors of a graph.
A minor of a graph $G$ is a graph obtained from $G$ by repeatedly deleting or contracting edges, or removing isolated vertices.
This statistic records the total number of (non-empty) non-isomorphic minors of a graph.
Matching statistic: St000086
Values
['A',1]
=> ([],1)
=> ([],1)
=> ([(0,1)],2)
=> 2
['A',2]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 8
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 12
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6
['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)
=> ([(0,6),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {24,48,48}
['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)
=> ([(0,9),(1,9),(2,7),(2,9),(3,5),(3,8),(3,9),(4,6),(4,8),(4,9),(5,6),(5,7),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? ∊ {24,48,48}
['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)
=> ([(0,9),(1,9),(2,7),(2,9),(3,5),(3,8),(3,9),(4,6),(4,8),(4,9),(5,6),(5,7),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? ∊ {24,48,48}
Description
The number of subgraphs.
Given a graph $G$, this is the number of graphs $H$ such that $H \hookrightarrow G$.
Matching statistic: St000081
Values
['A',1]
=> ([],1)
=> ([],1)
=> ([(0,1)],2)
=> 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)
=> 5 = 6 - 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)
=> 7 = 8 - 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)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 12 - 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)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? ∊ {24,48,48} - 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)
=> ([(0,8),(0,9),(1,7),(1,9),(2,6),(2,9),(3,6),(3,8),(3,9),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,9),(7,9),(8,9)],10)
=> ? ∊ {24,48,48} - 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)
=> ([(0,8),(0,9),(1,7),(1,9),(2,6),(2,9),(3,6),(3,8),(3,9),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,9),(7,9),(8,9)],10)
=> ? ∊ {24,48,48} - 1
Description
The number of edges of a graph.
Matching statistic: St001649
Values
['A',1]
=> ([],1)
=> ([],1)
=> ([(0,1)],2)
=> 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)
=> 5 = 6 - 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)
=> 7 = 8 - 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)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 12 - 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)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? ∊ {24,48,48} - 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)
=> ([(0,8),(0,9),(1,7),(1,9),(2,6),(2,9),(3,6),(3,8),(3,9),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,9),(7,9),(8,9)],10)
=> ? ∊ {24,48,48} - 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)
=> ([(0,8),(0,9),(1,7),(1,9),(2,6),(2,9),(3,6),(3,8),(3,9),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,9),(7,9),(8,9)],10)
=> ? ∊ {24,48,48} - 1
Description
The length of a longest trail in a graph.
A trail is a sequence of distinct edges, such that two consecutive edges share a vertex.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!