Your data matches 182 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000857
Mapping
St000857: 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]
 => 3 = 2 + 1
['B',2]
 => 4 = 3 + 1
['G',2]
 => 6 = 5 + 1
Description
The number of reflections of the Weyl group of a finite Cartan type. By the one-to-one correspondence between reflections and reflecting hyperplanes, this is also the number of reflecting hyperplanes. This is given by $nh/2$ where $n$ is the rank and $h$ is the Coxeter number.
Matching statistic: St000858
Mapping
St000858: 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]
 => 3 = 2 + 1
['B',2]
 => 4 = 3 + 1
['G',2]
 => 6 = 5 + 1
Description
The number of factorizations of any Coxeter element into reflections of a finite Cartan type. The number of such factorizations is given by $n!h^n / |W|$ where $n$ is the rank, $h$ is the Coxeter number and $W$ is the Weyl group of the given Cartan type. This was originally proven in a letter from Deligne to Looijenga in the 1970s, and then recovered in [2, Theorem 3.6]. As an example, consider the three ($=2!3^2/6$) factorizations of the Coxeter element $$(1,2,3) = (1,2)(2,3) = (1,3)(1,2) = (2,3)(1,3)$$ in type $A_2$.
Matching statistic: St001756
Mapping
St001756: 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]
 => 3 = 2 + 1
['B',2]
 => 4 = 3 + 1
['G',2]
 => 6 = 5 + 1
Description
The number of minimal subsets of reflections that generate the group. $~$ This counts the number of minimal subsets $S \subseteq R$ of reflections $R \subseteq W$ that generate the group $W$.
Matching statistic: St001945
Mapping
St001945: 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]
 => 4 = 2 + 2
['B',2]
 => 5 = 3 + 2
['G',2]
 => 7 = 5 + 2
Description
The number of non-isomorphic subgroups of the Weyl group of a finite Cartan type. This statistic returns the number of non-isomorphic abstract groups. See [[St001155]] for the number of conjugacy classes of subgroups.
Matching statistic: St001300
(load all 2 compositions to match this statistic)
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
St001300: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
 => ([],1)
 => 0
['A',2]
 => ([(0,2),(1,2)],3)
 => 2
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => 3
['G',2]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => 5
Description
The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset.
Matching statistic: St000189
(load all 2 compositions to match this statistic)
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
St000189: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
 => ([],1)
 => 1 = 0 + 1
['A',2]
 => ([(0,2),(1,2)],3)
 => 3 = 2 + 1
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => 4 = 3 + 1
['G',2]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => 6 = 5 + 1
Description
The number of elements in the poset.
Matching statistic: St001636
(load all 2 compositions to match this statistic)
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
St001636: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
 => ([],1)
 => 1 = 0 + 1
['A',2]
 => ([(0,2),(1,2)],3)
 => 3 = 2 + 1
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => 4 = 3 + 1
['G',2]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => 6 = 5 + 1
Description
The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset.
Matching statistic: St000081
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000081: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
 => ([],1)
 => ([],1)
 => 0
['A',2]
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 2
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 3
['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
Description
The number of edges of a graph.
Matching statistic: St000506
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St000506: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
 => ([],1)
 => [2]
 => 0
['A',2]
 => ([(0,2),(1,2)],3)
 => [3,2]
 => 2
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => 3
['G',2]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [6,2]
 => 5
Description
The number of standard desarrangement tableaux of shape equal to the given partition. A '''standard desarrangement tableau''' is a standard tableau whose first ascent is even. Here, an ascent of a standard tableau is an entry $i$ such that $i+1$ appears to the right or above $i$ in the tableau (with respect to English tableau notation). This is also the nullity of the random-to-random operator (and the random-to-top) operator acting on the simple module of the symmetric group indexed by the given partition. See also: * [[St000046]]: The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition * [[St000500]]: Eigenvalues of the random-to-random operator acting on the regular representation.
Matching statistic: St000987
(load all 5 compositions to match this statistic)
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000987: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
 => ([],1)
 => ([],1)
 => 0
['A',2]
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 2
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 3
['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
Description
The number of positive eigenvalues of the Laplacian matrix of the graph. This is the number of vertices minus the number of connected components of the graph.
The following 172 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001279The sum of the parts of an integer partition that are at least two. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001391The disjunction number of a graph. St001479The number of bridges of a graph. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001622The number of join-irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St001827The number of two-component spanning forests of a graph. St001869The maximum cut size of a graph. St000048The multinomial of the parts of a partition. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000228The size of a partition. St000459The hook length of the base cell of a partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000479The Ramsey number of a graph. St000517The Kreweras number of an integer partition. St000636The hull number of a graph. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St000939The number of characters of the symmetric group whose value on the partition is positive. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001342The number of vertices in the center of a graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001672The restrained domination number of a graph. St000059The inversion number of a standard tableau as defined by Haglund and Stevens. St000145The Dyson rank of a partition. St000169The cocharge of a standard tableau. St000171The degree of the graph. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000330The (standard) major index of a standard tableau. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000921The number of internal inversions of a binary word. St001117The game chromatic index of a graph. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001345The Hamming dimension of a graph. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001458The rank of the adjacency matrix of a graph. St001541The Gini index of an integer partition. St001697The shifted natural comajor index of a standard Young tableau. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000005The bounce statistic of a Dyck path. St000144The pyramid weight of the Dyck path. St000147The largest part of an integer partition. St000184The size of the centralizer of any permutation of given cycle type. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000271The chromatic index of a graph. St000293The number of inversions of a binary word. St000384The maximal part of the shifted composition of an integer partition. St000395The sum of the heights of the peaks of a Dyck path. St000531The leading coefficient of the rook polynomial of an integer partition. St000548The number of different non-empty partial sums of an integer partition. St000667The greatest common divisor of the parts of the partition. St000722The number of different neighbourhoods in a graph. St000734The last entry in the first row of a standard tableau. St000738The first entry in the last row of a standard tableau. St000784The maximum of the length and the largest part of the integer partition. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000947The major index east count of a Dyck path. St000992The alternating sum of the parts of an integer partition. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001034The area of the parallelogram polyomino associated with the Dyck path. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001102The number of words with multiplicities of the letters given by the composition, avoiding the consecutive pattern 132. St001118The acyclic chromatic index of a graph. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001176The size of a partition minus its first part. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001249Sum of the odd parts of a partition. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001312Number of parabolic noncrossing partitions indexed by the composition. St001360The number of covering relations in Young's lattice below a partition. St001389The number of partitions of the same length below the given integer partition. St001415The length of the longest palindromic prefix of a binary word. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001725The harmonious chromatic number of a graph. St001746The coalition number of a graph. St001917The order of toric promotion on the set of labellings of a graph. St000063The number of linear extensions of a certain poset defined for an integer partition. St000108The number of partitions contained in the given partition. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000532The total number of rook placements on a Ferrers board. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001400The total number of Littlewood-Richardson tableaux of given shape. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001814The number of partitions interlacing the given partition. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St000656The number of cuts of a poset. St000567The sum of the products of all pairs of parts. 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. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001128The exponens consonantiae of a partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St001875The number of simple modules with projective dimension at most 1. St000477The weight of a partition according to Alladi. St000770The major index of an integer partition when read from bottom to top. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St001651The Frankl number of a lattice. St000095The number of triangles of a graph. St000309The number of vertices with even degree. St000681The Grundy value of Chomp on Ferrers diagrams. St000693The modular (standard) major index of a standard tableau. St000741The Colin de Verdière graph invariant. St000937The number of positive values of the symmetric group character corresponding to the partition. St000946The sum of the skew hook positions in a Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001742The difference of the maximal and the minimal degree in a graph. St000327The number of cover relations in a poset. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000450The number of edges minus the number of vertices plus 2 of a graph. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000674The number of hills of a Dyck path. St000706The product of the factorials of the multiplicities of an integer partition. St000744The length of the path to the largest entry in a standard Young tableau. St000815The number of semistandard Young tableaux of partition weight of given shape. St000932The number of occurrences of the pattern UDU in a Dyck path. St000934The 2-degree of an integer partition. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001637The number of (upper) dissectors of a poset. St001645The pebbling number of a connected graph. St001668The number of points of the poset minus the width of the poset. St000806The semiperimeter of the associated bargraph. St000936The number of even values of the symmetric group character corresponding to the partition. St000941The number of characters of the symmetric group whose value on the partition is even. St000301The number of facets of the stable set polytope of a graph. St001706The number of closed sets in a graph.