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Your data matches 730 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000050
(load all 14 compositions to match this statistic)

Mapping

St000050: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[.,.]
 => 1
[.,[.,.]]
 => 2
[[.,.],.]
 => 2
[.,[.,[.,.]]]
 => 3
[.,[[.,.],.]]
 => 3
[[.,.],[.,.]]
 => 2
[[.,[.,.]],.]
 => 3
[[[.,.],.],.]
 => 3

Description

The depth or height of a binary tree. The depth (or height) of a binary tree is the maximal depth (or height) of one of its vertices. The '''height''' of a vertex is the number of edges on the longest path between that node and a leaf. The '''depth''' of a vertex is the number of edges from the vertex to the root. See [1] and [2] for this terminology. The depth (or height) of a tree

$T$ can be recursively defined:

$\operatorname{depth}(T) = 0$ if

$T$ is empty and

$\operatorname{depth}(T) = 1 + max(\operatorname{depth}(L),\operatorname{depth}(R))$ if

$T$ is nonempty with left and right subtrees

$L$ and

$R$ , respectively. The upper and lower bounds on the depth of a binary tree

$T$ of size

$n$ are

$log_2(n) \leq \operatorname{depth}(T) \leq n$ .

Matching statistic: St001554
(load all 13 compositions to match this statistic)

Mapping

St001554: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[.,.]
 => 1
[.,[.,.]]
 => 2
[[.,.],.]
 => 2
[.,[.,[.,.]]]
 => 3
[.,[[.,.],.]]
 => 3
[[.,.],[.,.]]
 => 2
[[.,[.,.]],.]
 => 3
[[[.,.],.],.]
 => 3

Description

The number of distinct nonempty subtrees of a binary tree.

Matching statistic: St000144
(load all 21 compositions to match this statistic)

Mapping

Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000144: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[.,.]
 => [1,0]
 => 1
[.,[.,.]]
 => [1,1,0,0]
 => 2
[[.,.],.]
 => [1,0,1,0]
 => 2
[.,[.,[.,.]]]
 => [1,1,1,0,0,0]
 => 3
[.,[[.,.],.]]
 => [1,1,0,1,0,0]
 => 2
[[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => 3
[[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => 3
[[[.,.],.],.]
 => [1,0,1,0,1,0]
 => 3

Description

The pyramid weight of the Dyck path. The pyramid weight of a Dyck path is the sum of the lengths of the maximal pyramids (maximal sequences of the form

$1^h0^h$ ) in the path. Maximal pyramids are called lower interactions by Le Borgne [2], see [[St000331]] and [[St000335]] for related statistics.

Matching statistic: St000166
(load all 2 compositions to match this statistic)

Mapping

Mp00008: Binary trees —to complete tree⟶ Ordered trees
St000166: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[.,.]
 => [[],[]]
 => 1
[.,[.,.]]
 => [[],[[],[]]]
 => 2
[[.,.],.]
 => [[[],[]],[]]
 => 2
[.,[.,[.,.]]]
 => [[],[[],[[],[]]]]
 => 3
[.,[[.,.],.]]
 => [[],[[[],[]],[]]]
 => 3
[[.,.],[.,.]]
 => [[[],[]],[[],[]]]
 => 2
[[.,[.,.]],.]
 => [[[],[[],[]]],[]]
 => 3
[[[.,.],.],.]
 => [[[[],[]],[]],[]]
 => 3

Description

The depth minus 1 of an ordered tree. The ordered trees of size

$n$ are bijection with the Dyck paths of size

$n-1$ , and this statistic then corresponds to [[St000013]].

Matching statistic: St000522
(load all 2 compositions to match this statistic)

Mapping

Mp00008: Binary trees —to complete tree⟶ Ordered trees
St000522: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[.,.]
 => [[],[]]
 => 1
[.,[.,.]]
 => [[],[[],[]]]
 => 2
[[.,.],.]
 => [[[],[]],[]]
 => 2
[.,[.,[.,.]]]
 => [[],[[],[[],[]]]]
 => 3
[.,[[.,.],.]]
 => [[],[[[],[]],[]]]
 => 3
[[.,.],[.,.]]
 => [[[],[]],[[],[]]]
 => 2
[[.,[.,.]],.]
 => [[[],[[],[]]],[]]
 => 3
[[[.,.],.],.]
 => [[[[],[]],[]],[]]
 => 3

Description

The number of 1-protected nodes of a rooted tree. This is the number of nodes with minimal distance one to a leaf.

Matching statistic: St000528
(load all 5 compositions to match this statistic)

Mapping

Mp00013: Binary trees —to poset⟶ Posets
St000528: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The height of a poset. This equals the rank of the poset [[St000080]] plus one.

Matching statistic: St000725
(load all 31 compositions to match this statistic)

Mapping

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000725: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[.,.]
 => [1] => 1
[.,[.,.]]
 => [2,1] => 2
[[.,.],.]
 => [1,2] => 2
[.,[.,[.,.]]]
 => [3,2,1] => 3
[.,[[.,.],.]]
 => [2,3,1] => 3
[[.,.],[.,.]]
 => [1,3,2] => 3
[[.,[.,.]],.]
 => [2,1,3] => 2
[[[.,.],.],.]
 => [1,2,3] => 3

Description

The smallest label of a leaf of the increasing binary tree associated to a permutation.

Matching statistic: St000863
(load all 29 compositions to match this statistic)

Mapping

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000863: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[.,.]
 => [1] => 1
[.,[.,.]]
 => [2,1] => 2
[[.,.],.]
 => [1,2] => 2
[.,[.,[.,.]]]
 => [3,2,1] => 3
[.,[[.,.],.]]
 => [2,3,1] => 3
[[.,.],[.,.]]
 => [3,1,2] => 2
[[.,[.,.]],.]
 => [2,1,3] => 3
[[[.,.],.],.]
 => [1,2,3] => 3

Description

The length of the first row of the shifted shape of a permutation. The diagram of a strict partition

$\lambda_1 < \lambda_2 < \dots < \lambda_\ell$ of

$n$ is a tableau with

$\ell$ rows, the

$i$ -th row being indented by

$i$ cells. A shifted standard Young tableau is a filling of such a diagram, where entries in rows and columns are strictly increasing. The shifted Robinson-Schensted algorithm [1] associates to a permutation a pair

$(P, Q)$ of standard shifted Young tableaux of the same shape, where off-diagonal entries in

$Q$ may be circled. This statistic records the length of the first row of

$P$ and

$Q$ .

Matching statistic: St000912
(load all 4 compositions to match this statistic)

Mapping

Mp00013: Binary trees —to poset⟶ Posets
St000912: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of maximal antichains in a poset.

Matching statistic: St000975
(load all 9 compositions to match this statistic)

Mapping

Mp00010: Binary trees —to ordered tree: left child = left brother⟶ Ordered trees
St000975: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[.,.]
 => [[]]
 => 1
[.,[.,.]]
 => [[[]]]
 => 2
[[.,.],.]
 => [[],[]]
 => 2
[.,[.,[.,.]]]
 => [[[[]]]]
 => 3
[.,[[.,.],.]]
 => [[[],[]]]
 => 3
[[.,.],[.,.]]
 => [[],[[]]]
 => 3
[[.,[.,.]],.]
 => [[[]],[]]
 => 3
[[[.,.],.],.]
 => [[],[],[]]
 => 2

Description

The length of the boundary minus the length of the trunk of an ordered tree. This is the size of the set of edges which are either on the left most path or on the right most path from the root.

The following 720 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001343The dimension of the reduced incidence algebra of a poset. St001717The largest size of an interval in a poset. St000080The rank of the poset. St000094The depth of an ordered tree. St000521The number of distinct subtrees of an ordered tree. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St000013The height of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000056The decomposition (or block) number of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000093The cardinality of a maximal independent set of vertices of a graph. St000147The largest part of an integer partition. St000153The number of adjacent cycles of a permutation. St000184The size of the centralizer of any permutation of given cycle type. St000203The number of external nodes of a binary tree. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000213The number of weak exceedances (also weak excedences) of a permutation. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000240The number of indices that are not small excedances. St000258The burning number of a graph. St000273The domination number of a graph. St000287The number of connected components of a graph. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000362The size of a minimal vertex cover of a graph. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000384The maximal part of the shifted composition of an integer partition. St000387The matching number of a graph. St000469The distinguishing number of a graph. St000482The (zero)-forcing number of a graph. St000544The cop number of a graph. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000553The number of blocks of a graph. St000568The hook number of a binary tree. St000636The hull number of a graph. St000740The last entry of a permutation. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000784The maximum of the length and the largest part of the integer partition. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000808The number of up steps of the associated bargraph. St000907The number of maximal antichains of minimal length in a poset. St000916The packing number of a graph. St000935The number of ordered refinements of an integer partition. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000991The number of right-to-left minima of a permutation. St001058The breadth of the ordered tree. St001183The maximum of

$projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001245The cyclic maximal difference between two consecutive entries of a permutation. St001246The maximal difference between two consecutive entries of a permutation. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001286The annihilation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001389The number of partitions of the same length below the given integer partition. St001461The number of topologically connected components of the chord diagram of a permutation. St001463The number of distinct columns in the nullspace of a graph. St001497The position of the largest weak excedence of a permutation. St001566The length of the longest arithmetic progression in a permutation. St001616The number of neutral elements in a lattice. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001720The minimal length of a chain of small intervals in a lattice. St001746The coalition number of a graph. St001757The number of orbits of toric promotion on a graph. St001820The size of the image of the pop stack sorting operator. St001828The Euler characteristic of a graph. St001829The common independence number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St000234The number of global ascents of a permutation. St000245The number of ascents of a permutation. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000442The maximal area to the right of an up step of a Dyck path. St000474Dyson's crank of a partition. St000672The number of minimal elements in Bruhat order not less than the permutation. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000778The metric dimension of a graph. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001096The size of the overlap set of a permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St001340The cardinality of a minimal non-edge isolating set of a graph. St001405The number of bonds in a permutation. St001581The achromatic number of a graph. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001619The number of non-isomorphic sublattices of a lattice. St001622The number of join-irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St001666The number of non-isomorphic subposets of a lattice which are lattices. St001670The connected partition number of a graph. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001949The rigidity index of a graph. St001958The degree of the polynomial interpolating the values of a permutation. St000003The number of standard Young tableaux of the partition. St000007The number of saliances of the permutation. St000010The length of the partition. St000011The number of touch points (or returns) of a Dyck path. St000015The number of peaks of a Dyck path. St000054The first entry of the permutation. St000058The order of a permutation. St000060The greater neighbor of the maximum. St000075The orbit size of a standard tableau under promotion. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000099The number of valleys of a permutation, including the boundary. St000105The number of blocks in the set partition. St000110The number of permutations less than or equal to a permutation in left weak order. St000134The size of the orbit of an alternating sign matrix under gyration. St000141The maximum drop size of a permutation. St000160The multiplicity of the smallest part of a partition. St000171The degree of the graph. St000172The Grundy number of a graph. St000189The number of elements in the poset. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000209Maximum difference of elements in cycles. St000210Minimum over maximum difference of elements in cycles. St000216The absolute length of a permutation. St000225Difference between largest and smallest parts in a partition. St000271The chromatic index of a graph. St000286The number of connected components of the complement of a graph. St000306The bounce count of a Dyck path. St000316The number of non-left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000346The number of coarsenings of a partition. St000377The dinv defect of an integer partition. St000378The diagonal inversion number of an integer partition. St000383The last part of an integer composition. St000392The length of the longest run of ones in a binary word. St000393The number of strictly increasing runs in a binary word. St000444The length of the maximal rise of a Dyck path. St000451The length of the longest pattern of the form k 1 2. St000470The number of runs in a permutation. St000475The number of parts equal to 1 in a partition. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000501The size of the first part in the decomposition of a permutation. St000507The number of ascents of a standard tableau. St000542The number of left-to-right-minima of a permutation. St000619The number of cyclic descents of a permutation. St000626The minimal period of a binary word. St000628The balance of a binary word. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000653The last descent of a permutation. St000676The number of odd rises of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000702The number of weak deficiencies of a permutation. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000722The number of different neighbourhoods in a graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000734The last entry in the first row of a standard tableau. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000780The size of the orbit under rotation of a perfect matching. St000785The number of distinct colouring schemes of a graph. St000822The Hadwiger number of the graph. St000831The number of indices that are either descents or recoils. St000839The largest opener of a set partition. St000877The depth of the binary word interpreted as a path. St000883The number of longest increasing subsequences of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000891The number of distinct diagonal sums of a permutation matrix. St000911The number of maximal antichains of maximal size in a poset. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000942The number of critical left to right maxima of the parking functions. St000945The number of matchings in the dihedral orbit of a perfect matching. St000955Number of times one has

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

$i>0$ for the corresponding LNakayama algebra. St000956The maximal displacement of a permutation. St000982The length of the longest constant subword. St000988The orbit size of a permutation under Foata's bijection. St000996The number of exclusive left-to-right maxima of a permutation. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001090The number of pop-stack-sorts needed to sort a permutation. St001112The 3-weak dynamic number of a graph. St001114The number of odd descents of a permutation. St001116The game chromatic number of a graph. St001118The acyclic chromatic index of a graph. St001120The length of a longest path in a graph. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001197The global dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n−1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a special CNakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n-1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a Dyck path as follows: 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

$eA$ . St001207The Lowey length of the algebra

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001263The index of the maximal parabolic seaweed algebra associated with the composition. St001267The length of the Lyndon factorization of the binary word. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001315The dissociation number of a graph. St001316The domatic number of a graph. St001330The hat guessing number of a graph. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001349The number of different graphs obtained from the given graph by removing an edge. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001372The length of a longest cyclic run of ones of a binary word. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001391The disjunction number of a graph. St001415The length of the longest palindromic prefix of a binary word. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001437The flex of a binary word. St001439The number of even weak deficiencies and of odd weak exceedences. St001462The number of factors of a standard tableaux under concatenation. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001471The magnitude of a Dyck path. St001486The number of corners of the ribbon associated with an integer composition. St001494The Alon-Tarsi number of a graph. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001512The minimum rank of a graph. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001517The length of a longest pair of twins in a permutation. St001530The depth of a Dyck path. St001580The acyclic chromatic number of a graph. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001642The Prague dimension of a graph. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001667The maximal size of a pair of weak twins for a permutation. St001726The number of visible inversions of a permutation. St001733The number of weak left to right maxima of a Dyck path. St001734The lettericity of a graph. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001780The order of promotion on the set of standard tableaux of given shape. St001807The lower middle entry of a permutation. St001812The biclique partition number of a graph. St001863The number of weak excedances of a signed permutation. 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). St001883The mutual visibility number of a graph. St001889The size of the connectivity set of a signed permutation. St001904The length of the initial strictly increasing segment of a parking function. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001917The order of toric promotion on the set of labellings of a graph. St001928The number of non-overlapping descents in a permutation. St001933The largest multiplicity of a part in an integer partition. St001937The size of the center of a parking function. St001963The tree-depth of a graph. St000004The major index of a permutation. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000021The number of descents of a permutation. St000023The number of inner peaks of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000052The number of valleys of a Dyck path not on the x-axis. St000053The number of valleys of the Dyck path. St000070The number of antichains in a poset. St000104The number of facets in the order polytope of this poset. St000151The number of facets in the chain polytope of the poset. St000155The number of exceedances (also excedences) of a permutation. St000156The Denert index of a permutation. St000224The sorting index of a permutation. St000235The number of indices that are not cyclical small weak excedances. St000237The number of small exceedances. St000238The number of indices that are not small weak excedances. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000272The treewidth of a graph. St000291The number of descents of a binary word. St000293The number of inversions of a binary word. St000299The number of nonisomorphic vertex-induced subtrees. St000310The minimal degree of a vertex of a graph. St000326The position of the first one in a binary word after appending a 1 at the end. St000331The number of upper interactions of a Dyck path. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000338The number of pixed points of a permutation. St000339The maf index of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000386The number of factors DDU in a Dyck path. St000395The sum of the heights of the peaks of a Dyck path. St000441The number of successions of a permutation. St000459The hook length of the base cell of a partition. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St000546The number of global descents of a permutation. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000646The number of big ascents of a permutation. St000647The number of big descents of a permutation. St000659The number of rises of length at least 2 of a Dyck path. St000662The staircase size of the code of a permutation. St000670The reversal length of a permutation. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000743The number of entries in a standard Young tableau such that the next integer is a neighbour. St000829The Ulam distance of a permutation to the identity permutation. St000834The number of right outer peaks of a permutation. St000837The number of ascents of distance 2 of a permutation. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000864The number of circled entries of the shifted recording tableau of a permutation. St000868The aid statistic in the sense of Shareshian-Wachs. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path)

$[c_0,c_1,...,c_{n-1}]$ by adding

$c_0$ to

$c_{n-1}$ . St000973The length of the boundary of an ordered tree. St000981The length of the longest zigzag subpath. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension 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. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001035The convexity degree 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. St001040The depth of the decreasing labelled binary unordered tree associated with the perfect matching. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001079The minimal length of a factorization of a permutation using the permutations (12)(34). St001091The number of parts in an integer partition whose next smaller part has the same size. St001110The 3-dynamic chromatic number of a graph. 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. 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. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. 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. St001176The size of a partition minus its first part. St001180Number of indecomposable injective modules with projective dimension at most 1. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero 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. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001278The number of indecomposable modules that are fixed by

$\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. 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. 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. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001323The independence gap of a graph. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001358The largest degree of a regular subgraph of a graph. St001375The pancake length of a permutation. St001424The number of distinct squares in a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001469The holeyness of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001569The maximal modular displacement of a permutation. 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. St001577The minimal number of edges to add or remove to make a graph a cograph. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001613The binary logarithm of the size of the center of a lattice. St001621The number of atoms of a lattice. St001631The number of simple modules

$S$ with

$dim Ext^1(S,A)=1$ in the incidence algebra

$A$ of the poset. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001644The dimension of a graph. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001651The Frankl number of a lattice. St001664The number of non-isomorphic subposets of a poset. St001668The number of points of the poset minus the width of the poset. St001674The number of vertices of the largest induced star graph in the graph. St001684The reduced word complexity of a permutation. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001727The number of invisible inversions of a permutation. St001742The difference of the maximal and the minimal degree in a graph. St001743The discrepancy of a graph. St001759The Rajchgot index of a permutation. St001760The number of prefix or suffix reversals needed to sort a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001777The number of weak descents in an integer composition. St001782The order of rowmotion on the set of order ideals of a poset. St001792The arboricity of a graph. St001935The number of ascents in a parking function. St001955The number of natural descents for set-valued two row standard Young tableaux. St001962The proper pathwidth of a graph. St000806The semiperimeter of the associated bargraph. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St000064The number of one-box pattern of a permutation. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000654The first descent of a permutation. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000402Half the size of the symmetry class of a permutation. St000530The number of permutations with the same descent word as the given permutation. St000627The exponent of a binary word. St000631The number of distinct palindromic decompositions of a binary word. St000741The Colin de Verdière graph invariant. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000876The number of factors in the Catalan decomposition of a binary word. St000922The minimal number such that all substrings of this length are unique. St000939The number of characters of the symmetric group whose value on the partition is positive. St001052The length of the exterior of a permutation. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001884The number of borders of a binary word. St001948The number of augmented double ascents of a permutation. St000295The length of the border of a binary word. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000928The sum of the coefficients of the character polynomial of an integer partition. St000941The number of characters of the symmetric group whose value on the partition is even. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001082The number of boxed occurrences of 123 in a permutation. St001130The number of two successive successions in a permutation. St001388The number of non-attacking neighbors of a permutation. St001413Half the length of the longest even length palindromic prefix of a binary word. St001524The degree of symmetry of a binary word. St001557The number of inversions of the second entry of a permutation. St001930The weak major index of a binary word. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000485The length of the longest cycle of a permutation. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000656The number of cuts of a poset. St000673The number of non-fixed points of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000925The number of topologically connected components of a set partition. St000990The first ascent of a permutation. St001060The distinguishing index of a graph. 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

$eA$ . St001927Sparre Andersen's number of positives of a signed permutation. St000020The rank of the permutation. St000040The number of regions of the inversion arrangement of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000100The number of linear extensions of a poset. St000109The number of elements less than or equal to the given element in Bruhat order. St000288The number of ones in a binary word. St000297The number of leading ones in a binary word. St000327The number of cover relations in a poset. St000354The number of recoils of a permutation. St000454The largest eigenvalue of a graph if it is integral. St000494The number of inversions of distance at most 3 of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000529The number of permutations whose descent word is the given binary word. St000539The number of odd inversions of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000543The size of the conjugacy class of a binary word. St000630The length of the shortest palindromic decomposition of a binary word. St000633The size of the automorphism group of a poset. St000638The number of up-down runs of a permutation. St000640The rank of the largest boolean interval in a poset. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000657The smallest part of an integer composition. St000675The number of centered multitunnels of a Dyck path. St000694The number of affine bounded permutations that project to a given permutation. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000744The length of the path to the largest entry in a standard Young tableau. St000753The Grundy value for the game of Kayles on a binary word. St000794The mak of a permutation. St000809The reduced reflection length of the permutation. St000886The number of permutations with the same antidiagonal sums. St000898The number of maximal entries in the last diagonal of the monotone triangle. St000910The number of maximal chains of minimal length in a poset. St000932The number of occurrences of the pattern UDU in a Dyck path. St000937The number of positive values of the symmetric group character corresponding to the partition. St000957The number of Bruhat lower covers of a permutation. St000983The length of the longest alternating subword. St000989The number of final rises of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001077The prefix exchange distance of a permutation. St001080The minimal length of a factorization of a permutation using the transposition (12) and the cycle (1,. 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. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001128The exponens consonantiae of a partition. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001220The width of a permutation. St001281The normalized isoperimetric number of a graph. St001285The number of primes in the column sums of the two line notation of a permutation. St001346The number of parking functions that give the same permutation. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001482The product of the prefix sums of a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001570The minimal number of edges to add to make a graph Hamiltonian. St001592The maximal number of simple paths between any two different vertices of a graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001637The number of (upper) dissectors of a poset. St001915The size of the component corresponding to a necklace in Bulgarian solitaire. St000055The inversion sum of a permutation. St000154The sum of the descent bottoms of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000305The inverse major index of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000352The Elizalde-Pak rank of a permutation. St000353The number of inner valleys of a permutation. St000379The number of Hamiltonian cycles in a graph. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000435The number of occurrences of the pattern 213 or of the pattern 231 in a permutation. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000446The disorder of a permutation. St000461The rix statistic of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000538The number of even inversions of a permutation. St000557The number of occurrences of the pattern {{1},{2},{3}} in a set partition. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000591The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal. St000592The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal. St000593The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000624The normalized sum of the minimal distances to a greater element. St000677The standardized bi-alternating inversion number of a permutation. St000691The number of changes of a binary word. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000779The tier of a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000836The number of descents of distance 2 of a permutation. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St000934The 2-degree of an integer partition. 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. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001171The vector space dimension of

$Ext_A^1(I_o,A)$ when

$I_o$ is the tilting module corresponding to the permutation

$o$ in the Auslander algebra

$A$ of

$K[x]/(x^n)$ . St001174The Gorenstein dimension of the algebra

$A/I$ when

$I$ is the tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001412Number of minimal entries in the Bruhat order matrix of a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001552The number of inversions between excedances and fixed points of a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001665The number of pure excedances of a permutation. St001671Haglund's hag of a permutation. St001729The number of visible descents of a permutation. St001731The factorization defect of a permutation. St001737The number of descents of type 2 in a permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001874Lusztig's a-function for the symmetric group. St001875The number of simple modules with projective dimension at most 1. St001960The number of descents of a permutation minus one if its first entry is not one. St000307The number of rowmotion orbits of a poset. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000259The diameter of a connected graph. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001645The pebbling number of a connected graph. St000487The length of the shortest cycle of a permutation. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001623The number of doubly irreducible elements of a lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000219The number of occurrences of the pattern 231 in a permutation. St000260The radius of a connected graph. St000455The second largest eigenvalue of a graph if it is integral. St000466The Gutman (or modified Schultz) index of a connected graph. St000795The mad of a permutation. St000796The stat' of a permutation. St000797The stat`` of a permutation. St000798The makl of a permutation. St000833The comajor index of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000264The girth of a graph, which is not a tree. St001095The number of non-isomorphic posets with precisely one further covering relation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001117The game chromatic index of 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. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001578The minimal number of edges to add or remove to make a graph a line graph. St001649The length of a longest trail in a graph. St001964The interval resolution global dimension of a poset. St001002Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St000117The number of centered tunnels of a Dyck path. St000317The cycle descent number of a permutation. St000363The number of minimal vertex covers of a graph. St000535The rank-width of a graph. St000644The number of graphs with given frequency partition. St000699The toughness times the least common multiple of 1,. St000739The first entry in the last row of a semistandard tableau. St000742The number of big ascents of a permutation after prepending zero. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000958The number of Bruhat factorizations of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001044The number of pairs whose larger element is at most one more than half the size of the perfect matching. 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. 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. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001350Half of the Albertson index of a graph. St001401The number of distinct entries in a semistandard tableau. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001589The nesting number of a perfect matching. St001686The order of promotion on a Gelfand-Tsetlin pattern. St001712The number of natural descents of a standard Young tableau. St001735The number of permutations with the same set of runs. St001739The number of graphs with the same edge polytope as the given graph. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St000002The number of occurrences of the pattern 123 in a permutation. St000039The number of crossings of a permutation. St000095The number of triangles of a graph. St000101The cocharge of a semistandard tableau. St000223The number of nestings in the permutation. St000241The number of cyclical small excedances. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length

$3$ . St000374The number of exclusive right-to-left minima of a permutation. St000427The number of occurrences of the pattern 123 or of the pattern 231 in a permutation. St000443The number of long tunnels of a Dyck path. St000450The number of edges minus the number of vertices plus 2 of a graph. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000703The number of deficiencies of a permutation. St000881The number of short braid edges in the graph of braid moves of a permutation. 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. St001057The Grundy value of the game of creating an independent set in a graph. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules 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. St001271The competition number of a graph. St001310The number of induced diamond graphs in a graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001352The number of internal nodes in the modular decomposition of a graph. St001353The number of prime nodes in the modular decomposition of a graph. St001395The number of strictly unfriendly partitions of a graph. St001470The cyclic holeyness of a permutation. St001488The number of corners of a skew partition. St001513The number of nested exceedences of a permutation. St001520The number of strict 3-descents. St001537The number of cyclic crossings of a permutation. St001556The number of inversions of the third entry of a permutation. St001638The book thickness of a graph. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001691The number of kings in a graph. St001703The villainy of a graph. St001765The number of connected components of the friends and strangers graph. St001783The number of odd automorphisms of a graph. St001811The Castelnuovo-Mumford regularity of a permutation. St001856The number of edges in the reduced word graph of a permutation. St001926Sparre Andersen's position of the maximum of a signed permutation. St000242The number of indices that are not cyclical small weak excedances. St000519The largest length of a factor maximising the subword complexity. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St001093The detour number of a graph. St001345The Hamming dimension of a graph. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001725The harmonious chromatic number of a graph. St000422The energy of a graph, if it is integral. St000637The length of the longest cycle in a graph. St000735The last entry on the main diagonal of a standard tableau.