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Your data matches 286 different statistics following compositions of up to 3 maps.
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Matching statistic: St001626
Values
([(0,1)],2)
=> 2
([(0,2),(2,1)],3)
=> 3
([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(2,1),(3,2)],4)
=> 4
Description
The number of maximal proper sublattices of a lattice.
Matching statistic: St001720
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(load all 2 compositions to match this statistic)
Values
([(0,1)],2)
=> 2
([(0,2),(2,1)],3)
=> 3
([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(2,1),(3,2)],4)
=> 4
Description
The minimal length of a chain of small intervals in a lattice.
An interval $[a, b]$ is small if $b$ is a join of elements covering $a$.
Matching statistic: St001820
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(load all 2 compositions to match this statistic)
Values
([(0,1)],2)
=> 1 = 2 - 1
([(0,2),(2,1)],3)
=> 2 = 3 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
Description
The size of the image of the pop stack sorting operator.
The pop stack sorting operator is defined by $Pop_L^\downarrow(x) = x\wedge\bigwedge\{y\in L\mid y\lessdot x\}$. This statistic returns the size of $Pop_L^\downarrow(L)\}$.
Matching statistic: St001651
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(load all 2 compositions to match this statistic)
Values
([(0,1)],2)
=> 0 = 2 - 2
([(0,2),(2,1)],3)
=> 1 = 3 - 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
([(0,3),(2,1),(3,2)],4)
=> 2 = 4 - 2
Description
The Frankl number of a lattice.
For a lattice $L$ on at least two elements, this is
$$
\max_x(|L|-2|[x, 1]|),
$$
where we maximize over all join irreducible elements and $[x, 1]$ denotes the interval from $x$ to the top element. Frankl's conjecture asserts that this number is non-negative, and zero if and only if $L$ is a Boolean lattice.
Matching statistic: St001846
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(load all 2 compositions to match this statistic)
Values
([(0,1)],2)
=> 0 = 2 - 2
([(0,2),(2,1)],3)
=> 1 = 3 - 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
([(0,3),(2,1),(3,2)],4)
=> 2 = 4 - 2
Description
The number of elements which do not have a complement in the lattice.
A complement of an element $x$ in a lattice is an element $y$ such that the meet of $x$ and $y$ is the bottom element and their join is the top element.
Matching statistic: St000907
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(load all 2 compositions to match this statistic)
Values
([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
Description
The number of maximal antichains of minimal length in a poset.
Matching statistic: St001782
Values
([(0,1)],2)
=> ([],1)
=> 2
([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 3
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],2)
=> 2
([(0,3),(2,1),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 4
Description
The order of rowmotion on the set of order ideals of a poset.
Matching statistic: St000528
Values
([(0,1)],2)
=> ([],1)
=> 1 = 2 - 1
([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],2)
=> 1 = 2 - 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 3 = 4 - 1
Description
The height of a poset.
This equals the rank of the poset [[St000080]] plus one.
Matching statistic: St000911
Values
([(0,1)],2)
=> ([],1)
=> 1 = 2 - 1
([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],2)
=> 1 = 2 - 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 3 = 4 - 1
Description
The number of maximal antichains of maximal size in a poset.
Matching statistic: St000912
Values
([(0,1)],2)
=> ([],1)
=> 1 = 2 - 1
([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],2)
=> 1 = 2 - 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 3 = 4 - 1
Description
The number of maximal antichains in a poset.
The following 276 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001343The dimension of the reduced incidence algebra of a poset. St001717The largest size of an interval in a poset. St000080The rank of the poset. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St000145The Dyson rank of a partition. St000309The number of vertices with even degree. St000315The number of isolated vertices of a graph. St000469The distinguishing number of a graph. St000531The leading coefficient of the rook polynomial of an integer partition. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St000992The alternating sum of the parts of an integer partition. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001571The Cartan determinant of the integer partition. St001619The number of non-isomorphic sublattices of a lattice. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001666The number of non-isomorphic subposets of a lattice which are lattices. St001691The number of kings in a graph. St001814The number of partitions interlacing the given partition. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St000086The number of subgraphs. St000093The cardinality of a maximal independent set of vertices of a graph. St000147The largest part of an integer partition. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000273The domination number of a graph. St000287The number of connected components of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000377The dinv defect of an integer partition. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000468The Hosoya index of a graph. St000482The (zero)-forcing number of a graph. St000544The cop number of a graph. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000553The number of blocks of a graph. St000667The greatest common divisor of the parts of the partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000916The packing number of a graph. St000935The number of ordered refinements of an integer partition. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001093The detour number of a graph. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001286The annihilation number of a graph. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001352The number of internal nodes in the modular decomposition of a graph. St001363The Euler characteristic of a graph according to Knill. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001389The number of partitions of the same length below the given integer partition. St001463The number of distinct columns in the nullspace of a graph. St001527The cyclic permutation representation number of an integer partition. St001616The number of neutral elements in a lattice. St001674The number of vertices of the largest induced star graph in the graph. St001725The harmonious chromatic number of a graph. St001828The Euler characteristic of a graph. St001829The common independence number of a graph. St000081The number of edges of a graph. St000142The number of even parts of a partition. St000148The number of odd parts of a partition. St000171The degree of the graph. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000549The number of odd partial sums of an integer partition. St000552The number of cut vertices of a graph. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001092The number of distinct even parts of a partition. 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. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001479The number of bridges of a graph. St001512The minimum rank of a graph. St001521Half the total irregularity 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. St001623The number of doubly irreducible elements of a lattice. St001649The length of a longest trail in a graph. St001692The number of vertices with higher degree than the average degree in 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. St001826The maximal number of leaves on a vertex of a graph. St001869The maximum cut size of a graph. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000013The height of a Dyck path. St000025The number of initial rises of a Dyck path. St000120The number of left tunnels of a Dyck path. St000160The multiplicity of the smallest part of a partition. St000312The number of leaves in a graph. St000443The number of long tunnels of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000445The number of rises of length 1 of a Dyck path. St000475The number of parts equal to 1 in a partition. St000636The hull number of a graph. St000674The number of hills of a Dyck path. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000904The maximal number of repetitions of an integer composition. St000917The open packing number of a graph. St000982The length of the longest constant subword. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001060The distinguishing index of a graph. St001091The number of parts in an integer partition whose next smaller part has the same size. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. 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. St001459The number of zero columns in the nullspace of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St001523The degree of symmetry of a Dyck path. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001672The restrained domination number of a graph. St001809The index of the step at the first peak of maximal height in a Dyck path. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001933The largest multiplicity of a part in an integer partition. St000010The length of the partition. St000024The number of double up and double down steps of a Dyck path. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000144The pyramid weight of the Dyck path. St000172The Grundy number of a graph. St000259The diameter of a connected graph. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000286The number of connected components of the complement of a graph. St000346The number of coarsenings of a partition. St000363The number of minimal vertex covers of a graph. St000439The position of the first down step of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000548The number of different non-empty partial sums of an integer partition. St000626The minimal period of a binary word. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000722The number of different neighbourhoods in a graph. St000741The Colin de Verdière graph invariant. St000744The length of the path to the largest entry in a standard Young tableau. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000822The Hadwiger number of the graph. St000874The position of the last double rise in a Dyck path. St000877The depth of the binary word interpreted as a path. St000885The number of critical steps in the Catalan decomposition of a binary word. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001116The game chromatic number of a graph. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001180Number of indecomposable injective modules with projective dimension at most 1. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001267The length of the Lyndon factorization of the binary word. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001316The domatic number of a graph. St001330The hat guessing number of a graph. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001494The Alon-Tarsi number of a graph. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001777The number of weak descents in an integer composition. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St001881The number of factors of a lattice as a Cartesian product of lattices. St001883The mutual visibility number of a graph. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001917The order of toric promotion on the set of labellings of a graph. St001955The number of natural descents for set-valued two row standard Young tableaux. St001963The tree-depth of a graph. St000137The Grundy value of an integer partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000272The treewidth of a graph. St000288The number of ones in a binary word. St000297The number of leading ones in a binary word. St000310The minimal degree of a vertex of a graph. St000313The number of degree 2 vertices of a graph. St000362The size of a minimal vertex cover of a graph. St000369The dinv deficit of a Dyck path. St000392The length of the longest run of ones in a binary word. St000448The number of pairs of vertices of a graph with distance 2. St000454The largest eigenvalue of a graph if it is integral. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000536The pathwidth of a graph. St000537The cutwidth of a graph. 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. St000753The Grundy value for the game of Kayles on a binary word. St000931The number of occurrences of the pattern UUU in a Dyck path. 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. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001176The size of a partition minus its first part. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. 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. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001308The number of induced paths on three vertices in a graph. St001354The number of series nodes in the modular decomposition of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001358The largest degree of a regular subgraph of a graph. St001372The length of a longest cyclic run of ones of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001484The number of singletons of an integer partition. St001525The number of symmetric hooks on the diagonal of a 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. St001644The dimension of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001743The discrepancy of a graph. St001792The arboricity of a graph. St001812The biclique partition number of a graph. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001962The proper pathwidth of a graph. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000643The size of the largest orbit of antichains under Panyushev complementation. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St000327The number of cover relations in a poset. St001668The number of points of the poset minus the width of the poset. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000668The least common multiple of the parts of the partition. 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. St000478Another weight of a partition according to Alladi. St000928The sum of the coefficients of the character polynomial of an integer partition. St000656The number of cuts of a poset. St000680The Grundy value for Hackendot on posets. St001637The number of (upper) dissectors of a poset. St001645The pebbling number of a connected graph. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001624The breadth of a lattice. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001281The normalized isoperimetric number of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000455The second largest eigenvalue of a graph if it is integral.
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