Your data matches 439 different statistics following compositions of up to 3 maps.
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Matching statistic: St000705
(load all 6 compositions to match this statistic)
Mapping
Mp00323: Integer partitions —Loehr-Warrington inverse⟶ Integer partitions
St000705: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1]
 => 1
[2]
 => [1,1]
 => 1
[1,1]
 => [2]
 => 3
[2,1]
 => [1,1,1]
 => 1
Description
The number of semistandard tableaux on a given integer partition of n with maximal entry n. This is, for an integer partition $\lambda = ( \lambda_1 \geq \cdots \geq \lambda_k \geq 0) \vdash n$, the number of semistandard tableaux of shape $\lambda$ with maximal entry $n$. Equivalently, this is the evaluation $s_\lambda(1,\ldots,1)$ of the Schur function $s_\lambda$ in $n$ variables, or, explicitly, $$\prod_{(i,j) \in \lambda} \frac{n+j-i}{\operatorname{hook}(i,j)}$$ where the product is over all cells $(i,j) \in \lambda$ and $\operatorname{hook}(i,j)$ is the hook length of a cell. See [Theorem 6.3, 1] for details.
Matching statistic: St000810
(load all 2 compositions to match this statistic)
Mapping
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000810: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1]
 => 1
[2]
 => [2]
 => 1
[1,1]
 => [1,1]
 => 3
[2,1]
 => [3]
 => 1
Description
The 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. For example, $p_{22} = 2m_{22} + m_4$, so the statistic on the partition $22$ is 3.
Matching statistic: St000964
(load all 11 compositions to match this statistic)
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000964: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0]
 => 1
[2]
 => [1,0,1,0]
 => 1
[1,1]
 => [1,1,0,0]
 => 3
[2,1]
 => [1,0,1,1,0,0]
 => 1
Description
Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra.
Matching statistic: St000965
(load all 11 compositions to match this statistic)
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000965: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0]
 => 1
[2]
 => [1,0,1,0]
 => 1
[1,1]
 => [1,1,0,0]
 => 3
[2,1]
 => [1,0,1,1,0,0]
 => 1
Description
The sum of the dimension of Ext^i(D(A),A) for i=1,...,g when g denotes the global dimension of the corresponding LNakayama algebra.
Matching statistic: St001564
Mapping
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St001564: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1]
 => 1
[2]
 => [2]
 => 1
[1,1]
 => [1,1]
 => 3
[2,1]
 => [3]
 => 1
Description
The value of the forgotten symmetric functions when all variables set to 1. Let $f_\lambda(x)$ denote the forgotten symmetric functions. Then the statistic associated with $\lambda$, where $\lambda$ has $\ell$ parts, is $f_\lambda(1,1,\dotsc,1)$ where there are $\ell$ variables substituted by $1$.
Matching statistic: St000511
Mapping
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000511: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1]
 => 2 = 1 + 1
[2]
 => [2]
 => 2 = 1 + 1
[1,1]
 => [1,1]
 => 4 = 3 + 1
[2,1]
 => [3]
 => 2 = 1 + 1
Description
The number of invariant subsets when acting with a permutation of given cycle type.
Matching statistic: St000995
(load all 7 compositions to match this statistic)
Mapping
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000995: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1]
 => 0 = 1 - 1
[2]
 => [1,1]
 => 0 = 1 - 1
[1,1]
 => [2]
 => 2 = 3 - 1
[2,1]
 => [3]
 => 0 = 1 - 1
Description
The largest even part of an integer partition.
Matching statistic: St001177
Mapping
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St001177: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1]
 => 0 = 1 - 1
[2]
 => [2]
 => 0 = 1 - 1
[1,1]
 => [1,1]
 => 2 = 3 - 1
[2,1]
 => [3]
 => 0 = 1 - 1
Description
Twice the mean value of the major index among all standard Young tableaux of a partition. For a partition $\lambda$ of $n$, this mean value is given in [1, Proposition 3.1] by $$\frac{1}{2}\Big(\binom{n}{2} - \sum_i\binom{\lambda_i}{2} + \sum_i\binom{\lambda_i'}{2}\Big),$$ where $\lambda_i$ is the size of the $i$-th row of $\lambda$ and $\lambda_i'$ is the size of the $i$-th column.
Matching statistic: St001248
(load all 6 compositions to match this statistic)
Mapping
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St001248: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1]
 => 0 = 1 - 1
[2]
 => [1,1]
 => 0 = 1 - 1
[1,1]
 => [2]
 => 2 = 3 - 1
[2,1]
 => [3]
 => 0 = 1 - 1
Description
Sum of the even parts of a partition.
Matching statistic: St001279
(load all 5 compositions to match this statistic)
Mapping
Mp00323: Integer partitions —Loehr-Warrington inverse⟶ Integer partitions
St001279: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1]
 => 0 = 1 - 1
[2]
 => [1,1]
 => 0 = 1 - 1
[1,1]
 => [2]
 => 2 = 3 - 1
[2,1]
 => [1,1,1]
 => 0 = 1 - 1
Description
The sum of the parts of an integer partition that are at least two.
The following 429 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001916The number of transient elements in the orbit of Bulgarian solitaire corresponding to a necklace. St000154The sum of the descent bottoms of a permutation. St000156The Denert index of a permutation. St000391The sum of the positions of the ones in a binary word. St000715The number of semistandard Young tableaux of given shape and entries at most 3. St000756The sum of the positions of the left to right maxima of a permutation. St000763The sum of the positions of the strong records of an integer composition. St000792The Grundy value for the game of ruler on a binary word. St000946The sum of the skew hook positions in a Dyck path. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. 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. St001468The smallest fixpoint of a permutation. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001721The degree of a binary word. St001778The largest greatest common divisor of an element and its image in a permutation. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St001806The upper middle entry of a permutation. St001838The number of nonempty primitive factors of a binary word. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St001915The size of the component corresponding to a necklace in Bulgarian solitaire. St000027The major index of a Dyck path. St000235The number of indices that are not cyclical small weak excedances. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000462The major index minus the number of excedences of a permutation. St000463The number of admissible inversions of a permutation. St000726The normalized sum of the leaf labels of the increasing binary tree associated to a permutation. St000979Half of MacMahon's equal index of a Dyck path. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001185The number of indecomposable injective modules of grade at least 2 in 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. St001274The number of indecomposable injective modules with projective dimension equal to two. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St000001The number of reduced words for a permutation. St000004The major index of a permutation. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000007The number of saliances of the permutation. St000008The major index of the composition. St000011The number of touch points (or returns) of a Dyck path. St000012The area of a Dyck path. St000014The number of parking functions supported by a Dyck path. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000047The number of standard immaculate tableaux of a given shape. St000054The first entry of the permutation. St000055The inversion sum of a permutation. St000061The number of nodes on the left branch of a binary tree. St000133The "bounce" of a permutation. St000141The maximum drop size of a permutation. St000163The size of the orbit of the set partition under rotation. St000176The total number of tiles in the Gelfand-Tsetlin pattern. St000210Minimum over maximum difference of elements in cycles. St000224The sorting index of a permutation. St000230Sum of the minimal elements of the blocks of a set partition. St000238The number of indices that are not small weak excedances. St000240The number of indices that are not small excedances. St000267The number of maximal spanning forests contained in a graph. St000289The decimal representation of a binary word. St000304The load of a permutation. St000305The inverse major index of a permutation. 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. St000332The positive inversions of an alternating sign matrix. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000335The difference of lower and upper interactions. St000339The maf index of a permutation. St000347The inversion sum of a binary word. St000348The non-inversion sum of a binary word. St000383The last part of an integer composition. St000416The number of inequivalent increasing trees of an ordered tree. St000420The number of Dyck paths that are weakly above a Dyck path. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000472The sum of the ascent bottoms of a permutation. St000487The length of the shortest cycle of a permutation. St000501The size of the first part in the decomposition of a permutation. St000505The biggest entry in the block containing the 1. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St000529The number of permutations whose descent word is the given binary word. St000539The number of odd inversions of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000543The size of the conjugacy class of a binary word. St000558The number of occurrences of the pattern {{1,2}} in a set partition. St000617The number of global maxima of a Dyck path. St000626The minimal period of a binary word. St000627The exponent of a binary word. St000651The maximal size of a rise in a permutation. St000652The maximal difference between successive positions of a permutation. St000655The length of the minimal rise of a Dyck path. St000657The smallest part of an integer composition. St000675The number of centered multitunnels of a Dyck path. St000678The number of up steps after the last double rise 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. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000691The number of changes of a binary word. St000692Babson and Steingrímsson's statistic of a permutation. St000740The last entry of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000762The sum of the positions of the weak records of an integer composition. St000765The number of weak records in an integer composition. St000767The number of runs in an integer composition. 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. 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. St000820The number of compositions obtained by rotating the composition. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000833The comajor index of a permutation. St000847The number of standard Young tableaux whose descent set is the binary word. St000868The aid statistic in the sense of Shareshian-Wachs. St000874The position of the last double rise in a Dyck path. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000877The depth of the binary word interpreted as a path. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000883The number of longest increasing subsequences of a permutation. St000898The number of maximal entries in the last diagonal of the monotone triangle. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St000903The number of different parts of an integer composition. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000947The major index east count of a Dyck path. St000976The sum of the positions of double up-steps of a Dyck path. St000983The length of the longest alternating subword. St000984The number of boxes below precisely one peak. St000988The orbit size of a permutation under Foata's bijection. St000990The first ascent of a permutation. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001030Half the number of non-boundary horizontal edges in the fully packed loop corresponding to the alternating sign matrix. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001041The depth of the label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001048The number of leaves in the subtree containing 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001077The prefix exchange distance of a permutation. 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. St001094The depth index of a set partition. St001102The number of words with multiplicities of the letters given by the composition, avoiding the consecutive pattern 132. St001132The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. 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 $K[x]/(x^n)$. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001220The width of a permutation. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001267The length of the Lyndon factorization of the binary word. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001312Number of parabolic noncrossing partitions indexed by the composition. St001313The number of Dyck paths above the lattice path given by a binary word. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001375The pancake length of a permutation. St001388The number of non-attacking neighbors of a permutation. St001412Number of minimal entries in the Bruhat order matrix of a permutation. St001415The length of the longest palindromic prefix of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001437The flex of a binary word. St001461The number of topologically connected components of the chord diagram of a permutation. St001481The minimal height of a peak of a Dyck path. St001500The global dimension of magnitude 1 Nakayama algebras. St001524The degree of symmetry of a binary word. St001546The number of monomials in the Tutte polynomial of a graph. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001639The number of alternating subsets such that applying the permutation does not yield an alternating subset. St001671Haglund's hag of a permutation. St001675The number of parts equal to the part in the reversed composition. St001694The number of maximal dissociation sets in a graph. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001733The number of weak left to right maxima of a Dyck path. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001807The lower middle entry of a permutation. St001808The box weight or horizontal decoration of a Dyck path. St001850The number of Hecke atoms of a permutation. St001874Lusztig's a-function for the symmetric group. St001884The number of borders of a binary word. St001930The weak major index of a binary word. St001931The weak major index of an integer composition regarded as a word. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000022The number of fixed points of a permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000038The product of the heights of the descending steps of a Dyck path. St000039The number of crossings of a permutation. St000040The number of regions of the inversion arrangement of a permutation. St000043The number of crossings plus two-nestings of a perfect matching. St000051The size of the left subtree of a binary tree. St000089The absolute variation of a composition. St000109The number of elements less than or equal to the given element in Bruhat order. St000111The sum of the descent tops (or Genocchi descents) of a permutation. St000119The number of occurrences of the pattern 321 in a permutation. St000120The number of left tunnels of a Dyck path. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000213The number of weak exceedances (also weak excedences) 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. St000222The number of alignments in the permutation. St000223The number of nestings in the permutation. St000237The number of small exceedances. St000268The number of strongly connected orientations of a graph. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000290The major index of a binary word. St000293The number of inversions of a binary word. St000295The length of the border of a binary word. St000296The length of the symmetric border of a binary word. St000297The number of leading ones in a binary word. St000338The number of pixed points of a permutation. St000352The Elizalde-Pak rank of a permutation. St000355The number of occurrences of the pattern 21-3. St000359The number of occurrences of the pattern 23-1. St000360The number of occurrences of the pattern 32-1. St000365The number of double ascents of a permutation. St000369The dinv deficit of a Dyck path. 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 $3$. St000376The bounce deficit of a Dyck path. St000389The number of runs of ones of odd length in a binary word. St000401The size of the symmetry class of a permutation. St000402Half the size of the symmetry class 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. 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. 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. St000434The number of occurrences of the pattern 213 or of the pattern 312 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. St000439The position of the first down step of a Dyck path. St000445The number of rises of length 1 of a Dyck path. St000457The number of occurrences of one of the patterns 132, 213 or 321 in a permutation. St000461The rix statistic of a permutation. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000495The number of inversions of distance at most 2 of a permutation. St000498The lcs statistic of a set partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000534The number of 2-rises of a permutation. St000538The number of even inversions of a permutation. St000542The number of left-to-right-minima of a permutation. St000546The number of global descents of a permutation. St000616The inversion index of a permutation. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000646The number of big ascents of a permutation. St000648The number of 2-excedences of a permutation. St000658The number of rises of length 2 of a Dyck path. St000663The number of right floats of a permutation. St000676The number of odd rises of a Dyck path. St000682The Grundy value of Welter's game on a binary word. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000702The number of weak deficiencies of a permutation. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000725The smallest label of a leaf 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. St000766The number of inversions of an integer composition. St000769The major index of a composition regarded as a word. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000811The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to Schur symmetric functions. 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. St000824The sum of the number of descents and the number of recoils of a permutation. St000825The sum of the major and the inverse major index of a permutation. St000828The spearman's rho of a permutation and the identity permutation. St000830The total displacement of a permutation. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000836The number of descents of distance 2 of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000873The aix statistic of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St000881The number of short braid edges in the graph of braid moves of a permutation. St000885The number of critical steps in the Catalan decomposition of a binary word. St000921The number of internal inversions of a binary word. St000932The number of occurrences of the pattern UDU in a Dyck path. St000933The number of multipartitions of sizes given by an integer partition. St000951The dimension of $Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra. St000953The largest degree of an irreducible factor of the Coxeter polynomial of the Dyck path over the rational numbers. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000961The shifted major index of a permutation. St000963The 2-shifted major index of a permutation. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St000973The length of the boundary of an ordered tree. St000981The length of the longest zigzag subpath. St000992The alternating sum of the parts of an integer partition. 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. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001034The area of the parallelogram polyomino associated with the Dyck path. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001058The breadth of the ordered tree. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001073The number of nowhere zero 3-flows of a graph. 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. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001130The number of two successive successions in a permutation. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001134The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. St001161The major index north count of a Dyck path. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. 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. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. 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. St001288The number of primes obtained by multiplying preimage and image of a permutation and adding one. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001351The Albertson index of a graph. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001367The smallest number which does not occur as degree of a vertex in a graph. St001371The length of the longest Yamanouchi prefix of a binary word. St001374The Padmakar-Ivan index of a graph. St001379The number of inversions plus the major index of a permutation. St001381The fertility of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001402The number of separators in a permutation. St001411The number of patterns 321 or 3412 in a permutation. St001413Half the length of the longest even length palindromic prefix of a binary word. St001423The number of distinct cubes in a binary word. St001433The flag major index of a signed permutation. St001439The number of even weak deficiencies and of odd weak exceedences. St001484The number of singletons of an integer partition. 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. St001531Number of partial orders contained in the poset determined by the Dyck path. St001552The number of inversions between excedances and fixed points of a permutation. St001557The number of inversions of the second entry of a permutation. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001669The number of single rises in a Dyck path. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. 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. 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. St001727The number of invisible inversions of a permutation. St001735The number of permutations with the same set of runs. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001809The index of the step at the first peak of maximal height in a Dyck path. St001810The number of fixed points of a permutation smaller than its largest moved point. St001856The number of edges in the reduced word graph of a permutation. St001892The flag excedance statistic of a signed permutation. St001893The flag descent of a signed permutation. St001902The number of potential covers of a poset. St001911A descent variant minus the number of inversions. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001959The product of the heights of the peaks of a Dyck path. St001966Half the global dimension of the stable Auslander algebra of a sincere Nakayama algebra (with associated Dyck path). St000037The sign of a permutation. St000458The number of permutations obtained by switching adjacencies or successions. St000978The sum of the positions of double down-steps of a Dyck path. St001377The major index minus the number of inversions of a permutation. St000977MacMahon's equal index of a Dyck path. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. 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. 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. St000674The number of hills of a Dyck path. St000260The radius of a connected graph. St000456The monochromatic index of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001404The number of distinct entries in a Gelfand Tsetlin pattern. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001498The normalised height of a Nakayama algebra with magnitude 1. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000227The osculating paths major index of an alternating sign matrix. St000247The number of singleton blocks of a set partition. St000248The number of anti-singletons of a set partition. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000422The energy of a graph, if it is integral. St000471The sum of the ascent tops of a permutation. St000673The number of non-fixed points of a permutation. St000894The trace of an alternating sign matrix. St000896The number of zeros on the main diagonal of an alternating sign matrix. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. 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 $eA$. 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 $eA$. 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 $eA$. St001268The size of the largest ordinal summand in the poset. St001713The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St001964The interval resolution global dimension of a poset.