Your data matches 249 different statistics following compositions of up to 3 maps.
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Mp00307: Posets promotion cycle typeInteger partitions
St000208: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> 1
([],2)
=> [2]
=> 2
([(0,1)],2)
=> [1]
=> 1
([],3)
=> [3,3]
=> 6
([(1,2)],3)
=> [3]
=> 3
([(0,1),(0,2)],3)
=> [2]
=> 2
([(0,2),(2,1)],3)
=> [1]
=> 1
([(0,2),(1,2)],3)
=> [2]
=> 2
Description
Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that $P_{\lambda,w}$ is integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has only integer lattice points as vertices. See also [[St000205]], [[St000206]] and [[St000207]].
Mp00307: Posets promotion cycle typeInteger partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> 1
([],2)
=> [2]
=> 2
([(0,1)],2)
=> [1]
=> 1
([],3)
=> [3,3]
=> 6
([(1,2)],3)
=> [3]
=> 3
([(0,1),(0,2)],3)
=> [2]
=> 2
([(0,2),(2,1)],3)
=> [1]
=> 1
([(0,2),(1,2)],3)
=> [2]
=> 2
Description
The size of a partition. This statistic is the constant statistic of the level sets.
Matching statistic: St000345
Mp00307: Posets promotion cycle typeInteger partitions
St000345: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> 1
([],2)
=> [2]
=> 2
([(0,1)],2)
=> [1]
=> 1
([],3)
=> [3,3]
=> 6
([(1,2)],3)
=> [3]
=> 3
([(0,1),(0,2)],3)
=> [2]
=> 2
([(0,2),(2,1)],3)
=> [1]
=> 1
([(0,2),(1,2)],3)
=> [2]
=> 2
Description
The number of refinements of a partition. A partition $\lambda$ refines a partition $\mu$ if the parts of $\mu$ can be subdivided to obtain the parts of $\lambda$.
Mp00307: Posets promotion cycle typeInteger partitions
St000531: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> 1
([],2)
=> [2]
=> 2
([(0,1)],2)
=> [1]
=> 1
([],3)
=> [3,3]
=> 6
([(1,2)],3)
=> [3]
=> 3
([(0,1),(0,2)],3)
=> [2]
=> 2
([(0,2),(2,1)],3)
=> [1]
=> 1
([(0,2),(1,2)],3)
=> [2]
=> 2
Description
The leading coefficient of the rook polynomial of an integer partition. Let $m$ be the minimum of the number of parts and the size of the first part of an integer partition $\lambda$. Then this statistic yields the number of ways to place $m$ non-attacking rooks on the Ferrers board of $\lambda$.
Mp00307: Posets promotion cycle typeInteger partitions
St001389: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> 1
([],2)
=> [2]
=> 2
([(0,1)],2)
=> [1]
=> 1
([],3)
=> [3,3]
=> 6
([(1,2)],3)
=> [3]
=> 3
([(0,1),(0,2)],3)
=> [2]
=> 2
([(0,2),(2,1)],3)
=> [1]
=> 1
([(0,2),(1,2)],3)
=> [2]
=> 2
Description
The number of partitions of the same length below the given integer partition. For a partition $\lambda_1 \geq \dots \lambda_k > 0$, this number is $$ \det\left( \binom{\lambda_{k+1-i}}{j-i+1} \right)_{1 \le i,j \le k}.$$
Mp00307: Posets promotion cycle typeInteger partitions
St001659: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> 1
([],2)
=> [2]
=> 2
([(0,1)],2)
=> [1]
=> 1
([],3)
=> [3,3]
=> 6
([(1,2)],3)
=> [3]
=> 3
([(0,1),(0,2)],3)
=> [2]
=> 2
([(0,2),(2,1)],3)
=> [1]
=> 1
([(0,2),(1,2)],3)
=> [2]
=> 2
Description
The number of ways to place as many non-attacking rooks as possible on a Ferrers board.
Matching statistic: St000071
Mp00195: Posets order idealsLattices
Mp00193: Lattices to posetPosets
St000071: Posets ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 6
([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 3
([(0,1),(0,2)],3)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
Description
The number of maximal chains in a poset.
Matching statistic: St000184
Mp00307: Posets promotion cycle typeInteger partitions
Mp00312: Integer partitions Glaisher-FranklinInteger partitions
St000184: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> [1]
=> 1
([],2)
=> [2]
=> [1,1]
=> 2
([(0,1)],2)
=> [1]
=> [1]
=> 1
([],3)
=> [3,3]
=> [6]
=> 6
([(1,2)],3)
=> [3]
=> [3]
=> 3
([(0,1),(0,2)],3)
=> [2]
=> [1,1]
=> 2
([(0,2),(2,1)],3)
=> [1]
=> [1]
=> 1
([(0,2),(1,2)],3)
=> [2]
=> [1,1]
=> 2
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 $1$.
Mp00307: Posets promotion cycle typeInteger partitions
Mp00095: Integer partitions to binary wordBinary words
St000293: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> 10 => 1
([],2)
=> [2]
=> 100 => 2
([(0,1)],2)
=> [1]
=> 10 => 1
([],3)
=> [3,3]
=> 11000 => 6
([(1,2)],3)
=> [3]
=> 1000 => 3
([(0,1),(0,2)],3)
=> [2]
=> 100 => 2
([(0,2),(2,1)],3)
=> [1]
=> 10 => 1
([(0,2),(1,2)],3)
=> [2]
=> 100 => 2
Description
The number of inversions of a binary word.
Matching statistic: St000378
Mp00307: Posets promotion cycle typeInteger partitions
Mp00321: Integer partitions 2-conjugateInteger partitions
St000378: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> [1]
=> 1
([],2)
=> [2]
=> [2]
=> 2
([(0,1)],2)
=> [1]
=> [1]
=> 1
([],3)
=> [3,3]
=> [3,2,1]
=> 6
([(1,2)],3)
=> [3]
=> [2,1]
=> 3
([(0,1),(0,2)],3)
=> [2]
=> [2]
=> 2
([(0,2),(2,1)],3)
=> [1]
=> [1]
=> 1
([(0,2),(1,2)],3)
=> [2]
=> [2]
=> 2
Description
The diagonal inversion number of an integer partition. The dinv of a partition is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \in \{0,1\}$. See also exercise 3.19 of [2]. This statistic is equidistributed with the length of the partition, see [3].
The following 239 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000384The maximal part of the shifted composition of an integer partition. St000395The sum of the heights of the peaks of a Dyck path. 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. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000784The maximum of the length and the largest part of the integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000909The number of maximal chains of maximal size in a poset. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001034The area of the parallelogram polyomino associated with the Dyck path. St001360The number of covering relations in Young's lattice below a partition. St001441The number of non-empty connected induced subgraphs of a graph. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001757The number of orbits of toric promotion on a graph. St001959The product of the heights of the peaks of a Dyck path. 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. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001400The total number of Littlewood-Richardson tableaux of given shape. St001834The number of non-isomorphic minors of a graph. 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. St001838The number of nonempty primitive factors of a binary word. St000290The major index of a binary word. St000391The sum of the positions of the ones in a binary word. St000682The Grundy value of Welter's game on a binary word. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001809The index of the step at the first peak of maximal height in a Dyck path. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St000012The area of a Dyck path. St000792The Grundy value for the game of ruler on a binary word. St000867The sum of the hook lengths in the first row of an integer partition. St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. St001259The vector space dimension of the double dual of D(A) in the corresponding Nakayama algebra. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001721The degree of a binary word. St000100The number of linear extensions of a poset. St000454The largest eigenvalue of a graph if it is integral. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001779The order of promotion on the set of linear extensions of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000907The number of maximal antichains of minimal length in a poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000172The Grundy number of a graph. St000363The number of minimal vertex covers of a graph. St000388The number of orbits of vertices of a graph under automorphisms. St000450The number of edges minus the number of vertices plus 2 of a graph. St000456The monochromatic index of a connected graph. St000537The cutwidth of a graph. St000553The number of blocks of a graph. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001271The competition number of a graph. St001281The normalized isoperimetric number of a graph. St001304The number of maximally independent sets of vertices of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001645The pebbling number of a connected graph. St001670The connected partition number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001725The harmonious chromatic number of a graph. St001963The tree-depth of a graph. St000171The degree of the graph. St000271The chromatic index of a graph. St000315The number of isolated vertices of a graph. St000362The size of a minimal vertex cover of a graph. St000387The matching number of a graph. St000469The distinguishing number of a graph. St000552The number of cut vertices of a graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000741The Colin de Verdière graph invariant. St000776The maximal multiplicity of an eigenvalue in a graph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001117The game chromatic index of 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. St001305The number of induced cycles on four vertices in a graph. 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. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001366The maximal multiplicity of a degree of a vertex of a graph. St001480The number of simple summands of the module J^2/J^3. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. 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. St001689The number of celebrities in a graph. St001691The number of kings in a graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001742The difference of the maximal and the minimal degree in a graph. St001812The biclique partition number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001883The mutual visibility number of a graph. St000144The pyramid weight of the Dyck path. St001180Number of indecomposable injective modules with projective dimension at most 1. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001877Number of indecomposable injective modules with projective dimension 2. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000455The second largest eigenvalue of a graph if it is integral. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001624The breadth of a lattice. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000706The product of the factorials of the multiplicities of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001118The acyclic chromatic index of a graph. St001330The hat guessing number of a graph. St000087The number of induced subgraphs. St000244The cardinality of the automorphism group of a graph. St000269The number of acyclic orientations of a graph. St000286The number of connected components of the complement of a graph. St000364The exponent of the automorphism group of a graph. St000438The position of the last up step in a Dyck path. St000636The hull number of a graph. St000667The greatest common divisor of the parts of the partition. St000674The number of hills of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000722The number of different neighbourhoods in a graph. St000822The Hadwiger number of the graph. St000914The sum of the values of the Möbius function of a poset. St000926The clique-coclique number of a graph. St001029The size of the core of a graph. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001109The number of proper colourings of a graph with as few colours as possible. St001249Sum of the odd parts of a partition. St001302The number of minimally dominating sets of vertices of a graph. St001316The domatic number of a graph. St001342The number of vertices in the center of a graph. St001368The number of vertices of maximal degree in a graph. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001494The Alon-Tarsi number of a graph. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. 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. St001746The coalition number of a graph. St001758The number of orbits of promotion on a graph. St001802The number of endomorphisms of a graph. St000466The Gutman (or modified Schultz) index of a connected graph. St001651The Frankl number of a lattice. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001545The second Elser number of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000467The hyper-Wiener index of a connected graph. St000939The number of characters of the symmetric group whose value on the partition is positive. 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 $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001200The number of simple modules in $eAe$ with projective dimension at most 2 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 $eA$. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001498The normalised height of a Nakayama algebra with magnitude 1. St001568The smallest positive integer that does not appear twice in the partition. St000070The number of antichains in a poset. St000104The number of facets in the order polytope of this poset. St000151The number of facets in the chain polytope of the poset. St000189The number of elements in the poset. St000327The number of cover relations in a poset. St000528The height of a poset. St000656The number of cuts of a poset. 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. St000911The number of maximal antichains of maximal size in a poset. St000912The number of maximal antichains in a poset. St001060The distinguishing index of a graph. St001343The dimension of the reduced incidence algebra of a poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001637The number of (upper) dissectors of a poset. St001664The number of non-isomorphic subposets of a poset. St001668The number of points of the poset minus the width of the poset. St001717The largest size of an interval in a poset. St001782The order of rowmotion on the set of order ideals of a poset. St001813The product of the sizes of the principal order filters in a poset. St001815The number of order preserving surjections from a poset to a total order. St000477The weight of a partition according to Alladi. St000478Another weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. 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. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000668The least common multiple of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000699The toughness times the least common multiple of 1,. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001564The value of the forgotten symmetric functions when all variables set to 1. St001592The maximal number of simple paths between any two different vertices of a graph. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition.