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Your data matches 362 different statistics following compositions of up to 3 maps.
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Matching statistic: St001441
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(load all 2 compositions to match this statistic)
Values
([],1)
=> 1
([],2)
=> 2
([(0,1)],2)
=> 3
([],3)
=> 3
([],4)
=> 4
([],5)
=> 5
([],6)
=> 6
Description
The number of non-empty connected induced subgraphs of a graph.
More precisely, this is the number of non-empty subsets of the set of vertices of a graph, such that the induced subgraph is connected.
Matching statistic: St000081
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(load all 3 compositions to match this statistic)
Values
([],1)
=> ([(0,1)],2)
=> 1
([],2)
=> ([(0,2),(1,2)],3)
=> 2
([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 5
([],6)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 6
Description
The number of edges of a graph.
Matching statistic: St000271
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(load all 2 compositions to match this statistic)
Values
([],1)
=> ([(0,1)],2)
=> 1
([],2)
=> ([(0,2),(1,2)],3)
=> 2
([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 5
([],6)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 6
Description
The chromatic index of a graph.
This is the minimal number of colours needed such that no two adjacent edges have the same colour.
Matching statistic: St000867
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(load all 4 compositions to match this statistic)
Mp00251: Graphs —clique sizes⟶ Integer partitions
St000867: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000867: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> 1
([],2)
=> [1,1]
=> 2
([(0,1)],2)
=> [2]
=> 3
([],3)
=> [1,1,1]
=> 3
([],4)
=> [1,1,1,1]
=> 4
([],5)
=> [1,1,1,1,1]
=> 5
([],6)
=> [1,1,1,1,1,1]
=> 6
Description
The sum of the hook lengths in the first row of an integer partition.
For a cell in the Ferrers diagram of a partition, the hook length is given by the number of boxes to its right plus the number of boxes below plus one. This statistic is the sum of the hook lengths of the first row of a partition.
Put differently, for a partition of size $n$ with first parth $\lambda_1$, this is $\binom{\lambda_1}{2} + n$.
Matching statistic: St001118
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(load all 4 compositions to match this statistic)
Values
([],1)
=> ([(0,1)],2)
=> 1
([],2)
=> ([(0,2),(1,2)],3)
=> 2
([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 5
([],6)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 6
Description
The acyclic chromatic index of a graph.
An acyclic edge coloring of a graph is a proper colouring of the edges of a graph such that the union of the edges colored with any two given colours is a forest.
The smallest number of colours such that such a colouring exists is the acyclic chromatic index.
Matching statistic: St000300
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(load all 3 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> 2 = 1 + 1
([],2)
=> ([(0,1)],2)
=> 3 = 2 + 1
([(0,1)],2)
=> ([],2)
=> 4 = 3 + 1
([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4 = 3 + 1
([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 5 = 4 + 1
([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6 = 5 + 1
([],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7 = 6 + 1
Description
The number of independent sets of vertices of a graph.
An independent set of vertices of a graph $G$ is a subset $U \subset V(G)$ such that no two vertices in $U$ are adjacent.
This is also the number of vertex covers of $G$ as the map $U \mapsto V(G)\setminus U$ is a bijection between independent sets of vertices and vertex covers.
The size of the largest independent set, also called independence number of $G$, is [[St000093]]
Matching statistic: St000301
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(load all 4 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> 2 = 1 + 1
([],2)
=> ([(0,1)],2)
=> 3 = 2 + 1
([(0,1)],2)
=> ([],2)
=> 4 = 3 + 1
([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4 = 3 + 1
([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 5 = 4 + 1
([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6 = 5 + 1
([],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7 = 6 + 1
Description
The number of facets of the stable set polytope of a graph.
The stable set polytope of a graph $G$ is the convex hull of the characteristic vectors of stable (or independent) sets of vertices of $G$ inside $\mathbb{R}^{V(G)}$.
Matching statistic: St000468
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(load all 2 compositions to match this statistic)
Values
([],1)
=> ([(0,1)],2)
=> 2 = 1 + 1
([],2)
=> ([(0,2),(1,2)],3)
=> 3 = 2 + 1
([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 4 = 3 + 1
([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 4 = 3 + 1
([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 5 = 4 + 1
([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 6 = 5 + 1
([],6)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 7 = 6 + 1
Description
The Hosoya index of a graph.
This is the total number of matchings in the graph.
Matching statistic: St000915
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(load all 3 compositions to match this statistic)
Values
([],1)
=> ([(0,1)],2)
=> 2 = 1 + 1
([],2)
=> ([(0,2),(1,2)],3)
=> 3 = 2 + 1
([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 4 = 3 + 1
([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 4 = 3 + 1
([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 5 = 4 + 1
([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 6 = 5 + 1
([],6)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 7 = 6 + 1
Description
The Ore degree of a graph.
This is the maximal Ore degree of an edge, which is the sum of the degrees of its two endpoints.
Matching statistic: St000926
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(load all 3 compositions to match this statistic)
Values
([],1)
=> ([],2)
=> 2 = 1 + 1
([],2)
=> ([],3)
=> 3 = 2 + 1
([(0,1)],2)
=> ([(1,2)],3)
=> 4 = 3 + 1
([],3)
=> ([],4)
=> 4 = 3 + 1
([],4)
=> ([],5)
=> 5 = 4 + 1
([],5)
=> ([],6)
=> 6 = 5 + 1
([],6)
=> ([],7)
=> 7 = 6 + 1
Description
The clique-coclique number of a graph.
This is the product of the size of a maximal clique [[St000097]] and the size of a maximal independent set [[St000093]].
The following 352 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001391The disjunction number of a graph. St000008The major index of the composition. St000087The number of induced subgraphs. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000147The largest part of an integer partition. St000172The Grundy number of a graph. St000184The size of the centralizer of any permutation of given cycle type. St000228The size of a partition. St000286The number of connected components of the complement of a graph. St000363The number of minimal vertex covers of a graph. St000384The maximal part of the shifted composition of an integer partition. St000459The hook length of the base cell of a partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000469The distinguishing number of a graph. St000531The leading coefficient of the rook polynomial of an integer partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000636The hull number of a graph. St000667The greatest common divisor of the parts of the partition. St000722The number of different neighbourhoods in a graph. St000784The maximum of the length and the largest part of the integer partition. St000822The Hadwiger number of the graph. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000992The alternating sum of the parts of an integer partition. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001249Sum of the odd parts of a partition. 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. St001342The number of vertices in the center of a graph. St001360The number of covering relations in Young's lattice below a partition. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001389The number of partitions of the same length below the given integer partition. St001494The Alon-Tarsi number of a graph. 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. St001622The number of join-irreducible elements of a lattice. St001645The pebbling number of a connected graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001670The connected partition 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. St001725The harmonious chromatic number of a graph. St001746The coalition 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. St001963The tree-depth of a graph. St000063The number of linear extensions of a certain poset defined for an integer partition. St000108The number of partitions contained in the given partition. St000145The Dyson rank of a partition. St000171The degree of the graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000272The treewidth of a graph. St000310The minimal degree of a vertex 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. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000454The largest eigenvalue of a graph if it is integral. St000532The total number of rook placements on a Ferrers board. St000536The pathwidth of a graph. St000741The Colin de Verdière graph invariant. St000778The metric dimension of a graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. 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. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001358The largest degree of a regular subgraph of a graph. 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. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001400The total number of Littlewood-Richardson tableaux of given shape. St001644The dimension of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001814The number of partitions interlacing the given partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001949The rigidity index of a graph. St001962The proper pathwidth of a graph. St000438The position of the last up step in a Dyck path. St000012The area of a Dyck path. St000231Sum of the maximal elements of the blocks of a set partition. St000391The sum of the positions of the ones in a binary word. St000567The sum of the products of all pairs of parts. St000734The last entry in the first row of a standard tableau. St000874The position of the last double rise in a Dyck path. St000984The number of boxes below precisely one peak. St001228The vector space dimension of the space of module homomorphisms between J and itself when J denotes the Jacobson radical of the corresponding Nakayama algebra. St001254The vector space dimension of the first extension-group between A/soc(A) and J when A is the corresponding Nakayama algebra with Jacobson radical J. St001415The length of the longest palindromic prefix of a binary word. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001721The degree of a binary word. St000027The major index of a Dyck path. St000294The number of distinct factors of a binary word. St000395The sum of the heights of the peaks of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000518The number of distinct subsequences in a binary word. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000770The major index of an integer partition when read from bottom to top. St000979Half of MacMahon's equal index of a Dyck path. St001034The area of the parallelogram polyomino associated with the Dyck path. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001365The number of lattice paths of the same length weakly above the path given by a binary word. 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. St001808The box weight or horizontal decoration of a Dyck path. St001809The index of the step at the first peak of maximal height in a Dyck path. St000477The weight of a partition according to Alladi. St000878The number of ones minus the number of zeros of a binary word. St001838The number of nonempty primitive factors of a binary word. St001437The flex of a binary word. St001834The number of non-isomorphic minors of a graph. St001117The game chromatic index of a graph. St001827The number of two-component spanning forests of a graph. St000086The number of subgraphs. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St001311The cyclomatic number of a graph. St001341The number of edges in the center of a graph. St000537The cutwidth of a graph. St000678The number of up steps after the last double rise of a Dyck path. St000681The Grundy value of Chomp on Ferrers diagrams. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001345The Hamming dimension of a graph. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001812The biclique partition number of a graph. St000932The number of occurrences of the pattern UDU in a Dyck path. St000224The sorting index of a permutation. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St000841The largest opener of a perfect matching. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St000444The length of the maximal rise of a Dyck path. St000625The sum of the minimal distances to a greater element. St000654The first descent of a permutation. St000675The number of centered multitunnels of a Dyck path. St000702The number of weak deficiencies of a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000729The minimal arc length of a set partition. St000833The comajor index of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000868The aid statistic in the sense of Shareshian-Wachs. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St000925The number of topologically connected components of a set partition. St000990The first ascent of a permutation. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. 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. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001959The product of the heights of the peaks of a Dyck path. St000083The number of left oriented leafs of a binary tree except the first one. St000354The number of recoils of a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000539The number of odd inversions of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000653The last descent of a permutation. St000794The mak of a permutation. St000795The mad of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000831The number of indices that are either descents or recoils. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St000989The number of final rises of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001480The number of simple summands of the module J^2/J^3. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St001060The distinguishing index of a graph. St000639The number of relations in a poset. St000641The number of non-empty boolean intervals in a poset. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St000993The multiplicity of the largest part of an integer partition. 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. St000450The number of edges minus the number of vertices plus 2 of a graph. St000806The semiperimeter of the associated bargraph. 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. St001933The largest multiplicity of a part in an integer partition. St000656The number of cuts of a poset. St001176The size of a partition minus its first part. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000014The number of parking functions supported by a Dyck path. St000144The pyramid weight of the Dyck path. St000393The number of strictly increasing runs in a binary word. St000420The number of Dyck paths that are weakly above a Dyck path. St000519The largest length of a factor maximising the subword complexity. St000529The number of permutations whose descent word is the given binary word. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St000922The minimal number such that all substrings of this length are unique. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001267The length of the Lyndon factorization of the binary word. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St001658The total number of rook placements on a Ferrers board. St000010The length of the partition. St000026The position of the first return of a Dyck path. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000148The number of odd parts of a partition. St000160The multiplicity of the smallest part of a partition. St000288The number of ones in a binary word. St000290The major index of a binary word. St000293The number of inversions of a binary word. St000296The length of the symmetric border of a binary word. St000297The number of leading ones in a binary word. St000335The difference of lower and upper interactions. St000392The length of the longest run of ones in a binary word. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000443The number of long tunnels of a Dyck path. St000475The number of parts equal to 1 in a partition. St000548The number of different non-empty partial sums of an integer partition. St000627The exponent of a binary word. St000733The row containing the largest entry of a standard tableau. St000738The first entry in the last row of a standard tableau. St000876The number of factors in the Catalan decomposition of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000982The length of the longest constant subword. 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. St001007Number of simple modules with projective dimension 1 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. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001127The sum of the squares of the parts of a partition. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. 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. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. 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. St001372The length of a longest cyclic run of ones of a binary word. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001485The modular major index of a binary word. St001498The normalised height of a Nakayama algebra with magnitude 1. St001523The degree of symmetry of a Dyck path. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001884The number of borders of a binary word. St000295The length of the border of a binary word. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000377The dinv defect of an integer partition. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000921The number of internal inversions of a binary word. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. 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. St001091The number of parts in an integer partition whose next smaller part has the same size. 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. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001432The order dimension of the partition. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001570The minimal number of edges to add to make a graph Hamiltonian. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001948The number of augmented double ascents of a permutation. St001955The number of natural descents for set-valued two row standard Young tableaux. St000225Difference between largest and smallest parts in a partition. St001280The number of parts of an integer partition that are at least two. St000474Dyson's crank of a 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 $eA$. St000937The number of positive values of the symmetric group character corresponding to the partition. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000744The length of the path to the largest entry in a standard Young tableau. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001931The weak major index of an integer composition regarded as a word. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001621The number of atoms of a lattice. St000299The number of nonisomorphic vertex-induced subtrees. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001093The detour number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001716The 1-improper chromatic number of a graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St001271The competition number of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001512The minimum rank of a graph. St001638The book thickness of a graph. St000422The energy of a graph, if it is integral. St000381The largest part of an integer composition. St000767The number of runs in an integer composition. St000808The number of up steps of the associated bargraph. St000820The number of compositions obtained by rotating the composition. St000903The number of different parts of an integer composition. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001642The Prague dimension of a graph. St000766The number of inversions of an integer composition. St000815The number of semistandard Young tableaux of partition weight of given shape. St001423The number of distinct cubes in a binary word. St001524The degree of symmetry of a binary word. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders.
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