St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St000676The number of odd rises of a Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows: St000521The number of distinct subtrees of an ordered tree. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St000052The number of valleys of a Dyck path not on the x-axis. St000121The number of occurrences of the contiguous pattern [.,[.,[.,[.,.]]]] in a binary tree. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000126The number of occurrences of the contiguous pattern [.,[.,[.,[.,[.,.]]]]] in a binary tree. St000245The number of ascents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001298The number of repeated entries in the Lehmer code of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000157The number of descents of a standard tableau. St000288The number of ones in a binary word. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000340The number of non-final maximal constant sub-paths of length greater than one. St000377The dinv defect of an integer partition. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000443The number of long tunnels of a Dyck path. St000691The number of changes of a binary word. St000730The maximal arc length of a set partition. St000744The length of the path to the largest entry in a standard Young tableau. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001176The size of a partition minus its first part. St001187The number of simple modules with grade at least one 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. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000010The length of the partition. St000011The number of touch points (or returns) of a Dyck path. St000093The cardinality of a maximal independent set of vertices of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000105The number of blocks in the set partition. St000147The largest part of an integer partition. St000384The maximal part of the shifted composition of an integer partition. St000507The number of ascents of a standard tableau. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000733The row containing the largest entry of a standard tableau. St000784The maximum of the length and the largest part of the integer partition. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000925The number of topologically connected components of a set partition. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001279The sum of the parts of an integer partition that are at least two. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001389The number of partitions of the same length below the given integer partition. St001494The Alon-Tarsi number of a graph. St001581The achromatic number of a graph. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000094The depth of an ordered tree. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St000172The Grundy number of a graph. St001029The size of the core of a graph. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000632The jump number of the poset. St000527The width of the poset. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St000167The number of leaves of an ordered tree. St000528The height of a poset. St000912The number of maximal antichains in a poset. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001343The dimension of the reduced incidence algebra of a poset. St001963The tree-depth of a graph. St000651The maximal size of a rise in a permutation. St000470The number of runs in a permutation. St000619The number of cyclic descents of a permutation. St000652The maximal difference between successive positions of a permutation. St001717The largest size of an interval in a poset. St000354The number of recoils of a permutation. St000795The mad of a permutation. St000809The reduced reflection length of the permutation. St000829The Ulam distance of a permutation to the identity permutation. St000831The number of indices that are either descents or recoils. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000957The number of Bruhat lower covers of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001489The maximum of the number of descents and the number of inverse descents. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St000306The bounce count of a Dyck path. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St000308The height of the tree associated to a permutation. St000451The length of the longest pattern of the form k 1 2. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000356The number of occurrences of the pattern 13-2. St000662The staircase size of the code of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000071The number of maximal chains in a poset. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000318The number of addable cells of the Ferrers diagram of an integer partition. St001044The number of pairs whose larger element is at most one more than half the size of the perfect matching. St001874Lusztig's a-function for the symmetric group. St000035The number of left outer peaks of a permutation. St000731The number of double exceedences of a permutation. St001557The number of inversions of the second entry of a permutation. St001928The number of non-overlapping descents in a permutation. St000141The maximum drop size of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St001091The number of parts in an integer partition whose next smaller part has the same size. St000159The number of distinct parts of the integer partition. St001726The number of visible inversions of a permutation. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St000702The number of weak deficiencies of a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St000956The maximal displacement of a permutation. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000183The side length of the Durfee square of an integer partition. St000028The number of stack-sorts needed to sort a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St000624The normalized sum of the minimal distances to a greater element. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St000332The positive inversions of an alternating sign matrix. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000982The length of the longest constant subword. St001644The dimension of a graph. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000862The number of parts of the shifted shape of a permutation. St000326The position of the first one in a binary word after appending a 1 at the end. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St001884The number of borders of a binary word. St000295The length of the border of a binary word. St000296The length of the symmetric border of a binary word. St000983The length of the longest alternating subword. St000214The number of adjacencies of a permutation. St001432The order dimension of the partition. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000489The number of cycles of a permutation of length at most 3. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001461The number of topologically connected components of the chord diagram of a permutation. St000007The number of saliances of the permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St000292The number of ascents of a binary word. St000164The number of short pairs. St000291The number of descents of a binary word. St000390The number of runs of ones in a binary word. 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. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St000021The number of descents of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000120The number of left tunnels of a Dyck path. St000168The number of internal nodes of an ordered tree. St000238The number of indices that are not small weak excedances. St000316The number of non-left-to-right-maxima of a permutation. St000331The number of upper interactions of a 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. 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. St000015The number of peaks of a Dyck path. St000166The depth minus 1 of an ordered tree. St000325The width of the tree associated to a permutation. St000991The number of right-to-left minima of a permutation. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St000317The cycle descent number of a permutation. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. 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. 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. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-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. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000080The rank of the poset. St000155The number of exceedances (also excedences) of a permutation. St000216The absolute length of a permutation. St000314The number of left-to-right-maxima of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000864The number of circled entries of the shifted recording tableau of a permutation. St001047The maximal number of arcs crossing a given arc of a perfect matching. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over 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. St001427The number of descents of a signed permutation. St001480The number of simple summands of the module J^2/J^3. 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. St000213The number of weak exceedances (also weak excedences) of a permutation. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000242The number of indices that are not cyclical small weak excedances. St000542The number of left-to-right-minima of a permutation. St000822The Hadwiger number of the graph. St000877The depth of the binary word interpreted as a 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. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001180Number of indecomposable injective modules with projective dimension at most 1. 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. St000255The number of reduced Kogan faces with the permutation as type. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001256Number of simple reflexive modules that are 2-stable reflexive. St000056The decomposition (or block) number of a permutation. St000078The number of alternating sign matrices whose left key is the permutation. St000570The Edelman-Greene number of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000889The number of alternating sign matrices with the same antidiagonal sums. St001006Number of simple modules with projective dimension equal to the global dimension of 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. St001162The minimum jump of a permutation. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. 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 $g$. 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. St001344The neighbouring number of a permutation. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000217The number of occurrences of the pattern 312 in a permutation. St000221The number of strong fixed points of a permutation. St000234The number of global ascents of a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000358The number of occurrences of the pattern 31-2. St000360The number of occurrences of the pattern 32-1. St000367The number of simsun double descents of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000664The number of right ropes of a permutation. St000666The number of right tethers of a permutation. St000674The number of hills of a Dyck path. St000709The number of occurrences of 14-2-3 or 14-3-2. St000732The number of double deficiencies of a permutation. St000750The number of occurrences of the pattern 4213 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. St000879The number of long braid edges in the graph of braid moves of a permutation. St000962The 3-shifted major index of a permutation. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001139The number of occurrences of hills of size 2 in a Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001381The fertility of a permutation. St001513The number of nested exceedences of a permutation. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001705The number of occurrences of the pattern 2413 in a permutation. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001741The largest integer such that all patterns of this size are contained in the permutation. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St001847The number of occurrences of the pattern 1432 in a permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000758The length of the longest staircase fitting into an integer composition. St000764The number of strong records in an integer composition. St001812The biclique partition number of a graph. St001965The number of decreasable positions in the corner sum matrix of an alternating sign matrix. St000455The second largest eigenvalue of a graph if it is integral. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. 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 $eA$. St000735The last entry on the main diagonal of a standard tableau. St000924The number of topologically connected components of a perfect matching. St000481The number of upper covers of a partition in dominance order. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St001782The order of rowmotion on the set of order ideals of a poset. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000383The last part of an integer composition. St000788The number of nesting-similar perfect matchings of a perfect matching. St000124The cardinality of the preimage of the Simion-Schmidt map. St000647The number of big descents of a permutation. St000787The number of flips required to make a perfect matching noncrossing. St000871The number of very big ascents of 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. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St000834The number of right outer peaks of a permutation. 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 $K[x]/(x^n)$. St000022The number of fixed points of a permutation. St000352The Elizalde-Pak rank of a permutation. St000405The number of occurrences of the pattern 1324 in a permutation. St000842The breadth of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001371The length of the longest Yamanouchi prefix of a binary word. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001556The number of inversions of the third entry of a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St000044The number of vertices of the unicellular map given by a perfect matching. St000017The number of inversions of a standard tableau. St001721The degree of a binary word. St000016The number of attacking pairs of a standard tableau. St000958The number of Bruhat factorizations of a permutation. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001481The minimal height of a peak of a Dyck path. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000153The number of adjacent cycles of a permutation. St000223The number of nestings in the permutation. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. 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. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. 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. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. 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. St001234The number of indecomposable three dimensional modules with projective dimension one. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St000891The number of distinct diagonal sums of a permutation matrix. St001958The degree of the polynomial interpolating the values of a permutation. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000181The number of connected components of the Hasse diagram for the poset. St000486The number of cycles of length at least 3 of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001081The number of minimal length factorizations of a permutation into star transpositions. St001590The crossing number of a perfect matching. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001890The maximum magnitude of the Möbius function of a poset. St000488The number of cycles of a permutation of length at most 2. St000950Number of tilting modules of the corresponding LNakayama algebra, where a tilting module is a generalised tilting module of projective dimension 1. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001520The number of strict 3-descents. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001569The maximal modular displacement of a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001850The number of Hecke atoms of a permutation. St000037The sign of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000209Maximum difference of elements in cycles. St000210Minimum over maximum difference of elements in cycles. St001002Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001077The prefix exchange distance of a permutation. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000501The size of the first part in the decomposition of a permutation. St000673The number of non-fixed points of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000949Gives the number of generalised tilting modules of the corresponding LNakayama algebra. St001003The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001468The smallest fixpoint of a permutation. St001243The sum of coefficients in the Schur basis of certain LLT polynomials associated with a Dyck path. St001731The factorization defect of a permutation. St000690The size of the conjugacy class of a permutation. St001242The toal dimension of certain Sn modules determined by LLT polynomials associated with a Dyck path. St001560The product of the cardinalities of the lower order ideal and upper order ideal generated by a permutation in weak order. St001528The number of permutations such that the product with the permutation has the same number of fixed points. St000023The number of inner peaks of a permutation. St000033The number of permutations greater than or equal to the given permutation in (strong) Bruhat order. St000061The number of nodes on the left branch of a binary tree. St000084The number of subtrees. St000096The number of spanning trees of a graph. St000230Sum of the minimal elements of the blocks of a set partition. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000287The number of connected components of a graph. St000310The minimal degree of a vertex of a graph. St000353The number of inner valleys of a permutation. St000450The number of edges minus the number of vertices plus 2 of a graph. St000545The number of parabolic double cosets with minimal element being the given permutation. St000627The exponent of a binary word. St000654The first descent of a permutation. St000655The length of the minimal rise of a Dyck path. St000660The number of rises of length at least 3 of a Dyck path. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000696The number of cycles in the breakpoint graph of a permutation. St000740The last entry of a permutation. St000741The Colin de Verdière graph invariant. St000756The sum of the positions of the left to right maxima of a permutation. St000779The tier of a permutation. St000843The decomposition number of a perfect matching. St000872The number of very big descents of a permutation. St000876The number of factors in the Catalan decomposition of a binary word. St000884The number of isolated descents of a permutation. St000963The 2-shifted major index 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. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001048The number of leaves in the subtree containing 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 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. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001260The permanent of an alternating sign matrix. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001470The cyclic holeyness of a permutation. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding 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. St001589The nesting number of a perfect matching. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001665The number of pure excedances of a permutation. St001737The number of descents of type 2 in a permutation. St001828The Euler characteristic of a graph. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. 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. St000002The number of occurrences of the pattern 123 in a permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000042The number of crossings of a perfect matching. St000051The size of the left subtree of a binary tree. St000092The number of outer peaks of a permutation. St000095The number of triangles of a graph. St000099The number of valleys of a permutation, including the boundary. St000117The number of centered tunnels of a Dyck path. St000133The "bounce" of a permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000241The number of cyclical small excedances. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000315The number of isolated vertices of a graph. St000357The number of occurrences of the pattern 12-3. St000365The number of double ascents of a permutation. St000441The number of successions of a permutation. St000461The rix statistic of a permutation. St000508Eigenvalues of the random-to-random operator acting on a simple module. St000516The number of stretching pairs of a permutation. St000534The number of 2-rises of a permutation. St000629The defect of a binary word. St000649The number of 3-excedences of a permutation. St000650The number of 3-rises of a permutation. St000665The number of rafts of a permutation. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000819The propagating number of a perfect matching. St000873The aix statistic of a permutation. St000878The number of ones minus the number of zeros of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000886The number of permutations with the same antidiagonal sums. St000895The number of ones on the main diagonal of an alternating sign matrix. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000989The number of final rises of a permutation. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001025Number of simple modules with projective dimension 4 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. St001049The smallest label in the subtree not containing 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001082The number of boxed occurrences of 123 in a permutation. St001130The number of two successive successions in a permutation. St001131The number of trivial trees on the path to label one in the decreasing labelled binary unordered tree associated with the perfect matching. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001220The width of a permutation. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001411The number of patterns 321 or 3412 in a permutation. St001429The number of negative entries in a signed permutation. St001471The magnitude of a Dyck path. St001530The depth of a Dyck path. St001536The number of cyclic misalignments of a permutation. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001640The number of ascent tops in the permutation such that all smaller elements appear before. 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. St001729The number of visible descents of a permutation. St001806The upper middle entry of a permutation. St001856The number of edges in the reduced word graph of a permutation. St001871The number of triconnected components of a graph. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St000454The largest eigenvalue of a graph if it is integral. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St001090The number of pop-stack-sorts needed to sort a permutation. St001517The length of a longest pair of twins in a permutation. St001667The maximal size of a pair of weak twins for a permutation. St001668The number of points of the poset minus the width of the poset. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000327The number of cover relations in a poset. St000402Half the size of the symmetry class of a permutation. St000538The number of even inversions of a permutation. St000638The number of up-down runs of a permutation. St000863The length of the first row of the shifted shape of a permutation. St000953The largest degree of an irreducible factor of the Coxeter polynomial of the Dyck path over the rational numbers. St001439The number of even weak deficiencies and of odd weak exceedences. St001497The position of the largest weak excedence of a permutation. St001637The number of (upper) dissectors of a poset. St001703The villainy of a graph. St001727The number of invisible inversions of a permutation. St001925The minimal number of zeros in a row of an alternating sign matrix. St000067The inversion number of the alternating sign matrix. St000144The pyramid weight of the Dyck path. St000197The number of entries equal to positive one in the alternating sign matrix. St000653The last descent of a permutation. St000890The number of nonzero entries in an alternating sign matrix. 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. St001245The cyclic maximal difference between two consecutive entries of a permutation. St001246The maximal difference between two consecutive entries of a permutation. St001827The number of two-component spanning forests of a graph. St001869The maximum cut size of a graph. St000189The number of elements in the poset. St000229Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. St000231Sum of the maximal elements of the blocks of a set partition. St000235The number of indices that are not cyclical small weak excedances. St000240The number of indices that are not small excedances. St000336The leg major index of a standard tableau. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. 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. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001778The largest greatest common divisor of an element and its image in a permutation. St000026The position of the first return of a Dyck path. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St000401The size of the symmetry class of a permutation. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St000134The size of the orbit of an alternating sign matrix under gyration. St000423The number of occurrences of the pattern 123 or of the pattern 132 in a permutation. St000428The number of occurrences of the pattern 123 or of the pattern 213 in a permutation. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St000830The total displacement of a permutation. St000301The number of facets of the stable set polytope of a graph. St000109The number of elements less than or equal to the given element in Bruhat order. St000343The number of spanning subgraphs of a graph. St000302The determinant of the distance matrix of a connected graph. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St000959The number of strong Bruhat factorizations of a permutation. St001757The number of orbits of toric promotion on a graph. St000880The number of connected components 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. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St000307The number of rowmotion orbits of a poset. St000546The number of global descents of a permutation. St000648The number of 2-excedences of a permutation. St001060The distinguishing index of a graph. St001394The genus of a permutation. St000837The number of ascents of distance 2 of a permutation. St000746The number of pairs with odd minimum in a perfect matching. 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. St001712The number of natural descents of a standard Young tableau. St000635The number of strictly order preserving maps of a poset into itself. St001462The number of factors of a standard tableaux under concatenation. St000529The number of permutations whose descent word is the given binary word. St000669The number of permutations obtained by switching ascents or descents of size 2. St000670The reversal length of a permutation. St000742The number of big ascents of a permutation after prepending zero. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000054The first entry of the permutation. 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. St000218The number of occurrences of the pattern 213 in a permutation. St000220The number of occurrences of the pattern 132 in a permutation. St000297The number of leading ones in a binary word. St000359The number of occurrences of the pattern 23-1. St000389The number of runs of ones of odd length in a binary word. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St001083The number of boxed occurrences of 132 in a permutation. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001434The number of negative sum pairs of a signed permutation. St001964The interval resolution global dimension of a poset. St000187The determinant of an alternating sign matrix. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St001330The hat guessing number of a graph. St001428The number of B-inversions of a signed permutation. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001613The binary logarithm of the size of the center of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St001881The number of factors of a lattice as a Cartesian product of lattices. St000215The number of adjacencies of a permutation, zero appended. St000312The number of leaves in a graph. St000313The number of degree 2 vertices of a graph. St000366The number of double descents of a permutation. St000701The protection number of a binary tree. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001095The number of non-isomorphic posets with precisely one further covering relation. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001469The holeyness of a permutation. St001597The Frobenius rank of a skew partition. St001615The number of join prime elements of a lattice. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001846The number of elements which do not have a complement in the lattice. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000840The number of closers smaller than the largest opener in a perfect matching. St000200The row of the unique '1' in the last column of the alternating sign matrix. St001040The depth of the decreasing labelled binary unordered tree associated with the perfect matching. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St000841The largest opener of a perfect matching. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St000232The number of crossings of a set partition. St000233The number of nestings of a set partition. St000496The rcs statistic of a set partition. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St001801Half the number of preimage-image pairs of different parity in a permutation. St000039The number of crossings of a permutation. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St000001The number of reduced words for a permutation. St000091The descent variation of a composition. St000409The number of pitchforks in a binary tree. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St001050The number of terminal closers of a set partition. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001335The cardinality of a minimal cycle-isolating set of a graph. St001722The number of minimal chains with small intervals between a binary word and the top element. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St001114The number of odd descents of a permutation. 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. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001263The index of the maximal parabolic seaweed algebra associated with the composition. St001516The number of cyclic bonds of a permutation. St001565The number of arithmetic progressions of length 2 in a permutation. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001566The length of the longest arithmetic progression in a permutation. St000381The largest part of an integer composition. St000519The largest length of a factor maximising the subword complexity. St000922The minimal number such that all substrings of this length are unique. St001041The depth of the label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001966Half the global dimension of the stable Auslander algebra of a sincere Nakayama algebra (with associated Dyck path).