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Your data matches 838 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000019
(load all 25 compositions to match this statistic)

Mapping

St000019: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The cardinality of the support of a permutation. A permutation

$\sigma$ may be written as a product

$\sigma = s_{i_1}\dots s_{i_k}$ with

$k$ minimal, where

$s_i = (i,i+1)$ denotes the simple transposition swapping the entries in positions

$i$ and

$i+1$ . The set of indices

$\{i_1,\dots,i_k\}$ is the '''support''' of

$\sigma$ and independent of the chosen way to write

$\sigma$ as such a product. See [2], Definition 1 and Proposition 10. The '''connectivity set''' of

$\sigma$ of length

$n$ is the set of indices

$1 \leq i < n$ such that

$\sigma(k) < i$ for all

$k < i$ . Thus, the connectivity set is the complement of the support.

Matching statistic: St000029
(load all 23 compositions to match this statistic)

Mapping

St000029: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The depth of a permutation. This is given by

$\operatorname{dp}(\sigma) = \sum_{\sigma_i>i} (\sigma_i-i) = |\{ i \leq j : \sigma_i > j\}|.$ The depth is half of the total displacement [4], Problem 5.1.1.28, or Spearman’s disarray [3]

$\sum_i |\sigma_i-i|$ . Permutations with depth at most

$1$ are called ''almost-increasing'' in [5].

Matching statistic: St000030
(load all 25 compositions to match this statistic)

Mapping

St000030: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The sum of the descent differences of a permutations. This statistic is given by

$\pi \mapsto \sum_{i\in\operatorname{Des}(\pi)} (\pi_i-\pi_{i+1}).$ See [[St000111]] and [[St000154]] for the sum of the descent tops and the descent bottoms, respectively. This statistic was studied in [1] and [2] where is was called the ''drop'' of a permutation.

Matching statistic: St000209
(load all 23 compositions to match this statistic)

Mapping

St000209: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

Maximum difference of elements in cycles. Given a cycle

$C$ in a permutation, we can compute the maximum distance between elements in the cycle, that is

$\max \{ a_i-a_j | a_i, a_j \in C \}$ . The statistic is then the maximum of this value over all cycles in the permutation.

Matching statistic: St000240
(load all 17 compositions to match this statistic)

Mapping

St000240: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of indices that are not small excedances. A small excedance is an index

$i$ for which

$\pi_i = i+1$ .

Matching statistic: St000021
(load all 8 compositions to match this statistic)

Mapping

Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of descents of a permutation. This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].

Matching statistic: St000141
(load all 22 compositions to match this statistic)

Mapping

Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The maximum drop size of a permutation. The maximum drop size of a permutation

$\pi$ of

$[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of

$i-\pi(i)$ .

Matching statistic: St000156
(load all 7 compositions to match this statistic)

Mapping

Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000156: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The Denert index of a permutation. It is defined as

$\begin{align*} den(\sigma) &= \#\{ 1\leq l < k \leq n : \sigma(k) < \sigma(l) \leq k \} \\ &+ \#\{ 1\leq l < k \leq n : \sigma(l) \leq k < \sigma(k) \} \\ &+ \#\{ 1\leq l < k \leq n : k < \sigma(k) < \sigma(l) \} \end{align*}$ where

$n$ is the size of

$\sigma$ . It was studied by Denert in [1], and it was shown by Foata and Zeilberger in [2] that the bistatistic

$(exc,den)$ is [[Permutations/Descents-Major#Euler-Mahonian_statistics|Euler-Mahonian]]. Here,

$exc$ is the number of weak exceedences, see [[St000155]].

Matching statistic: St000171
(load all 7 compositions to match this statistic)

Mapping

Mp00160: Permutations —graph of inversions⟶ Graphs
St000171: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The degree of the graph. This is the maximal vertex degree of a graph.

Matching statistic: St000214
(load all 5 compositions to match this statistic)

Mapping

Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000214: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of adjacencies of a permutation. An adjacency of a permutation

$\pi$ is an index

$i$ such that

$\pi(i)-1 = \pi(i+1)$ . Adjacencies are also known as ''small descents''. This can be also described as an occurrence of the bivincular pattern ([2,1], {((0,1),(1,0),(1,1),(1,2),(2,1)}), i.e., the middle row and the middle column are shaded, see [3].

The following 828 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000224The sorting index of a permutation. St000238The number of indices that are not small weak excedances. St000246The number of non-inversions of a permutation. St000304The load of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000651The maximal size of a rise in a permutation. St000662The staircase size of the code of a permutation. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001120The length of a longest path in a graph. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001489The maximum of the number of descents and the number of inverse descents. St001671Haglund's hag of a permutation. St001726The number of visible inversions of a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001861The number of Bruhat lower covers of a permutation. St001869The maximum cut size of a graph. St001894The depth of a signed permutation. St000325The width of the tree associated to a permutation. St000378The diagonal inversion number of an integer partition. St000470The number of runs in a permutation. St000722The number of different neighbourhoods in a graph. St000734The last entry in the first row of a standard tableau. St000738The first entry in the last row of a standard tableau. St000883The number of longest increasing subsequences of a permutation. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001268The size of the largest ordinal summand in the poset. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001463The number of distinct columns in the nullspace of a graph. St001725The harmonious chromatic number of a graph. St001779The order of promotion on the set of linear extensions of a poset. St001806The upper middle entry of a permutation. St001807The lower middle entry of a permutation. 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. St000004The major index of a permutation. St000009The charge of a standard tableau. St000018The number of inversions of a permutation. St000024The number of double up and double down steps of a Dyck path. St000028The number of stack-sorts needed to sort a permutation. St000067The inversion number of the alternating sign matrix. St000081The number of edges of a graph. St000091The descent variation of a composition. St000133The "bounce" of a permutation. St000136The dinv of a parking function. St000155The number of exceedances (also excedences) of a permutation. St000157The number of descents of a standard tableau. St000194The number of primary dinversion pairs of a labelled dyck path corresponding to a parking function. St000237The number of small exceedances. St000245The number of ascents of a permutation. St000272The treewidth of a graph. St000293The number of inversions of a binary word. St000305The inverse major index of a permutation. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. 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. St000362The size of a minimal vertex cover of a graph. St000374The number of exclusive right-to-left minima of a permutation. St000377The dinv defect of an integer partition. St000441The number of successions of a permutation. St000446The disorder of a permutation. St000454The largest eigenvalue of a graph if it is integral. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St000546The number of global descents of a permutation. St000632The jump number of the poset. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000670The reversal length of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000692Babson and Steingrímsson's statistic of a permutation. St000703The number of deficiencies of a permutation. St000840The number of closers smaller than the largest opener in a perfect matching. St000864The number of circled entries of the shifted recording tableau of a permutation. St000868The aid statistic in the sense of Shareshian-Wachs. St000996The number of exclusive left-to-right maxima of a permutation. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001079The minimal length of a factorization of a permutation using the permutations (12)(34). St001090The number of pop-stack-sorts needed to sort a permutation. St001094The depth index of a set partition. St001117The game chromatic index of a graph. St001176The size of a partition minus its first part. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001209The pmaj statistic of a parking function. 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. 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. St001287The number of primes obtained by multiplying preimage and image of a permutation and subtracting one. St001298The number of repeated entries in the Lehmer code of a permutation. St001345The Hamming dimension of a graph. St001358The largest degree of a regular subgraph of a graph. St001375The pancake length of a permutation. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001391The disjunction number of a graph. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001427The number of descents of a signed permutation. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001479The number of bridges of a graph. St001512The minimum rank of a graph. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001644The dimension of a graph. St001649The length of a longest trail in a graph. St001684The reduced word complexity of a permutation. St001742The difference of the maximal and the minimal degree in a graph. St001743The discrepancy of a graph. St001760The number of prefix or suffix reversals needed to sort a permutation. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001792The arboricity of a graph. St001812The biclique partition number of a graph. St001821The sorting index of a signed permutation. St001823The Stasinski-Voll length of a signed permutation. St001826The maximal number of leaves on a vertex of a graph. St001896The number of right descents of a signed permutations. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001946The number of descents in a parking function. St001962The proper pathwidth of a graph. St000006The dinv of a Dyck path. St000007The number of saliances of the permutation. St000010The length of the partition. St000013The height of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000054The first entry of the permutation. St000058The order of a permutation. St000060The greater neighbor of the maximum. St000062The length of the longest increasing subsequence of the permutation. St000071The number of maximal chains in a poset. St000086The number of subgraphs. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000110The number of permutations less than or equal to a permutation in left weak order. St000147The largest part of an integer partition. St000172The Grundy number of a graph. St000189The number of elements in the poset. 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. St000239The number of small weak excedances. St000299The number of nonisomorphic vertex-induced subtrees. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000321The number of integer partitions of n that are dominated by an integer partition. St000335The difference of lower and upper interactions. St000345The number of refinements of a partition. St000346The number of coarsenings of a partition. St000363The number of minimal vertex covers of a graph. St000381The largest part of an integer composition. St000443The number of long tunnels of a Dyck path. St000451The length of the longest pattern of the form k 1 2. St000453The number of distinct Laplacian eigenvalues of a graph. St000468The Hosoya index of a graph. St000479The Ramsey number of a graph. St000501The size of the first part in the decomposition of a permutation. St000505The biggest entry in the block containing the 1. St000527The width of the poset. St000542The number of left-to-right-minima of a permutation. St000548The number of different non-empty partial sums of an integer partition. St000626The minimal period of a binary word. St000636The hull number of a graph. St000638The number of up-down runs of a permutation. St000653The last descent of a permutation. St000667The greatest common divisor of the parts of the partition. St000740The last entry of a permutation. St000784The maximum of the length and the largest part of the integer partition. St000808The number of up steps of the associated bargraph. St000822The Hadwiger number of the graph. St000839The largest opener of a set partition. St000909The number of maximal chains of maximal size in a poset. St000935The number of ordered refinements of an integer partition. 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. St001045The number of leaves in the subtree not containing one in the decreasing labelled binary unordered tree associated with the perfect matching. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001093The detour number of a graph. St001116The game chromatic number of a graph. St001136The largest label with larger sister in the leaf labelled binary unordered tree associated with the perfect matching. 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. St001235The global dimension of the corresponding Comp-Nakayama algebra. 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. 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. St001330The hat guessing number of a graph. St001342The number of vertices in the center of a graph. St001389The number of partitions of the same length below the given integer partition. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001399The distinguishing number of a poset. St001415The length of the longest palindromic prefix of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001441The number of non-empty connected induced subgraphs of a graph. St001461The number of topologically connected components of the chord diagram of a permutation. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001494The Alon-Tarsi number of a graph. St001497The position of the largest weak excedence of a permutation. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001652The length of a longest interval of consecutive numbers. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001662The length of the longest factor of consecutive numbers in a permutation. St001670The connected partition number of a graph. St001672The restrained domination number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001746The coalition number of a graph. St001757The number of orbits of toric promotion on a graph. St001883The mutual visibility number of a graph. St001917The order of toric promotion on the set of labellings of a graph. St001963The tree-depth of a graph. 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. St001458The rank of the adjacency matrix of a graph. St001814The number of partitions interlacing the given partition. St001834The number of non-isomorphic minors of a graph. St000005The bounce statistic of a Dyck path. St000008The major index of the composition. St000012The area of a Dyck path. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000039The number of crossings of a permutation. St000051The size of the left subtree of a binary tree. St000053The number of valleys of the Dyck path. St000059The inversion number of a standard tableau as defined by Haglund and Stevens. St000080The rank of the poset. St000089The absolute variation of a composition. St000095The number of triangles of a graph. St000119The number of occurrences of the pattern 321 in a permutation. St000120The number of left tunnels of a Dyck path. St000148The number of odd parts of a partition. St000154The sum of the descent bottoms of a permutation. St000160The multiplicity of the smallest part of a partition. St000161The sum of the sizes of the right subtrees of a binary tree. St000168The number of internal nodes of an ordered tree. St000169The cocharge of a standard tableau. St000173The segment statistic of a semistandard tableau. St000211The rank of the set partition. St000222The number of alignments in the permutation. St000223The number of nestings in the permutation. St000228The size of a partition. St000234The number of global ascents of a permutation. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000290The major index of a binary word. St000295The length of the border of a binary word. St000306The bounce count of a Dyck path. St000310The minimal degree of a vertex of a graph. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000330The (standard) major index of a standard tableau. 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. St000339The maf index of a permutation. St000340The number of non-final maximal constant sub-paths of length greater than one. St000369The dinv deficit of a Dyck path. 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

$3$ . St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length

$3$ . St000376The bounce deficit of a Dyck path. St000384The maximal part of the shifted composition of an integer partition. St000429The number of occurrences of the pattern 123 or of the pattern 321 in a permutation. St000433The number of occurrences of the pattern 132 or of the pattern 321 in a permutation. St000448The number of pairs of vertices of a graph with distance 2. St000459The hook length of the base cell of a partition. St000463The number of admissible inversions of a permutation. 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. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000519The largest length of a factor maximising the subword complexity. St000682The Grundy value of Welter's game on a binary word. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000691The number of changes of a binary word. St000731The number of double exceedences of a permutation. St000741The Colin de Verdière graph invariant. St000742The number of big ascents of a permutation after prepending zero. St000778The metric dimension of a graph. St000800The number of occurrences of the vincular pattern |231 in 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. St000867The sum of the hook lengths in the first row of an integer partition. St000879The number of long braid edges in the graph of braid moves of a permutation. St000954Number of times the corresponding LNakayama algebra has

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

$i>0$ . 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. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001082The number of boxed occurrences of 123 in a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001091The number of parts in an integer partition whose next smaller part has the same size. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001115The number of even descents of a permutation. St001119The length of a shortest maximal path in a graph. St001127The sum of the squares of the parts of a partition. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001152The number of pairs with even minimum in a perfect matching. 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. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001192The maximal dimension of

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

$S$ over the corresponding Nakayama algebra

$A$ . St001194The injective dimension of

$A/AfA$ in the corresponding Nakayama algebra

$A$ when

$Af$ is the minimal faithful projective-injective left

$A$ -module St001197The global dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001205The number of non-simple indecomposable projective-injective modules of the algebra

$eAe$ in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001274The number of indecomposable injective modules with projective dimension equal to two. St001280The number of parts of an integer partition that are at least two. 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. St001311The cyclomatic number 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. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001319The minimal number of occurrences of the star-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001357The maximal degree of a regular spanning subgraph of a graph. St001372The length of a longest cyclic run of ones of a binary word. St001397Number of pairs of incomparable elements in a finite poset. St001403The number of vertical separators in a permutation. St001411The number of patterns 321 or 3412 in a permutation. St001428The number of B-inversions of a signed permutation. St001429The number of negative entries in a signed permutation. St001434The number of negative sum pairs of a signed permutation. St001480The number of simple summands of the module J^2/J^3. St001485The modular major index of a binary word. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. 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. 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. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001584The area statistic between a Dyck path and its bounce path. 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. 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. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001697The shifted natural comajor index of a standard Young tableau. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001721The degree of a binary word. St001727The number of invisible inversions of a permutation. St001759The Rajchgot index of a permutation. St001769The reflection length of a signed permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001777The number of weak descents in an integer composition. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001822The number of alignments of a signed permutation. St001862The number of crossings of a signed permutation. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001866The nesting alignments of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001892The flag excedance statistic of a signed permutation. St001893The flag descent of a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001935The number of ascents in a parking function. St001948The number of augmented double ascents of a permutation. St001949The rigidity index of a graph. St001956The comajor index for set-valued two-row standard Young tableaux. St000011The number of touch points (or returns) of a Dyck path. St000015The number of peaks of a Dyck path. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000056The decomposition (or block) number of a permutation. St000063The number of linear extensions of a certain poset defined for an integer partition. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000068The number of minimal elements in a poset. St000069The number of maximal elements of a poset. St000087The number of induced subgraphs. St000093The cardinality of a maximal independent set of vertices of a graph. St000105The number of blocks in the set partition. St000108The number of partitions contained in the given partition. St000153The number of adjacent cycles of a permutation. St000166The depth minus 1 of an ordered tree. St000167The number of leaves of an ordered tree. St000181The number of connected components of the Hasse diagram for the poset. St000193The row of the unique '1' in the first column of the alternating sign matrix. 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. St000216The absolute length of a permutation. St000286The number of connected components of the complement of a graph. St000288The number of ones in a binary word. St000328The maximum number of child nodes in a tree. St000382The first part of an integer composition. St000383The last part of an integer composition. St000388The number of orbits of vertices of a graph under automorphisms. St000392The length of the longest run of ones in a binary word. St000442The maximal area to the right of an up step of a Dyck path. St000450The number of edges minus the number of vertices plus 2 of a graph. St000452The number of distinct eigenvalues of a graph. St000460The hook length of the last cell along the main diagonal of an integer partition. St000469The distinguishing number of a graph. St000482The (zero)-forcing number of a graph. St000507The number of ascents of a standard tableau. St000528The height of a poset. St000532The total number of rook placements on a Ferrers board. St000539The number of odd inversions of a permutation. St000553The number of blocks of a graph. St000617The number of global maxima of a Dyck path. St000628The balance of a binary word. 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. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000733The row containing the largest entry of a standard tableau. St000736The last entry in the first row of a semistandard tableau. St000739The first entry in the last row of a semistandard tableau. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000797The stat`` of a permutation. St000809The reduced reflection length of the permutation. St000847The number of standard Young tableaux whose descent set is the binary word. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000876The number of factors in the Catalan decomposition of a binary word. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000885The number of critical steps in the Catalan decomposition of a binary word. St000907The number of maximal antichains of minimal length in a poset. St000908The length of the shortest maximal antichain in a poset. St000911The number of maximal antichains of maximal size in a poset. St000912The number of maximal antichains in a poset. St000917The open packing number of a graph. St000926The clique-coclique 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. St000947The major index east count of a Dyck path. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St000971The smallest closer of a set partition. St000982The length of the longest constant subword. St000983The length of the longest alternating subword. St000984The number of boxes below precisely one peak. St000988The orbit size of a permutation under Foata's bijection. 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. St001000Number of indecomposable modules with projective 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. St001041The depth of the label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001048The number of leaves in the subtree containing 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001050The number of terminal closers of a set partition. St001058The breadth of the ordered tree. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001080The minimal length of a factorization of a permutation using the transposition (12) and the cycle (1,. St001102The number of words with multiplicities of the letters given by the composition, avoiding the consecutive pattern 132. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module

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

$L=[c_0,c_1,...,c_{n−1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a special CNakayama algebra. 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: St001207The Lowey length of the algebra

$A/T$ when

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

$K[x]/(x^n)$ . 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. St001220The width of a permutation. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001285The number of primes in the column sums of the two line notation of a permutation. St001286The annihilation number of a graph. St001312Number of parabolic noncrossing partitions indexed by the composition. St001313The number of Dyck paths above the lattice path given by a binary word. St001316The domatic number of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001343The dimension of the reduced incidence algebra of a poset. St001352The number of internal nodes in the modular decomposition of a graph. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001400The total number of Littlewood-Richardson tableaux of given shape. 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. St001462The number of factors of a standard tableaux under concatenation. St001481The minimal height of a peak of a Dyck path. St001486The number of corners of the ribbon associated with an integer composition. St001500The global dimension of magnitude 1 Nakayama algebras. St001530The depth of a Dyck path. St001555The order of a signed permutation. St001589The nesting number of a perfect matching. St001645The pebbling number 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. St001717The largest size of an interval in a poset. St001733The number of weak left to right maxima of a Dyck path. 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. St001778The largest greatest common divisor of an element and its image in a permutation. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001809The index of the step at the first peak of maximal height in a Dyck path. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001855The number of signed permutations less than or equal to a signed permutation in left weak order. St001933The largest multiplicity of a part in an integer partition. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000094The depth of an ordered tree. St000235The number of indices that are not cyclical small weak excedances. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St000439The position of the first down step of a Dyck path. St000521The number of distinct subtrees of an ordered tree. St000673The number of non-fixed points of a permutation. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000844The size of the largest block in the direct sum decomposition of a permutation. St000896The number of zeros on the main diagonal of an alternating sign matrix. 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

$c_{n-1}$ . 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. St001180Number of indecomposable injective modules with projective dimension at most 1. St001267The length of the Lyndon factorization of the binary word. St001288The number of primes obtained by multiplying preimage and image of a permutation and adding one. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001439The number of even weak deficiencies and of odd weak exceedences. 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. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. 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. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. 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. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St000831The number of indices that are either descents or recoils. St000327The number of cover relations in a poset. St000354The number of recoils of a permutation. St000503The maximal difference between two elements in a common block. St000728The dimension of a set partition. St000795The mad of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001061The number of indices that are both descents and recoils of a permutation. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St000494The number of inversions of distance at most 3 of a permutation. St000495The number of inversions of distance at most 2 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. St000579The number of occurrences of the pattern {{1},{2}} such that 2 is a maximal element. St000693The modular (standard) major index of a standard tableau. St000730The maximal arc length of a set partition. St000753The Grundy value for the game of Kayles on a binary word. St000794The mak of a permutation. St000796The stat' of a permutation. St000798The makl of a permutation. St000833The comajor index of a permutation. St000877The depth of the binary word interpreted as a path. 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. St001592The maximal number of simple paths between any two different vertices of a graph. St000326The position of the first one in a binary word after appending a 1 at the end. St000444The length of the maximal rise of a Dyck path. St000485The length of the longest cycle of a permutation. St000489The number of cycles of a permutation of length at most 3. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000910The number of maximal chains of minimal length in a poset. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000990The first ascent of a permutation. St001346The number of parking functions that give the same permutation. St001926Sparre Andersen's position of the maximum of a signed permutation. St001959The product of the heights of the peaks of a Dyck path. St000083The number of left oriented leafs of a binary tree except the first one. St000289The decimal representation of a binary word. St000297The number of leading ones in a binary word. St000391The sum of the positions of the ones in a binary word. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000472The sum of the ascent bottoms of a permutation. St000490The intertwining number of a set partition. St000492The rob statistic of a set partition. St000493The los statistic of a set partition. St000499The rcb statistic of a set partition. St000502The number of successions of a set partitions. St000574The number of occurrences of the pattern {{1},{2}} such that 1 is a minimal and 2 a maximal element. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000792The Grundy value for the game of ruler on a binary word. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000874The position of the last double rise in a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000946The sum of the skew hook positions in a Dyck path. St000989The number of final rises of a permutation. St001281The normalized isoperimetric number of a graph. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001668The number of points of the poset minus the width of the poset. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001927Sparre Andersen's number of positives of a signed permutation. St000061The number of nodes on the left branch of a binary tree. St000082The number of elements smaller than a binary tree in Tamari order. St000100The number of linear extensions of a poset. St000420The number of Dyck paths that are weakly above a Dyck path. St000504The cardinality of the first block of a set partition. St000654The first descent of a permutation. St000675The number of centered multitunnels of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000729The minimal arc length of a set partition. St000823The number of unsplittable factors of the set partition. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St000898The number of maximal entries in the last diagonal of the monotone triangle. St000914The sum of the values of the Möbius function of a poset. St000925The number of topologically connected components of a set partition. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. 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. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001808The box weight or horizontal decoration of a Dyck path. St001877Number of indecomposable injective modules with projective dimension 2. St001498The normalised height of a Nakayama algebra with magnitude 1. St000258The burning number of a graph. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000478Another weight of a partition according to Alladi. St000918The 2-limited packing number of a graph. St001118The acyclic chromatic index of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St001315The dissociation number of a graph. 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. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000259The diameter of a connected graph. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. 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. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001360The number of covering relations in Young's lattice below a partition. St001378The product of the cohook lengths of the integer partition. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001642The Prague dimension of a graph. St000455The second largest eigenvalue of a graph if it is integral. St001624The breadth of a lattice. St001626The number of maximal proper sublattices of a lattice. St000260The radius of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000038The product of the heights of the descending steps of a Dyck path. St000088The row sums of the character table of the symmetric group. St000145The Dyson rank of a partition. St000296The length of the symmetric border of a binary word. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000389The number of runs of ones of odd length in a binary word. St000418The number of Dyck paths that are weakly below a Dyck path. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000549The number of odd partial sums of an integer partition. St000644The number of graphs with given frequency partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000706The product of the factorials of the multiplicities of an integer partition. St000707The product of the factorials of the parts. St000759The smallest missing part in an integer partition. 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. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000921The number of internal inversions of a binary word. 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. St000992The alternating sum of the parts of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001471The magnitude of a Dyck path. 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). St001523The degree of symmetry of a Dyck path. St001531Number of partial orders contained in the poset determined by the Dyck path. St001568The smallest positive integer that does not appear twice in the partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001955The number of natural descents for set-valued two row standard Young tableaux. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in 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

$eA$ . St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000137The Grundy value of an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000567The sum of the products of all pairs of parts. St000618The number of self-evacuating tableaux of given shape. St000699The toughness times the least common multiple of 1,. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000934The 2-degree of an integer partition. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae 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. St001383The BG-rank of an integer partition. St001432The order dimension of the partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001541The Gini index of an integer partition. St001587Half of the largest even part of an integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001637The number of (upper) dissectors of a poset. St001657The number of twos in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. 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. St001961The sum of the greatest common divisors of all pairs of parts. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000464The Schultz index of a connected graph. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000929The constant term of the character polynomial of an integer partition. St001060The distinguishing index of a graph. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001198The number of simple modules in the algebra

$eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001206The maximal dimension of an indecomposable projective

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

$eA$ . St001545The second Elser number of a connected graph. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000177The number of free tiles in the pattern. St000178Number of free entries. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000284The Plancherel distribution on integer partitions. St000302The determinant of the distance matrix of a connected graph. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000456The monochromatic index of a connected graph. St000477The weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000674The number of hills of a Dyck path. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000781The number of proper colouring schemes of a Ferrers diagram. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000815The number of semistandard Young tableaux of partition weight of given shape. 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. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000927The alternating 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. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001139The number of occurrences of hills of size 2 in a Dyck path. 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. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001520The number of strict 3-descents. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001763The Hurwitz number of an integer partition. St001811The Castelnuovo-Mumford regularity of a permutation. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St000958The number of Bruhat factorizations of a permutation. St001208The number of connected components of the quiver of

$A/T$ when

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

$A$ of

$K[x]/(x^n)$ . St001569The maximal modular displacement of a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000075The orbit size of a standard tableau under promotion.