Your data matches 674 different statistics following compositions of up to 3 maps.
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St001327: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> 0
([],2)
=> 0
([(0,1)],2)
=> 0
([],3)
=> 0
([(1,2)],3)
=> 0
([(0,2),(1,2)],3)
=> 0
([(0,1),(0,2),(1,2)],3)
=> 0
([],4)
=> 0
([(2,3)],4)
=> 0
([(1,3),(2,3)],4)
=> 0
([(0,3),(1,3),(2,3)],4)
=> 0
([(0,3),(1,2)],4)
=> 1
([(0,3),(1,2),(2,3)],4)
=> 0
([(1,2),(1,3),(2,3)],4)
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
Description
The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. A graph is a split graph if and only if in any linear ordering of its vertices, there are no three vertices $a < b < c$ such that $(a,b)$ is an edge and $(b,c)$ is not an edge. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.
St000785: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> 1 = 0 + 1
([],2)
=> 1 = 0 + 1
([(0,1)],2)
=> 1 = 0 + 1
([],3)
=> 1 = 0 + 1
([(1,2)],3)
=> 1 = 0 + 1
([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
([],4)
=> 1 = 0 + 1
([(2,3)],4)
=> 2 = 1 + 1
([(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
([(0,3),(1,2)],4)
=> 1 = 0 + 1
([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
([(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> 1 = 0 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
Description
The number of distinct colouring schemes of a graph. To any proper colouring with the minimal number of colours possible we associate the integer partition recording how often each colour is used. This statistic records the number of distinct partitions that occur. For example, the graph on six vertices consisting of a square together with two attached triangles - ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) in the list of values - is three-colourable and admits two colouring schemes, $[2,2,2]$ and $[3,2,1]$.
Mp00276: Graphs to edge-partition of biconnected componentsInteger partitions
St000142: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> []
=> 0
([],2)
=> []
=> 0
([(0,1)],2)
=> [1]
=> 0
([],3)
=> []
=> 0
([(1,2)],3)
=> [1]
=> 0
([(0,2),(1,2)],3)
=> [1,1]
=> 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([],4)
=> []
=> 0
([(2,3)],4)
=> [1]
=> 0
([(1,3),(2,3)],4)
=> [1,1]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 0
([(0,3),(1,2)],4)
=> [1,1]
=> 0
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 0
([(1,2),(1,3),(2,3)],4)
=> [3]
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 1
Description
The number of even parts of a partition.
Mp00152: Graphs Laplacian multiplicitiesInteger compositions
St000768: Integer compositions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => 0
([],2)
=> [2] => 0
([(0,1)],2)
=> [1,1] => 0
([],3)
=> [3] => 0
([(1,2)],3)
=> [1,2] => 0
([(0,2),(1,2)],3)
=> [1,1,1] => 0
([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0
([],4)
=> [4] => 0
([(2,3)],4)
=> [1,3] => 0
([(1,3),(2,3)],4)
=> [1,1,2] => 0
([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => 1
([(0,3),(1,2)],4)
=> [2,2] => 0
([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => 0
([(1,2),(1,3),(2,3)],4)
=> [2,2] => 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => 0
Description
The number of peaks in an integer composition. A peak is an ascent followed by a descent, i.e., a subsequence $c_{i-1} c_i c_{i+1}$ with $c_i > \max(c_{i-1}, c_{i+1})$.
Mp00276: Graphs to edge-partition of biconnected componentsInteger partitions
St001092: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> []
=> 0
([],2)
=> []
=> 0
([(0,1)],2)
=> [1]
=> 0
([],3)
=> []
=> 0
([(1,2)],3)
=> [1]
=> 0
([(0,2),(1,2)],3)
=> [1,1]
=> 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([],4)
=> []
=> 0
([(2,3)],4)
=> [1]
=> 0
([(1,3),(2,3)],4)
=> [1,1]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 0
([(0,3),(1,2)],4)
=> [1,1]
=> 0
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 0
([(1,2),(1,3),(2,3)],4)
=> [3]
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 1
Description
The number of distinct even parts of a partition. See Section 3.3.1 of [1].
Mp00250: Graphs clique graphGraphs
St001311: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> 0
([],2)
=> ([],2)
=> 0
([(0,1)],2)
=> ([],1)
=> 0
([],3)
=> ([],3)
=> 0
([(1,2)],3)
=> ([],2)
=> 0
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> 0
([],4)
=> ([],4)
=> 0
([(2,3)],4)
=> ([],3)
=> 0
([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0
([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,3),(1,2)],4)
=> ([],2)
=> 0
([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 0
([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> 0
Description
The cyclomatic number of a graph. This is the minimum number of edges that must be removed from the graph so that the result is a forest. This is also the first Betti number of the graph. It can be computed as $c + m - n$, where $c$ is the number of connected components, $m$ is the number of edges and $n$ is the number of vertices.
Mp00250: Graphs clique graphGraphs
St001317: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> 0
([],2)
=> ([],2)
=> 0
([(0,1)],2)
=> ([],1)
=> 0
([],3)
=> ([],3)
=> 0
([(1,2)],3)
=> ([],2)
=> 0
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> 0
([],4)
=> ([],4)
=> 0
([(2,3)],4)
=> ([],3)
=> 0
([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0
([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,3),(1,2)],4)
=> ([],2)
=> 0
([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 0
([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> 0
Description
The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. A graph is a forest if and only if in any linear ordering of its vertices, there are no three vertices $a < b < c$ such that $(a,c)$ and $(b,c)$ are edges. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.
Mp00250: Graphs clique graphGraphs
St001331: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> 0
([],2)
=> ([],2)
=> 0
([(0,1)],2)
=> ([],1)
=> 0
([],3)
=> ([],3)
=> 0
([(1,2)],3)
=> ([],2)
=> 0
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> 0
([],4)
=> ([],4)
=> 0
([(2,3)],4)
=> ([],3)
=> 0
([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0
([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,3),(1,2)],4)
=> ([],2)
=> 0
([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 0
([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> 0
Description
The size of the minimal feedback vertex set. A feedback vertex set is a set of vertices whose removal results in an acyclic graph.
Mp00250: Graphs clique graphGraphs
St001335: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> 0
([],2)
=> ([],2)
=> 0
([(0,1)],2)
=> ([],1)
=> 0
([],3)
=> ([],3)
=> 0
([(1,2)],3)
=> ([],2)
=> 0
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> 0
([],4)
=> ([],4)
=> 0
([(2,3)],4)
=> ([],3)
=> 0
([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0
([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,3),(1,2)],4)
=> ([],2)
=> 0
([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 0
([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> 0
Description
The cardinality of a minimal cycle-isolating set of a graph. Let $\mathcal F$ be a set of graphs. A set of vertices $S$ is $\mathcal F$-isolating, if the subgraph induced by the vertices in the complement of the closed neighbourhood of $S$ does not contain any graph in $\mathcal F$. This statistic returns the cardinality of the smallest isolating set when $\mathcal F$ contains all cycles.
Mp00250: Graphs clique graphGraphs
St001521: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> 0
([],2)
=> ([],2)
=> 0
([(0,1)],2)
=> ([],1)
=> 0
([],3)
=> ([],3)
=> 0
([(1,2)],3)
=> ([],2)
=> 0
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> 0
([],4)
=> ([],4)
=> 0
([(2,3)],4)
=> ([],3)
=> 0
([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,3),(1,2)],4)
=> ([],2)
=> 0
([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1
([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> 0
Description
Half the total irregularity of a graph. This is half the sum of the absolute values of the degree differences of all pairs of vertices: $$ \frac{1}{4}\sum_{u,v} |d_u-d_v| $$ It is easy to show by induction on the number of edges that this is an integer.
The following 664 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001574The minimal number of edges to add or remove to make a graph regular. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001638The book thickness of a graph. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001736The total number of cycles in a graph. St001742The difference of the maximal and the minimal degree in a graph. St000388The number of orbits of vertices of a graph under automorphisms. St000897The number of different multiplicities of parts of an integer partition. St001282The number of graphs with the same chromatic polynomial. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000052The number of valleys of a Dyck path not on the x-axis. St000089The absolute variation of a composition. St000091The descent variation of a composition. St000143The largest repeated part of a partition. St000150The floored half-sum of the multiplicities of a partition. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000257The number of distinct parts of a partition that occur at least twice. 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. St000362The size of a minimal vertex cover of a graph. St000370The genus of a graph. St000387The matching number of a graph. St000535The rank-width of a graph. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St000671The maximin edge-connectivity for choosing a subgraph. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. 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. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000761The number of ascents in an integer composition. St000766The number of inversions of an integer composition. St000769The major index of a composition regarded as a word. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. St001175The size of a partition minus the hook length of the base cell. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001270The bandwidth of a graph. St001271The competition number of a graph. St001277The degeneracy of a graph. St001307The number of induced stars on four vertices in a graph. St001309The number of four-cliques in a graph. St001323The independence gap of a graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001350Half of the Albertson index of a graph. St001353The number of prime nodes in the modular decomposition of a graph. St001354The number of series nodes in the modular decomposition of a graph. St001358The largest degree of a regular subgraph of a graph. St001393The induced matching number of a graph. St001395The number of strictly unfriendly partitions of a graph. St001578The minimal number of edges to add or remove to make a graph a line graph. St001584The area statistic between a Dyck path and its bounce path. St001644The dimension of a graph. St001673The degree of asymmetry of an integer composition. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001743The discrepancy of a graph. St001792The arboricity of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St001812The biclique partition number of a graph. St001871The number of triconnected components of a graph. St001962The proper pathwidth of a graph. St000047The number of standard immaculate tableaux of a given shape. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000172The Grundy number of a graph. St000183The side length of the Durfee square of an integer partition. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000273The domination number of a graph. St000277The number of ribbon shaped standard tableaux. St000286The number of connected components of the complement of a graph. St000383The last part of an integer composition. St000482The (zero)-forcing number of a graph. St000657The smallest part of an integer composition. St000722The number of different neighbourhoods in a graph. St000758The length of the longest staircase fitting into an integer composition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000765The number of weak records in an integer composition. St000767The number of runs in an integer composition. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000816The number of standard composition tableaux of the composition. St000820The number of compositions obtained by rotating the composition. St000822The Hadwiger number of the graph. St000899The maximal number of repetitions of an integer composition. St000903The number of different parts of an integer composition. St000905The number of different multiplicities of parts of an integer composition. St000913The number of ways to refine the partition into singletons. St000916The packing number of a graph. St000917The open packing number of a graph. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001109The number of proper colourings of a graph with as few colours as possible. St001111The weak 2-dynamic chromatic number of a graph. St001112The 3-weak dynamic number of a graph. St001116The game chromatic number of a graph. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001261The Castelnuovo-Mumford regularity of a graph. St001286The annihilation number of a graph. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. 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. St001322The size of a minimal independent dominating set in a graph. St001330The hat guessing number of a graph. St001339The irredundance number of a graph. St001352The number of internal nodes in the modular decomposition of a graph. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001494The Alon-Tarsi number of a graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001591The number of graphs with the given composition of multiplicities of Laplacian eigenvalues. St001642The Prague dimension of a graph. St001670The connected partition number of a graph. St001672The restrained domination number of a graph. St001716The 1-improper chromatic number of a graph. St001722The number of minimal chains with small intervals between a binary word and the top element. St001734The lettericity of 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. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001829The common independence number of a graph. St001883The mutual visibility number of a graph. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001963The tree-depth of a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St000008The major index of the composition. St000017The number of inversions of a standard tableau. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000057The Shynar inversion number of a standard tableau. St000090The variation of a composition. St000095The number of triangles of a graph. St000121The number of occurrences of the contiguous pattern [.,[.,[.,[.,.]]]] in a binary tree. St000122The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]] in a binary tree. St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000149The number of cells of the partition whose leg is zero and arm is odd. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000185The weighted size of a partition. St000217The number of occurrences of the pattern 312 in a permutation. St000225Difference between largest and smallest parts in a partition. St000252The number of nodes of degree 3 of a binary tree. St000256The number of parts from which one can substract 2 and still get an integer partition. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000317The cycle descent number of a permutation. St000322The skewness of a graph. St000323The minimal crossing number of a graph. St000358The number of occurrences of the pattern 31-2. St000366The number of double descents of a permutation. St000367The number of simsun double descents of a permutation. St000369The dinv deficit of a Dyck path. St000377The dinv defect of an integer partition. St000386The number of factors DDU in a Dyck path. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000405The number of occurrences of the pattern 1324 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St000516The number of stretching pairs of a permutation. St000547The number of even non-empty partial sums of an integer partition. St000552The number of cut vertices of a graph. St000562The number of internal points of a set partition. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000648The number of 2-excedences of a permutation. St000650The number of 3-rises of a permutation. St000660The number of rises of length at least 3 of a Dyck path. St000661The number of rises of length 3 of a Dyck path. St000664The number of right ropes of a permutation. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000807The sum of the heights of the valleys of the associated bargraph. St000871The number of very big ascents of a permutation. St000921The number of internal inversions of a binary word. St000931The number of occurrences of the pattern UUU in a Dyck path. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three 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. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001214The aft of an integer partition. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. 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. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001263The index of the maximal parabolic seaweed algebra associated with the composition. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001310The number of induced diamond graphs in a graph. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001319The minimal number of occurrences of the star-pattern in a linear ordering of the vertices of the graph. St001320The minimal number of occurrences of the path-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. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001394The genus of a permutation. St001411The number of patterns 321 or 3412 in a permutation. St001423The number of distinct cubes in a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001470The cyclic holeyness of a permutation. St001513The number of nested exceedences of a permutation. St001520The number of strict 3-descents. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001556The number of inversions of the third entry of a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001565The number of arithmetic progressions of length 2 in a permutation. 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. St001586The number of odd parts smaller than the largest even part in an integer partition. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001596The number of two-by-two squares inside a skew partition. St001649The length of a longest trail in a graph. St001689The number of celebrities in a graph. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001705The number of occurrences of the pattern 2413 in a permutation. St001712The number of natural descents of a standard Young tableau. St001727The number of invisible inversions of a permutation. St001728The number of invisible descents of a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001777The number of weak descents in an integer composition. St001797The number of overfull subgraphs of a graph. St001856The number of edges in the reduced word graph of a permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001931The weak major index of an integer composition regarded as a word. St001932The number of pairs of singleton blocks in the noncrossing set partition corresponding to a Dyck path, that can be merged to create another noncrossing set partition. St000001The number of reduced words for a permutation. St000003The number of standard Young tableaux of the partition. St000021The number of descents of a permutation. St000024The number of double up and double down steps of a Dyck path. St000035The number of left outer peaks of a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000053The number of valleys of the Dyck path. St000124The cardinality of the preimage of the Simion-Schmidt map. St000159The number of distinct parts of the integer partition. St000179The product of the hook lengths of the integer partition. St000184The size of the centralizer of any permutation of given cycle type. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000306The bounce count of a Dyck path. St000321The number of integer partitions of n that are dominated by an integer partition. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000340The number of non-final maximal constant sub-paths of length greater than one. St000345The number of refinements of a partition. St000346The number of coarsenings of a partition. St000352The Elizalde-Pak rank of a permutation. St000382The first part of an integer composition. St000442The maximal area to the right of an up step of a Dyck path. St000531The leading coefficient of the rook polynomial of an integer partition. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000617The number of global maxima of a Dyck path. St000628The balance of a binary word. St000655The length of the minimal rise of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000662The staircase size of the code of a permutation. St000667The greatest common divisor of the parts of the partition. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000783The side length of the largest staircase partition fitting into a partition. St000805The number of peaks 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. St000834The number of right outer peaks of a permutation. St000847The number of standard Young tableaux whose descent set is the binary word. St000884The number of isolated descents of a permutation. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St000904The maximal number of repetitions of an integer composition. St000920The logarithmic height of a Dyck path. St000935The number of ordered refinements of an integer partition. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St000968We 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 $c_{n−1}$. St001006Number of simple modules with projective dimension equal to the global dimension of 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. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001052The length of the exterior of a permutation. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001096The size of the overlap set of a permutation. St001102The number of words with multiplicities of the letters given by the composition, avoiding the consecutive pattern 132. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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 $eA$. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001312Number of parabolic noncrossing partitions indexed by the composition. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001432The order dimension of the partition. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001481The minimal height of a peak of a Dyck path. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001595The number of standard Young tableaux of the skew partition. St001597The Frobenius rank of a skew partition. St001612The number of coloured multisets of cycles such that the multiplicities of colours are given by a partition. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001675The number of parts equal to the part in the reversed composition. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. St001729The number of visible descents of a permutation. St001732The number of peaks visible from the left. St001737The number of descents of type 2 in a permutation. St001780The order of promotion on the set of standard tableaux of given shape. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001928The number of non-overlapping descents in a permutation. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000013The height of a Dyck path. St000015The number of peaks of a Dyck path. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000325The width of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama 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: 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. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001471The magnitude of a Dyck path. St001530The depth of a Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001964The interval resolution global dimension of a poset. St000454The largest eigenvalue of a graph if it is integral. St001305The number of induced cycles on four vertices in a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001765The number of connected components of the friends and strangers graph. St000376The bounce deficit of a Dyck path. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000711The number of big exceedences of a permutation. St000732The number of double deficiencies of a permutation. St000872The number of very big descents of a permutation. St000881The number of short braid edges in the graph of braid moves of a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001141The number of occurrences of hills of size 3 in a Dyck path. St001731The factorization defect of a permutation. St000570The Edelman-Greene number of a permutation. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St001162The minimum jump of a permutation. St001220The width of a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000842The breadth of a permutation. 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 $eA$. St000422The energy of a graph, if it is integral. St001846The number of elements which do not have a complement in the lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. 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 $eA$. St001845The number of join irreducibles minus the rank of a lattice. St000069The number of maximal elements of a poset. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000379The number of Hamiltonian cycles in a graph. St001176The size of a partition minus its first part. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001961The sum of the greatest common divisors of all pairs of parts. St000010The length of the partition. St000012The area of a Dyck path. 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. St000292The number of ascents of a binary word. St000295The length of the border of a binary word. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000378The diagonal inversion number of an integer partition. St000384The maximal part of the shifted composition of an integer partition. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000447The number of pairs of vertices of a graph with distance 3. St000448The number of pairs of vertices of a graph with distance 2. St000449The number of pairs of vertices of a graph with distance 4. St000459The hook length of the base cell of a partition. St000475The number of parts equal to 1 in a partition. St000519The largest length of a factor maximising the subword complexity. St000548The number of different non-empty partial sums of an integer partition. St000549The number of odd 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. St000869The sum of the hook lengths of an integer partition. St000877The depth of the binary word interpreted as a path. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St000944The 3-degree 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. St001091The number of parts in an integer partition whose next smaller part has the same size. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001127The sum of the squares of the parts of a partition. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001280The number of parts of an integer partition that are at least two. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001308The number of induced paths on three vertices in a graph. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001371The length of the longest Yamanouchi prefix of a binary word. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001413Half the length of the longest even length palindromic prefix of a binary word. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001424The number of distinct squares in a binary word. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001484The number of singletons of an integer partition. St001524The degree of symmetry of a binary word. St001541The Gini index of an integer partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001692The number of vertices with higher degree than the average degree in a graph. St001697The shifted natural comajor index of a standard Young tableau. St001721The degree of a binary word. St001764The number of non-convex subsets of vertices in a graph. St001793The difference between the clique number and the chromatic number of a graph. St001799The number of proper separations of a graph. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001930The weak major index of a binary word. St000068The number of minimal elements in a poset. St000553The number of blocks of a graph. St000775The multiplicity of the largest eigenvalue in a graph. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001739The number of graphs with the same edge polytope as the given graph. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000016The number of attacking pairs of a standard tableau. St000120The number of left tunnels of a Dyck path. St000171The degree of the graph. St000263The Szeged index of a graph. St000265The Wiener index of a graph. St000274The number of perfect matchings of a graph. St000291The number of descents of a binary word. St000331The number of upper interactions of a Dyck path. St000361The second Zagreb index of a graph. St000478Another weight of a partition according to Alladi. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000674The number of hills of a Dyck path. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000936The number of even values of the symmetric group character corresponding to the partition. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001071The beta invariant of the graph. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. 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. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001139The number of occurrences of hills of size 2 in a Dyck path. St001161The major index north count of a Dyck path. St001185The number of indecomposable injective modules of grade at least 2 in 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. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. 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. St001341The number of edges in the center of a graph. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001357The maximal degree of a regular spanning subgraph of a graph. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001479The number of bridges of a graph. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001512The minimum rank of a graph. St001647The number of edges that can be added without increasing the clique number. St001648The number of edges that can be added without increasing the chromatic number. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001783The number of odd automorphisms of a graph. St001826The maximal number of leaves on a vertex of a graph. St001869The maximum cut size of a graph. St001910The height of the middle non-run of a Dyck path. St000117The number of centered tunnels of a Dyck path. St000260The radius of a connected graph. St001306The number of induced paths on four vertices in a graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001577The minimal number of edges to add or remove to make a graph a cograph. St000259The diameter of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001204Call 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. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000455The second largest eigenvalue of a graph if it is integral. St001301The first Betti number of the order complex associated with the poset. St000181The number of connected components of the Hasse diagram for the poset. St000908The length of the shortest maximal antichain in a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000929The constant term of the character polynomial of an integer partition. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St000567The sum of the products of all pairs of parts. St001099The 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 leaf labelled binary trees. St001100The 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 leaf labelled trees. 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. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000658The number of rises of length 2 of a Dyck path. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000693The modular (standard) major index of a standard tableau. St000874The position of the last double rise in a Dyck path. St000934The 2-degree of an integer partition. St000946The sum of the skew hook positions in a Dyck path. St000976The sum of the positions of double up-steps of a Dyck path. St000984The number of boxes below precisely one peak. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001281The normalized isoperimetric number of a graph. St001480The number of simple summands of the module J^2/J^3. St000287The number of connected components of a graph. St001272The number of graphs with the same degree sequence. St001496The number of graphs with the same Laplacian spectrum as the given graph. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000947The major index east count of a Dyck path. St001498The normalised height of a Nakayama algebra with magnitude 1. 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. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000699The toughness times the least common multiple of 1,. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. 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. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001095The number of non-isomorphic posets with precisely one further covering relation. 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. St000347The inversion sum of a binary word. St000348The non-inversion sum of a binary word. St000629The defect of a binary word. St000682The Grundy value of Welter's game on a binary word. St000691The number of changes of a binary word. St000995The largest even part of an integer partition. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001485The modular major index of a binary word. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001699The major index of a standard tableau minus the weighted size of its shape. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001651The Frankl number of a lattice. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000137The Grundy value of an integer partition. St000145The Dyson rank of a partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001383The BG-rank of an 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. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. 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. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000782The indicator function of whether a given perfect matching is an L & P matching. St001720The minimal length of a chain of small intervals in a lattice. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd.