Your data matches 532 different statistics following compositions of up to 3 maps.
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Matching statistic: St000950
(load all 7 compositions to match this statistic)
Mapping
St000950: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => 2 = 1 + 1
[1,0,1,0]
 => 2 = 1 + 1
[1,1,0,0]
 => 5 = 4 + 1
Description
Number of tilting modules of the corresponding LNakayama algebra, where a tilting module is a generalised tilting module of projective dimension 1.
Matching statistic: St001002
(load all 2 compositions to match this statistic)
Mapping
St001002: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => 3 = 1 + 2
[1,0,1,0]
 => 3 = 1 + 2
[1,1,0,0]
 => 6 = 4 + 2
Description
Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000712
Mapping
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000712: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => []
 => 1
[1,0,1,0]
 => [1]
 => 4
[1,1,0,0]
 => []
 => 1
Description
The number of semistandard Young tableau of given shape, with entries at most 4. This is also the dimension of the corresponding irreducible representation of $GL_4$.
Matching statistic: St001255
(load all 4 compositions to match this statistic)
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001255: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,1,0,0]
 => 1
[1,0,1,0]
 => [1,1,0,1,0,0]
 => 4
[1,1,0,0]
 => [1,1,1,0,0,0]
 => 1
Description
The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J.
Matching statistic: St001473
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Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001473: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,1,0,0]
 => 1
[1,0,1,0]
 => [1,1,0,1,0,0]
 => 4
[1,1,0,0]
 => [1,1,1,0,0,0]
 => 1
Description
The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra.
Matching statistic: St001872
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Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001872: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,1,0,0]
 => 1
[1,0,1,0]
 => [1,1,0,1,0,0]
 => 4
[1,1,0,0]
 => [1,1,1,0,0,0]
 => 1
Description
The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra.
Matching statistic: St000027
(load all 3 compositions to match this statistic)
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000027: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,1,0,0]
 => 0 = 1 - 1
[1,0,1,0]
 => [1,1,0,1,0,0]
 => 3 = 4 - 1
[1,1,0,0]
 => [1,1,1,0,0,0]
 => 0 = 1 - 1
Description
The major index of a Dyck path. This is the sum over all $i+j$ for which $(i,j)$ is a valley of $D$. The generating function of the major index yields '''MacMahon''' 's $q$-Catalan numbers $$\sum_{D \in \mathfrak{D}_n} q^{\operatorname{maj}(D)} = \frac{1}{[n+1]_q}\begin{bmatrix} 2n \\ n \end{bmatrix}_q,$$ where $[k]_q := 1+q+\ldots+q^{k-1}$ is the usual $q$-extension of the integer $k$, $[k]_q!:= [1]_q[2]_q \cdots [k]_q$ is the $q$-factorial of $k$ and $\left[\begin{smallmatrix} k \\ l \end{smallmatrix}\right]_q:=[k]_q!/[l]_q![k-l]_q!$ is the $q$-binomial coefficient. The major index was first studied by P.A.MacMahon in [1], where he proved this generating function identity. There is a bijection $\psi$ between Dyck paths and '''noncrossing permutations''' which simultaneously sends the area of a Dyck path [[St000012]] to the number of inversions [[St000018]], and the major index of the Dyck path to $n(n-1)$ minus the sum of the major index and the major index of the inverse [2]. For the major index on other collections, see [[St000004]] for permutations and [[St000290]] for binary words.
Matching statistic: St000226
(load all 28 compositions to match this statistic)
Mapping
Mp00201: Dyck paths —Ringel⟶ Permutations
St000226: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [2,1] => 0 = 1 - 1
[1,0,1,0]
 => [3,1,2] => 0 = 1 - 1
[1,1,0,0]
 => [2,3,1] => 3 = 4 - 1
Description
The convexity of a permutation. It is given by the maximal value of $2x_i-x_{i-1}-x_{i+1}$ over all $i \in \{2,\ldots,n-1\}$.
Matching statistic: St000235
(load all 6 compositions to match this statistic)
Mapping
Mp00201: Dyck paths —Ringel⟶ Permutations
St000235: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [2,1] => 0 = 1 - 1
[1,0,1,0]
 => [3,1,2] => 3 = 4 - 1
[1,1,0,0]
 => [2,3,1] => 0 = 1 - 1
Description
The number of indices that are not cyclical small weak excedances. A cyclical small weak excedance is an index $i < n$ such that $\pi_i = i+1$, or the index $i = n$ if $\pi_n = 1$.
Matching statistic: St000242
(load all 13 compositions to match this statistic)
Mapping
Mp00201: Dyck paths —Ringel⟶ Permutations
St000242: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [2,1] => 0 = 1 - 1
[1,0,1,0]
 => [3,1,2] => 3 = 4 - 1
[1,1,0,0]
 => [2,3,1] => 0 = 1 - 1
Description
The number of indices that are not cyclical small weak excedances. A cyclical small weak excedance is an index $i$ such that $\pi_i \in \{ i,i+1 \}$ considered cyclically.
The following 522 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001519The pinnacle sum of a permutation. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St000014The number of parking functions supported by a Dyck path. St000033The number of permutations greater than or equal to the given permutation in (strong) Bruhat order. St000038The product of the heights of the descending steps of a Dyck path. St000040The number of regions of the inversion arrangement of a permutation. St000055The inversion sum of a permutation. St000078The number of alternating sign matrices whose left key is the permutation. St000109The number of elements less than or equal to the given element in Bruhat order. St000154The sum of the descent bottoms of a permutation. St000341The non-inversion sum of a permutation. St000418The number of Dyck paths that are weakly below a Dyck path. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000543The size of the conjugacy class of a binary word. St000545The number of parabolic double cosets with minimal element being the given permutation. St000635The number of strictly order preserving maps of a poset into itself. St000669The number of permutations obtained by switching ascents or descents of size 2. St000692Babson and Steingrímsson's statistic of a permutation. St000756The sum of the positions of the left to right maxima of a permutation. St000958The number of Bruhat factorizations of a permutation. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St000983The length of the longest alternating subword. St001079The minimal length of a factorization of a permutation using the permutations (12)(34). St001081The number of minimal length factorizations of a permutation into star transpositions. St001129The product of the squares of the parts of a partition. 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. 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 $K[x]/(x^n)$. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001267The length of the Lyndon factorization of the binary word. St001412Number of minimal entries in the Bruhat order matrix of a permutation. St001437The flex of a binary word. St001468The smallest fixpoint of a permutation. St001500The global dimension of magnitude 1 Nakayama algebras. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001531Number of partial orders contained in the poset determined by the Dyck path. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001684The reduced word complexity of a permutation. St001735The number of permutations with the same set of runs. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St001808The box weight or horizontal decoration of a Dyck path. St001809The index of the step at the first peak of maximal height in a Dyck path. St001959The product of the heights of the peaks of a Dyck path. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000012The area of a Dyck path. St000020The rank of the permutation. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000111The sum of the descent tops (or Genocchi descents) of a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000224The sorting index of a permutation. St000238The number of indices that are not small weak excedances. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000339The maf index of a permutation. St000347The inversion sum of a binary word. St000348The non-inversion sum of a binary word. St000360The number of occurrences of the pattern 32-1. St000367The number of simsun double descents of a permutation. St000420The number of Dyck paths that are weakly above a Dyck path. St000445The number of rises of length 1 of a Dyck path. St000471The sum of the ascent tops of a permutation. St000616The inversion index of a permutation. St000651The maximal size of a rise in a permutation. St000673The number of non-fixed points of a permutation. St000674The number of hills of a Dyck path. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000691The number of changes of a binary word. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000825The sum of the major and the inverse major index of a permutation. St000868The aid statistic in the sense of Shareshian-Wachs. St000874The position of the last double rise in a Dyck path. St000946The sum of the skew hook positions in a Dyck path. St000947The major index east count of a Dyck path. St000963The 2-shifted major index of a permutation. St000976The sum of the positions of double up-steps of a Dyck path. St000978The sum of the positions of double down-steps of a Dyck path. St000979Half of MacMahon's equal index of a Dyck path. St000984The number of boxes below precisely one peak. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001077The prefix exchange distance of a permutation. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. 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. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. 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. 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. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001371The length of the longest Yamanouchi prefix of a binary word. St001375The pancake length of a permutation. St001379The number of inversions plus the major index of a permutation. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001695The natural comajor index of a standard Young tableau. St001696The natural major index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a 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. St000652The maximal difference between successive positions of a permutation. St000715The number of semistandard Young tableaux of given shape and entries at most 3. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. 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. 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. St000002The number of occurrences of the pattern 123 in a permutation. St000004The major index of a permutation. St000018The number of inversions of a permutation. St000026The position of the first return of a Dyck path. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000047The number of standard immaculate tableaux of a given shape. St000048The multinomial of the parts of a partition. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000054The first entry of the permutation. St000058The order of a permutation. St000076The rank of the alternating sign matrix in the alternating sign matrix poset. St000089The absolute variation of a composition. St000096The number of spanning trees of a graph. St000110The number of permutations less than or equal to a permutation in left weak order. St000133The "bounce" of a permutation. St000141The maximum drop size of a permutation. St000156The Denert index of a permutation. St000163The size of the orbit of the set partition under rotation. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000230Sum of the minimal elements of the blocks of a set partition. St000240The number of indices that are not small excedances. St000263The Szeged index of a graph. St000265The Wiener index of a graph. St000266The number of spanning subgraphs of a graph with the same connected components. St000267The number of maximal spanning forests contained in a graph. St000269The number of acyclic orientations of a graph. St000270The number of forests contained in a graph. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St000284The Plancherel distribution on integer partitions. St000289The decimal representation of a binary word. St000290The major index of a binary word. St000305The inverse major index of a permutation. St000307The number of rowmotion orbits of a poset. St000316The number of non-left-to-right-maxima of a permutation. St000326The position of the first one in a binary word after appending a 1 at the end. St000335The difference of lower and upper interactions. St000342The cosine of a permutation. St000343The number of spanning subgraphs of a graph. St000361The second Zagreb index of a graph. St000391The sum of the positions of the ones in a binary word. St000392The length of the longest run of ones in a binary word. St000395The sum of the heights of the peaks of a Dyck path. St000401The size of the symmetry class of a permutation. St000402Half the size of the symmetry class of a permutation. St000423The number of occurrences of the pattern 123 or of the pattern 132 in a permutation. St000427The number of occurrences of the pattern 123 or of the pattern 231 in a permutation. St000428The number of occurrences of the pattern 123 or of the pattern 213 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. St000431The number of occurrences of the pattern 213 or of the pattern 321 in a permutation. St000433The number of occurrences of the pattern 132 or of the pattern 321 in a permutation. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000446The disorder of a permutation. St000451The length of the longest pattern of the form k 1 2. St000456The monochromatic index of a connected graph. St000463The number of admissible inversions of a permutation. St000472The sum of the ascent bottoms of a permutation. St000485The length of the longest cycle of a permutation. St000487The length of the shortest 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. St000501The size of the first part in the decomposition of a permutation. St000505The biggest entry in the block containing the 1. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000517The Kreweras number of an integer partition. St000537The cutwidth of a graph. St000539The number of odd inversions of a permutation. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000626The minimal period of a binary word. St000638The number of up-down runs of a permutation. St000655The length of the minimal rise of a Dyck path. St000677The standardized bi-alternating inversion number of a permutation. St000682The Grundy value of Welter's game on a binary word. St000694The number of affine bounded permutations that project to a given permutation. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000735The last entry on the main diagonal of a standard tableau. St000738The first entry in the last row of a standard tableau. St000740The last entry of a permutation. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000762The sum of the positions of the weak records of an integer composition. St000763The sum of the positions of the strong records of an integer composition. St000765The number of weak records in an integer composition. St000770The major index of an integer partition when read from bottom to top. St000794The mak of a permutation. St000795The mad of a permutation. St000796The stat' of a permutation. St000797The stat`` of a permutation. St000798The makl of a permutation. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000815The number of semistandard Young tableaux of partition weight of given shape. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. 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. St000833The comajor index of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000847The number of standard Young tableaux whose descent set is the binary word. St000867The sum of the hook lengths in the first row of an integer partition. St000876The number of factors in the Catalan decomposition of a binary word. St000877The depth of the binary word interpreted as a path. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. 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. St000933The number of multipartitions of sizes given by an integer partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000948The chromatic discriminant of a graph. St000961The shifted major index of a permutation. St000972The composition number of a graph. St000982The length of the longest constant subword. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001030Half the number of non-boundary horizontal edges in the fully packed loop corresponding to the alternating sign matrix. St001080The minimal length of a factorization of a permutation using the transposition (12) and the cycle (1,. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001090The number of pop-stack-sorts needed to sort a permutation. St001095The number of non-isomorphic posets with precisely one further covering relation. 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. 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. St001109The number of proper colourings of a graph with as few colours as possible. St001112The 3-weak dynamic number of a graph. St001118The acyclic chromatic index of a graph. St001127The sum of the squares of the parts of a partition. 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 $g$. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001285The number of primes in the column sums of the two line notation of a permutation. St001287The number of primes obtained by multiplying preimage and image of a permutation and subtracting one. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001312Number of parabolic noncrossing partitions indexed by the composition. St001341The number of edges in the center 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. 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. St001378The product of the cohook lengths of the integer partition. St001415The length of the longest palindromic prefix of a binary word. St001459The number of zero columns in the nullspace of a graph. 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. St001474The evaluation of the Tutte polynomial of the graph at (x,y) equal to (2,-1). St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001523The degree of symmetry of a Dyck path. St001527The cyclic permutation representation number of an integer partition. St001546The number of monomials in the Tutte polynomial of a graph. St001555The order of a signed permutation. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001565The number of arithmetic progressions of length 2 in a permutation. St001592The maximal number of simple paths between any two different vertices of a graph. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001612The number of coloured multisets of cycles such that the multiplicities of colours are given by a partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001639The number of alternating subsets such that applying the permutation does not yield an alternating subset. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001671Haglund's hag of a permutation. St001675The number of parts equal to the part in the reversed composition. St001688The sum of the squares of the heights of the peaks of a Dyck path. St001697The shifted natural comajor index of a standard Young tableau. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. St001721The degree of a binary word. St001726The number of visible inversions of a permutation. St001760The number of prefix or suffix reversals needed to sort a permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001802The number of endomorphisms of a graph. St001806The upper middle entry of a permutation. St001807The lower middle entry of a permutation. St001813The product of the sizes of the principal order filters in a poset. St001814The number of partitions interlacing the given partition. St001821The sorting index of a signed permutation. St001838The number of nonempty primitive factors of a binary word. St001848The atomic length of a signed permutation. St001865The number of alignments of a signed permutation. St001874Lusztig's a-function for the symmetric group. St001885The number of binary words with the same proper border set. St001911A descent variant minus the number of inversions. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000008The major index of the composition. St000016The number of attacking pairs of a standard tableau. St000019The cardinality of the support of a permutation. St000022The number of fixed points of a permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000041The number of nestings of a perfect matching. St000057The Shynar inversion number of a standard tableau. St000059The inversion number of a standard tableau as defined by Haglund and Stevens. St000067The inversion number of the alternating sign matrix. 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. St000153The number of adjacent cycles of a permutation. St000161The sum of the sizes of the right subtrees of a binary tree. St000169The cocharge of a standard tableau. St000209Maximum difference of elements in cycles. St000210Minimum over maximum difference of elements in cycles. St000215The number of adjacencies of a permutation, zero appended. St000216The absolute length of a permutation. St000217The number of occurrences of the pattern 312 in a permutation. St000218The number of occurrences of the pattern 213 in a permutation. St000219The number of occurrences of the pattern 231 in a permutation. St000220The number of occurrences of the pattern 132 in a permutation. St000221The number of strong fixed points of a permutation. St000227The osculating paths major index of an alternating sign matrix. St000228The size of a partition. St000236The number of cyclical small weak excedances. St000237The number of small exceedances. St000239The number of small weak excedances. St000241The number of cyclical small excedances. St000246The number of non-inversions of a permutation. St000247The number of singleton blocks of a set partition. St000248The number of anti-singletons of a set partition. St000293The number of inversions of a binary word. St000296The length of the symmetric border of a binary word. St000297The number of leading ones in a binary word. St000304The load of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000330The (standard) major index of a standard tableau. St000332The positive inversions of an alternating sign matrix. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000338The number of pixed points of a permutation. St000340The number of non-final maximal constant sub-paths of length greater than one. St000346The number of coarsenings of a partition. St000357The number of occurrences of the pattern 12-3. St000374The number of exclusive right-to-left minima of a permutation. St000378The diagonal inversion number of an integer partition. St000379The number of Hamiltonian cycles in a graph. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000406The number of occurrences of the pattern 3241 in a permutation. St000424The number of occurrences of the pattern 132 or of the pattern 231 in a permutation. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000426The number of occurrences of the pattern 132 or of the pattern 312 in a permutation. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000435The number of occurrences of the pattern 213 or of the pattern 231 in a permutation. St000459The hook length of the base cell of a partition. St000461The rix statistic of a permutation. St000462The major index minus the number of excedences of a permutation. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000488The number of cycles of a permutation of length at most 2. 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. St000500Eigenvalues of the random-to-random operator acting on the regular representation. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000532The total number of rook placements on a Ferrers board. St000548The number of different non-empty partial sums of an integer partition. St000556The 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. 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. St000567The sum of the products of all pairs of parts. 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. St000595The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000637The length of the longest cycle in a graph. St000644The number of graphs with given frequency partition. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000653The last descent of a permutation. St000693The modular (standard) major index of a standard tableau. St000703The number of deficiencies of a permutation. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000719The number of alignments in a perfect matching. St000726The normalized sum of the leaf labels of the increasing binary tree associated to a permutation. St000747A variant of the major index of a set partition. St000748The major index of the permutation obtained by flattening the set partition. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000792The Grundy value for the game of ruler on a binary word. St000801The number of occurrences of the vincular pattern |312 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. St000809The reduced reflection length of the 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. St000831The number of indices that are either descents or recoils. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000869The sum of the hook lengths of an integer partition. St000873The aix statistic of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000894The trace of an alternating sign matrix. St000895The number of ones on the main diagonal of an alternating sign matrix. St000896The number of zeros on the main diagonal of an alternating sign matrix. St000915The Ore degree of a graph. St000921The number of internal inversions of a binary word. St000937The number of positive values of the symmetric group character corresponding to the partition. St000949Gives the number of generalised tilting modules of the corresponding LNakayama algebra. St000951The dimension of $Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St000974The length of the trunk of an ordered tree. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. 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. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001094The depth index of a set partition. 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. St001131The number of trivial trees on the path to label one in the decreasing labelled binary unordered tree associated with the perfect matching. St001160The number of proper blocks (or intervals) of a permutations. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. 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. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001249Sum of the odd parts of a partition. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001279The sum of the parts of an integer partition that are at least two. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001311The cyclomatic number of a graph. St001313The number of Dyck paths above the lattice path given by a binary word. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001362The normalized Knill dimension of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001388The number of non-attacking neighbors of a permutation. St001397Number of pairs of incomparable elements in a finite poset. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001428The number of B-inversions of a signed permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001478The number of nowhere zero 4-flows of a graph. St001480The number of simple summands of the module J^2/J^3. St001485The modular major index of a binary word. St001486The number of corners of the ribbon associated with an integer composition. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. 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. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001511The minimal number of transpositions needed to sort a permutation in either direction. St001521Half the total irregularity of a graph. St001541The Gini index of an integer partition. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001584The area statistic between a Dyck path and its bounce path. St001591The number of graphs with the given composition of multiplicities of Laplacian eigenvalues. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001649The length of a longest trail in a graph. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001727The number of invisible inversions of a permutation. St001736The total number of cycles in a graph. St001754The number of tolerances of a finite lattice. St001759The Rajchgot index of a permutation. St001764The number of non-convex subsets of vertices in a graph. St001765The number of connected components of the friends and strangers graph. St001777The number of weak descents in an integer composition. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St001819The flag Denert index of a signed permutation. St001842The major index of a set partition. St001843The Z-index of a set partition. St001850The number of Hecke atoms of a permutation. St001855The number of signed permutations less than or equal to a signed permutation in left weak order. St001910The height of the middle non-run of a Dyck path. St001931The weak major index of an integer composition regarded as a word. St001956The comajor index for set-valued two-row standard Young tableaux. St000231Sum of the maximal elements of the blocks of a set partition. St000400The path length of an ordered tree. St000474Dyson's crank of a partition. St000639The number of relations in a poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St000690The size of the conjugacy class of a permutation. 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. St000827The decimal representation of a binary word with a leading 1. St001003The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001138The number of indecomposable modules with projective dimension or injective dimension at most one in the corresponding Nakayama 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. St001709The number of homomorphisms to the three element chain of a poset. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001833The number of linear intervals in a lattice. St001259The vector space dimension of the double dual of D(A) in the corresponding Nakayama algebra.