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Your data matches 88 different statistics following compositions of up to 3 maps.
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Matching statistic: St001154
St001154: Finite Cartan types ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
=> 2 = 0 + 2
['A',2]
=> 3 = 1 + 2
['B',2]
=> 3 = 1 + 2
['G',2]
=> 4 = 2 + 2
['A',3]
=> 4 = 2 + 2
['B',3]
=> 5 = 3 + 2
['C',3]
=> 4 = 2 + 2
Description
The dual Coxeter number of a finite Cartan type.
Matching statistic: St000319
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000319: 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]
=> 2
['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]
=> [2,2,1,1]
=> 1
['A',3]
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [3,3]
=> 3
['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 spin of an integer partition.
The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape.
The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$
The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross.
This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.
Matching statistic: St000320
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000320: 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]
=> 2
['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]
=> [2,2,1,1]
=> 1
['A',3]
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [3,3]
=> 3
['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 dinv adjustment of an integer partition.
The Ferrers shape of an integer partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For $0 \leq j < \lambda_1$ let $n_j$ be the length of the border strip starting at $(\lambda_1-j,0)$.
The dinv adjustment is then defined by
$$\sum_{j:n_j > 0}(\lambda_1-1-j).$$
The following example is taken from Appendix B in [2]: Let $\lambda=(5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$$
and we obtain $(n_0,\ldots,n_4) = (10,7,0,3,1)$.
The dinv adjustment is thus $4+3+1+0 = 8$.
Matching statistic: St000010
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
=> ([],1)
=> [1]
=> [1]
=> 1 = 0 + 1
['A',2]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> [1,1,1]
=> 3 = 2 + 1
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [3,1]
=> 2 = 1 + 1
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [5,1]
=> [5,1]
=> 2 = 1 + 1
['A',3]
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [3,1,1,1]
=> 4 = 3 + 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]
=> [5,3,1]
=> 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)
=> [5,3,1]
=> [5,3,1]
=> 3 = 2 + 1
Description
The length of the partition.
Matching statistic: St000148
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St000148: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St000148: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
=> ([],1)
=> [1]
=> [1]
=> 1 = 0 + 1
['A',2]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> [1,1,1]
=> 3 = 2 + 1
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [3,1]
=> 2 = 1 + 1
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [5,1]
=> [5,1]
=> 2 = 1 + 1
['A',3]
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [3,1,1,1]
=> 4 = 3 + 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]
=> [5,3,1]
=> 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)
=> [5,3,1]
=> [5,3,1]
=> 3 = 2 + 1
Description
The number of odd parts of a partition.
Matching statistic: St000992
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000992: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000992: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
=> ([],1)
=> [1]
=> [1]
=> 1 = 0 + 1
['A',2]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> [3]
=> 3 = 2 + 1
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [2,1,1]
=> 2 = 1 + 1
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 2 = 1 + 1
['A',3]
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [5,1]
=> 4 = 3 + 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]
=> [3,2,2,1,1]
=> 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)
=> [5,3,1]
=> [3,2,2,1,1]
=> 3 = 2 + 1
Description
The alternating sum of the parts of an integer partition.
For a partition $\lambda = (\lambda_1,\ldots,\lambda_k)$, this is $\lambda_1 - \lambda_2 + \cdots \pm \lambda_k$.
Matching statistic: St000675
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000675: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 86%●distinct values known / distinct values provided: 75%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000675: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 86%●distinct values known / distinct values provided: 75%
Values
['A',1]
=> ([],1)
=> [1]
=> [1,0]
=> ? = 0
['A',2]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 2
['A',3]
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
['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]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> 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]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> 2
Description
The number of centered multitunnels of a Dyck path.
This is the number of factorisations $D = A B C$ of a Dyck path, such that $B$ is a Dyck path and $A$ and $B$ have the same length.
Matching statistic: St000260
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Values
['A',1]
=> ([],1)
=> ([],1)
=> 0
['A',2]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],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)
=> 2
['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
['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)
=> ? ∊ {2,3}
['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)
=> ? ∊ {2,3}
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St000362
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Values
['A',1]
=> ([],1)
=> ([],1)
=> 0
['A',2]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],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)
=> 2
['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
['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)
=> ? ∊ {2,3}
['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)
=> ? ∊ {2,3}
Description
The size of a minimal vertex cover of a graph.
A '''vertex cover''' of a graph $G$ is a subset $S$ of the vertices of $G$ such that each edge of $G$ contains at least one vertex of $S$. Finding a minimal vertex cover is an NP-hard optimization problem.
Matching statistic: St000387
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Values
['A',1]
=> ([],1)
=> ([],1)
=> 0
['A',2]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],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)
=> 2
['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
['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)
=> ? ∊ {2,3}
['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)
=> ? ∊ {2,3}
Description
The matching number of a graph.
For a graph $G$, this is defined as the maximal size of a '''matching''' or '''independent edge set''' (a set of edges without common vertices) contained in $G$.
The following 78 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001393The induced matching number of a graph. St001459The number of zero columns in the nullspace of a graph. St001812The biclique partition number of a graph. St001949The rigidity index of a graph. St000172The Grundy number of a graph. St000258The burning number of a graph. St000917The open packing number of a graph. St001116The game chromatic number of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001734The lettericity of a graph. St001924The number of cells in an integer partition whose arm and leg length coincide. St001963The tree-depth of a graph. St000171The degree of the graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000535The rank-width of a graph. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001349The number of different graphs obtained from the given graph by removing an edge. St001621The number of atoms of a lattice. St001792The arboricity of a graph. St000918The 2-limited packing number of a graph. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001270The bandwidth of a graph. St001286The annihilation number of a graph. St001316The domatic number of a graph. St001352The number of internal nodes in the modular decomposition of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001642The Prague dimension of a graph. St001654The monophonic hull number of a graph. St001672The restrained domination number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001716The 1-improper chromatic number of a graph. St001655The general position number of a graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001118The acyclic chromatic index of a graph. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001613The binary logarithm of the size of the center of a lattice. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000455The second largest eigenvalue of a graph if it is integral. St000928The sum of the coefficients of the character polynomial of an integer partition. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000095The number of triangles of a graph. St000259The diameter of a connected graph. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000454The largest eigenvalue of a graph if it is integral. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001638The book thickness of a graph. St001644The dimension of a graph. St001742The difference of the maximal and the minimal degree in a graph. St000450The number of edges minus the number of vertices plus 2 of a graph. St000741The Colin de Verdière graph invariant. St000936The number of even values of the symmetric group character corresponding to the partition. St001330The hat guessing number of a graph. St001391The disjunction number of a graph. St000286The number of connected components of the complement of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000822The Hadwiger number of the graph. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders.
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