Your data matches 505 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001160
(load all 57 compositions to match this statistic)
Mapping
St001160: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0 = 1 - 1
[1,2] => 0 = 1 - 1
[2,1] => 0 = 1 - 1
[1,2,3] => 2 = 3 - 1
[1,3,2] => 1 = 2 - 1
[2,1,3] => 1 = 2 - 1
[2,3,1] => 1 = 2 - 1
[3,1,2] => 1 = 2 - 1
[3,2,1] => 2 = 3 - 1
Description
The number of proper blocks (or intervals) of a permutations. Let $\pi = [\pi_1,\ldots,\pi_n]$ be a permutation. A block (or interval) of $\pi$ is then a consecutive subpattern $\pi_i,\ldots,\pi_{i+k}$ whose values form a set of contiguous integers.
Matching statistic: St001807
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Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001807: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1
[1,2] => [1,2] => 1
[2,1] => [1,2] => 1
[1,2,3] => [1,2,3] => 2
[1,3,2] => [1,2,3] => 2
[2,1,3] => [1,2,3] => 2
[2,3,1] => [1,2,3] => 2
[3,1,2] => [1,3,2] => 3
[3,2,1] => [1,3,2] => 3
Description
The lower middle entry of a permutation. This is the entry $\sigma(\frac{n+1}{2})$ when $n$ is odd, and $\sigma(\frac{n}{2})$ when $n$ is even, where $n$ is the size of the permutation $\sigma$.
Matching statistic: St000484
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Mapping
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000484: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0 = 1 - 1
[1,2] => [1,2] => 0 = 1 - 1
[2,1] => [2,1] => 0 = 1 - 1
[1,2,3] => [1,2,3] => 1 = 2 - 1
[1,3,2] => [1,3,2] => 2 = 3 - 1
[2,1,3] => [2,1,3] => 2 = 3 - 1
[2,3,1] => [3,2,1] => 1 = 2 - 1
[3,1,2] => [3,2,1] => 1 = 2 - 1
[3,2,1] => [3,2,1] => 1 = 2 - 1
Description
The sum of St000483 over all subsequences of length at least three.
Matching statistic: St001626
(load all 2 compositions to match this statistic)
Mapping
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001626: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([(0,1)],2)
 => 2 = 1 + 1
[1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => 4 = 3 + 1
[1,3,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 3 = 2 + 1
[2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 3 = 2 + 1
[2,3,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 3 = 2 + 1
[3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 3 = 2 + 1
[3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => 4 = 3 + 1
Description
The number of maximal proper sublattices of a lattice.
Matching statistic: St000058
(load all 6 compositions to match this statistic)
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
St000058: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 1
[1,2] => [1,2] => [1,2] => 1
[2,1] => [1,2] => [1,2] => 1
[1,2,3] => [1,2,3] => [3,2,1] => 2
[1,3,2] => [1,2,3] => [3,2,1] => 2
[2,1,3] => [1,2,3] => [3,2,1] => 2
[2,3,1] => [1,2,3] => [3,2,1] => 2
[3,1,2] => [1,3,2] => [2,3,1] => 3
[3,2,1] => [1,3,2] => [2,3,1] => 3
Description
The order of a permutation. $\operatorname{ord}(\pi)$ is given by the minimial $k$ for which $\pi^k$ is the identity permutation.
Matching statistic: St000182
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Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000182: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
 => 1
[1,2] => [1,2] => [2]
 => 1
[2,1] => [1,2] => [2]
 => 1
[1,2,3] => [1,2,3] => [3]
 => 2
[1,3,2] => [1,2,3] => [3]
 => 2
[2,1,3] => [1,2,3] => [3]
 => 2
[2,3,1] => [1,2,3] => [3]
 => 2
[3,1,2] => [1,3,2] => [2,1]
 => 3
[3,2,1] => [1,3,2] => [2,1]
 => 3
Description
The number of permutations whose cycle type is the given integer partition. This number is given by $$\{ \pi \in \mathfrak{S}_n : \text{type}(\pi) = \lambda\} = \frac{n!}{\lambda_1 \cdots \lambda_k \mu_1(\lambda)! \cdots \mu_n(\lambda)!}$$ where $\mu_j(\lambda)$ denotes the number of parts of $\lambda$ equal to $j$. All permutations with the same cycle type form a [[wikipedia:Conjugacy class]].
Matching statistic: St001464
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Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
St001464: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 1
[1,2] => [1,2] => [1,2] => 1
[2,1] => [1,2] => [1,2] => 1
[1,2,3] => [1,2,3] => [3,2,1] => 2
[1,3,2] => [1,2,3] => [3,2,1] => 2
[2,1,3] => [1,2,3] => [3,2,1] => 2
[2,3,1] => [1,2,3] => [3,2,1] => 2
[3,1,2] => [1,3,2] => [2,3,1] => 3
[3,2,1] => [1,3,2] => [2,3,1] => 3
Description
The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise.
Matching statistic: St001497
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Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St001497: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 1
[1,2] => [1,2] => [2,1] => 1
[2,1] => [1,2] => [2,1] => 1
[1,2,3] => [1,2,3] => [2,3,1] => 2
[1,3,2] => [1,2,3] => [2,3,1] => 2
[2,1,3] => [1,2,3] => [2,3,1] => 2
[2,3,1] => [1,2,3] => [2,3,1] => 2
[3,1,2] => [1,3,2] => [2,1,3] => 3
[3,2,1] => [1,3,2] => [2,1,3] => 3
Description
The position of the largest weak excedence of a permutation.
Matching statistic: St001528
(load all 14 compositions to match this statistic)
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St001528: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 1
[1,2] => [1,2] => [2,1] => 1
[2,1] => [1,2] => [2,1] => 1
[1,2,3] => [1,2,3] => [2,3,1] => 2
[1,3,2] => [1,2,3] => [2,3,1] => 2
[2,1,3] => [1,2,3] => [2,3,1] => 2
[2,3,1] => [1,2,3] => [2,3,1] => 2
[3,1,2] => [1,3,2] => [3,2,1] => 3
[3,2,1] => [1,3,2] => [3,2,1] => 3
Description
The number of permutations such that the product with the permutation has the same number of fixed points. More formally, given a permutation $\pi$, this is the number of permutations $\sigma$ such that $\pi$ and $\pi\sigma$ have the same number of fixed points. Note that the number of permutations $\sigma$ such that $\pi$ and $\pi\sigma$ have the same cycle type is the size of the conjugacy class of $\pi$, [[St000690]].
Matching statistic: St001599
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Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001599: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
 => 1
[1,2] => [1,2] => [2]
 => 1
[2,1] => [1,2] => [2]
 => 1
[1,2,3] => [1,2,3] => [3]
 => 2
[1,3,2] => [1,2,3] => [3]
 => 2
[2,1,3] => [1,2,3] => [3]
 => 2
[2,3,1] => [1,2,3] => [3]
 => 2
[3,1,2] => [1,3,2] => [2,1]
 => 3
[3,2,1] => [1,3,2] => [2,1]
 => 3
Description
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees.
The following 495 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001806The upper middle entry of a permutation. St001820The size of the image of the pop stack sorting operator. St001957The number of Hasse diagrams with a given underlying undirected graph. St000028The number of stack-sorts needed to sort a permutation. St000155The number of exceedances (also excedences) of a permutation. St000246The number of non-inversions of a permutation. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000651The maximal size of a rise in a permutation. St000670The reversal length of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St001375The pancake length of a permutation. St001402The number of separators in a permutation. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001651The Frankl number of a lattice. St001760The number of prefix or suffix reversals needed to sort a permutation. St001956The comajor index for set-valued two-row standard Young tableaux. St000007The number of saliances of the permutation. St000010The length of the partition. St000031The number of cycles in the cycle decomposition of a permutation. St000054The first entry of the permutation. St000056The decomposition (or block) number of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000110The number of permutations less than or equal to a permutation in left weak order. St000147The largest part of an integer partition. St000153The number of adjacent cycles of a permutation. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000240The number of indices that are not small excedances. St000271The chromatic index of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000321The number of integer partitions of n that are dominated by an integer partition. St000325The width of the tree associated to a permutation. St000345The number of refinements of a partition. St000346The number of coarsenings of a partition. St000378The diagonal inversion number of an integer partition. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000451The length of the longest pattern of the form k 1 2. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000470The number of runs in a permutation. St000501The size of the first part in the decomposition of a permutation. St000539The number of odd inversions of a permutation. St000542The number of left-to-right-minima of a permutation. St000638The number of up-down runs of a permutation. St000734The last entry in the first row of a standard tableau. St000737The last entry on the main diagonal of a semistandard tableau. St000740The last entry of a permutation. St000810The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to monomial symmetric functions. 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. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000904The maximal number of repetitions of an integer composition. 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. 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. St001093The detour number of a graph. St001111The weak 2-dynamic chromatic number of a graph. St001112The 3-weak dynamic number of a graph. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001386The number of prime labellings of a graph. St001389The number of partitions of the same length below the given integer partition. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001424The number of distinct squares in a binary word. St001461The number of topologically connected components of the chord diagram of a permutation. St001468The smallest fixpoint of a permutation. St001480The number of simple summands of the module J^2/J^3. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001555The order of a signed permutation. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001674The number of vertices of the largest induced star graph in the graph. St001778The largest greatest common divisor of an element and its image in a permutation. St001823The Stasinski-Voll length of a signed permutation. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St000002The number of occurrences of the pattern 123 in a permutation. St000004The major index of a permutation. St000005The bounce statistic of a Dyck path. St000006The dinv 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. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000120The number of left tunnels of a Dyck path. St000133The "bounce" of a permutation. St000136The dinv of a parking function. St000141The maximum drop size of a permutation. St000156The Denert index of a permutation. St000172The Grundy number of a graph. St000194The number of primary dinversion pairs of a labelled dyck path corresponding to a parking function. St000203The number of external nodes of a binary tree. St000209Maximum difference of elements in cycles. St000211The rank of the set partition. St000223The number of nestings in the permutation. St000224The sorting index of a permutation. St000234The number of global ascents of a permutation. St000235The number of indices that are not cyclical small weak excedances. St000237The number of small exceedances. St000238The number of indices that are not small weak excedances. St000245The number of ascents of a permutation. St000304The load of a permutation. St000305The inverse major index of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000333The dez statistic, the number of descents 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. St000357The number of occurrences of the pattern 12-3. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000374The number of exclusive right-to-left minima of a permutation. St000427The number of occurrences of the pattern 123 or of the pattern 231 in a permutation. St000429The number of occurrences of the pattern 123 or of the pattern 321 in a permutation. St000430The number of occurrences of the pattern 123 or of the pattern 312 in a permutation. St000441The number of successions of a permutation. St000446The disorder of a permutation. St000463The number of admissible inversions of a permutation. St000546The number of global descents of a permutation. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000631The number of distinct palindromic decompositions of a binary word. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000662The staircase size of the code of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000703The number of deficiencies of a permutation. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000742The number of big ascents of a permutation after prepending zero. St000837The number of ascents of distance 2 of a permutation. St000846The maximal number of elements covering an element of a poset. St000864The number of circled entries of the shifted recording tableau of a permutation. St000868The aid statistic in the sense of Shareshian-Wachs. St000918The 2-limited packing number of a graph. St000922The minimal number such that all substrings of this length are unique. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001079The minimal length of a factorization of a permutation using the permutations (12)(34). St001082The number of boxed occurrences of 123 in a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001176The size of a partition minus its first part. St001180Number of indecomposable injective modules with projective dimension at most 1. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001274The number of indecomposable injective modules with projective dimension equal to two. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001287The number of primes obtained by multiplying preimage and image of a permutation and subtracting one. St001298The number of repeated entries in the Lehmer code of a permutation. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001311The cyclomatic number of a graph. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001323The independence gap of a graph. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001423The number of distinct cubes in a binary word. St001439The number of even weak deficiencies and of odd weak exceedences. St001470The cyclic holeyness of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001512The minimum rank of a graph. 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. St001581The achromatic number of a graph. St001584The area statistic between a Dyck path and its bounce path. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001671Haglund's hag of a permutation. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001684The reduced word complexity of a permutation. St001689The number of celebrities in a graph. St001692The number of vertices with higher degree than the average degree in a graph. St001726The number of visible inversions of a permutation. St001759The Rajchgot index of a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001769The reflection length of a signed permutation. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001864The number of excedances of a signed permutation. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001930The weak major index of a binary word. St001672The restrained domination number of a graph. St000625The sum of the minimal distances to a greater element. St000294The number of distinct factors of a binary word. St000518The number of distinct subsequences in a binary word. St000520The number of patterns in a permutation. St000485The length of the longest cycle of a permutation. St000494The number of inversions of distance at most 3 of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000509The diagonal index (content) of a partition. St000690The size of the conjugacy class of a permutation. St000794The mak of a permutation. St000795The mad of a permutation. St001346The number of parking functions that give the same permutation. St001915The size of the component corresponding to a necklace in Bulgarian solitaire. St001926Sparre Andersen's position of the maximum of a signed permutation. St000189The number of elements in the poset. St000216The absolute length of a permutation. St000656The number of cuts of a poset. St000770The major index of an integer partition when read from bottom to top. St000809The reduced reflection length of the permutation. St001030Half the number of non-boundary horizontal edges in the fully packed loop corresponding to the alternating sign matrix. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001077The prefix exchange distance of a permutation. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001717The largest size of an interval in a poset. St000231Sum of the maximal elements of the blocks of a set partition. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St000289The decimal representation of a binary word. St000391The sum of the positions of the ones in a binary word. St000420The number of Dyck paths that are weakly above a Dyck path. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000504The cardinality of the first block of a set partition. St000567The sum of the products of all pairs of parts. St000654The first descent of a permutation. St000668The least common multiple of the parts of the partition. St000675The number of centered multitunnels of a Dyck path. St000699The toughness times the least common multiple of 1,. St000702The number of weak deficiencies of a permutation. St000708The product of the parts of an integer partition. St000727The largest label of a leaf in the binary search tree associated with the permutation. 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. St000823The number of unsplittable factors of the set partition. St000833The comajor index of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000849The number of 1/3-balanced pairs in a poset. St000867The sum of the hook lengths in the first row of an integer partition. St000874The position of the last double rise in a Dyck path. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000984The number of boxes below precisely one peak. St000990The first ascent of a permutation. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001062The maximal size of a block of a set partition. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001118The acyclic chromatic index of a graph. St001228The vector space dimension of the space of module homomorphisms between J and itself when J denotes the Jacobson radical of the corresponding Nakayama algebra. St001242The toal dimension of certain Sn modules determined by LLT polynomials associated with a Dyck path. St001243The sum of coefficients in the Schur basis of certain LLT polynomials associated with a 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. St001441The number of non-empty connected induced subgraphs of a graph. St001500The global dimension of magnitude 1 Nakayama algebras. St001721The degree of a binary word. St001808The box weight or horizontal decoration of a Dyck path. St000180The number of chains of a poset. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St000288The number of ones in a binary word. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St000327The number of cover relations in a poset. St000354The number of recoils of a permutation. St000392The length of the longest run of ones in a binary word. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000438The position of the last up step in a Dyck path. St000462The major index minus the number of excedences of a permutation. St000472The sum of the ascent bottoms of a permutation. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000639The number of relations in a poset. St000641The number of non-empty boolean intervals in a poset. St000653The last descent of a permutation. St000753The Grundy value for the game of Kayles on a binary word. St000779The tier of a permutation. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000829The Ulam distance of a permutation to the identity permutation. St000831The number of indices that are either descents or recoils. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000872The number of very big descents of a permutation. St000947The major index east count of a Dyck path. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St000961The shifted major index of a permutation. St000981The length of the longest zigzag subpath. St000989The number of final rises of a permutation. St001080The minimal length of a factorization of a permutation using the transposition (12) and the cycle (1,. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. St001259The vector space dimension of the double dual of D(A) in the corresponding Nakayama algebra. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001372The length of a longest cyclic run of ones of a binary word. St001391The disjunction number of a graph. St001406The number of nonzero entries in a Gelfand Tsetlin pattern. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001419The length of the longest palindromic factor beginning with a one of 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. St001834The number of non-isomorphic minors of a graph. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001948The number of augmented double ascents of a permutation. St001959The product of the heights of the peaks of a Dyck path. St000477The weight of a partition according to Alladi. St000928The sum of the coefficients of the character polynomial of an integer partition. St001875The number of simple modules with projective dimension at most 1. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000095The number of triangles of a graph. St000259The diameter of a connected graph. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001742The difference of the maximal and the minimal degree in a graph. St000045The number of linear extensions of a binary tree. St000219The number of occurrences of the pattern 231 in a permutation. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000487The length of the shortest cycle of a permutation. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000677The standardized bi-alternating inversion number of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001114The number of odd descents of a permutation. St001168The vector space dimension of the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001569The maximal modular displacement of a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000460The hook length of the last cell along the main diagonal of an integer partition. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000643The size of the largest orbit of antichains under Panyushev complementation. St000827The decimal representation of a binary word with a leading 1. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001060The distinguishing index of a graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001645The pebbling number of a connected graph. St000082The number of elements smaller than a binary tree in Tamari order. St000260The radius of a connected graph. St000326The position of the first one in a binary word after appending a 1 at the end. St000418The number of Dyck paths that are weakly below a Dyck path. St000444The length of the maximal rise of a Dyck path. St000466The Gutman (or modified Schultz) index of a connected graph. St000678The number of up steps after the last double rise of a Dyck path. St000706The product of the factorials of the multiplicities of an integer partition. St000707The product of the factorials of the parts. St000717The number of ordinal summands of a poset. St000729The minimal arc length of a set partition. St000815The number of semistandard Young tableaux of partition weight of given shape. 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. St000906The length of the shortest maximal chain 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. St000993The multiplicity of the largest part of an integer partition. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. 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 $eA$. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001531Number of partial orders contained in the poset determined by the Dyck path. St001568The smallest positive integer that does not appear twice in the partition. St001858The number of covering elements of a signed permutation in absolute order. St000083The number of left oriented leafs of a binary tree except the first one. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000297The number of leading ones in a binary word. St000389The number of runs of ones of odd length in a binary word. St000390The number of runs of ones in a binary word. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000442The maximal area to the right of an up step of a Dyck path. St000478Another weight of a partition according to Alladi. St000490The intertwining number of a set partition. St000492The rob statistic of a set partition. St000493The los statistic of a set partition. St000498The lcs statistic of a set partition. St000499The rcb statistic of a set partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000564The 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. 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. 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. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000640The rank of the largest boolean interval in a poset. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000681The Grundy value of Chomp on Ferrers diagrams. 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. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000877The depth of the binary word interpreted as a path. St000929The constant term of the character polynomial of an integer partition. St000932The number of occurrences of the pattern UDU in a Dyck path. St000934The 2-degree of an integer partition. St000946The sum of the skew hook positions in a Dyck path. St000976The sum of the positions of double up-steps of a Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. 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. St001139The number of occurrences of hills of size 2 in a Dyck path. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001281The normalized isoperimetric number of a graph. St001371The length of the longest Yamanouchi prefix of a binary word. St001498The normalised height of a Nakayama algebra with magnitude 1. St001592The maximal number of simple paths between any two different vertices of a graph. St001668The number of points of the poset minus the width of the poset. St000455The second largest eigenvalue of a graph if it is integral. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000741The Colin de Verdière graph invariant. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001625The Möbius invariant of a lattice. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000464The Schultz index of a connected graph. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001545The second Elser number of a connected graph. St000467The hyper-Wiener index of a connected graph. St000181The number of connected components of the Hasse diagram for the poset. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001890The maximum magnitude of the Möbius function of a poset. St000102The charge of a semistandard tableau. St001964The interval resolution global dimension of a poset. St000096The number of spanning trees of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000287The number of connected components of a graph. St000309The number of vertices with even degree. St000310The minimal degree of a vertex of a graph. St000450The number of edges minus the number of vertices plus 2 of a graph. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000736The last entry in the first row of a semistandard tableau. St000739The first entry in the last row of a semistandard tableau. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001518The number of graphs with the same ordinary spectrum as the given graph. St001613The binary logarithm of the size of the center of a lattice. St001621The number of atoms of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001828The Euler characteristic of a graph. St001881The number of factors of a lattice as a Cartesian product of lattices. St000101The cocharge of a semistandard tableau. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000274The number of perfect matchings of a graph. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000315The number of isolated vertices of a graph. St000822The Hadwiger number 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. St001556The number of inversions of the third entry of 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. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001734The lettericity of a graph. St001783The number of odd automorphisms of a graph. St001845The number of join irreducibles minus the rank of a lattice. St001856The number of edges in the reduced word graph of a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001871The number of triconnected components of a graph. St001960The number of descents of a permutation minus one if its first entry is not one. St001738The minimal order of a graph which is not an induced subgraph of the given graph.