Your data matches 222 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',3]
 => 1 = 0 + 1
['B',3]
 => 2 = 1 + 1
['C',3]
 => 2 = 1 + 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: St000860
Mapping
St000860: Finite Cartan types ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',3]
 => 1 = 0 + 1
['B',3]
 => 2 = 1 + 1
['C',3]
 => 2 = 1 + 1
Description
The size of the center of the Weyl group of a finite Cartan type.
Matching statistic: St001369
Mapping
St001369: Finite Cartan types ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',3]
 => 1 = 0 + 1
['B',3]
 => 2 = 1 + 1
['C',3]
 => 2 = 1 + 1
Description
The largest coefficient in the highest root in the root system of a Cartan type.
Matching statistic: St001886
Mapping
St001886: Finite Cartan types ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',3]
 => 3 = 0 + 3
['B',3]
 => 4 = 1 + 3
['C',3]
 => 4 = 1 + 3
Description
The number of orbits of the rowmotion operator on the root poset of a finite Cartan type.
Matching statistic: St001158
Mapping
St001158: Finite Cartan types ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',3]
 => 4 = 0 + 4
['B',3]
 => 5 = 1 + 4
['C',3]
 => 5 = 1 + 4
Description
The size of the mutation class of quivers of given type.
Matching statistic: St001953
Mapping
St001953: Finite Cartan types ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',3]
 => 4 = 0 + 4
['B',3]
 => 5 = 1 + 4
['C',3]
 => 5 = 1 + 4
Description
The number of distinct orders of elements in a Weyl group of finite Cartan type.
Matching statistic: St001788
Mapping
St001788: Finite Cartan types ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',3]
 => 5 = 0 + 5
['B',3]
 => 6 = 1 + 5
['C',3]
 => 6 = 1 + 5
Description
The number of types of parabolic subgroups of the associated Weyl group. Let $W$ be a Weyl group with simple generators $\mathcal{S} \subseteq W$. A subgroup of $W$ generated by a subset $X \subseteq \mathcal{S}$ is called standard parabolic subgroup. A parabolic subgroup is a subgroup of $W$ that is conjugate to a standard parabolic subgroup. This is the number of all pairwise different types of subgroups of $W$ obtained as (standard) parabolic subgroups (including type $A_0$).
Matching statistic: St000137
(load all 6 compositions to match this statistic)
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000137: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => [3,2,1]
 => 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]
 => 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
Description
The Grundy value of an integer partition. Consider the two-player game on an integer partition. In each move, a player removes either a box, or a 2x2-configuration of boxes such that the resulting diagram is still a partition. The first player that cannot move lose. This happens exactly when the empty partition is reached. The grundy value of an integer partition is defined as the grundy value of this two-player game as defined in [1]. This game was described to me during Norcom 2013, by Urban Larsson, and it seems to be quite difficult to give a good description of the partitions with Grundy value 0.
Matching statistic: St000150
(load all 3 compositions to match this statistic)
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St000150: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => [8,4,2]
 => 0
['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]
 => 1
['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]
 => 1
Description
The floored half-sum of the multiplicities of a partition. This statistic is equidistributed with [[St000143]] and [[St000149]], see [1].
Matching statistic: St000257
(load all 3 compositions to match this statistic)
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St000257: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => [8,4,2]
 => 0
['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]
 => 1
['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]
 => 1
Description
The number of distinct parts of a partition that occur at least twice. See Section 3.3.1 of [2].
The following 212 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000312The number of leaves in a graph. St000477The weight of a partition according to Alladi. St000478Another weight of a partition according to Alladi. St000811The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to Schur symmetric functions. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St001175The size of a partition minus the hook length of the base cell. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000148The number of odd parts of a partition. St000183The side length of the Durfee square of an integer partition. St000992The alternating sum of the parts of an integer partition. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St000142The number of even parts of a partition. St000271The chromatic index of a graph. St000549The number of odd partial sums of an integer partition. St001176The size of a partition minus its first part. St001214The aft of an integer partition. St000185The weighted size of a partition. St000378The diagonal inversion number of an integer partition. St000548The number of different non-empty partial sums of an integer partition. St000143The largest repeated part of a partition. St000145The Dyson rank of a partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000256The number of parts from which one can substract 2 and still get an integer partition. St000376The bounce deficit of a Dyck path. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St000618The number of self-evacuating tableaux of given shape. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000659The number of rises of length at least 2 of a Dyck path. St000660The number of rises of length at least 3 of a Dyck path. St000661The number of rises of length 3 of a Dyck path. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000931The number of occurrences of the pattern UUU in a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001037The number of inner corners of the upper path 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. St001091The number of parts in an integer partition whose next smaller part has the same size. St001092The number of distinct even parts of a partition. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001335The cardinality of a minimal cycle-isolating set of a graph. St001383The BG-rank of an integer partition. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001413Half the length of the longest even length palindromic prefix of a binary word. St001423The number of distinct cubes in a binary word. St001424The number of distinct squares in a binary word. St001484The number of singletons of an integer partition. St001524The degree of symmetry of a binary word. St001587Half of the largest even part of an integer partition. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001730The number of times the path corresponding to a binary word crosses the base line. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001910The height of the middle non-run of a Dyck path. St001932The number of pairs of singleton blocks in the noncrossing set partition corresponding to a Dyck path, that can be merged to create another noncrossing set partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000052The number of valleys of a Dyck path not on the x-axis. St000088The row sums of the character table of the symmetric group. St000160The multiplicity of the smallest part of a partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St000295The length of the border of a binary word. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000475The number of parts equal to 1 in a partition. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000531The leading coefficient of the rook polynomial of an integer partition. St000547The number of even non-empty partial sums of an integer partition. St000617The number of global maxima of a Dyck path. St000674The number of hills of a Dyck path. St000706The product of the factorials of the multiplicities of an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000897The number of different multiplicities of parts of an integer partition. St000920The logarithmic height of a Dyck path. St000928The sum of the coefficients of the character polynomial of an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000982The length of the longest constant subword. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001280The number of parts of an integer partition that are at least two. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001568The smallest positive integer that does not appear twice in the partition. St001571The Cartan determinant of the integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001657The number of twos in an integer partition. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001933The largest multiplicity of a part in an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000003The number of standard Young tableaux of the partition. St000010The length of the partition. St000011The number of touch points (or returns) of a Dyck path. St000071The number of maximal chains in a poset. St000147The largest part of an integer partition. St000159The number of distinct parts of the integer partition. St000184The size of the centralizer of any permutation of given cycle type. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000288The number of ones in a binary word. St000306The bounce count of a Dyck path. St000345The number of refinements of a partition. St000381The largest part of an integer composition. St000384The maximal part of the shifted composition of an integer partition. St000393The number of strictly increasing runs in a binary word. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000567The sum of the products of all pairs of parts. St000626The minimal period of a binary word. St000668The least common multiple of the parts of the partition. St000678The number of up steps after the last double rise of a Dyck path. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. 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. St000783The side length of the largest staircase partition fitting into a partition. St000784The maximum of the length and the largest part of the integer partition. St000808The number of up steps of the associated bargraph. St000876The number of factors in the Catalan decomposition of a binary word. St000913The number of ways to refine the partition into singletons. St000922The minimal number such that all substrings of this length are unique. St000933The number of multipartitions of sizes given by an integer partition. St000935The number of ordered refinements of an integer partition. St000944The 3-degree of an integer partition. St000997The even-odd crank of an integer partition. St001128The exponens consonantiae of a partition. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001267The length of the Lyndon factorization of the binary word. St001279The sum of the parts of an integer partition that are at least two. St001372The length of a longest cyclic run of ones of a binary word. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001389The number of partitions of the same length below the given integer partition. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001432The order dimension of the partition. St001437The flex of a binary word. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001780The order of promotion on the set of standard tableaux of given shape. St001884The number of borders of a binary word. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. 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. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000459The hook length of the base cell of a partition. St000517The Kreweras number of an integer partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows: St001485The modular major index of a binary word. St001581The achromatic number of a graph. St001955The number of natural descents for set-valued two row standard Young tableaux. St000169The cocharge of a standard tableau. St000330The (standard) major index of a standard tableau. St000336The leg major index of a standard tableau. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000738The first entry in the last row of a standard tableau. 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. St000984The number of boxes below precisely one peak. St001249Sum of the odd parts of a partition. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St000519The largest length of a factor maximising the subword complexity. 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. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St000006The dinv of a Dyck path. St000321The number of integer partitions of n that are dominated by an integer partition. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001127The sum of the squares of the parts of a partition.