Your data matches 23 different statistics following compositions of up to 3 maps.
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Matching statistic: St000107
St000107: 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]
=> 5 = 4 + 1
['G',2]
=> 7 = 6 + 1
Description
The dimension of the representation $V(\Lambda_1)$. The sizes of $E_6$ and $E_7$ can be seen in [1].
Mp00148: Finite Cartan types to root posetPosets
Mp00074: Posets to graphGraphs
St000448: Graphs ⟶ 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)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 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)
=> 5 = 6 - 1
Description
The number of pairs of vertices of a graph with distance 2. This is the coefficient of the quadratic term of the Wiener polynomial.
Matching statistic: St001308
Mp00148: Finite Cartan types to root posetPosets
Mp00074: Posets to graphGraphs
St001308: Graphs ⟶ 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)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 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)
=> 5 = 6 - 1
Description
The number of induced paths on three vertices in a graph.
Matching statistic: St000377
Mp00148: Finite Cartan types to root posetPosets
Mp00306: Posets rowmotion cycle typeInteger partitions
Mp00044: Integer partitions conjugateInteger partitions
St000377: 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]
=> 2
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [2,2,1,1]
=> 4
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [6,2]
=> [2,2,1,1,1,1]
=> 6
Description
The dinv defect of an integer partition. This is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \not\in \{0,1\}$.
Matching statistic: St000476
Mp00148: Finite Cartan types to root posetPosets
Mp00110: Posets Greene-Kleitman invariantInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
St000476: 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]
=> 2
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 4
['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]
=> 6
Description
The sum of the semi-lengths of tunnels before a valley of a Dyck path. For each valley $v$ in a Dyck path $D$ there is a corresponding tunnel, which is the factor $T_v = s_i\dots s_j$ of $D$ where $s_i$ is the step after the first intersection of $D$ with the line $y = ht(v)$ to the left of $s_j$. This statistic is $$ \sum_v (j_v-i_v)/2. $$
Matching statistic: St001967
Mp00148: Finite Cartan types to root posetPosets
Mp00110: Posets Greene-Kleitman invariantInteger partitions
Mp00313: Integer partitions Glaisher-Franklin inverseInteger partitions
St001967: 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]
=> [1,1,1]
=> 2
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [3,1]
=> 4
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [5,1]
=> [5,1]
=> 6
Description
The coefficient of the monomial corresponding to the integer partition in a certain power series. Let $f(x) = 1 - c_1 x - c_2 x^2 - \dots$ and let $F(x) = x/f^{(-1)}(x) = 1 - C_1 x - C_2 x^2 - \dots$, where $f^{(-1)}$ denotes the compositional inverse of $f$. Then $C_n$ is a polynomial in $c_1,\dots,c_n$, whose monomials are indexed by integer partitions of $n$. This statistic records its coefficients.
Matching statistic: St001968
Mp00148: Finite Cartan types to root posetPosets
Mp00110: Posets Greene-Kleitman invariantInteger partitions
Mp00313: Integer partitions Glaisher-Franklin inverseInteger partitions
St001968: 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]
=> [1,1,1]
=> 2
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [3,1]
=> 4
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [5,1]
=> [5,1]
=> 6
Description
The coefficient of the monomial corresponding to the integer partition in a certain power series. Let $f(x) = 1 - c_1 x/1! - c_2 x^2/2! - \dots$ and let $F(x) = f'(x)/f(x) = 1 + C_1 x/1! + C_2 x^2/2! + \dots$. Then $C_{n-1}$ is a polynomial in $c_1,\dots,c_n$, whose monomials are indexed by integer partitions of $n$. This statistic records its coefficients.
Matching statistic: St001117
Mp00148: Finite Cartan types to root posetPosets
Mp00074: Posets to graphGraphs
Mp00111: Graphs complementGraphs
St001117: Graphs ⟶ 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)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 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)
=> 5 = 6 - 1
Description
The game chromatic index of a graph. Two players, Alice and Bob, take turns colouring properly any uncolored edge of the graph. Alice begins. If it is not possible for either player to colour a edge, then Bob wins. If the graph is completely colored, Alice wins. The game chromatic index is the smallest number of colours such that Alice has a winning strategy.
Matching statistic: St001310
Mp00148: Finite Cartan types to root posetPosets
Mp00074: Posets to graphGraphs
Mp00203: Graphs coneGraphs
St001310: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
['A',1]
=> ([],1)
=> ([],1)
=> ([(0,1)],2)
=> 0 = 1 - 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 = 2 - 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)
=> 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)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 6 - 1
Description
The number of induced diamond graphs in a graph. A diamond graph is a cycle on four vertices, with an additional edge connecting two of the non-adjacent vertices.
Matching statistic: St000550
Mp00148: Finite Cartan types to root posetPosets
Mp00205: Posets maximal antichainsLattices
Mp00197: Lattices lattice of congruencesLattices
St000550: Lattices ⟶ ℤResult quality: 75% values known / values provided: 75%distinct values known / distinct values provided: 75%
Values
['A',1]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
['A',2]
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 6
Description
The number of modular elements of a lattice. A pair $(x, y)$ of elements of a lattice $L$ is a modular pair if for every $z\geq y$ we have that $(y\vee x) \wedge z = y \vee (x \wedge z)$. An element $x$ is left-modular if $(x, y)$ is a modular pair for every $y\in L$, and is modular if both $(x, y)$ and $(y, x)$ are modular pairs for every $y\in L$.
The following 13 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000551The number of left modular elements of a lattice. St001616The number of neutral elements in a lattice. St001754The number of tolerances of a finite lattice. St000668The least common multiple of the parts of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St001118The acyclic chromatic index of a graph. St001128The exponens consonantiae of a partition. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001619The number of non-isomorphic sublattices of a lattice. St001666The number of non-isomorphic subposets of a lattice which are lattices.