Your data matches 859 different statistics following compositions of up to 3 maps.
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St000212: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 1 = 2 - 1
[2]
=> 0 = 1 - 1
[1,1]
=> 1 = 2 - 1
[3]
=> 0 = 1 - 1
[2,1]
=> 1 = 2 - 1
[1,1,1]
=> 1 = 2 - 1
Description
The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. Summing over all partitions of $n$ yields the sequence $$1, 1, 1, 2, 4, 9, 22, 59, 170, 516, 1658, \dots$$ which is [[oeis:A237770]]. The references in this sequence of the OEIS indicate a connection with Baxter permutations.
St001121: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 1 = 2 - 1
[2]
=> 1 = 2 - 1
[1,1]
=> 0 = 1 - 1
[3]
=> 1 = 2 - 1
[2,1]
=> 1 = 2 - 1
[1,1,1]
=> 0 = 1 - 1
Description
The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. The Kronecker coefficient is the multiplicity $g_{\mu,\nu}^\lambda$ of the Specht module $S^\lambda$ in $S^\mu\otimes S^\nu$: $$ S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda $$ This statistic records the Kronecker coefficient $g_{\lambda,\lambda}^\lambda$.
St001442: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 1 = 2 - 1
[2]
=> 1 = 2 - 1
[1,1]
=> 0 = 1 - 1
[3]
=> 1 = 2 - 1
[2,1]
=> 0 = 1 - 1
[1,1,1]
=> 1 = 2 - 1
Description
The number of standard Young tableaux whose major index is divisible by the size of a given integer partition.
St001593: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 1 = 2 - 1
[2]
=> 1 = 2 - 1
[1,1]
=> 0 = 1 - 1
[3]
=> 1 = 2 - 1
[2,1]
=> 1 = 2 - 1
[1,1,1]
=> 0 = 1 - 1
Description
This is the number of standard Young tableaux of the given shifted shape. For an integer partition $\lambda = (\lambda_1,\dots,\lambda_k)$, the shifted diagram is obtained by moving the $i$-th row in the diagram $i-1$ boxes to the right, i.e., $$\lambda^∗ = \{(i, j) | 1 \leq i \leq k, i \leq j \leq \lambda_i + i − 1 \}.$$ In particular, this statistic is zero if and only if $\lambda_{i+1} = \lambda_i$ for some $i$.
St001601: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 1 = 2 - 1
[2]
=> 1 = 2 - 1
[1,1]
=> 0 = 1 - 1
[3]
=> 1 = 2 - 1
[2,1]
=> 1 = 2 - 1
[1,1,1]
=> 0 = 1 - 1
Description
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees.
Mp00317: Integer partitions odd partsBinary words
St001885: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 1 => 2
[2]
=> 0 => 2
[1,1]
=> 11 => 1
[3]
=> 1 => 2
[2,1]
=> 01 => 2
[1,1,1]
=> 111 => 1
Description
The number of binary words with the same proper border set. The proper border set of a binary word $w$ is the set of proper prefixes which are also suffixes of $w$. For example, $0010000010$, $0010100010$ and $0010110010$ are the words with proper border set $\{0, 0010\}$, whereas $0010010010$ has proper border set $\{0, 0010, 0010010\}$.
Mp00043: Integer partitions to Dyck pathDyck paths
St001159: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> 1 = 2 - 1
[2]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[3]
=> [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[2,1]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
Description
Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra.
Mp00043: Integer partitions to Dyck pathDyck paths
St001185: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> 1 = 2 - 1
[2]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1]
=> [1,0,1,1,0,0]
=> 0 = 1 - 1
[3]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[2,1]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
Description
The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra.
Mp00043: Integer partitions to Dyck pathDyck paths
St001204: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> 1 = 2 - 1
[2]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[3]
=> [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[2,1]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
Description
Call 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. Associate to this special CNakayama algebra a Dyck path as follows: In the list L delete the first entry $c_0$ and substract from all other entries $n$−1 and then append the last element 1. The result is a Kupisch series of an LNakayama algebra. The statistic gives the $(t-1)/2$ when $t$ is the projective dimension of the simple module $S_{n-2}$.
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00129: Dyck paths to 321-avoiding permutation (Billey-Jockusch-Stanley)Permutations
St000007: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [2,1] => 2
[2]
=> [1,1,0,0,1,0]
=> [1,3,2] => 2
[1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 2
[2,1]
=> [1,0,1,0,1,0]
=> [2,3,1] => 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1
Description
The number of saliances of the permutation. A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
The following 849 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000025The number of initial rises of a Dyck path. St000054The first entry of the permutation. St000288The number of ones in a binary word. St000326The position of the first one in a binary word after appending a 1 at the end. St000542The number of left-to-right-minima of a permutation. St000630The length of the shortest palindromic decomposition of a binary word. St000678The number of up steps after the last double rise of a Dyck path. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000759The smallest missing part in an integer partition. St000982The length of the longest constant subword. St000990The first ascent of a permutation. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. 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. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001733The number of weak left to right maxima of a Dyck path. St001884The number of borders of a binary word. St000153The number of adjacent cycles of a permutation. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000297The number of leading ones in a binary word. St000352The Elizalde-Pak rank of a permutation. St000389The number of runs of ones of odd length in a binary word. St000390The number of runs of ones in a binary word. St000392The length of the longest run of ones in a binary word. St000439The position of the first down 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. St000658The number of rises of length 2 of a Dyck path. St000864The number of circled entries of the shifted recording tableau of a permutation. St000877The depth of the binary word interpreted as a path. St000932The number of occurrences of the pattern UDU in a Dyck path. 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 $c_{n-1}$. 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. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001114The number of odd descents of a permutation. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension 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. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001273The projective dimension of the first term in an injective coresolution of the regular module. 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. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001413Half the length of the longest even length palindromic prefix of a binary word. St001484The number of singletons of an integer partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St000950Number of tilting modules of the corresponding LNakayama algebra, where a tilting module is a generalised tilting module of projective dimension 1. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000011The number of touch points (or returns) of a Dyck path. St000061The number of nodes on the left branch of a binary tree. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000068The number of minimal elements in a poset. St000069The number of maximal elements of a poset. St000099The number of valleys of a permutation, including the boundary. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000314The number of left-to-right-maxima of a permutation. St000335The difference of lower and upper interactions. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000383The last part of an integer composition. St000504The cardinality of the first block of a set partition. St000654The first descent of a permutation. St000657The smallest part of an integer composition. St000675The number of centered multitunnels of a Dyck path. St000694The number of affine bounded permutations that project to a given permutation. St000696The number of cycles in the breakpoint graph of a permutation. St000733The row containing the largest entry of a standard tableau. St000740The last entry of a permutation. St000753The Grundy value for the game of Kayles on a binary word. St000757The length of the longest weakly inreasing subsequence of parts of 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. St000765The number of weak records in an integer composition. St000808The number of up steps of the associated bargraph. 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. St000823The number of unsplittable factors of the set partition. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000876The number of factors in the Catalan decomposition of a binary word. St000883The number of longest increasing subsequences of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000917The open packing number of a graph. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. 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 $c_{n−1}$. St000971The smallest closer of a set partition. St000983The length of the longest alternating subword. St000991The number of right-to-left minima 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. St001041The depth of the label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001050The number of terminal closers of a set partition. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. 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. St001235The global 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. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001313The number of Dyck paths above the lattice path given by a binary word. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001372The length of a longest cyclic run of ones of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001439The number of even weak deficiencies and of odd weak exceedences. St001461The number of topologically connected components of the chord diagram of a permutation. St001471The magnitude of a Dyck path. St001481The minimal height of a peak of a Dyck path. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001516The number of cyclic bonds of a permutation. St001568The smallest positive integer that does not appear twice in the partition. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001732The number of peaks visible from the left. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001806The upper middle entry of a permutation. St000022The number of fixed points of a permutation. St000023The number of inner peaks of a permutation. St000051The size of the left subtree of a binary tree. St000091The descent variation of a composition. St000096The number of spanning trees of a graph. St000120The number of left tunnels of a Dyck path. St000124The cardinality of the preimage of the Simion-Schmidt map. St000133The "bounce" of a permutation. St000137The Grundy value of an integer partition. St000143The largest repeated part of a partition. St000148The number of odd parts of a partition. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000236The number of cyclical small weak excedances. St000237The number of small exceedances. St000239The number of small weak excedances. St000241The number of cyclical small excedances. St000257The number of distinct parts of a partition that occur at least twice. St000271The chromatic index of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000338The number of pixed points of a permutation. St000344The number of strongly connected outdegree sequences of a graph. St000348The non-inversion sum of a binary word. St000356The number of occurrences of the pattern 13-2. St000374The number of exclusive right-to-left minima of a permutation. St000391The sum of the positions of the ones in a binary word. St000409The number of pitchforks in a binary tree. St000441The number of successions of a permutation. St000444The length of the maximal rise of a Dyck path. St000445The number of rises of length 1 of a Dyck path. St000450The number of edges minus the number of vertices plus 2 of a graph. St000461The rix statistic of a permutation. St000475The number of parts equal to 1 in a partition. St000486The number of cycles of length at least 3 of a permutation. St000523The number of 2-protected nodes of a rooted tree. St000534The number of 2-rises of a permutation. St000546The number of global descents of a permutation. St000573The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element. St000624The normalized sum of the minimal distances to a greater element. St000628The balance of a binary word. St000660The number of rises of length at least 3 of a Dyck path. St000661The number of rises of length 3 of a Dyck path. St000663The number of right floats of a permutation. St000665The number of rafts of a permutation. St000682The Grundy value of Welter's game on a binary word. 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. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. St000738The first entry in the last row of a standard tableau. St000754The Grundy value for the game of removing nestings in a perfect matching. St000761The number of ascents in an integer composition. St000769The major index of a composition regarded as a word. St000779The tier of a permutation. St000792The Grundy value for the game of ruler on a binary word. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. 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. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000884The number of isolated descents of a permutation. St000891The number of distinct diagonal sums of a permutation matrix. St000929The constant term of the character polynomial of an integer partition. St000931The number of occurrences of the pattern UUU in a Dyck path. St000948The chromatic discriminant of a graph. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. 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. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. 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. St001057The Grundy value of the game of creating an independent set in a graph. St001061The number of indices that are both descents and recoils of a permutation. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001071The beta invariant of the graph. St001083The number of boxed occurrences of 132 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001115The number of even descents of a permutation. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. 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. St001153The number of blocks with even minimum in a set partition. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. 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. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. 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 $g$. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra 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. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. 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. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. 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. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. 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. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001260The permanent of an alternating sign matrix. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. 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. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001383The BG-rank of an integer partition. St001403The number of vertical separators in a permutation. St001405The number of bonds in a permutation. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001415The length of the longest palindromic prefix of a binary word. St001420Half the length of a longest factor which is its own reverse-complement 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. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001479The number of bridges of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001510The number of self-evacuating linear extensions of a finite poset. St001524The degree of symmetry of a binary word. St001525The number of symmetric hooks on the diagonal of a partition. St001530The depth of a Dyck path. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001673The degree of asymmetry of an integer composition. St001675The number of parts equal to the part in the reversed composition. St001712The number of natural descents of a standard Young tableau. St001721The degree of a binary word. St001737The number of descents of type 2 in a permutation. 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. St001777The number of weak descents in an integer composition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001801Half the number of preimage-image pairs of different parity in a permutation. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001810The number of fixed points of a permutation smaller than its largest moved point. St001826The maximal number of leaves on a vertex of a graph. St001850The number of Hecke atoms of a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001931The weak major index of an integer composition regarded as a word. 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. 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. St000090The variation of a composition. St000146The Andrews-Garvan crank of a partition. St001377The major index minus the number of inversions of a permutation. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St000442The maximal area to the right of an up step of a Dyck path. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000659The number of rises of length at least 2 of a Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. 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. St000418The number of Dyck paths that are weakly below a Dyck path. St000485The length of the longest cycle of a permutation. St000489The number of cycles of a permutation of length at most 3. St000668The least common multiple of the parts of the partition. St000702The number of weak deficiencies of a permutation. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001062The maximal size of a block of a set partition. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001500The global dimension of magnitude 1 Nakayama algebras. St001531Number of partial orders contained in the poset determined by the Dyck path. St001959The product of the heights of the peaks of a Dyck path. St000083The number of left oriented leafs of a binary tree except the first one. St000216The absolute length of 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. St000354The number of recoils of a permutation. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000539The number of odd inversions of a permutation. St000558The number of occurrences of the pattern {{1,2}} in a set partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000677The standardized bi-alternating inversion number of a permutation. St000693The modular (standard) major index of a standard tableau. St000730The maximal arc length of a set partition. St000809The reduced reflection length of the permutation. St000829The Ulam distance of a permutation to the identity permutation. St000833The comajor index of a permutation. St000848The balance constant multiplied with the number of linear extensions of a poset. St000874The position of the last double rise in a Dyck path. St000919The number of maximal left branches of a binary tree. St000934The 2-degree of an integer partition. St000946The sum of the skew hook positions in a Dyck path. St000947The major index east count of a Dyck path. St000976The sum of the positions of double up-steps of a Dyck path. St000984The number of boxes below precisely one peak. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001077The prefix exchange distance 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. St001569The maximal modular displacement of a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St000082The number of elements smaller than a binary tree in Tamari order. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St000420The number of Dyck paths that are weakly above a Dyck path. St000633The size of the automorphism group of a poset. St000717The number of ordinal summands of a poset. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000844The size of the largest block in the direct sum decomposition of a permutation. 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. St000898The number of maximal entries in the last diagonal of the monotone triangle. St000906The length of the shortest maximal chain in a poset. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St000925The number of topologically connected components of a set partition. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001346The number of parking functions that give the same permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001808The box weight or horizontal decoration of a Dyck path. St000219The number of occurrences of the pattern 231 in a permutation. St000289The decimal representation of a binary word. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000472The sum of the ascent bottoms of a permutation. St000490The intertwining number of a set partition. St000492The rob statistic of a set partition. St000493The los statistic of a set partition. St000494The number of inversions of distance at most 3 of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000499The rcb statistic of a set partition. St000502The number of successions of a set partitions. St000503The maximal difference between two elements in a common block. St000574The number of occurrences of the pattern {{1},{2}} such that 1 is a minimal and 2 a maximal element. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000640The rank of the largest boolean interval in a poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St000653The last descent of a permutation. St000728The dimension of a set partition. St000794The mak of a permutation. St000795The mad of a permutation. St000796The stat' of a permutation. St000797The stat`` of a permutation. St000798The makl of a permutation. St000831The number of indices that are either descents or recoils. St000850The number of 1/2-balanced pairs in a poset. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St001080The minimal length of a factorization of a permutation using the transposition (12) and the cycle (1,. 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 $eA$. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001281The normalized isoperimetric number of a graph. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001587Half of the largest even part of an integer partition. St001592The maximal number of simple paths between any two different vertices of a graph. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001668The number of points of the poset minus the width of the poset. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001250The number of parts of a partition that are not congruent 0 modulo 3. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000038The product of the heights of the descending steps of a Dyck path. St000063The number of linear extensions of a certain poset defined for an integer partition. St000108The number of partitions contained in the given partition. 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. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000443The number of long tunnels of a Dyck path. St000460The hook length of the last cell along the main diagonal of an integer partition. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000532The total number of rook placements on a Ferrers board. St000667The greatest common divisor of the parts of the partition. St000706The product of the factorials of the multiplicities of an integer partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. 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. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000885The number of critical steps in the Catalan decomposition of a binary word. St000937The number of positive values of the symmetric group character corresponding to the partition. St000993The multiplicity of the largest part of an integer partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. 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. St001118The acyclic chromatic index of a graph. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001224Let X be the direct sum of all simple modules 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. 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. 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. St001389The number of partitions of the same length below the given integer partition. St001400The total number of Littlewood-Richardson tableaux of given shape. St001432The order dimension of the partition. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001515The vector space dimension of the socle of the first syzygy module of the regular module (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. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. 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. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001722The number of minimal chains with small intervals between a binary word and the top element. St001814The number of partitions interlacing the given partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St000145The Dyson rank of a partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000260The radius of a connected graph. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000478Another weight of a partition according to Alladi. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000567The sum of the products of all pairs of parts. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) 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. St000618The number of self-evacuating tableaux of given shape. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000873The aix statistic of a permutation. 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. St001176The size of a partition minus its first part. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001541The Gini index of an integer partition. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001657The number of twos in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000302The determinant of the distance matrix of a connected graph. St000928The sum of the coefficients of the character polynomial of an integer partition. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St000010The length of the partition. St000015The number of peaks of a Dyck path. St000026The position of the first return of a Dyck path. St000048The multinomial of the parts of a partition. St000088The row sums of the character table of the symmetric group. St000117The number of centered tunnels of a Dyck path. St000144The pyramid weight of the Dyck path. St000147The largest part of an integer partition. St000160The multiplicity of the smallest part of a partition. St000179The product of the hook lengths of the integer partition. St000184The size of the centralizer of any permutation of given cycle type. St000228The size of a partition. St000290The major index of a binary word. St000293The number of inversions of a binary word. St000296The length of the symmetric border of a binary word. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000346The number of coarsenings of a partition. St000378The diagonal inversion number of an integer partition. St000384The maximal part of the shifted composition of an integer partition. St000393The number of strictly increasing runs in a binary word. St000395The sum of the heights of the peaks of a Dyck path. St000459The hook length of the base cell of a partition. St000519The largest length of a factor maximising the subword complexity. St000531The leading coefficient of the rook polynomial of an integer partition. St000548The number of different non-empty partial sums of an integer partition. St000627The exponent of a binary word. St000631The number of distinct palindromic decompositions of a binary word. St000644The number of graphs with given frequency partition. St000655The length of the minimal rise of a Dyck path. St000674The number of hills of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000784The maximum of the length and the largest part of the integer partition. St000812The sum of the entries in the column specified by the partition of the change of basis matrix from complete homogeneous symmetric functions to monomial symmetric functions. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000867The sum of the hook lengths in the first row of an integer partition. St000878The number of ones minus the number of zeros of a binary word. St000922The minimal number such that all substrings of this length are unique. St000935The number of ordered refinements of an integer partition. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St000953The largest degree of an irreducible factor of the Coxeter polynomial of the Dyck path over the rational numbers. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000992The alternating sum of the parts of an integer partition. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001127The sum of the squares of the parts of a partition. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. 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: St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001267The length of the Lyndon factorization of the binary word. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001437The flex of a binary word. St001485The modular major index of a binary word. St001488The number of corners of a skew partition. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001523The degree of symmetry of a Dyck path. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001612The number of coloured multisets of cycles such that the multiplicities of colours are given by a partition. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. St001809The index of the step at the first peak of maximal height in a Dyck path. St001910The height of the middle non-run of a Dyck path. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St001966Half the global dimension of the stable Auslander algebra of a sincere Nakayama algebra (with associated Dyck path). St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000008The major index of the composition. St000012The area of a Dyck path. St000013The height of a Dyck path. St000014The number of parking functions supported by a Dyck path. St000019The cardinality of the support of a permutation. St000020The rank of the permutation. St000027The major index of a Dyck path. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000033The number of permutations greater than or equal to the given permutation in (strong) Bruhat order. St000040The number of regions of the inversion arrangement of a permutation. St000045The number of linear extensions of a binary tree. St000047The number of standard immaculate tableaux of a given shape. St000050The depth or height of a binary tree. St000056The decomposition (or block) number of a permutation. St000058The order of a permutation. St000060The greater neighbor of the maximum. St000062The length of the longest increasing subsequence of the permutation. St000064The number of one-box pattern of a permutation. St000075The orbit size of a standard tableau under promotion. St000079The number of alternating sign matrices for a given Dyck path. St000084The number of subtrees. St000092The number of outer peaks of a permutation. St000105The number of blocks in the set partition. St000109The number of elements less than or equal to the given element in Bruhat order. St000110The number of permutations less than or equal to a permutation in left weak order. St000111The sum of the descent tops (or Genocchi descents) of a permutation. St000134The size of the orbit of an alternating sign matrix under gyration. St000155The number of exceedances (also excedences) of a permutation. St000157The number of descents of a standard tableau. St000161The sum of the sizes of the right subtrees of a binary tree. St000164The number of short pairs. St000166The depth minus 1 of an ordered tree. St000167The number of leaves of an ordered tree. St000169The cocharge of a standard tableau. St000186The sum of the first row in a Gelfand-Tsetlin pattern. St000189The number of elements in the poset. St000197The number of entries equal to positive one in the alternating sign matrix. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000201The number of leaf nodes in a binary tree. St000203The number of external nodes of a binary tree. St000209Maximum difference of elements in cycles. St000210Minimum over maximum difference of elements in cycles. St000213The number of weak exceedances (also weak excedences) of a permutation. St000221The number of strong fixed points of a permutation. St000224The sorting index of a permutation. 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. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000247The number of singleton blocks of a set partition. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000255The number of reduced Kogan faces with the permutation as type. St000277The number of ribbon shaped standard tableaux. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000294The number of distinct factors of a binary word. St000295The length of the border of a binary word. St000304The load of a permutation. St000308The height of the tree associated to a permutation. St000325The width of the tree associated to a permutation. St000328The maximum number of child nodes in a tree. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000330The (standard) major index of a standard tableau. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000336The leg major index of a standard tableau. St000339The maf index of a permutation. St000340The number of non-final maximal constant sub-paths of length greater than one. St000385The number of vertices with out-degree 1 in a binary tree. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000396The register function (or Horton-Strahler number) of a binary tree. St000397The Strahler number of a rooted tree. St000398The sum of the depths of the vertices (or total internal path length) of a binary tree. St000401The size of the symmetry class of a permutation. St000402Half the size of the symmetry class of a permutation. St000412The number of binary trees with the same underlying unordered tree. St000413The number of ordered trees with the same underlying unordered tree. St000414The binary logarithm of the number of binary trees with the same underlying unordered tree. St000415The size of the automorphism group of the rooted tree underlying the ordered tree. St000417The size of the automorphism group of the ordered tree. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000446The disorder of a permutation. St000458The number of permutations obtained by switching adjacencies or successions. St000470The number of runs in a permutation. St000487The length of the shortest cycle of a permutation. St000488The number of cycles of a permutation of length at most 2. St000498The lcs statistic of a set partition. St000501The size of the first part in the decomposition of a permutation. St000505The biggest entry in the block containing the 1. St000507The number of ascents of a standard tableau. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000518The number of distinct subsequences in a binary word. St000522The number of 1-protected nodes of a rooted tree. St000528The height of a poset. St000529The number of permutations whose descent word is the given binary word. St000530The number of permutations with the same descent word as the given permutation. St000543The size of the conjugacy class of a binary word. St000545The number of parabolic double cosets with minimal element being the given permutation. St000564The number of occurrences of the pattern {{1},{2}} in a set partition. St000568The hook number of a binary tree. St000578The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton. St000616The inversion index of a permutation. St000617The number of global maxima of a Dyck path. St000619The number of cyclic descents of a permutation. St000626The minimal period of a binary word. St000638The number of up-down runs of a permutation. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000651The maximal size of a rise in a permutation. St000652The maximal difference between successive positions of a permutation. St000670The reversal length of a permutation. St000673The number of non-fixed points of a permutation. St000676The number of odd rises of a Dyck path. St000681The Grundy value of Chomp on Ferrers diagrams. St000690The size of the conjugacy class of a permutation. St000691The number of changes of a binary word. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000700The protection number of an ordered tree. St000701The protection number of a binary tree. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000719The number of alignments in a perfect matching. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000729The minimal arc length of a set partition. St000734The last entry in the first row of a standard tableau. St000743The number of entries in a standard Young tableau such that the next integer is a neighbour. St000744The length of the path to the largest entry in a standard Young tableau. St000746The number of pairs with odd minimum in a perfect matching. St000758The length of the longest staircase fitting into an integer composition. St000767The number of runs in an integer composition. St000780The size of the orbit under rotation of a perfect matching. St000789The number of crossing-similar perfect matchings of a perfect matching. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000819The propagating number of a perfect matching. St000820The number of compositions obtained by rotating the composition. St000824The sum of the number of descents and the number of recoils of a permutation. St000826The stopping time of the decimal representation of the binary word for the 3x+1 problem. St000828The spearman's rho of a permutation and the identity permutation. St000830The total displacement of a permutation. St000839The largest opener of a set partition. St000841The largest opener of a perfect matching. St000842The breadth of a permutation. St000843The decomposition number of a perfect matching. St000847The number of standard Young tableaux whose descent set is the binary word. St000862The number of parts of the shifted shape of a permutation. St000863The length of the first row of the shifted shape of a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000868The aid statistic in the sense of Shareshian-Wachs. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000886The number of permutations with the same antidiagonal sums. St000893The number of distinct diagonal sums of an alternating sign matrix. St000894The trace of an alternating sign matrix. St000895The number of ones on the main diagonal of an alternating sign matrix. St000899The maximal number of repetitions of an integer composition. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St000903The number of different parts of an integer composition. St000904The maximal number of repetitions of an integer composition. St000907The number of maximal antichains of minimal length in a poset. St000911The number of maximal antichains of maximal size in a poset. St000912The number of maximal antichains in a poset. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St000924The number of topologically connected components of a perfect matching. St000945The number of matchings in the dihedral orbit of a perfect matching. St000958The number of Bruhat factorizations of a permutation. St000959The number of strong Bruhat factorizations of a permutation. St000974The length of the trunk of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St000979Half of MacMahon's equal index of a Dyck path. St000981The length of the longest zigzag subpath. St000988The orbit size of a permutation under Foata's bijection. St000997The even-odd crank of an integer partition. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001034The area of the parallelogram polyomino associated with the Dyck path. St001042The size of the automorphism group of the leaf labelled binary unordered tree associated with the perfect matching. St001045The number of leaves in the subtree not containing one in the decreasing labelled binary unordered tree associated with the perfect matching. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001052The length of the exterior of a permutation. St001058The breadth of the ordered tree. St001081The number of minimal length factorizations of a permutation into star transpositions. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001102The number of words with multiplicities of the letters given by the composition, avoiding the consecutive pattern 132. 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. St001128The exponens consonantiae of a partition. St001132The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001134The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. St001136The largest label with larger sister in the leaf labelled binary unordered tree associated with the perfect matching. St001161The major index north count of a Dyck path. St001162The minimum jump of a permutation. 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. St001180Number of indecomposable injective modules with projective dimension at most 1. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001220The width of a permutation. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001228The vector space dimension of the space of module homomorphisms between J and itself when J denotes the Jacobson radical of the corresponding Nakayama algebra. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001242The toal dimension of certain Sn modules determined by LLT polynomials associated with a Dyck path. St001243The sum of coefficients in the Schur basis of certain LLT polynomials associated with a Dyck path. St001245The cyclic maximal difference between two consecutive entries of a permutation. St001246The maximal difference between two consecutive entries of a permutation. St001254The vector space dimension of the first extension-group between A/soc(A) and J when A is the corresponding Nakayama algebra with Jacobson radical J. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001285The number of primes in the column sums of the two line notation of a permutation. St001287The number of primes obtained by multiplying preimage and image of a permutation and subtracting one. St001288The number of primes obtained by multiplying preimage and image of a permutation and adding one. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001312Number of parabolic noncrossing partitions indexed by the composition. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001343The dimension of the reduced incidence algebra of a poset. St001371The length of the longest Yamanouchi prefix of a binary word. St001375The pancake length of a permutation. St001379The number of inversions plus the major index of a permutation. St001381The fertility of a permutation. St001399The distinguishing number of a poset. St001424The number of distinct squares in a binary word. St001462The number of factors of a standard tableaux under concatenation. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001482The product of the prefix sums of a permutation. St001486The number of corners of the ribbon associated with an integer composition. 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. St001497The position of the largest weak excedence of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001528The number of permutations such that the product with the permutation has the same number of fixed points. St001545The second Elser number of a connected graph. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001554The number of distinct nonempty subtrees of a binary tree. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001561The value of the elementary symmetric function evaluated at 1. St001566The length of the longest arithmetic progression in a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001589The nesting number of a perfect matching. St001595The number of standard Young tableaux of the skew partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001614The cyclic permutation representation number 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. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001641The number of ascent tops in the flattened set partition such that all smaller elements appear before. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001684The reduced word complexity of a permutation. St001688The sum of the squares of the heights of the peaks of a Dyck path. St001696The natural major index of a standard Young tableau. St001697The shifted natural comajor index of a standard Young tableau. St001699The major index of a standard tableau minus the weighted size of its shape. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001717The largest size of an interval in a poset. St001726The number of visible inversions of a permutation. St001735The number of permutations with the same set of runs. St001741The largest integer such that all patterns of this size are contained in the permutation. St001759The Rajchgot index of a permutation. St001760The number of prefix or suffix reversals needed to sort a permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St001780The order of promotion on the set of standard tableaux of given shape. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St001807The lower middle entry of a permutation. St001813The product of the sizes of the principal order filters in a poset. St001815The number of order preserving surjections from a poset to a total order. 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. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001915The size of the component corresponding to a necklace in Bulgarian solitaire. St001916The number of transient elements in the orbit of Bulgarian solitaire corresponding to a necklace. St001924The number of cells in an integer partition whose arm and leg length coincide. St001925The minimal number of zeros in a row of an alternating sign matrix. St001930The weak major index of a binary word. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001955The number of natural descents for set-valued two row standard Young tableaux. St001956The comajor index for set-valued two-row standard Young tableaux. St001958The degree of the polynomial interpolating the values of a permutation. St000782The indicator function of whether a given perfect matching is an L & P matching. St001964The interval resolution global dimension of a poset. St001168The vector space dimension of the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.