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Your data matches 667 different statistics following compositions of up to 3 maps.
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Matching statistic: St001071
(load all 7 compositions to match this statistic)

Mapping

St001071: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => 0 = -1 + 1
([],2)
 => 0 = -1 + 1
([(0,1)],2)
 => 1 = 0 + 1
([],3)
 => 0 = -1 + 1
([(1,2)],3)
 => 1 = 0 + 1
([(0,2),(1,2)],3)
 => 0 = -1 + 1
([(0,1),(0,2),(1,2)],3)
 => 1 = 0 + 1

Description

The beta invariant of the graph. The beta invariant was introduced by Crapo [1] for matroids. For graphs with

$n$ vertices the beta invariant is

$\beta(G) = (-1)^{n-c}\sum_{S\subseteq E} (-1)^{|S|} (n-c(S)),$ where

$c(S)$ is the number of connected components of the subgraph of

$G$ with edge set

$S$ . For graphs with at least one edge the beta invariant equals the absolute value of the derivative of the chromatic polynomial at

$1$ . [2] The beta invariant also coincides with the coefficient of the monomial

$x$ , and also with the coefficient of the monomial

$y$ , of the Tutte polynomial.

Matching statistic: St001271
(load all 10 compositions to match this statistic)

Mapping

St001271: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => 0 = -1 + 1
([],2)
 => 0 = -1 + 1
([(0,1)],2)
 => 1 = 0 + 1
([],3)
 => 0 = -1 + 1
([(1,2)],3)
 => 0 = -1 + 1
([(0,2),(1,2)],3)
 => 1 = 0 + 1
([(0,1),(0,2),(1,2)],3)
 => 1 = 0 + 1

Description

The competition number of a graph. The competition graph of a digraph

$D$ is a (simple undirected) graph which has the same vertex set as

$D$ and has an edge between

$x$ and

$y$ if and only if there exists a vertex

$v$ in

$D$ such that

$(x, v)$ and

$(y, v)$ are arcs of

$D$ . For any graph,

$G$ together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number

$k(G)$ is the smallest number of such isolated vertices.

Matching statistic: St000148
(load all 3 compositions to match this statistic)

Mapping

Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000148: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => []
 => 0 = -1 + 1
([],2)
 => []
 => 0 = -1 + 1
([(0,1)],2)
 => [1]
 => 1 = 0 + 1
([],3)
 => []
 => 0 = -1 + 1
([(1,2)],3)
 => [1]
 => 1 = 0 + 1
([(0,2),(1,2)],3)
 => [2]
 => 0 = -1 + 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1

Description

The number of odd parts of a partition.

Matching statistic: St000149
(load all 10 compositions to match this statistic)

Mapping

Mp00251: Graphs —clique sizes⟶ Integer partitions
St000149: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => [1]
 => 0 = -1 + 1
([],2)
 => [1,1]
 => 0 = -1 + 1
([(0,1)],2)
 => [2]
 => 1 = 0 + 1
([],3)
 => [1,1,1]
 => 0 = -1 + 1
([(1,2)],3)
 => [2,1]
 => 0 = -1 + 1
([(0,2),(1,2)],3)
 => [2,2]
 => 1 = 0 + 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1

Description

The number of cells of the partition whose leg is zero and arm is odd. This statistic is equidistributed with [[St000143]], see [1].

Matching statistic: St000150
(load all 11 compositions to match this statistic)

Mapping

Mp00251: Graphs —clique sizes⟶ Integer partitions
St000150: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => [1]
 => 0 = -1 + 1
([],2)
 => [1,1]
 => 1 = 0 + 1
([(0,1)],2)
 => [2]
 => 0 = -1 + 1
([],3)
 => [1,1,1]
 => 1 = 0 + 1
([(1,2)],3)
 => [2,1]
 => 0 = -1 + 1
([(0,2),(1,2)],3)
 => [2,2]
 => 1 = 0 + 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 0 = -1 + 1

Description

The floored half-sum of the multiplicities of a partition. This statistic is equidistributed with [[St000143]] and [[St000149]], see [1].

Matching statistic: St000256
(load all 11 compositions to match this statistic)

Mapping

Mp00251: Graphs —clique sizes⟶ Integer partitions
St000256: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => [1]
 => 0 = -1 + 1
([],2)
 => [1,1]
 => 0 = -1 + 1
([(0,1)],2)
 => [2]
 => 1 = 0 + 1
([],3)
 => [1,1,1]
 => 0 = -1 + 1
([(1,2)],3)
 => [2,1]
 => 0 = -1 + 1
([(0,2),(1,2)],3)
 => [2,2]
 => 1 = 0 + 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1

Description

The number of parts from which one can substract 2 and still get an integer partition.

Matching statistic: St000257
(load all 12 compositions to match this statistic)

Mapping

Mp00251: Graphs —clique sizes⟶ Integer partitions
St000257: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => [1]
 => 0 = -1 + 1
([],2)
 => [1,1]
 => 1 = 0 + 1
([(0,1)],2)
 => [2]
 => 0 = -1 + 1
([],3)
 => [1,1,1]
 => 1 = 0 + 1
([(1,2)],3)
 => [2,1]
 => 0 = -1 + 1
([(0,2),(1,2)],3)
 => [2,2]
 => 1 = 0 + 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 0 = -1 + 1

Description

The number of distinct parts of a partition that occur at least twice. See Section 3.3.1 of [2].

Matching statistic: St000274
(load all 2 compositions to match this statistic)

Mapping

Mp00154: Graphs —core⟶ Graphs
St000274: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => ([],1)
 => 0 = -1 + 1
([],2)
 => ([],1)
 => 0 = -1 + 1
([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 0 + 1
([],3)
 => ([],1)
 => 0 = -1 + 1
([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 0 + 1
([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1 = 0 + 1
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 0 = -1 + 1

Description

The number of perfect matchings of a graph. A matching of a graph

$G$ is a subset

$F \subset E(G)$ such that no two edges in

$F$ share a vertex in common. A perfect matching

$F'$ is then a matching such that every vertex in

$V(G)$ is incident with exactly one edge in

$F'$ .

Matching statistic: St000481
(load all 10 compositions to match this statistic)

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000481: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => [1]
 => 0 = -1 + 1
([],2)
 => [1,1]
 => 1 = 0 + 1
([(0,1)],2)
 => [2]
 => 0 = -1 + 1
([],3)
 => [1,1,1]
 => 1 = 0 + 1
([(1,2)],3)
 => [2,1]
 => 1 = 0 + 1
([(0,2),(1,2)],3)
 => [3]
 => 0 = -1 + 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 0 = -1 + 1

Description

The number of upper covers of a partition in dominance order.

Matching statistic: St000506
(load all 10 compositions to match this statistic)

Mapping

Mp00251: Graphs —clique sizes⟶ Integer partitions
St000506: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => [1]
 => 0 = -1 + 1
([],2)
 => [1,1]
 => 1 = 0 + 1
([(0,1)],2)
 => [2]
 => 0 = -1 + 1
([],3)
 => [1,1,1]
 => 0 = -1 + 1
([(1,2)],3)
 => [2,1]
 => 1 = 0 + 1
([(0,2),(1,2)],3)
 => [2,2]
 => 1 = 0 + 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 0 = -1 + 1

Description

The number of standard desarrangement tableaux of shape equal to the given partition. A '''standard desarrangement tableau''' is a standard tableau whose first ascent is even. Here, an ascent of a standard tableau is an entry

$i$ such that

$i+1$ appears to the right or above

$i$ in the tableau (with respect to English tableau notation). This is also the nullity of the random-to-random operator (and the random-to-top) operator acting on the simple module of the symmetric group indexed by the given partition. See also: * [[St000046]]: The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition * [[St000500]]: Eigenvalues of the random-to-random operator acting on the regular representation.

The following 657 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000549The number of odd partial sums of an integer partition. St000552The number of cut vertices of a graph. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001092The number of distinct even parts of a partition. St001395The number of strictly unfriendly partitions of a graph. St001479The number of bridges of a graph. St001484The number of singletons of an integer partition. St001587Half of the largest even part of an integer partition. St001691The number of kings in a graph. St001826The maximal number of leaves on a vertex of a graph. St001827The number of two-component spanning forests of a graph. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000759The smallest missing part in an integer partition. St000917The open packing number of a graph. St001057The Grundy value of the game of creating an independent set in a graph. St001642The Prague dimension of a graph. St001672The restrained domination number of a graph. St000455The second largest eigenvalue of a graph if it is integral. St000024The number of double up and double down steps of a Dyck path. St000142The number of even parts of a partition. St000143The largest repeated part of a partition. St000147The largest part of an integer partition. St000159The number of distinct parts of the integer partition. St000183The side length of the Durfee square of an integer partition. St000260The radius of a connected graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000272The treewidth of a graph. St000310The minimal degree of a vertex of a graph. St000340The number of non-final maximal constant sub-paths of length greater than one. St000362The size of a minimal vertex cover of a graph. St000378The diagonal inversion number of an integer partition. St000386The number of factors DDU in a Dyck path. St000387The matching number of a graph. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000475The number of parts equal to 1 in a partition. St000480The number of lower covers of a partition in dominance order. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000535The rank-width of a graph. St000536The pathwidth of a graph. St000547The number of even non-empty partial sums of an integer partition. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000761The number of ascents in an integer composition. St000783The side length of the largest staircase partition fitting into a partition. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000897The number of different multiplicities of parts of an integer partition. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000992The alternating sum of the parts of an integer partition. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001194The injective dimension of

$A/AfA$ in the corresponding Nakayama algebra

$A$ when

$Af$ is the minimal faithful projective-injective left

$A$ -module St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001263The index of the maximal parabolic seaweed algebra associated with the composition. St001274The number of indecomposable injective modules with projective dimension equal to two. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001277The degeneracy of a graph. St001280The number of parts of an integer partition that are at least two. St001333The cardinality of a minimal edge-isolating set of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001358The largest degree of a regular subgraph of a graph. St001393The induced matching number of a graph. St001413Half the length of the longest even length palindromic prefix of a binary word. St001424The number of distinct squares in a binary word. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001459The number of zero columns in the nullspace of a graph. St001512The minimum rank of a graph. St001524The degree of symmetry of a binary word. St001615The number of join prime elements of a lattice. St001657The number of twos in an integer partition. St001673The degree of asymmetry of an integer composition. St001689The number of celebrities in a graph. St001692The number of vertices with higher degree than the average degree in a graph. St001730The number of times the path corresponding to a binary word crosses the base line. St001743The discrepancy of a graph. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001777The number of weak descents in an integer composition. St001792The arboricity of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001797The number of overfull subgraphs of a graph. St001812The biclique partition number of a graph. 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). St001931The weak major index of an integer composition regarded as a word. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000013The height of a Dyck path. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000172The Grundy number of a graph. St000258The burning number of a graph. St000259The diameter of a connected graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000286The number of connected components of the complement of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000388The number of orbits of vertices of a graph under automorphisms. St000443The number of long tunnels of a Dyck path. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000630The length of the shortest palindromic decomposition of a binary word. St000722The number of different neighbourhoods in a graph. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000765The number of weak records in an integer composition. St000767The number of runs in an integer composition. St000822The Hadwiger number of the graph. St000899The maximal number of repetitions of an integer composition. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St000903The number of different parts of an integer composition. St000904The maximal number of repetitions of an integer composition. St000918The 2-limited packing number of a graph. St001007Number of simple modules with projective dimension 1 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. St001029The size of the core of a graph. St001093The detour number of a graph. St001109The number of proper colourings of a graph with as few colours as possible. St001111The weak 2-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001261The Castelnuovo-Mumford regularity of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001316The domatic number of a graph. St001330The hat guessing number of a graph. St001352The number of internal nodes in the modular decomposition of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. 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. St001674The number of vertices of the largest induced star graph in the graph. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. St001716The 1-improper chromatic number of a graph. St001734The lettericity of a graph. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001814The number of partitions interlacing the given partition. St001828The Euler characteristic of a graph. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St001955The number of natural descents for set-valued two row standard Young tableaux. St001963The tree-depth of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected 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. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000010The length of the partition. St000011The number of touch points (or returns) of a Dyck path. St000012The area of a Dyck path. St000021The number of descents of a permutation. St000025The number of initial rises of a Dyck path. St000028The number of stack-sorts needed to sort a permutation. St000035The number of left outer peaks of a permutation. St000039The number of crossings of a permutation. St000052The number of valleys of a Dyck path not on the x-axis. St000053The number of valleys of the Dyck path. St000120The number of left tunnels of a Dyck path. St000133The "bounce" of a permutation. St000141The maximum drop size of a permutation. St000145The Dyson rank of a partition. St000155The number of exceedances (also excedences) of a permutation. St000160The multiplicity of the smallest part of a partition. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000168The number of internal nodes of an ordered tree. St000185The weighted size of a partition. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000211The rank of the set partition. St000214The number of adjacencies of a permutation. St000237The number of small exceedances. St000238The number of indices that are not small weak excedances. St000245The number of ascents of a permutation. St000271The chromatic index of a graph. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000295The length of the border of a binary word. St000304The load of a permutation. St000306The bounce count of a Dyck path. St000309The number of vertices with even degree. St000313The number of degree 2 vertices of a graph. St000316The number of non-left-to-right-maxima of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000331The number of upper interactions of a Dyck path. St000332The positive inversions of an alternating sign matrix. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000352The Elizalde-Pak rank of a permutation. St000353The number of inner valleys of a permutation. St000355The number of occurrences of the pattern 21-3. St000356The number of occurrences of the pattern 13-2. St000369The dinv deficit of a Dyck path. St000374The number of exclusive right-to-left minima of a permutation. St000376The bounce deficit of a Dyck path. St000377The dinv defect of an integer partition. St000389The number of runs of ones of odd length in a binary word. St000390The number of runs of ones in a binary word. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000445The number of rises of length 1 of a Dyck path. St000461The rix statistic of a permutation. St000507The number of ascents of a standard tableau. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000546The number of global descents of a permutation. St000548The number of different non-empty partial sums of an integer partition. St000628The balance of a binary word. St000648The number of 2-excedences of a permutation. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000660The number of rises of length at least 3 of a Dyck path. St000661The number of rises of length 3 of a Dyck path. St000662The staircase size of the code of a permutation. St000665The number of rafts of a permutation. St000670The reversal length of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000676The number of odd rises of a Dyck path. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. 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. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000766The number of inversions of an integer composition. St000769The major index of a composition regarded as a word. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000834The number of right outer peaks of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000868The aid statistic in the sense of Shareshian-Wachs. St000873The aix statistic of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000877The depth of the binary word interpreted as a path. St000884The number of isolated descents of a permutation. St000920The logarithmic height of a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000954Number of times the corresponding LNakayama algebra has

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

$i>0$ . St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. 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. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001083The number of boxed occurrences of 132 in a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. 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. St001090The number of pop-stack-sorts needed to sort a permutation. St001091The number of parts in an integer partition whose next smaller part has the same size. St001114The number of odd descents of a permutation. St001139The number of occurrences of hills of size 2 in a Dyck path. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001153The number of blocks with even minimum in a set partition. St001163The number of simple modules with dominant dimension at least three 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. St001172The number of 1-rises at odd height of a Dyck path. St001176The size of a partition minus its first part. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. 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. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001192The maximal dimension of

$Ext_A^2(S,A)$ for a simple module

$S$ over the corresponding Nakayama algebra

$A$ . St001197The global dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n−1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a special CNakayama algebra. 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

$eA$ . St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001234The number of indecomposable three dimensional modules with projective dimension one. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. 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. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001278The number of indecomposable modules that are fixed by

$\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001298The number of repeated entries in the Lehmer code of a permutation. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001341The number of edges in the center of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001371The length of the longest Yamanouchi prefix of a binary word. St001375The pancake length of a permutation. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001394The genus of a permutation. St001414Half the length of the longest odd length 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. St001423The number of distinct cubes in a binary word. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001435The number of missing boxes in the first row. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001438The number of missing boxes of a skew partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001557The number of inversions of the second entry of a permutation. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001584The area statistic between a Dyck path and its bounce path. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001665The number of pure excedances of a permutation. St001671Haglund's hag of a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001721The degree of a binary word. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in 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. St001783The number of odd automorphisms of a graph. St001801Half the number of preimage-image pairs of different parity in a permutation. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001874Lusztig's a-function for the symmetric group. St001928The number of non-overlapping descents in a permutation. St001930The weak major index of a binary word. St001932The number of pairs of singleton blocks in the noncrossing set partition corresponding to a Dyck path, that can be merged to create another noncrossing set partition. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001961The sum of the greatest common divisors of all pairs of parts. St000007The number of saliances of the permutation. St000015The number of peaks 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. St000054The first entry of the permutation. St000058The order of a permutation. St000061The number of nodes on the left branch of a binary tree. St000062The length of the longest increasing subsequence of the permutation. St000088The row sums of the character table of the symmetric group. St000092The number of outer peaks of a permutation. St000166The depth minus 1 of an ordered tree. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000201The number of leaf nodes in a binary tree. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000297The number of leading ones in a binary word. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000326The position of the first one in a binary word after appending a 1 at the end. St000335The difference of lower and upper interactions. St000346The number of coarsenings of a partition. St000354The number of recoils of a permutation. St000363The number of minimal vertex covers of a graph. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000383The last part of an integer composition. St000392The length of the longest run of ones in a binary word. St000393The number of strictly increasing runs in a binary word. St000396The register function (or Horton-Strahler number) of a binary tree. St000442The maximal area to the right of an up step of a Dyck path. St000451The length of the longest pattern of the form k 1 2. St000470The number of runs in a permutation. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000531The leading coefficient of the rook polynomial of an integer partition. St000542The number of left-to-right-minima of a permutation. St000568The hook number of a binary tree. St000617The number of global maxima of a Dyck path. St000627The exponent of a binary word. St000638The number of up-down runs of a permutation. St000657The smallest part of an integer composition. St000667The greatest common divisor of the parts of the partition. St000678The number of up steps after the last double rise of a Dyck path. 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. St000691The number of changes of a binary word. St000701The protection number of a binary tree. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000758The length of the longest staircase fitting into an integer composition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000796The stat' of a permutation. St000808The number of up steps of the associated bargraph. 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. 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. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000847The number of standard Young tableaux whose descent set is the binary word. St000862The number of parts of the shifted shape of a permutation. St000874The position of the last double rise in a Dyck path. St000876The number of factors in the Catalan decomposition of a binary word. St000878The number of ones minus the number of zeros of a binary word. St000883The number of longest increasing subsequences of a permutation. St000885The number of critical steps in the Catalan decomposition of a binary word. St000905The number of different multiplicities of parts of an integer composition. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. 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. 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

$c_{n−1}$ . St000982The length of the longest constant subword. St000988The orbit size of a permutation under Foata's bijection. St000990The first ascent of a permutation. St000991The number of right-to-left minima of a permutation. St000999Number of indecomposable projective module with injective dimension equal to the global dimension 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. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001081The number of minimal length factorizations of a permutation into star transpositions. 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. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001201The grade of the simple module

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

$L=[c_0,c_1,...,c_{n−1}]$ such that

$n=c_0 < c_i$ for all

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

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

$n=c_0 < c_i$ for all

$i > 0$ a Dyck path as follows: St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001267The length of the Lyndon factorization of the binary word. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001315The dissociation 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. St001342The number of vertices in the center of a graph. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001368The number of vertices of maximal degree in a graph. St001372The length of a longest cyclic run of ones of a binary word. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001437The flex of a binary word. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001471The magnitude of a Dyck path. St001481The minimal height of a peak of a Dyck path. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001486The number of corners of the ribbon associated with an integer composition. St001487The number of inner corners of a skew partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001523The degree of symmetry of a Dyck path. St001527The cyclic permutation representation number of an integer partition. St001530The depth of a Dyck path. St001571The Cartan determinant of the integer partition. St001589The nesting number of a perfect matching. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. 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. St001732The number of peaks visible from the left. St001733The number of weak left to right maxima of a Dyck path. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. 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. St001884The number of borders of a binary word. St001933The largest multiplicity of a part in an integer partition. St000094The depth of an ordered tree. St000144The pyramid weight of the Dyck path. St000439The position of the first down step of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000521The number of distinct subtrees of an ordered tree. St000797The stat`` of 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

$c_{n-1}$ . St000983The length of the longest alternating subword. 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. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001049The smallest label in the subtree not containing 1 in the decreasing labelled binary unordered tree associated with the perfect matching. 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. 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. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001285The number of primes in the column sums of the two line notation of a permutation. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. 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. 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. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St001060The distinguishing index of a graph. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000929The constant term of the character polynomial of an integer partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001568The smallest positive integer that does not appear twice in the partition. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St000083The number of left oriented leafs of a binary tree except the first one. St000216The absolute length of a permutation. St000251The number of nonsingleton blocks of a set partition. St000253The crossing number of a set partition. St000254The nesting number of a set partition. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000454The largest eigenvalue of a graph if it is integral. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000539The number of odd inversions of a permutation. St000558The number of occurrences of the pattern {{1,2}} in a set partition. St000677The standardized bi-alternating inversion number of a permutation. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000693The modular (standard) major index of a standard tableau. St000809The reduced reflection length of the permutation. St000829The Ulam distance of a permutation to the identity permutation. St000833The comajor index of a permutation. St000919The number of maximal left branches of a binary tree. St000947The major index east count of a Dyck path. St000976The sum of the positions of double up-steps of a Dyck path. St000984The number of boxes below precisely one peak. St000989The number of final rises of a permutation. St001061The number of indices that are both descents and recoils 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. 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. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001480The number of simple summands of the module J^2/J^3. St001569The maximal modular displacement of a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000418The number of Dyck paths that are weakly below a Dyck path. St000485The length of the longest cycle of a permutation. St000654The first descent of a permutation. St000675The number of centered multitunnels of a Dyck path. St000702The number of weak deficiencies of a permutation. 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. St000898The number of maximal entries in the last diagonal of the monotone triangle. St000939The number of characters of the symmetric group whose value on the partition is positive. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001062The maximal size of a block of a set partition. St001500The global dimension of magnitude 1 Nakayama algebras. St001531Number of partial orders contained in the poset determined by the Dyck path. St001959The product of the heights of the peaks of a Dyck path. St000741The Colin de Verdière graph invariant. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001645The pebbling number of a connected graph. 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

$eA$ . St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000090The variation of a composition. St000146The Andrews-Garvan crank of a partition. St000422The energy of a graph, if it is integral. St001651The Frankl number of a lattice. St001845The number of join irreducibles minus the rank of a lattice. St000477The weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000928The sum of the coefficients of the character polynomial of an integer partition. St000219The number of occurrences of the pattern 231 in a permutation. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000699The toughness times the least common multiple of 1,. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000379The number of Hamiltonian cycles in a graph. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000302The determinant of the distance matrix of a connected graph. St000467The hyper-Wiener index of a connected graph. St001626The number of maximal proper sublattices of a lattice. St000016The number of attacking pairs of a standard tableau. St000017The number of inversions of a standard tableau. St000117The number of centered tunnels of a Dyck path. St000290The major index of a binary word. St000293The number of inversions of a binary word. St000296The length of the symmetric border of a binary word. St000347The inversion sum of a binary word. St000348The non-inversion sum of a binary word. St000629The defect of a binary word. St000682The Grundy value of Welter's game on a binary word. St000687The dimension of

$Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000921The number of internal inversions of a binary word. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St000995The largest even part of an integer partition. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001141The number of occurrences of hills of size 3 in a Dyck path. St001161The major index north count of a Dyck path. 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. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001193The dimension of

$Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra

$A$ such that

$eA$ is a minimal faithful projective-injective module. St001214The aft of an integer partition. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001485The modular major index of a binary word. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001596The number of two-by-two squares inside a skew partition. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St001910The height of the middle non-run of a Dyck path.