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Your data matches 525 different statistics following compositions of up to 3 maps.
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Matching statistic: St001300
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Values
([],1)
=> 0
([],2)
=> 0
([(0,1)],2)
=> 1
([],3)
=> 0
([(1,2)],3)
=> 1
([(0,1),(0,2)],3)
=> 2
([(0,2),(2,1)],3)
=> 2
([(0,2),(1,2)],3)
=> 2
Description
The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset.
Matching statistic: St000081
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(load all 2 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> 0
([],2)
=> ([],2)
=> 0
([(0,1)],2)
=> ([(0,1)],2)
=> 1
([],3)
=> ([],3)
=> 0
([(1,2)],3)
=> ([(1,2)],3)
=> 1
([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
Description
The number of edges of a graph.
Matching statistic: St000171
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(load all 5 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> 0
([],2)
=> ([],2)
=> 0
([(0,1)],2)
=> ([(0,1)],2)
=> 1
([],3)
=> ([],3)
=> 0
([(1,2)],3)
=> ([(1,2)],3)
=> 1
([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
Description
The degree of the graph.
This is the maximal vertex degree of a graph.
Matching statistic: St000987
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(load all 5 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> 0
([],2)
=> ([],2)
=> 0
([(0,1)],2)
=> ([(0,1)],2)
=> 1
([],3)
=> ([],3)
=> 0
([(1,2)],3)
=> ([(1,2)],3)
=> 1
([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
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.
Matching statistic: St001117
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(load all 2 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> 0
([],2)
=> ([],2)
=> 0
([(0,1)],2)
=> ([(0,1)],2)
=> 1
([],3)
=> ([],3)
=> 0
([(1,2)],3)
=> ([(1,2)],3)
=> 1
([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
Description
The game chromatic index of a graph.
Two players, Alice and Bob, take turns colouring properly any uncolored edge of the graph. Alice begins. If it is not possible for either player to colour a edge, then Bob wins. If the graph is completely colored, Alice wins.
The game chromatic index is the smallest number of colours such that Alice has a winning strategy.
Matching statistic: St001120
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(load all 5 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> 0
([],2)
=> ([],2)
=> 0
([(0,1)],2)
=> ([(0,1)],2)
=> 1
([],3)
=> ([],3)
=> 0
([(1,2)],3)
=> ([(1,2)],3)
=> 1
([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
Description
The length of a longest path in a graph.
Matching statistic: St001479
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(load all 2 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> 0
([],2)
=> ([],2)
=> 0
([(0,1)],2)
=> ([(0,1)],2)
=> 1
([],3)
=> ([],3)
=> 0
([(1,2)],3)
=> ([(1,2)],3)
=> 1
([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
Description
The number of bridges of a graph.
A bridge is an edge whose removal increases the number of connected components of the graph.
Matching statistic: St001512
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(load all 3 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> 0
([],2)
=> ([],2)
=> 0
([(0,1)],2)
=> ([(0,1)],2)
=> 1
([],3)
=> ([],3)
=> 0
([(1,2)],3)
=> ([(1,2)],3)
=> 1
([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
Description
The minimum rank of a graph.
The minimum rank of a simple graph G is the smallest possible rank over all symmetric real matrices whose entry in row $i$ and column $j$ (for $i\neq j$) is nonzero whenever $\{i, j\}$ is an edge in
$G$, and zero otherwise.
Matching statistic: St001649
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(load all 2 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> 0
([],2)
=> ([],2)
=> 0
([(0,1)],2)
=> ([(0,1)],2)
=> 1
([],3)
=> ([],3)
=> 0
([(1,2)],3)
=> ([(1,2)],3)
=> 1
([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
Description
The length of a longest trail in a graph.
A trail is a sequence of distinct edges, such that two consecutive edges share a vertex.
Matching statistic: St001826
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(load all 2 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> 0
([],2)
=> ([],2)
=> 0
([(0,1)],2)
=> ([(0,1)],2)
=> 1
([],3)
=> ([],3)
=> 0
([(1,2)],3)
=> ([(1,2)],3)
=> 1
([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
Description
The maximal number of leaves on a vertex of a graph.
The following 515 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001869The maximum cut size of a graph. St000086The number of subgraphs. St000299The number of nonisomorphic vertex-induced subtrees. St000378The diagonal inversion number 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. St000722The number of different neighbourhoods in a graph. St001093The detour number of a graph. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001725The harmonious chromatic number of a graph. St000010The length of the partition. St000095The number of triangles of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000142The number of even parts of a partition. St000143The largest repeated part of a partition. St000147The largest part of an integer partition. St000148The number of odd parts of a partition. St000160The multiplicity of the smallest part of a partition. St000228The size of a partition. St000271The chromatic index of a graph. St000272The treewidth of a graph. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000362The size of a minimal vertex cover of a graph. St000377The dinv defect of an integer partition. St000384The maximal part of the shifted composition of an integer partition. St000454The largest eigenvalue of a graph if it is integral. St000459The hook length of the base cell of a partition. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000475The number of parts equal to 1 in a partition. St000479The Ramsey number of a graph. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St000548The number of different non-empty partial sums of an integer partition. 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. St000867The sum of the hook lengths in the first row of an integer partition. 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. St001127The sum of the squares of the parts of a partition. St001176The size of a partition minus its first part. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001280The number of parts of an integer partition that are at least two. 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. St001319The minimal number of occurrences of the star-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. St001341The number of edges in the center of a graph. St001358The largest degree of a regular subgraph of a graph. St001391The disjunction number of a graph. 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. St001613The binary logarithm of the size of the center of a lattice. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001621The number of atoms of a lattice. St001622The number of join-irreducible elements of a lattice. St001644The dimension of a graph. St001657The number of twos in an integer partition. St001742The difference of the maximal and the minimal degree in a graph. St001743The discrepancy of a graph. St001792The arboricity of a graph. St001812The biclique partition number of a graph. St001827The number of two-component spanning forests of a graph. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001962The proper pathwidth of a graph. St000006The dinv 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. St000172The Grundy number of a graph. St000189The number of elements in the poset. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000300The number of independent sets of vertices of a graph. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000346The number of coarsenings of a partition. 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. St000532The total number of rook placements on a Ferrers board. St000636The hull number of a graph. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000822The Hadwiger number of the graph. St000935The number of ordered refinements of an integer partition. St001029The size of the core of a graph. St001116The game chromatic number of a graph. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001330The hat guessing number of a graph. St001342The number of vertices in the center of a graph. St001352The number of internal nodes in the modular decomposition of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001389The number of partitions of the same length below the given integer partition. St001400The total number of Littlewood-Richardson tableaux of given shape. St001463The number of distinct columns in the nullspace of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001670The connected partition number of a graph. St001746The coalition number of a graph. St001814The number of partitions interlacing the given partition. St001883The mutual visibility number of a graph. St001917The order of toric promotion on the set of labellings of a graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St001963The tree-depth of a graph. St001458The rank of the adjacency matrix 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. St000005The bounce statistic of a Dyck path. St000012The area of a Dyck path. St000024The number of double up and double down steps of a Dyck path. St000053The number of valleys of the Dyck path. St000093The cardinality of a maximal independent set of vertices of a graph. St000120The number of left tunnels of a Dyck path. St000154The sum of the descent bottoms of a permutation. St000157The number of descents of a standard tableau. St000185The weighted size of a partition. St000237The number of small exceedances. St000290The major index of a binary word. St000293The number of inversions of a binary word. St000295The length of the border of a binary word. St000305The inverse major index of a permutation. St000306The bounce count of a Dyck path. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000331The number of upper interactions of a Dyck path. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000340The number of non-final maximal constant sub-paths of length greater than one. St000369The dinv deficit of a Dyck path. St000374The number of exclusive right-to-left minima of a permutation. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000448The number of pairs of vertices of a graph with distance 2. St000463The number of admissible inversions of a permutation. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000651The maximal size of a rise in a permutation. St000670The reversal length of a permutation. St000682The Grundy value of Welter's game on a binary word. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000691The number of changes of a binary word. St000692Babson and Steingrímsson's statistic of a permutation. St000703The number of deficiencies of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000766The number of inversions of an integer composition. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001090The number of pop-stack-sorts needed to sort a permutation. St001094The depth index of a set partition. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001161The major index north count of a Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. 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. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001274The number of indecomposable injective modules with projective dimension equal to two. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001345The Hamming dimension of a graph. St001375The pancake length of a permutation. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001485The modular major index of a binary word. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001524The degree of symmetry of a binary word. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001671Haglund's hag of a permutation. St001760The number of prefix or suffix reversals needed to sort a permutation. St001777The number of weak descents in an integer composition. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001956The comajor index for set-valued two-row standard Young tableaux. St001961The sum of the greatest common divisors of all pairs of parts. St000011The number of touch points (or returns) of a Dyck path. St000013The height of a Dyck path. St000015The number of peaks of a Dyck path. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000047The number of standard immaculate tableaux of a given shape. St000058The order of a permutation. St000288The number of ones in a binary word. St000297The number of leading ones in a binary word. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000383The last part of an integer composition. St000392The length of the longest run of ones in a binary word. St000443The number of long tunnels of a Dyck path. St000451The length of the longest pattern of the form k 1 2. St000460The hook length of the last cell along the main diagonal of an integer partition. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000482The (zero)-forcing number of a graph. St000501The size of the first part in the decomposition of a permutation. St000507The number of ascents of a standard tableau. St000553The number of blocks of a graph. St000617The number of global maxima of a Dyck path. St000626The minimal period of a binary word. St000638The number of up-down runs of a permutation. St000653The last descent of a permutation. St000676The number of odd rises of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000733The row containing the largest entry of a standard tableau. St000734The last entry in the first row of a standard tableau. St000738The first entry in the last row of a standard tableau. St000740The last entry of a permutation. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000753The Grundy value for the game of Kayles on a binary word. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000808The number of up steps of the associated bargraph. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St000847The number of standard Young tableaux whose descent set is the binary word. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000917The open packing number of a graph. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000947The major index east count of a Dyck path. St000983The length of the longest alternating subword. St000984The number of boxes below precisely one peak. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of 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. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001041The depth of the label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001048The number of leaves in the subtree containing 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001102The number of words with multiplicities of the letters given by the composition, avoiding the consecutive pattern 132. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001202Call 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 special CNakayama algebra. 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:
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. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001285The number of primes in the column sums of the two line notation of a permutation. St001286The annihilation number of a graph. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001312Number of parabolic noncrossing partitions indexed by the composition. St001313The number of Dyck paths above the lattice path given by a binary word. 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. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001372The length of a longest cyclic run of ones of a binary word. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001415The length of the longest palindromic prefix of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001437The flex of a binary word. St001441The number of non-empty connected induced subgraphs of a graph. St001462The number of factors of a standard tableaux under concatenation. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001481The minimal height of a peak of a Dyck path. St001486The number of corners of the ribbon associated with an integer composition. St001497The position of the largest weak excedence of a permutation. St001500The global dimension of magnitude 1 Nakayama algebras. St001530The depth of a Dyck path. St001672The restrained domination number of a graph. St001733The number of weak left to right maxima of a Dyck path. St001757The number of orbits of toric promotion on a graph. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001806The upper middle entry of a permutation. St001807The lower middle entry of a permutation. St001809The index of the step at the first peak of maximal height in a Dyck path. St001949The rigidity index of a graph. St000439The position of the first down step of a Dyck path. St000469The distinguishing number of a graph. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001180Number of indecomposable injective modules with projective dimension at most 1. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001366The maximal multiplicity of a degree of a vertex of a graph. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001834The number of non-isomorphic minors of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St000327The number of cover relations in a poset. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000946The sum of the skew hook positions in a Dyck path. 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. St001592The maximal number of simple paths between any two different vertices of a graph. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. 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. St000939The number of characters of the symmetric group whose value on the partition is positive. St000216The absolute length of a permutation. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000442The maximal area to the right of an up step of a Dyck path. St000472The sum of the ascent bottoms of a permutation. St000730The maximal arc length of a set partition. St000796The stat' of a permutation. St000797The stat`` of a permutation. St000798The makl of a permutation. St000809The reduced reflection length of the permutation. St000874The position of the last double rise in a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000934The 2-degree of an integer partition. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001077The prefix exchange distance of a permutation. St001080The minimal length of a factorization of a permutation using the transposition (12) and the cycle (1,. St001350Half of the Albertson index of a graph. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001480The number of simple summands of the module J^2/J^3. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St000420The number of Dyck paths that are weakly above a Dyck path. St000444The length of the maximal rise of a Dyck path. St000485The length of the longest cycle of a permutation. St000678The number of up steps after the last double rise of a Dyck path. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001808The box weight or horizontal decoration of a Dyck path. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000741The Colin de Verdière graph invariant. St001645The pebbling number of a connected graph. St000259The diameter of a connected graph. St000680The Grundy value for Hackendot on posets. St001498The normalised height of a Nakayama algebra with magnitude 1. St001118The acyclic chromatic index of a graph. St000087The number of induced subgraphs. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000244The cardinality of the automorphism group of a graph. St000258The burning number of a graph. St000269The number of acyclic orientations of a graph. St000270The number of forests contained in a graph. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000286The number of connected components of the complement of a graph. St000311The number of vertices of odd degree in a graph. St000343The number of spanning subgraphs of a graph. St000350The sum of the vertex degrees of a graph. St000364The exponent of the automorphism group of a graph. St000422The energy of a graph, if it is integral. St000465The first Zagreb index of a graph. St000467The hyper-Wiener index of a connected graph. St000571The F-index (or forgotten topological index) of a graph. St000667The greatest common divisor of the parts of the partition. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000915The Ore degree of a graph. St000918The 2-limited packing number of a graph. St000926The clique-coclique number of a graph. St000972The composition number of a graph. St001109The number of proper colourings of a graph with as few colours as possible. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001248Sum of the even parts of a partition. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001261The Castelnuovo-Mumford regularity of a graph. St001279The sum of the parts of an integer partition that are at least two. St001315The dissociation number of a graph. St001316The domatic number of a graph. St001360The number of covering relations in Young's lattice below a partition. St001368The number of vertices of maximal degree in a graph. St001378The product of the cohook lengths of the integer partition. St001459The number of zero columns in the nullspace of a graph. St001474The evaluation of the Tutte polynomial of the graph at (x,y) equal to (2,-1). St001527The cyclic permutation representation number of an integer partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St001571The Cartan determinant of the integer partition. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. 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. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. 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. St001758The number of orbits of promotion on a graph. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001802The number of endomorphisms of a graph. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St000273The domination number of a graph. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000287The number of connected components of a graph. St000309The number of vertices with even degree. St000315The number of isolated vertices of a graph. St000544The cop number of a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St000916The packing number of a graph. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001642The Prague dimension of a graph. St001651The Frankl number of a lattice. St001691The number of kings in a graph. St001765The number of connected components of the friends and strangers graph. St001828The Euler characteristic of a graph. St001829The common independence number of a graph. St000455The second largest eigenvalue of a graph if it is integral. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000706The product of the factorials of the multiplicities of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001568The smallest positive integer that does not appear twice in the partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. 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 $eA$. St001060The distinguishing index of a graph. St000080The rank of the poset. St000464The Schultz index of a connected graph. St000477The weight of a partition according to Alladi. 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. St000528The height of a poset. St000707The product of the factorials of the parts. 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. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000907The number of maximal antichains of minimal length in a poset. St000911The number of maximal antichains of maximal size in a poset. St000912The number of maximal antichains in a poset. St000997The even-odd crank of an integer partition. St001098The 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 vertex labelled trees. St001343The dimension of the reduced incidence algebra of a poset. St001545The second Elser number of a connected graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001637The number of (upper) dissectors 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. 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. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000100The number of linear extensions of a poset. St000137The Grundy value of an integer partition. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000379The number of Hamiltonian cycles in a graph. St000478Another weight of a partition according to Alladi. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. 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. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000681The Grundy value of Chomp on Ferrers diagrams. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St000910The number of maximal chains of minimal length in a poset. St000928The sum of the coefficients of the character polynomial of an integer partition. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001281The normalized isoperimetric number 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. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001525The number of symmetric hooks on the diagonal of a 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. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. 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.
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