- FindStat

Your data matches 445 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000273
(load all 3 compositions to match this statistic)

Mapping

St000273: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => 1
([],2)
 => 2
([(0,1)],2)
 => 1
([],3)
 => 3
([(1,2)],3)
 => 2
([(0,2),(1,2)],3)
 => 1
([(0,1),(0,2),(1,2)],3)
 => 1
([],4)
 => 4
([(2,3)],4)
 => 3
([(1,3),(2,3)],4)
 => 2
([(0,3),(1,3),(2,3)],4)
 => 1
([(0,3),(1,2)],4)
 => 2
([(0,3),(1,2),(2,3)],4)
 => 2
([(1,2),(1,3),(2,3)],4)
 => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1

Description

The domination number of a graph. The domination number of a graph is given by the minimum size of a dominating set of vertices. A dominating set of vertices is a subset of the vertex set of such that every vertex is either in this subset or adjacent to an element of this subset.

Matching statistic: St001316
(load all 6 compositions to match this statistic)

Mapping

St001316: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => 1
([],2)
 => 1
([(0,1)],2)
 => 2
([],3)
 => 1
([(1,2)],3)
 => 1
([(0,2),(1,2)],3)
 => 2
([(0,1),(0,2),(1,2)],3)
 => 3
([],4)
 => 1
([(2,3)],4)
 => 1
([(1,3),(2,3)],4)
 => 1
([(0,3),(1,3),(2,3)],4)
 => 2
([(0,3),(1,2)],4)
 => 2
([(0,3),(1,2),(2,3)],4)
 => 2
([(1,2),(1,3),(2,3)],4)
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4

Description

The domatic number of a graph. This is the maximal size of a partition of the vertices into dominating sets.

Matching statistic: St001322
(load all 3 compositions to match this statistic)

Mapping

St001322: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => 1
([],2)
 => 2
([(0,1)],2)
 => 1
([],3)
 => 3
([(1,2)],3)
 => 2
([(0,2),(1,2)],3)
 => 1
([(0,1),(0,2),(1,2)],3)
 => 1
([],4)
 => 4
([(2,3)],4)
 => 3
([(1,3),(2,3)],4)
 => 2
([(0,3),(1,3),(2,3)],4)
 => 1
([(0,3),(1,2)],4)
 => 2
([(0,3),(1,2),(2,3)],4)
 => 2
([(1,2),(1,3),(2,3)],4)
 => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1

Description

The size of a minimal independent dominating set in a graph.

Matching statistic: St001339
(load all 3 compositions to match this statistic)

Mapping

St001339: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => 1
([],2)
 => 2
([(0,1)],2)
 => 1
([],3)
 => 3
([(1,2)],3)
 => 2
([(0,2),(1,2)],3)
 => 1
([(0,1),(0,2),(1,2)],3)
 => 1
([],4)
 => 4
([(2,3)],4)
 => 3
([(1,3),(2,3)],4)
 => 2
([(0,3),(1,3),(2,3)],4)
 => 1
([(0,3),(1,2)],4)
 => 2
([(0,3),(1,2),(2,3)],4)
 => 2
([(1,2),(1,3),(2,3)],4)
 => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1

Description

The irredundance number of a graph. A set

$S$ of vertices is irredundant, if there is no vertex in

$S$ , whose closed neighbourhood is contained in the union of the closed neighbourhoods of the other vertices of

$S$ . The irredundance number is the smallest size of a maximal irredundant set.

Matching statistic: St001829
(load all 3 compositions to match this statistic)

Mapping

St001829: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => 1
([],2)
 => 2
([(0,1)],2)
 => 1
([],3)
 => 3
([(1,2)],3)
 => 2
([(0,2),(1,2)],3)
 => 1
([(0,1),(0,2),(1,2)],3)
 => 1
([],4)
 => 4
([(2,3)],4)
 => 3
([(1,3),(2,3)],4)
 => 2
([(0,3),(1,3),(2,3)],4)
 => 1
([(0,3),(1,2)],4)
 => 2
([(0,3),(1,2),(2,3)],4)
 => 2
([(1,2),(1,3),(2,3)],4)
 => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1

Description

The common independence number of a graph. The common independence number of a graph

$G$ is the greatest integer

$r$ such that every vertex of

$G$ belongs to some independent set

$X$ of vertices of cardinality at least

$r$ .

Matching statistic: St000093
(load all 4 compositions to match this statistic)

Mapping

Mp00250: Graphs —clique graph⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => ([],1)
 => 1
([],2)
 => ([],2)
 => 2
([(0,1)],2)
 => ([],1)
 => 1
([],3)
 => ([],3)
 => 3
([(1,2)],3)
 => ([],2)
 => 2
([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1
([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => 1
([],4)
 => ([],4)
 => 4
([(2,3)],4)
 => ([],3)
 => 3
([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => 2
([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
([(0,3),(1,2)],4)
 => ([],2)
 => 2
([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 2
([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 1

Description

The cardinality of a maximal independent set of vertices of a graph. An independent set of a graph is a set of pairwise non-adjacent vertices. A maximum independent set is an independent set of maximum cardinality. This statistic is also called the independence number or stability number

$\alpha(G)$ of

$G$ .

Matching statistic: St000482
(load all 6 compositions to match this statistic)

Mapping

Mp00250: Graphs —clique graph⟶ Graphs
St000482: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => ([],1)
 => 1
([],2)
 => ([],2)
 => 2
([(0,1)],2)
 => ([],1)
 => 1
([],3)
 => ([],3)
 => 3
([(1,2)],3)
 => ([],2)
 => 2
([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1
([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => 1
([],4)
 => ([],4)
 => 4
([(2,3)],4)
 => ([],3)
 => 3
([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => 2
([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
([(0,3),(1,2)],4)
 => ([],2)
 => 2
([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 1
([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 1

Description

The (zero)-forcing number of a graph. This is the minimal number of vertices initially coloured black, such that eventually all vertices of the graph are coloured black when using the following rule: when

$u$ is a black vertex of

$G$ , and exactly one neighbour

$v$ of

$u$ is white, then colour

$v$ black.

Matching statistic: St000774
(load all 4 compositions to match this statistic)

Mapping

Mp00250: Graphs —clique graph⟶ Graphs
St000774: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => ([],1)
 => 1
([],2)
 => ([],2)
 => 2
([(0,1)],2)
 => ([],1)
 => 1
([],3)
 => ([],3)
 => 3
([(1,2)],3)
 => ([],2)
 => 2
([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1
([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => 1
([],4)
 => ([],4)
 => 4
([(2,3)],4)
 => ([],3)
 => 3
([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => 2
([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
([(0,3),(1,2)],4)
 => ([],2)
 => 2
([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 1
([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 1

Description

The maximal multiplicity of a Laplacian eigenvalue in a graph.

Matching statistic: St000786
(load all 4 compositions to match this statistic)

Mapping

Mp00250: Graphs —clique graph⟶ Graphs
St000786: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => ([],1)
 => 1
([],2)
 => ([],2)
 => 2
([(0,1)],2)
 => ([],1)
 => 1
([],3)
 => ([],3)
 => 3
([(1,2)],3)
 => ([],2)
 => 2
([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1
([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => 1
([],4)
 => ([],4)
 => 4
([(2,3)],4)
 => ([],3)
 => 3
([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => 2
([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
([(0,3),(1,2)],4)
 => ([],2)
 => 2
([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 2
([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 1

Description

The maximal number of occurrences of a colour in a proper colouring 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 largest part occurring in any of these partitions. 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

$[3,2,1]$ . Therefore, the statistic on this graph is

$3$ .

Matching statistic: St001286
(load all 2 compositions to match this statistic)

Mapping

Mp00250: Graphs —clique graph⟶ Graphs
St001286: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => ([],1)
 => 1
([],2)
 => ([],2)
 => 2
([(0,1)],2)
 => ([],1)
 => 1
([],3)
 => ([],3)
 => 3
([(1,2)],3)
 => ([],2)
 => 2
([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1
([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => 1
([],4)
 => ([],4)
 => 4
([(2,3)],4)
 => ([],3)
 => 3
([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => 2
([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
([(0,3),(1,2)],4)
 => ([],2)
 => 2
([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 2
([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 1

Description

The annihilation number of a graph. For a graph on

$m$ edges with degree sequence

$d_1\leq\dots\leq d_n$ , this is the largest number

$k\leq n$ such that

$\sum_{i=1}^k d_i \leq m$ .

The following 435 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St000258The burning number of a graph. St000310The minimal degree of a vertex of a graph. St000469The distinguishing number of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000160The multiplicity of the smallest part of a partition. St000172The Grundy number of a graph. St000261The edge connectivity of a graph. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000765The number of weak records in an integer composition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000808The number of up steps of the associated bargraph. St000822The Hadwiger number of the graph. St001029The size of the core of a graph. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001116The game chromatic number of a graph. St001330The hat guessing number of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001883The mutual visibility number of a graph. St001949The rigidity index of a graph. St001963The tree-depth of a graph. St000272The treewidth of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000536The pathwidth of a graph. St001270The bandwidth of a graph. St001277The degeneracy 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. St001323The independence gap of a graph. St001358The largest degree of a regular subgraph of a graph. St001587Half of the largest even part of an integer partition. St001644The dimension of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001962The proper pathwidth of a graph. St000087The number of induced subgraphs. St000120The number of left tunnels of a Dyck path. St000286The number of connected components of the complement of a graph. St000297The number of leading ones in a binary word. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000363The number of minimal vertex covers of a graph. St000617The number of global maxima of a Dyck path. St000636The hull number of a graph. St000722The number of different neighbourhoods in a graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000733The row containing the largest entry of a standard tableau. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000904The maximal number of repetitions of an integer composition. St000926The clique-coclique number of a graph. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module

$S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n−1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a special CNakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001342The number of vertices in the center of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001481The minimal height of a peak of a Dyck path. St001571The Cartan determinant of the integer partition. 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. 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. St001733The number of weak left to right maxima of a Dyck path. St001746The coalition number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St000171The degree of the graph. St000262The vertex connectivity of a graph. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St000326The position of the first one in a binary word after appending a 1 at the end. St000362The size of a minimal vertex cover of a graph. St000454The largest eigenvalue of a graph if it is integral. St000674The number of hills of a Dyck path. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000741The Colin de Verdière graph invariant. St000778The metric dimension of a graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. 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. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001172The number of 1-rises at odd height of a Dyck path. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. 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. St001233The number of indecomposable 2-dimensional modules with projective dimension one. 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. St001274The number of indecomposable injective modules with projective dimension equal to two. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001391The disjunction number of a graph. St001777The number of weak descents in an integer composition. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001812The biclique partition number of a graph. St000993The multiplicity of the largest part of an integer partition. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St000378The diagonal inversion number of an integer partition. St000667The greatest common divisor of the parts of the partition. St000903The number of different parts of an integer composition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001498The normalised height of a Nakayama algebra with magnitude 1. St001672The restrained domination 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. St000675The number of centered multitunnels of a Dyck path. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001471The magnitude of a Dyck path. St001933The largest multiplicity of a part in an integer partition. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St001716The 1-improper chromatic number of a graph. St000006The dinv of a Dyck path. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. 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

$c_{n−1}$ . St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. 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. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001568The smallest positive integer that does not appear twice in the partition. 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). St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000914The sum of the values of the Möbius function of a poset. St001281The normalized isoperimetric number of a graph. St000444The length of the maximal rise of a Dyck path. St000707The product of the factorials of the parts. St000770The major index of an integer partition when read from bottom to top. St000933The number of multipartitions of sizes given by an integer partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000420The number of Dyck paths that are weakly above a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001500The global dimension of magnitude 1 Nakayama algebras. St001621The number of atoms of a lattice. St001808The box weight or horizontal decoration of a Dyck path. St000260The radius of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001592The maximal number of simple paths between any two different vertices of a graph. St000259The diameter of a connected graph. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000934The 2-degree of an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000460The hook length of the last cell along the main diagonal of an integer partition. St000640The rank of the largest boolean interval in a poset. St000699The toughness times the least common multiple of 1,. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000910The number of maximal chains of minimal length in a poset. St001060The distinguishing index of a graph. St001118The acyclic chromatic index of a graph. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001360The number of covering relations in Young's lattice below a partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001389The number of partitions of the same length below the given integer partition. St001527The cyclic permutation representation number of an integer partition. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001890The maximum magnitude of the Möbius function of a poset. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000003The number of standard Young tableaux of the partition. St000005The bounce statistic of a Dyck path. St000010The length of the partition. St000014The number of parking functions supported by a Dyck path. St000015The number of peaks of a Dyck path. St000026The position of the first return of a Dyck path. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000038The product of the heights of the descending steps of a Dyck path. St000048The multinomial of the parts of a partition. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000063The number of linear extensions of a certain poset defined for an integer partition. St000079The number of alternating sign matrices for a given Dyck path. St000108The number of partitions contained in the given partition. St000144The pyramid weight of the Dyck path. St000147The largest part of an integer partition. St000159The number of distinct parts of the integer partition. St000179The product of the hook lengths of the integer partition. St000182The number of permutations whose cycle type is the given integer partition. St000184The size of the centralizer of any permutation of given cycle type. St000228The size of a partition. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000288The number of ones in a binary word. St000290The major index of a binary word. St000293The number of inversions of a binary word. St000294The number of distinct factors of a binary word. St000296The length of the symmetric border of a binary word. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000321The number of integer partitions of n that are dominated by an integer partition. St000335The difference of lower and upper interactions. St000345The number of refinements of a partition. St000346The number of coarsenings of a partition. St000384The maximal part of the shifted composition of an integer partition. St000392The length of the longest run of ones in a binary word. St000393The number of strictly increasing runs in a binary word. St000395The sum of the heights of the peaks of a Dyck path. 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. St000459The hook length of the base cell of a partition. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000517The Kreweras number of an integer partition. St000518The number of distinct subsequences in a binary word. St000519The largest length of a factor maximising the subword complexity. St000531The leading coefficient of the rook polynomial of an integer partition. St000532The total number of rook placements on a Ferrers board. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000548The number of different non-empty partial sums of an integer partition. St000627The exponent of a binary word. St000631The number of distinct palindromic decompositions of a binary word. St000655The length of the minimal rise of a Dyck path. St000681The Grundy value of Chomp on Ferrers diagrams. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000705The number of semistandard tableaux on a given integer partition of n with maximal entry n. St000734The last entry in the first row of a standard tableau. St000738The first entry in the last row of a standard tableau. St000753The Grundy value for the game of Kayles on a binary word. St000759The smallest missing part in an integer partition. St000762The sum of the positions of the weak records of an integer composition. St000783The side length of the largest staircase partition fitting into a partition. St000784The maximum of the length and the largest part of the integer partition. St000792The Grundy value for the game of ruler on a binary word. St000810The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to monomial symmetric functions. St000811The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to Schur symmetric functions. St000812The sum of the entries in the column specified by the partition of the change of basis matrix from complete homogeneous symmetric functions to monomial symmetric functions. 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. St000815The number of semistandard Young tableaux of partition weight of given shape. St000867The sum of the hook lengths in the first row of an integer partition. 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. St000922The minimal number such that all substrings of this length are unique. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000935The number of ordered refinements of an integer partition. St000947The major index east count of a Dyck path. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St000955Number of times one has

$Ext^i(D(A),A)>0$ for

$i>0$ for the corresponding LNakayama algebra. St000982The length of the longest constant subword. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension 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. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple 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. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001127The sum of the squares of the parts of a partition. St001129The product of the squares of the parts of a partition. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001161The major index north count of a 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. St001183The maximum of

$projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. 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

$g$ . 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. 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. 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. St001239The largest vector space dimension of the double dual of a simple module 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. 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. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001267The length of the Lyndon factorization of the binary word. 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. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. 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. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. 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. St001372The length of a longest cyclic run of ones of a binary word. St001378The product of the cohook lengths of the integer partition. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001400The total number of Littlewood-Richardson tableaux of given shape. 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. St001432The order dimension of the partition. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001437The flex of a binary word. St001462The number of factors of a standard tableaux under concatenation. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001485The modular major index of a binary word. St001488The number of corners of a skew partition. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001523The degree of symmetry of a Dyck path. St001530The depth of a Dyck path. 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. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001612The number of coloured multisets of cycles such that the multiplicities of colours are given by a partition. St001614The cyclic permutation representation number of a skew partition. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001721The degree of a binary word. St001732The number of peaks visible from the left. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. 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. St001809The index of the step at the first peak of maximal height in a Dyck path. St001814The number of partitions interlacing the given partition. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St001884The number of borders of a binary word. St001885The number of binary words with the same proper border set. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001955The number of natural descents for set-valued two row standard Young tableaux. St001959The product of the heights of the peaks of a Dyck path. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St001877Number of indecomposable injective modules with projective dimension 2. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000456The monochromatic index of a connected graph. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St000706The product of the factorials of the multiplicities of an integer partition. St001176The size of a partition minus its first part. 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

$eA$ . St001200The number of simple modules in

$eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001206The maximal dimension of an indecomposable projective

$eAe$ -module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ . St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St000284The Plancherel distribution on integer partitions. St000379The number of Hamiltonian cycles in a graph. St000464The Schultz index of a connected graph. St000478Another weight of a partition according to Alladi. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000928The sum of the coefficients of the character polynomial of an integer partition. St001128The exponens consonantiae of a partition. St001545The second Elser number of a connected graph. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000455The second largest eigenvalue of a graph if it is integral. 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. St000618The number of self-evacuating tableaux of given shape. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000781The number of proper colouring schemes of a Ferrers diagram. St000929The constant term of the character polynomial of an integer partition. St000941The number of characters of the symmetric group whose value on the partition is even. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St000422The energy of a graph, if it is integral. St001617The dimension of the space of valuations of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. 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. St001845The number of join irreducibles minus the rank of a lattice. St000418The number of Dyck paths that are weakly below a Dyck path. St000438The position of the last up step in a Dyck path. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000735The last entry on the main diagonal of a standard tableau. St000744The length of the path to the largest entry in a standard Young tableau. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001531Number of partial orders contained in the poset determined by the Dyck path. St001626The number of maximal proper sublattices of a lattice. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St001561The value of the elementary symmetric function evaluated at 1.