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Your data matches 1084 different statistics following compositions of up to 3 maps.
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Matching statistic: St000743
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(load all 7 compositions to match this statistic)
St000743: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> 0
[[1,2]]
=> 1
[[1],[2]]
=> 1
[[1,2,3]]
=> 2
[[1,3],[2]]
=> 1
[[1,2],[3]]
=> 1
[[1],[2],[3]]
=> 2
Description
The number of entries in a standard Young tableau such that the next integer is a neighbour.
Matching statistic: St001096
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(load all 22 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001096: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001096: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => 0
[[1,2]]
=> [1,2] => 1
[[1],[2]]
=> [2,1] => 1
[[1,2,3]]
=> [1,2,3] => 2
[[1,3],[2]]
=> [2,1,3] => 1
[[1,2],[3]]
=> [3,1,2] => 1
[[1],[2],[3]]
=> [3,2,1] => 2
Description
The size of the overlap set of a permutation.
For a permutation $\pi\in\mathfrak S_n$ this is the number of indices $i < n$ such that the standardisation of $\pi_1\dots\pi_{n-i}$ equals the standardisation of $\pi_{i+1}\dots\pi_n$. In particular, for $n > 1$, the statistic is at least one, because the standardisations of $\pi_1$ and $\pi_n$ are both $1$.
For example, for $\pi=2143$, the standardisations of $21$ and $43$ are equal, and so are the standardisations of $2$ and $3$. Thus, the statistic on $\pi$ is $2$.
Matching statistic: St001405
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(load all 22 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001405: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001405: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => 0
[[1,2]]
=> [1,2] => 1
[[1],[2]]
=> [2,1] => 1
[[1,2,3]]
=> [1,2,3] => 2
[[1,3],[2]]
=> [2,1,3] => 1
[[1,2],[3]]
=> [3,1,2] => 1
[[1],[2],[3]]
=> [3,2,1] => 2
Description
The number of bonds in a permutation.
For a permutation $\pi$, the pair $(\pi_i, \pi_{i+1})$ is a bond if $|\pi_i-\pi_{i+1}| = 1$.
Matching statistic: St001958
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(load all 25 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001958: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001958: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => 0
[[1,2]]
=> [1,2] => 1
[[1],[2]]
=> [2,1] => 1
[[1,2,3]]
=> [1,2,3] => 1
[[1,3],[2]]
=> [2,1,3] => 2
[[1,2],[3]]
=> [3,1,2] => 2
[[1],[2],[3]]
=> [3,2,1] => 1
Description
The degree of the polynomial interpolating the values of a permutation.
Given a permutation $\pi\in\mathfrak S_n$ there is a polynomial $p$ of minimal degree such that $p(n)=\pi(n)$ for $n\in\{1,\dots,n\}$.
This statistic records the degree of $p$.
Matching statistic: St000384
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(load all 6 compositions to match this statistic)
Mp00083: Standard tableaux —shape⟶ Integer partitions
St000384: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000384: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1]
=> 1 = 0 + 1
[[1,2]]
=> [2]
=> 2 = 1 + 1
[[1],[2]]
=> [1,1]
=> 2 = 1 + 1
[[1,2,3]]
=> [3]
=> 3 = 2 + 1
[[1,3],[2]]
=> [2,1]
=> 2 = 1 + 1
[[1,2],[3]]
=> [2,1]
=> 2 = 1 + 1
[[1],[2],[3]]
=> [1,1,1]
=> 3 = 2 + 1
Description
The maximal part of the shifted composition of an integer partition.
A partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ is shifted into a composition by adding $i-1$ to the $i$-th part.
The statistic is then $\operatorname{max}_i\{ \lambda_i + i - 1 \}$.
See also [[St000380]].
Matching statistic: St000507
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Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000507: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [[1]]
=> 1 = 0 + 1
[[1,2]]
=> [[1,2]]
=> 2 = 1 + 1
[[1],[2]]
=> [[1,2]]
=> 2 = 1 + 1
[[1,2,3]]
=> [[1,2,3]]
=> 3 = 2 + 1
[[1,3],[2]]
=> [[1,2],[3]]
=> 2 = 1 + 1
[[1,2],[3]]
=> [[1,2,3]]
=> 3 = 2 + 1
[[1],[2],[3]]
=> [[1,2],[3]]
=> 2 = 1 + 1
Description
The number of ascents of a standard tableau.
Entry $i$ of a standard Young tableau is an '''ascent''' if $i+1$ appears to the right or above $i$ in the tableau (with respect to the English notation for tableaux).
Matching statistic: St000725
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(load all 19 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000725: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000725: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => 1 = 0 + 1
[[1,2]]
=> [1,2] => 2 = 1 + 1
[[1],[2]]
=> [2,1] => 2 = 1 + 1
[[1,2,3]]
=> [1,2,3] => 3 = 2 + 1
[[1,3],[2]]
=> [2,1,3] => 2 = 1 + 1
[[1,2],[3]]
=> [3,1,2] => 2 = 1 + 1
[[1],[2],[3]]
=> [3,2,1] => 3 = 2 + 1
Description
The smallest label of a leaf of the increasing binary tree associated to a permutation.
Matching statistic: St000734
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(load all 2 compositions to match this statistic)
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000734: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [[1]]
=> 1 = 0 + 1
[[1,2]]
=> [[1,2]]
=> 2 = 1 + 1
[[1],[2]]
=> [[1,2]]
=> 2 = 1 + 1
[[1,2,3]]
=> [[1,2,3]]
=> 3 = 2 + 1
[[1,3],[2]]
=> [[1,2],[3]]
=> 2 = 1 + 1
[[1,2],[3]]
=> [[1,2,3]]
=> 3 = 2 + 1
[[1],[2],[3]]
=> [[1,2],[3]]
=> 2 = 1 + 1
Description
The last entry in the first row of a standard tableau.
Matching statistic: St000784
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(load all 7 compositions to match this statistic)
Mp00083: Standard tableaux —shape⟶ Integer partitions
St000784: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000784: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1]
=> 1 = 0 + 1
[[1,2]]
=> [2]
=> 2 = 1 + 1
[[1],[2]]
=> [1,1]
=> 2 = 1 + 1
[[1,2,3]]
=> [3]
=> 3 = 2 + 1
[[1,3],[2]]
=> [2,1]
=> 2 = 1 + 1
[[1,2],[3]]
=> [2,1]
=> 2 = 1 + 1
[[1],[2],[3]]
=> [1,1,1]
=> 3 = 2 + 1
Description
The maximum of the length and the largest part of the integer partition.
This is the side length of the smallest square the Ferrers diagram of the partition fits into. It is also the minimal number of colours required to colour the cells of the Ferrers diagram such that no two cells in a column or in a row have the same colour, see [1].
See also [[St001214]].
Matching statistic: St001439
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(load all 8 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001439: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001439: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => 1 = 0 + 1
[[1,2]]
=> [1,2] => 2 = 1 + 1
[[1],[2]]
=> [2,1] => 2 = 1 + 1
[[1,2,3]]
=> [1,2,3] => 3 = 2 + 1
[[1,3],[2]]
=> [2,1,3] => 3 = 2 + 1
[[1,2],[3]]
=> [3,1,2] => 2 = 1 + 1
[[1],[2],[3]]
=> [3,2,1] => 2 = 1 + 1
Description
The number of even weak deficiencies and of odd weak exceedences.
For a permutation $\sigma$, this is the number of indices $i$ such that $\sigma(i) \leq i$ if $i$ is even and $\sigma(i) \geq i$ if $i$ is odd.
According to [1], $\sigma$ is a '''D-permutation''' if all indices have this property and the coefficients of the characteristic polynomial of the homogenized linial arrangement are given by the number of D-permutations with a given number of cycles.
The following 1074 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001462The number of factors of a standard tableaux under concatenation. St001486The number of corners of the ribbon associated with an integer composition. St001566The length of the longest arithmetic progression in a permutation. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000007The number of saliances of the permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000074The number of special entries. St000153The number of adjacent cycles of a permutation. St000157The number of descents of a standard tableau. St000211The rank of the set partition. St000234The number of global ascents of a permutation. St000245The number of ascents of a permutation. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000441The number of successions of a permutation. St000651The maximal size of a rise in a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000778The metric dimension of a graph. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. 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. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001949The rigidity index of a graph. St001955The number of natural descents for set-valued two row standard Young tableaux. St000050The depth or height of a binary tree. St000056The decomposition (or block) number of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000134The size of the orbit of an alternating sign matrix under gyration. St000147The largest part of an integer partition. St000184The size of the centralizer of any permutation of given cycle type. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000213The number of weak exceedances (also weak excedences) of a permutation. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000240The number of indices that are not small excedances. 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. St000345The number of refinements of a partition. St000378The diagonal inversion number of an integer partition. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000469The distinguishing number of a graph. St000505The biggest entry in the block containing the 1. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000738The first entry in the last row of a standard tableau. St000740The last entry of a permutation. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000808The number of up steps of the associated bargraph. St000863The length of the first row of the shifted shape of a permutation. St000891The number of distinct diagonal sums of a permutation matrix. St000907The number of maximal antichains of minimal length in a poset. St000918The 2-limited packing number of a graph. St000935The number of ordered refinements of an integer partition. St000971The smallest closer of a set partition. St000991The number of right-to-left minima of a permutation. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001288The number of primes obtained by multiplying preimage and image of a permutation and adding one. St001315The dissociation number of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001389The number of partitions of the same length below the given integer partition. St001461The number of topologically connected components of the chord diagram of a permutation. St001488The number of corners of a skew partition. St001497The position of the largest weak excedence of a permutation. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001554The number of distinct nonempty subtrees of a binary tree. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001806The upper middle entry of a permutation. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St000144The pyramid weight of the Dyck path. St000393The number of strictly increasing runs in a binary word. St000395The sum of the heights of the peaks of a Dyck path. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. 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. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001267The length of the Lyndon factorization of the binary word. 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. 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. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St000004The major index of a permutation. St000010The length of the partition. St000011The number of touch points (or returns) of a Dyck path. St000013The height of a Dyck path. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000021The number of descents of a permutation. St000024The number of double up and double down steps of a Dyck path. St000025The number of initial rises of a Dyck path. St000028The number of stack-sorts needed to sort a permutation. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000053The number of valleys of the Dyck path. St000069The number of maximal elements of a poset. St000080The rank of the poset. St000093The cardinality of a maximal independent set of vertices of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000133The "bounce" of a permutation. St000141The maximum drop size of a permutation. St000154The sum of the descent bottoms of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000156The Denert index of a permutation. St000160The multiplicity of the smallest part of a partition. St000209Maximum difference of elements in cycles. St000214The number of adjacencies of a permutation. St000224The sorting index of a permutation. St000237The number of small exceedances. St000238The number of indices that are not small weak excedances. St000246The number of non-inversions of a permutation. St000293The number of inversions of a binary word. St000305The inverse major index of a permutation. St000306The bounce count of a Dyck path. 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. 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. St000339The maf index of a permutation. St000340The number of non-final maximal constant sub-paths of length greater than one. St000374The number of exclusive right-to-left minima of a permutation. St000377The dinv defect of an integer partition. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000442The maximal area to the right of an up step of a Dyck path. St000445The number of rises of length 1 of a Dyck path. St000446The disorder of a permutation. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000546The number of global descents of a permutation. St000548The number of different non-empty partial sums of an integer partition. St000647The number of big descents of a permutation. St000662The staircase size of the code of a permutation. St000670The reversal length of a 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. St000691The number of changes of a binary word. St000692Babson and Steingrímsson's statistic of a permutation. St000703The number of deficiencies of a permutation. St000741The Colin de Verdière graph invariant. St000742The number of big ascents of a permutation after prepending zero. St000819The propagating number of a perfect matching. St000850The number of 1/2-balanced pairs in a poset. St000864The number of circled entries of the shifted recording tableau of a permutation. St000868The aid statistic in the sense of Shareshian-Wachs. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000996The number of exclusive left-to-right maxima of a permutation. 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. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001079The minimal length of a factorization of a permutation using the permutations (12)(34). St001090The number of pop-stack-sorts needed to sort a permutation. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the 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. 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. 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. 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. St001233The number of indecomposable 2-dimensional modules with projective dimension one. 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. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001340The cardinality of a minimal non-edge isolating set of a graph. St001375The pancake length of a permutation. 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. St001424The number of distinct squares 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. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001489The maximum of the number of descents and the number of inverse descents. 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. 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. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001671Haglund's hag of a permutation. St001684The reduced word complexity of a permutation. St001726The number of visible inversions of a permutation. St001759The Rajchgot index of a permutation. St001760The number of prefix or suffix reversals needed to sort a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. 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. St001827The number of two-component spanning forests of a graph. St001863The number of weak excedances of a signed permutation. St001935The number of ascents in a parking function. St000006The dinv of a Dyck path. St000015The number of peaks of a Dyck path. St000026The position of the first return of a Dyck path. St000054The first entry of the permutation. St000058The order of a permutation. St000075The orbit size of a standard tableau under promotion. St000105The number of blocks in the set partition. St000110The number of permutations less than or equal to a permutation in left weak order. St000166The depth minus 1 of an ordered tree. 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. St000203The number of external nodes of a binary tree. St000258The burning number of a graph. St000273The domination number of a graph. St000287The number of connected components of a graph. St000288The number of ones in a binary word. St000290The major index of a binary word. St000325The width of the tree associated to a permutation. St000335The difference of lower and upper interactions. St000346The number of coarsenings of a partition. St000383The last part of an integer composition. St000443The number of long tunnels of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000451The length of the longest pattern of the form k 1 2. St000470The number of runs in a permutation. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000482The (zero)-forcing number of a graph. St000495The number of inversions of distance at most 2 of a permutation. St000501The size of the first part in the decomposition of a permutation. St000519The largest length of a factor maximising the subword complexity. St000522The number of 1-protected nodes of a rooted tree. St000528The height of a poset. St000530The number of permutations with the same descent word as the given permutation. St000542The number of left-to-right-minima of a permutation. St000544The cop number of a graph. St000553The number of blocks of a graph. St000626The minimal period of a binary word. St000636The hull number of a graph. St000638The number of up-down runs of a permutation. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000696The number of cycles in the breakpoint graph of a permutation. 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. 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. St000780The size of the orbit under rotation of a perfect matching. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000831The number of indices that are either descents or recoils. St000839The largest opener of a set partition. St000883The number of longest increasing subsequences of a permutation. St000911The number of maximal antichains of maximal size in a poset. St000912The number of maximal antichains in a poset. St000916The packing number of a graph. St000917The open packing number of a graph. St000922The minimal number such that all substrings of this length are unique. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000942The number of critical left to right maxima of the parking functions. St000945The number of matchings in the dihedral orbit of a perfect matching. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St000982The length of the longest constant subword. St000983The length of the longest alternating subword. 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. 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. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001050The number of terminal closers of a set partition. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001119The length of a shortest maximal path in a graph. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. 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. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001245The cyclic maximal difference between two consecutive entries of a permutation. St001246The maximal difference between two consecutive entries of a permutation. 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. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001343The dimension of the reduced incidence algebra of a poset. St001363The Euler characteristic of a graph according to Knill. St001368The number of vertices of maximal degree in a graph. St001372The length of a longest cyclic run of ones of a binary word. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001437The flex of a binary word. St001463The number of distinct columns in the nullspace of a graph. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001485The modular major index of a binary word. St001530The depth of a Dyck path. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001622The number of join-irreducible elements of a lattice. St001641The number of ascent tops in the flattened set partition such that all smaller elements appear before. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001717The largest size of an interval in a poset. St001733The number of weak left to right maxima of a Dyck path. St001746The coalition number of a graph. St001757The number of orbits of toric promotion on 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. St001807The lower middle entry of a permutation. St001809The index of the step at the first peak of maximal height in a Dyck path. St001814The number of partitions interlacing the given partition. St001828The Euler characteristic of a graph. St001829The common independence number of a graph. St001889The size of the connectivity set of a signed permutation. St001904The length of the initial strictly increasing segment of a parking function. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001925The minimal number of zeros in a row of an alternating sign matrix. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St001937The size of the center of a parking function. St000014The number of parking functions supported by a Dyck path. St000094The depth of an ordered tree. St000197The number of entries equal to positive one in the alternating sign matrix. St000210Minimum over maximum difference of elements in cycles. St000216The absolute length of a permutation. St000229Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St000336The leg major index of a standard tableau. St000439The position of the first down step of a Dyck path. St000521The number of distinct subtrees of an ordered tree. St000744The length of the path to the largest entry in a standard Young tableau. St000809The reduced reflection length of the permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000876The number of factors in the Catalan decomposition of a binary word. St000890The number of nonzero entries in an alternating sign matrix. St000906The length of the shortest maximal chain in a poset. St000924The number of topologically connected components of a perfect matching. 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 $c_{n-1}$. St000981The length of the longest zigzag subpath. St001012Number of simple modules with projective dimension at most 2 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. St001034The area of the parallelogram polyomino associated with the Dyck path. St001040The depth of the decreasing labelled binary unordered tree associated with the perfect matching. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001077The prefix exchange distance of a permutation. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least 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. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001480The number of simple summands of the module J^2/J^3. 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. St000044The number of vertices of the unicellular map given by a perfect matching. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000673The number of non-fixed points of a permutation. St000806The semiperimeter of the associated bargraph. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001254The vector space dimension of the first extension-group between A/soc(A) and J when A is the corresponding Nakayama algebra with Jacobson radical J. St001468The smallest fixpoint of a permutation. St000060The greater neighbor of the maximum. St000402Half the size of the symmetry class of a permutation. St000529The number of permutations whose descent word is the given binary word. St000543The size of the conjugacy class of a binary word. St000619The number of cyclic descents of a permutation. St000627The exponent of a binary word. St000630The length of the shortest palindromic decomposition of a binary word. St000631The number of distinct palindromic decompositions of a binary word. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St001052The length of the exterior of a permutation. St001128The exponens consonantiae of a partition. St001415The length of the longest palindromic prefix of a binary word. St001686The order of promotion on a Gelfand-Tsetlin pattern. St001884The number of borders of a binary word. St000064The number of one-box pattern of a permutation. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000295The length of the border of a binary word. St000353The number of inner valleys of a permutation. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000538The number of even inversions of a permutation. St000628The balance of a binary word. St000711The number of big exceedences of a permutation. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000836The number of descents of distance 2 of a permutation. St000837The number of ascents of distance 2 of a permutation. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001388The number of non-attacking neighbors of a permutation. St001413Half the length of the longest even length palindromic prefix of a binary word. St001524The degree of symmetry of a binary word. St000020The rank of the permutation. St000033The number of permutations greater than or equal to the given permutation in (strong) Bruhat order. St000040The number of regions of the inversion arrangement of a permutation. St000071The number of maximal chains in a poset. St000100The number of linear extensions of a poset. St000109The number of elements less than or equal to the given element in Bruhat order. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000392The length of the longest run of ones in a binary word. St000502The number of successions of a set partitions. St000503The maximal difference between two elements in a common block. St000527The width of the poset. St000545The number of parabolic double cosets with minimal element being the given permutation. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000652The maximal difference between successive positions of a permutation. St000657The smallest part of an integer composition. St000690The size of the conjugacy class of a permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000728The dimension of a set partition. St000753The Grundy value for the game of Kayles on a binary word. St000877The depth of the binary word interpreted as a path. St000886The number of permutations with the same antidiagonal sums. 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. St000904The maximal number of repetitions of an integer composition. St000909The number of maximal chains of maximal size in a poset. St000910The number of maximal chains of minimal length in a poset. St000925The number of topologically connected components of a set partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000988The orbit size of a permutation under Foata's bijection. St000989The number of final rises of a permutation. St000990The first ascent of a permutation. St001081The number of minimal length factorizations of a permutation into star transpositions. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001220The width of a permutation. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001268The size of the largest ordinal summand in the poset. St001312Number of parabolic noncrossing partitions indexed by the composition. St001313The number of Dyck paths above the lattice path given by a binary word. St001399The distinguishing number of a poset. St001482The product of the prefix sums of a permutation. 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. St001510The number of self-evacuating linear extensions of a finite poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001675The number of parts equal to the part in the reversed composition. St001779The order of promotion on the set of linear extensions of a poset. St001915The size of the component corresponding to a necklace in Bulgarian solitaire. St001948The number of augmented double ascents of a permutation. St000008The major index of the composition. St000055The inversion sum of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000247The number of singleton blocks of a set partition. St000289The decimal representation of a binary word. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000304The load of a permutation. St000326The position of the first one in a binary word after appending a 1 at the end. St000341The non-inversion sum of a permutation. St000347The inversion sum of a binary word. St000348The non-inversion sum of a binary word. St000352The Elizalde-Pak rank of a permutation. St000354The number of recoils of a permutation. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000389The number of runs of ones of odd length in a binary word. St000390The number of runs of ones in a binary word. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000435The number of occurrences of the pattern 213 or of the pattern 231 in a permutation. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000462The major index minus the number of excedences of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000490The intertwining number of a set partition. St000491The number of inversions of a set partition. St000493The los statistic of a set partition. St000494The number of inversions of distance at most 3 of a permutation. St000497The lcb statistic of a set partition. St000498The lcs statistic of a set partition. St000504The cardinality of the first block of a set partition. St000508Eigenvalues of the random-to-random operator acting on a simple module. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000539The number of odd inversions of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000554The number of occurrences of the pattern {{1,2},{3}} in a set partition. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000565The major index of a set partition. St000572The dimension exponent of a set partition. St000574The number of occurrences of the pattern {{1},{2}} such that 1 is a minimal and 2 a maximal element. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000624The normalized sum of the minimal distances to a greater element. St000632The jump number of the poset. St000646The number of big ascents of a permutation. St000653The last descent of a permutation. St000654The first descent of a permutation. St000661The number of rises of length 3 of a Dyck path. St000668The least common multiple of the parts of the partition. St000677The standardized bi-alternating inversion number of a permutation. St000682The Grundy value of Welter's game on a binary word. St000693The modular (standard) major index of a standard tableau. St000702The number of weak deficiencies of a permutation. St000708The product of the parts of an integer partition. St000710The number of big deficiencies of a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000732The number of double deficiencies of a permutation. St000747A variant of the major index of a set partition. St000794The mak of a permutation. St000795The mad of a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000823The number of unsplittable factors of the set partition. St000829The Ulam distance of a permutation to the identity permutation. St000833The comajor index of a permutation. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000928The sum of the coefficients of the character polynomial of an integer partition. St000931The number of occurrences of the pattern UUU in a Dyck path. St000933The number of multipartitions of sizes given by an integer partition. St000941The number of characters of the symmetric group whose value on the partition is even. St000961The shifted major index of a permutation. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001061The number of indices that are both descents and recoils of a permutation. St001062The maximal size of a block of a set partition. St001080The minimal length of a factorization of a permutation using the transposition (12) and the cycle (1,. St001082The number of boxed occurrences of 123 in a permutation. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. 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. St001114The number of odd descents of a permutation. St001130The number of two successive successions in a permutation. St001141The number of occurrences of hills of size 3 in a Dyck path. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001412Number of minimal entries in the Bruhat order matrix of a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001552The number of inversions between excedances and fixed points of a permutation. St001557The number of inversions of the second entry of a permutation. St001569The maximal modular displacement of a permutation. St001665The number of pure excedances of a permutation. St001713The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern. St001729The number of visible descents of a permutation. St001731The factorization defect of a permutation. St001737The number of descents of type 2 in a permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. 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. St001931The weak major index of an integer composition regarded as a word. St001960The number of descents of a permutation minus one if its first entry is not one. St001404The number of distinct entries in a Gelfand Tsetlin pattern. 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. St000061The number of nodes on the left branch of a binary tree. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000068The number of minimal elements in a poset. St000083The number of left oriented leafs of a binary tree except the first one. St000086The number of subgraphs. St000172The Grundy number of a graph. 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. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000269The number of acyclic orientations of a graph. St000270The number of forests contained in a graph. St000277The number of ribbon shaped standard tableaux. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St000286The number of connected components of the complement of a graph. St000297The number of leading ones in a binary word. St000299The number of nonisomorphic vertex-induced subtrees. St000327The number of cover relations in a poset. St000343The number of spanning subgraphs of a graph. St000363The number of minimal vertex covers of a graph. St000391The sum of the positions of the ones in a binary word. St000418The number of Dyck paths that are weakly below a Dyck path. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000468The Hosoya index of a graph. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000568The hook number of a binary tree. St000617The number of global maxima of a Dyck path. St000655The length of the minimal rise of a Dyck path. St000667The greatest common divisor of the parts of the partition. St000675The number of centered multitunnels of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000694The number of affine bounded permutations that project to a given permutation. St000707The product of the factorials of the parts. St000722The number of different neighbourhoods in a graph. St000729The minimal arc length of a set partition. St000730The maximal arc length of a set partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000767The number of runs in an integer composition. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000792The Grundy value for the game of ruler on a binary word. St000796The stat' of a permutation. St000797The stat`` of a permutation. St000798The makl of a permutation. 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. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St000820The number of compositions obtained by rotating the composition. St000822The Hadwiger number of the graph. St000874The position of the last double rise in a Dyck path. St000898The number of maximal entries in the last diagonal of the monotone triangle. St000903The number of different parts of an integer composition. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St000932The number of occurrences of the pattern UDU in a Dyck path. St000937The number of positive values of the symmetric group character corresponding to the partition. St000972The composition number of a graph. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001093The detour number of a graph. St001102The number of words with multiplicities of the letters given by the composition, avoiding the consecutive pattern 132. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001108The 2-dynamic chromatic number of a graph. St001109The number of proper colourings of a graph with as few colours as possible. St001110The 3-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. 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. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. 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. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. 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. St001346The number of parking functions that give the same permutation. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001471The magnitude of a Dyck path. St001474The evaluation of the Tutte polynomial of the graph at (x,y) equal to (2,-1). St001481The minimal height of a peak of a Dyck path. St001494The Alon-Tarsi number of a graph. St001500The global dimension of magnitude 1 Nakayama algebras. St001527The cyclic permutation representation number of an integer partition. St001531Number of partial orders contained in the poset determined by the Dyck path. St001555The order of a signed permutation. St001571The Cartan determinant of the integer partition. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001614The cyclic permutation representation number of a skew partition. St001642The Prague dimension of a graph. St001668The number of points of the poset minus the width of the poset. St001670The connected partition number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001721The degree of a binary word. St001725The harmonious chromatic number of a graph. St001758The number of orbits of promotion on a graph. St001765The number of connected components of the friends and strangers graph. St001770The number of facets of a certain subword complex associated with the signed permutation. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St001813The product of the sizes of the principal order filters in a poset. St001852The size of the conjugacy class of the signed permutation. St001855The number of signed permutations less than or equal to a signed permutation in left weak order. St001883The mutual visibility number of a graph. St001933The largest multiplicity of a part in an integer partition. St001959The product of the heights of the peaks of a Dyck path. St001963The tree-depth of a graph. St000005The bounce statistic of a Dyck path. St000012The area of a Dyck path. St000016The number of attacking pairs of a standard tableau. St000051The size of the left subtree of a binary tree. St000067The inversion number of the alternating sign matrix. St000076The rank of the alternating sign matrix in the alternating sign matrix poset. St000081The number of edges of a graph. St000089The absolute variation of a composition. St000120The number of left tunnels of a Dyck path. St000142The number of even parts of a partition. St000143The largest repeated part of a partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000150The floored half-sum of the multiplicities of a partition. St000161The sum of the sizes of the right subtrees of a binary tree. St000171The degree of the graph. St000185The weighted size of a partition. St000219The number of occurrences of the pattern 231 in a permutation. St000248The number of anti-singletons of a set partition. St000251The number of nonsingleton blocks of a set partition. St000253The crossing number of a set partition. St000254The nesting number of a set partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St000257The number of distinct parts of a partition that occur at least twice. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000263The Szeged index of a graph. St000265The Wiener index of a graph. St000272The treewidth of a graph. St000274The number of perfect matchings of a graph. St000310The minimal degree of a vertex of a graph. St000332The positive inversions of an alternating sign matrix. St000361The second Zagreb index of a graph. St000362The size of a minimal vertex cover of a graph. St000379The number of Hamiltonian cycles in a graph. St000387The matching number of a graph. 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. St000461The rix statistic of a permutation. St000471The sum of the ascent tops of a permutation. St000472The sum of the ascent bottoms of a permutation. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000492The rob statistic of a set partition. St000499The rcb statistic of a set partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000535The rank-width of a graph. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St000556The number of occurrences of the pattern {{1},{2,3}} in a set partition. St000557The number of occurrences of the pattern {{1},{2},{3}} in a set partition. St000558The number of occurrences of the pattern {{1,2}} in a set partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000573The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element. St000578The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton. St000579The number of occurrences of the pattern {{1},{2}} such that 2 is a maximal element. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000586The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000591The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal. St000592The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal. St000593The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal. St000595The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000598The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, 3 is maximal, (2,3) are consecutive in a block. St000599The number of occurrences of the pattern {{1},{2,3}} such that (2,3) are consecutive in a block. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000605The number of occurrences of the pattern {{1},{2,3}} such that 3 is maximal, (2,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000607The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000612The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) are consecutive in a block. St000614The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. 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. St000680The Grundy value for Hackendot on posets. 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. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000699The toughness times the least common multiple of 1,. St000717The number of ordinal summands of a poset. St000779The tier of a permutation. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000873The aix statistic of a permutation. 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. St000893The number of distinct diagonal sums of an alternating sign matrix. St000919The number of maximal left branches of a binary tree. St000934The 2-degree of an integer partition. St000946The sum of the skew hook positions in a Dyck path. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000984The number of boxes below precisely one peak. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001001The number of indecomposable modules with projective and injective dimension equal to 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. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001071The beta invariant of the graph. St001091The number of parts in an integer partition whose next smaller part has the same size. St001092The number of distinct even parts of a partition. St001094The depth index of a set partition. St001117The game chromatic index of a graph. St001120The length of a longest path in a graph. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001139The number of occurrences of hills of size 2 in a Dyck path. St001153The number of blocks with even minimum in a set partition. St001161The major index north count of a Dyck path. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. 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. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. 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 $eA$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. 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. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001270The bandwidth of a graph. St001271The competition number of a graph. 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. St001281The normalized isoperimetric number of a graph. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001333The cardinality of a minimal edge-isolating set of a graph. St001341The number of edges in the center of a graph. St001347The number of pairs of vertices of a graph having the same neighbourhood. 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. St001357The maximal degree of a regular spanning subgraph of a graph. St001358The largest degree of a regular subgraph of a graph. St001362The normalized Knill dimension of a graph. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001393The induced matching number of a graph. St001395The number of strictly unfriendly partitions of a graph. St001427The number of descents of a signed permutation. St001428The number of B-inversions of a signed permutation. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001479The number of bridges of a graph. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001512The minimum rank of a graph. St001541The Gini index of an integer partition. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001587Half of the largest even part of an integer partition. St001592The maximal number of simple paths between any two different vertices of a graph. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001644The dimension of a graph. St001649The length of a longest trail in a graph. St001657The number of twos in an integer partition. St001673The degree of asymmetry of an integer composition. St001697The shifted natural comajor index of a standard Young tableau. 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. St001743The discrepancy of a graph. St001769The reflection length of a signed permutation. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001782The order of rowmotion on the set of order ideals of a poset. St001783The number of odd automorphisms of a graph. St001792The arboricity of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001799The number of proper separations of a graph. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. 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. St001848The atomic length of a signed permutation. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001861The number of Bruhat lower covers of a permutation. St001864The number of excedances of a signed permutation. St001869The maximum cut size 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). St001894The depth of a signed permutation. St001896The number of right descents of a signed permutations. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. 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. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001956The comajor index for set-valued two-row standard Young tableaux. St001961The sum of the greatest common divisors of all pairs of parts. St001962The proper pathwidth of a graph. St000643The size of the largest orbit of antichains under Panyushev complementation. St001875The number of simple modules with projective dimension at most 1. St000259The diameter of a connected graph. St001118The acyclic chromatic index of a graph. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000045The number of linear extensions of a binary tree. St000260The radius of a connected graph. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001360The number of covering relations in Young's lattice below a partition. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001498The normalised height of a Nakayama algebra with magnitude 1. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. 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. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001168The vector space dimension of the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000063The number of linear extensions of a certain poset defined for an integer partition. St000082The number of elements smaller than a binary tree in Tamari order. St000088The row sums of the character table of the symmetric group. St000108The number of partitions contained in the given partition. St000117The number of centered tunnels of a Dyck path. St000148The number of odd parts of a partition. St000179The product of the hook lengths of the integer partition. St000228The size 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. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000420The number of Dyck paths that are weakly above a Dyck path. St000459The hook length of the base cell of a partition. St000475The number of parts equal to 1 in a partition. 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. St000531The leading coefficient of the rook polynomial of an integer partition. St000532The total number of rook placements on a Ferrers board. 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. St000759The smallest missing part in an integer partition. St000770The major index of an integer partition when read from bottom to top. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000782The indicator function of whether a given perfect matching is an L & P matching. St000811The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to Schur symmetric functions. St000812The sum of the entries in the column specified by the partition of the change of basis matrix from complete homogeneous symmetric functions to monomial symmetric functions. St000815The number of semistandard Young tableaux of partition weight of given shape. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000867The sum of the hook lengths in the first row of an integer partition. St000878The number of ones minus the number of zeros of a binary word. St000885The number of critical steps in the Catalan decomposition 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. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001008Number of indecomposable injective modules with projective dimension 1 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. St001075The minimal size of a block of a set partition. St001127The sum of the squares of the parts of a partition. St001400The total number of Littlewood-Richardson tableaux of given shape. St001432The order dimension of the partition. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001523The degree of symmetry of a Dyck path. St001568The smallest positive integer that does not appear twice in the partition. St001570The minimal number of edges to add to make a graph Hamiltonian. 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. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001612The number of coloured multisets of cycles such that the multiplicities of colours are given by a partition. St001637The number of (upper) dissectors of a poset. St001645The pebbling number of a connected graph. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. St001732The number of peaks visible from the left. St001780The order of promotion on the set of standard tableaux of given shape. St001808The box weight or horizontal decoration of a Dyck path. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. 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. St001910The height of the middle non-run of a Dyck path. 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. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000438The position of the last up step in a Dyck path. St000466The Gutman (or modified Schultz) index of a connected graph. St000478Another weight of a partition according to Alladi. St000564The number of occurrences of the pattern {{1},{2}} in a set partition. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000639The number of relations in a poset. St000641The number of non-empty boolean intervals in a poset. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000827The decimal representation of a binary word with a leading 1. St000929The constant term of the character polynomial of an integer partition. St000947The major index east count of a Dyck path. St000976The sum of the positions of double up-steps of a Dyck path. St001030Half the number of non-boundary horizontal edges in the fully packed loop corresponding to the alternating sign matrix. St001095The number of non-isomorphic posets with precisely one further covering relation. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. 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 $eA$. 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. 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 $eA$. St001371The length of the longest Yamanouchi prefix of a binary word. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000634The number of endomorphisms of a poset. St001858The number of covering elements of a signed permutation in absolute order. St000101The cocharge of a semistandard tableau. St000456The monochromatic index of a connected graph. St000618The number of self-evacuating tableaux of given shape. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000781The number of proper colouring schemes of a Ferrers diagram. 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. St001593This is the number of standard Young tableaux of the given shifted shape. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001763The Hurwitz number of an integer partition. 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. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St000739The first entry in the last row of a semistandard tableau. St001401The number of distinct entries in a semistandard tableau.
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