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. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001730The number of times the path corresponding to a binary word crosses the base line. St000120The number of left tunnels of a Dyck path. St000335The difference of lower and upper interactions. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001481The minimal height of a peak of a Dyck path. St001732The number of peaks visible from the left. St000023The number of inner peaks of a permutation. St000142The number of even parts of a partition. St000143The largest repeated part of a partition. St000150The floored half-sum of the multiplicities of a partition. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000214The number of adjacencies of a permutation. St000237The number of small exceedances. St000257The number of distinct parts of a partition that occur at least twice. St000291The number of descents of a binary word. St000310The minimal degree of a vertex of a graph. St000352The Elizalde-Pak rank of a permutation. St000358The number of occurrences of the pattern 31-2. St000360The number of occurrences of the pattern 32-1. St000366The number of double descents of a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000386The number of factors DDU in a Dyck path. St000461The rix statistic of a permutation. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000475The number of parts equal to 1 in a partition. St000534The number of 2-rises of a permutation. St000546The number of global descents of a permutation. St000624The normalized sum of the minimal distances to a greater element. 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. St000682The Grundy value of Welter's game on a binary word. St000711The number of big exceedences of a permutation. St000761The number of ascents in an integer composition. St000766The number of inversions of an integer composition. St000769The major index of a composition regarded as a word. St000779The tier of a permutation. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000864The number of circled entries of the shifted recording tableau of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000884The number of isolated descents of a permutation. St000931The number of occurrences of the pattern UUU in a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001037The number of inner corners of the upper path 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. St001061The number of indices that are both descents and recoils of a permutation. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001083The number of boxed occurrences of 132 in a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. 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. St001092The number of distinct even parts of a partition. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. St001175The size of a partition minus the hook length of the base cell. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001252Half the sum of the even parts of a partition. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001271The competition number of a graph. St001274The number of indecomposable injective modules with projective dimension equal to two. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001394The genus of a permutation. St001413Half the length of the longest even length palindromic prefix of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001423The number of distinct cubes in a binary word. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001484The number of singletons of an integer partition. St001587Half of the largest even part of an integer partition. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001657The number of twos in an integer partition. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001673The degree of asymmetry of an integer composition. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001727The number of invisible inversions of a permutation. St001728The number of invisible descents of a permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000007The number of saliances of the permutation. St000011The number of touch points (or returns) of a Dyck path. St000024The number of double up and double down steps of a Dyck path. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000053The number of valleys of the Dyck path. St000099The number of valleys of a permutation, including the boundary. St000124The cardinality of the preimage of the Simion-Schmidt map. St000141The maximum drop size of a permutation. St000183The side length of the Durfee square of an integer partition. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000201The number of leaf nodes in a binary tree. St000297The number of leading ones in a binary word. St000307The number of rowmotion orbits of a poset. St000316The number of non-left-to-right-maxima of a permutation. St000331The number of upper interactions of a Dyck path. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000390The number of runs of ones in a binary word. St000392The length of the longest run of ones in a binary word. St000396The register function (or Horton-Strahler number) of a binary tree. St000442The maximal area to the right of an up step 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. St000542The number of left-to-right-minima of a permutation. St000617The number of global maxima of a Dyck path. St000630The length of the shortest palindromic decomposition of a binary word. St000659The number of rises of length at least 2 of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000700The protection number of an ordered tree. St000703The number of deficiencies of a permutation. St000733The row containing the largest entry of a standard tableau. St000742The number of big ascents of a permutation after prepending zero. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000758The length of the longest staircase fitting into an integer composition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000767The number of runs in an integer composition. St000820The number of compositions obtained by rotating the composition. St000862The number of parts of the shifted shape of a permutation. St000883The number of longest increasing subsequences of a permutation. St000899The maximal number of repetitions of an integer composition. St000903The number of different parts of an integer composition. St000908The length of the shortest maximal antichain in a poset. St000920The logarithmic height of a Dyck path. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000990The first ascent of a permutation. St000993The multiplicity of the largest part of an integer partition. St001011Number of simple modules of projective dimension 2 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. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. 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. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001280The number of parts of an integer partition that are at least two. St001316The domatic number of a graph. St001372The length of a longest cyclic run of ones of a binary word. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001591The number of graphs with the given composition of multiplicities of Laplacian eigenvalues. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St000013The height of a Dyck path. St000015The number of peaks of a Dyck path. St000439The position of the first down step of a Dyck path. St000443The number of long tunnels of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000451The length of the longest pattern of the form k 1 2. 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 $c_{n-1}$. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. 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. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001568The smallest positive integer that does not appear twice in the partition. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000989The number of final rises of a permutation. St001114The number of odd descents of a permutation. St001139The number of occurrences of hills of size 2 in a Dyck path. St000326The position of the first one in a binary word after appending a 1 at the end. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St001737The number of descents of type 2 in a permutation. St000117The number of centered tunnels of a Dyck path. St000241The number of cyclical small excedances. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000646The number of big ascents of a permutation. St000650The number of 3-rises of a permutation. St000664The number of right ropes of a permutation. St000754The Grundy value for the game of removing nestings in a perfect matching. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001513The number of nested exceedences of a permutation. St001549The number of restricted non-inversions between exceedances. 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. St001811The Castelnuovo-Mumford regularity of a permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000245The number of ascents of a permutation. St000314The number of left-to-right-maxima of a permutation. St000654The first descent of a permutation. St000834The number of right outer peaks of a permutation. St000991The number of right-to-left minima of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001597The Frobenius rank of a skew partition. St001665The number of pure excedances of a permutation. St001729The number of visible descents of a permutation. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001928The number of non-overlapping descents in a permutation. St000740The last entry of a permutation. 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 $eA$. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000256The number of parts from which one can substract 2 and still get an integer partition. St000929The constant term of the character polynomial of an integer partition. St001498The normalised height of a Nakayama algebra with magnitude 1. 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. St000768The number of peaks in an integer composition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001586The number of odd parts smaller than the largest even part in an integer partition. St000259The diameter of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000137The Grundy value of an integer partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St001525The number of symmetric hooks on the diagonal of a partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000016The number of attacking pairs of a standard tableau. St000149The number of cells of the partition whose leg is zero and arm is odd. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000547The number of even non-empty partial sums of an integer partition. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. St000877The depth of the binary word interpreted as a path. St000944The 3-degree of an integer partition. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. 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. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. 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. St001176The size of a partition minus its first part. St001185The number of indecomposable injective modules of grade at least 2 in 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. St001383The BG-rank of an integer partition. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001524The degree of symmetry of a binary word. St001584The area statistic between a Dyck path and its bounce path. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. 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. St001961The sum of the greatest common divisors of all pairs of parts. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000008The major index of the composition. St000010The length of the partition. St000012The area of a Dyck path. St000051The size of the left subtree of a binary tree. St000057The Shynar inversion number of a standard tableau. St000074The number of special entries. St000089The absolute variation of a composition. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000133The "bounce" of a permutation. St000147The largest part of an integer partition. St000148The number of odd parts of a partition. St000159The number of distinct parts of the integer partition. St000160The multiplicity of the smallest part of a partition. St000210Minimum over maximum difference of elements in cycles. St000211The rank of the set partition. St000225Difference between largest and smallest parts in a partition. St000228The size of a partition. St000234The number of global ascents of a permutation. St000242The number of indices that are not cyclical small weak excedances. St000252The number of nodes of degree 3 of a binary tree. St000288The number of ones in a binary word. St000290The major index of a binary word. St000293The number of inversions of a binary word. St000295The length of the border of a binary word. St000304The load of a permutation. St000306The bounce count of a Dyck path. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000338The number of pixed points of a permutation. St000340The number of non-final maximal constant sub-paths of length greater than one. St000341The non-inversion sum of a permutation. St000347The inversion sum of a binary word. St000348The non-inversion sum of a binary word. St000353The number of inner valleys of a permutation. St000369The dinv deficit of a Dyck path. St000377The dinv defect of an integer partition. St000378The diagonal inversion number of an integer partition. St000384The maximal part of the shifted composition of an integer partition. St000391The sum of the positions of the ones in a binary word. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000424The number of occurrences of the pattern 132 or of the pattern 231 in a permutation. St000426The number of occurrences of the pattern 132 or of the pattern 312 in a permutation. St000428The number of occurrences of the pattern 123 or of the pattern 213 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. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000445The number of rises of length 1 of a Dyck path. St000459The hook length of the base cell of a partition. 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. St000486The number of cycles of length at least 3 of a permutation. St000488The number of cycles of a permutation of length at most 2. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St000519The largest length of a factor maximising the subword complexity. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000538The number of even inversions of a permutation. St000548The number of different non-empty partial sums of an integer partition. St000549The number of odd partial sums of an integer partition. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000605The number of occurrences of the pattern {{1},{2,3}} such that 3 is maximal, (2,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000614The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000628The balance of a binary word. St000651The maximal size of a rise in a permutation. St000674The number of hills of a Dyck path. St000691The number of changes of a binary word. St000710The number of big deficiencies of a permutation. St000732The number of double deficiencies of a permutation. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000753The Grundy value for the game of Kayles on a binary word. St000783The side length of the largest staircase partition fitting into a partition. St000784The maximum of the length and the largest part of the integer partition. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000792The Grundy value for the game of ruler on a binary word. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000836The number of descents of distance 2 of a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000867The sum of the hook lengths in the first row of an integer partition. St000869The sum of the hook lengths of an integer partition. St000872The number of very big descents of a permutation. St000873The aix statistic of a permutation. St000897The number of different multiplicities of parts of an integer partition. St000921The number of internal inversions of a binary word. St000938The number of zeros of the symmetric group character corresponding to the partition. St000947The major index east count of a Dyck path. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension 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. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001091The number of parts in an integer partition whose next smaller part has the same size. St001094The depth index of a set partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001127The sum of the squares of the parts of a partition. St001141The number of occurrences of hills of size 3 in a Dyck path. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. 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. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001234The number of indecomposable three dimensional modules with projective dimension one. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001263The index of the maximal parabolic seaweed algebra associated with the composition. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001298The number of repeated entries in the Lehmer code of a permutation. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001377The major index minus the number of inversions of a permutation. St001402The number of separators in a permutation. St001403The number of vertical separators in a permutation. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001424The number of distinct squares in a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001469The holeyness of a permutation. St001470The cyclic holeyness of a permutation. St001485The modular major index of a binary word. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. 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. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001639The number of alternating subsets such that applying the permutation does not yield an alternating subset. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001697The shifted natural comajor index of a standard Young tableau. St001712The number of natural descents of a standard Young tableau. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001721The degree of a binary word. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001810The number of fixed points of a permutation smaller than its largest moved point. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001842The major index of a set partition. St001850The number of Hecke atoms of a permutation. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001487The number of inner corners of a skew partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000807The sum of the heights of the valleys of the associated bargraph. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000805The number of peaks of the associated bargraph. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001569The maximal modular displacement of a permutation. St000021The number of descents of a permutation. St000354The number of recoils of a permutation. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St001964The interval resolution global dimension of a poset. St000454The largest eigenvalue of a graph if it is integral. St000260The radius of a connected graph. St000383The last part of an integer composition. St000478Another weight of a partition according to Alladi. St000934The 2-degree of an integer partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000376The bounce deficit of a Dyck path. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. 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. St000083The number of left oriented leafs of a binary tree except the first one. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000219The number of occurrences of the pattern 231 in a permutation. St000251The number of nonsingleton blocks of a set partition. St000253The crossing number of a set partition. St000254The nesting number of a set partition. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000435The number of occurrences of the pattern 213 or of the pattern 231 in a permutation. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000472The sum of the ascent bottoms of a permutation. St000491The number of inversions of a set partition. St000497The lcb statistic of a set partition. St000502The number of successions of a set partitions. St000503The maximal difference between two elements in a common block. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000554The number of occurrences of the pattern {{1,2},{3}} in a set partition. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. 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. St000565The major index of a set partition. St000572The dimension exponent of a set partition. 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. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000586The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000592The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000595The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal. St000598The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, 3 is maximal, (2,3) are consecutive in a block. St000599The number of occurrences of the pattern {{1},{2,3}} such that (2,3) are consecutive in a block. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000607The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000612The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) are consecutive in a block. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000649The number of 3-excedences of a permutation. St000677The standardized bi-alternating inversion number of a permutation. St000693The modular (standard) major index of a standard tableau. St000728The dimension of a set partition. St000730The maximal arc length of a set partition. St000747A variant of the major index of a set partition. St000799The number of occurrences of the vincular pattern |213 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. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000837The number of ascents of distance 2 of a permutation. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St000874The position of the last double rise in a Dyck path. St000881The number of short braid edges in the graph of braid moves of a permutation. St000928The sum of the coefficients of the character polynomial of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000946The sum of the skew hook positions in a Dyck path. St000961The shifted major index of a permutation. St000976The sum of the positions of double up-steps of a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St000984The number of boxes below precisely one peak. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001082The number of boxed occurrences of 123 in a permutation. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001130The number of two successive successions in a permutation. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001371The length of the longest Yamanouchi prefix of a binary word. St001388The number of non-attacking neighbors of a permutation. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001480The number of simple summands of the module J^2/J^3. St001541The Gini index of an integer partition. St001552The number of inversions between excedances and fixed points of 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. St001731The factorization defect of a permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000455The second largest eigenvalue of a graph if it is integral. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St000017The number of inversions of a standard tableau. St000088The row sums of the character table of the symmetric group. St000145The Dyson rank of a partition. St000185The weighted size of a partition. St000296The length of the symmetric border of a binary word. St000618The number of self-evacuating tableaux of given shape. St000629The defect of a binary word. St000644The number of graphs with given frequency partition. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000811The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to Schur symmetric functions. St000995The largest even part of an integer partition. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. 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. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001214The aft of an integer partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001561The value of the elementary symmetric function evaluated at 1. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001596The number of two-by-two squares inside a skew partition. St001695The natural comajor 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. St001910The height of the middle non-run of a Dyck path. St001151The number of blocks with odd minimum. St001162The minimum jump of a permutation. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000456The monochromatic index of a connected graph. St000667The greatest common divisor of the parts of the partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000782The indicator function of whether a given perfect matching is an L & P matching. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001571The Cartan determinant of the integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000741The Colin de Verdière graph invariant. St000460The hook length of the last cell along the main diagonal of an integer partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001128The exponens consonantiae of a partition. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001360The number of covering relations in Young's lattice below a partition. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001527The cyclic permutation representation number of an integer partition. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001763The Hurwitz number of an integer partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. 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. St001943The sum of the squares of the hook lengths of an integer partition. St000356The number of occurrences of the pattern 13-2. St000054The first entry of the permutation. 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. St000045The number of linear extensions of a binary tree. St000091The descent variation of a composition. St000284The Plancherel distribution on integer partitions. St000365The number of double ascents of a permutation. St000562The number of internal points of a set partition. St000567The sum of the products of all pairs of parts. St000709The number of occurrences of 14-2-3 or 14-3-2. St000901The cube of the number of standard Young tableaux with shape given by the 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. 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. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St000090The variation of a composition. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000492The rob statistic of a set partition. St000498The lcs statistic of a set partition. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001722The number of minimal chains with small intervals between a binary word and the top element. St001896The number of right descents of a signed permutations. St001904The length of the initial strictly increasing segment of a parking function. St001935The number of ascents in a parking function. St001946The number of descents in a parking function. St000075The orbit size of a standard tableau under promotion. St000166The depth minus 1 of an ordered tree. St000308The height of the tree associated to a permutation. St000522The number of 1-protected nodes of a rooted tree. St000839The largest opener of a set partition. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St000230Sum of the minimal elements of the blocks of a set partition. St000521The number of distinct subtrees of an ordered tree. St001516The number of cyclic bonds of a permutation. St000735The last entry on the main diagonal of a standard tableau. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000177The number of free tiles in the pattern. St000178Number of free entries. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000508Eigenvalues of the random-to-random operator acting on a simple module. St000509The diagonal index (content) of a partition. St000706The product of the factorials of the multiplicities of an integer partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. 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. St000939The number of characters of the symmetric group whose value on the partition is positive. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001095The number of non-isomorphic posets with precisely one further covering relation. 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. St000632The jump number of the poset. St000736The last entry in the first row of a semistandard tableau. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001235The global dimension of the corresponding Comp-Nakayama algebra. St000717The number of ordinal summands of a poset.