Your data matches 699 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Mp00047: Ordered trees to posetPosets
St000068: Posets ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[]
=> ([],1)
=> 1
[[]]
=> ([(0,1)],2)
=> 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> 2
[[[]]]
=> ([(0,2),(2,1)],3)
=> 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 2
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
Description
The number of minimal elements in a poset.
Mp00047: Ordered trees to posetPosets
St000071: Posets ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[]
=> ([],1)
=> 1
[[]]
=> ([(0,1)],2)
=> 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> 2
[[[]]]
=> ([(0,2),(2,1)],3)
=> 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 2
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
Description
The number of maximal chains in a poset.
Mp00047: Ordered trees to posetPosets
St000527: Posets ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[]
=> ([],1)
=> 1
[[]]
=> ([(0,1)],2)
=> 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> 2
[[[]]]
=> ([(0,2),(2,1)],3)
=> 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 2
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
Description
The width of the poset. This is the size of the poset's longest antichain, also called Dilworth number.
Mp00051: Ordered trees to Dyck pathDyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[]
=> []
=> 0 = 1 - 1
[[]]
=> [1,0]
=> 0 = 1 - 1
[[],[]]
=> [1,0,1,0]
=> 0 = 1 - 1
[[[]]]
=> [1,1,0,0]
=> 1 = 2 - 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> 2 = 3 - 1
Description
The number of double up and double down steps of a Dyck path. In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.
Mp00047: Ordered trees to posetPosets
St000632: Posets ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[]
=> ([],1)
=> 0 = 1 - 1
[[]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[[[]]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 2 - 1
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 2 - 1
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 1 = 2 - 1
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
Description
The jump number of the poset. A jump in a linear extension $e_1, \dots, e_n$ of a poset $P$ is a pair $(e_i, e_{i+1})$ so that $e_{i+1}$ does not cover $e_i$ in $P$. The jump number of a poset is the minimal number of jumps in linear extensions of a poset.
Mp00047: Ordered trees to posetPosets
Mp00110: Posets Greene-Kleitman invariantInteger partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[]
=> ([],1)
=> [1]
=> 1
[[]]
=> ([(0,1)],2)
=> [2]
=> 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> 2
[[[]]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> 3
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 2
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 2
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> 2
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
Description
The length of the partition.
Mp00051: Ordered trees to Dyck pathDyck paths
Mp00199: Dyck paths prime Dyck pathDyck paths
St000015: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[]
=> []
=> [1,0]
=> 1
[[]]
=> [1,0]
=> [1,1,0,0]
=> 1
[[],[]]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[[[]]]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
Description
The number of peaks of a Dyck path.
Matching statistic: St000069
Mp00047: Ordered trees to posetPosets
Mp00125: Posets dual posetPosets
St000069: Posets ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[]
=> ([],1)
=> ([],1)
=> 1
[[]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> ([(0,1),(0,2)],3)
=> 2
[[[]]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3)],4)
=> 3
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(3,1)],4)
=> 2
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(3,1)],4)
=> 2
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(3,1),(3,2)],4)
=> 2
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
Description
The number of maximal elements of a poset.
Mp00047: Ordered trees to posetPosets
Mp00198: Posets incomparability graphGraphs
St000097: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[]
=> ([],1)
=> ([],1)
=> 1
[[]]
=> ([(0,1)],2)
=> ([],2)
=> 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[[]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 2
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1
Description
The order of the largest clique of the graph. A clique in a graph $G$ is a subset $U \subseteq V(G)$ such that any pair of vertices in $U$ are adjacent. I.e. the subgraph induced by $U$ is a complete graph.
Mp00047: Ordered trees to posetPosets
Mp00198: Posets incomparability graphGraphs
St000098: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[]
=> ([],1)
=> ([],1)
=> 1
[[]]
=> ([(0,1)],2)
=> ([],2)
=> 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[[]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 2
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1
Description
The chromatic number of a graph. The minimal number of colors needed to color the vertices of the graph such that no two vertices which share an edge have the same color.
The following 689 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000172The Grundy number of a graph. St000213The number of weak exceedances (also weak excedences) of a permutation. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000451The length of the longest pattern of the form k 1 2. St000676The number of odd rises of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000822The Hadwiger number of the graph. St000991The number of right-to-left minima of a permutation. St001029The size of the core of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001116The game chromatic number of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001330The hat guessing number of a graph. St001494The Alon-Tarsi number of a graph. St001530The depth of a Dyck path. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001820The size of the image of the pop stack sorting operator. St001883The mutual visibility number of a graph. St001963The tree-depth of a graph. St000021The number of descents of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000039The number of crossings of a permutation. St000052The number of valleys of a Dyck path not on the x-axis. St000053The number of valleys of the Dyck path. St000120The number of left tunnels of a Dyck path. St000159The number of distinct parts of the integer partition. St000160The multiplicity of the smallest part of a partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000245The number of ascents of a permutation. St000272The treewidth of a graph. St000317The cycle descent number of a permutation. St000331The number of upper interactions of a Dyck path. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000358The number of occurrences of the pattern 31-2. St000362The size of a minimal vertex cover of a graph. St000374The number of exclusive right-to-left minima of a permutation. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St000662The staircase size of the code of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000703The number of deficiencies of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000783The side length of the largest staircase partition fitting into a partition. St000996The number of exclusive left-to-right maxima of 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. St001090The number of pop-stack-sorts needed to sort a permutation. 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. St001176The size of a partition minus its first part. 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. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001280The number of parts of an integer partition that are at least two. 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. St001358The largest degree of a regular subgraph of a graph. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001623The number of doubly irreducible elements of a lattice. St001644The dimension of a graph. St001743The discrepancy of a graph. St001792The arboricity of a graph. St001812The biclique partition number of a graph. St001846The number of elements which do not have a complement in the lattice. St001962The proper pathwidth of a graph. St000007The number of saliances of the permutation. St000013The height of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000062The length of the longest increasing subsequence of the permutation. St000063The number of linear extensions of a certain poset defined for an integer partition. St000087The number of induced subgraphs. St000092The number of outer peaks of a permutation. St000093The cardinality of a maximal independent set of vertices of a graph. St000105The number of blocks in the set partition. St000108The number of partitions contained in the given partition. St000147The largest part of an integer partition. St000157The number of descents of a standard tableau. St000164The number of short pairs. St000167The number of leaves of an ordered tree. St000239The number of small weak excedances. St000286The number of connected components of the complement of a graph. St000288The number of ones in a binary word. St000291The number of descents of a binary word. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000321The number of integer partitions of n that are dominated by an integer partition. 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. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000345The number of refinements of a partition. St000346The number of coarsenings of a partition. St000354The number of recoils of a permutation. St000363The number of minimal vertex covers of a graph. St000378The diagonal inversion number of an integer partition. St000390The number of runs of ones in a binary word. St000443The number of long tunnels of a Dyck path. St000469The distinguishing number of a graph. St000470The number of runs in a permutation. St000482The (zero)-forcing number of a graph. St000532The total number of rook placements on a Ferrers board. St000542The number of left-to-right-minima of a permutation. St000636The hull number of a graph. St000638The number of up-down runs of a permutation. St000686The finitistic dominant dimension of a Dyck path. St000702The number of weak deficiencies of a permutation. St000722The number of different neighbourhoods in a graph. St000733The row containing the largest entry of a standard tableau. St000738The first entry in the last row of a standard tableau. St000740The last entry of a permutation. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000746The number of pairs with odd minimum in a perfect matching. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000820The number of compositions obtained by rotating the composition. St000864The number of circled entries of the shifted recording tableau of a permutation. St000876The number of factors in the Catalan decomposition of a binary word. St000883The number of longest increasing subsequences of a permutation. St000885The number of critical steps in the Catalan decomposition of a binary word. St000926The clique-coclique number of a graph. St000935The number of ordered refinements of an integer partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001058The breadth of the ordered tree. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one 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: 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. 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. 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. St001316The domatic number of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001342The number of vertices in the center of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001389The number of partitions of the same length below the given integer partition. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001400The total number of Littlewood-Richardson tableaux of given shape. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001461The number of topologically connected components of the chord diagram of a permutation. St001471The magnitude of a Dyck path. St001497The position of the largest weak excedence of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001642The Prague dimension of a graph. St001645The pebbling number of a connected graph. St001652The length of a longest interval of consecutive numbers. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001662The length of the longest factor of consecutive numbers in a permutation. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001725The harmonious chromatic number of a graph. St001733The number of weak left to right maxima of a Dyck path. St001746The coalition number of a graph. St001778The largest greatest common divisor of an element and its image in a permutation. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001863The number of weak excedances of a signed permutation. St001881The number of factors of a lattice as a Cartesian product of lattices. 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. St000004The major index of a permutation. St000012The area of a Dyck path. St000018The number of inversions of a permutation. St000025The number of initial rises of a Dyck path. St000035The number of left outer peaks of a permutation. St000065The number of entries equal to -1 in an alternating sign matrix. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000141The maximum drop size of a permutation. St000143The largest repeated part of a partition. St000148The number of odd parts of a partition. St000154The sum of the descent bottoms of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000171The degree of the graph. St000204The number of internal nodes of a binary tree. St000214The number of adjacencies of a permutation. St000222The number of alignments in the permutation. St000223The number of nestings in the permutation. St000225Difference between largest and smallest parts in a partition. St000228The size of a partition. St000237The number of small exceedances. St000242The number of indices that are not cyclical small weak excedances. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000292The number of ascents of a binary word. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St000306The bounce count of a Dyck path. St000310The minimal degree of a vertex of a graph. St000316The number of non-left-to-right-maxima of a permutation. 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. St000352The Elizalde-Pak rank of a permutation. St000353The number of inner valleys of a permutation. St000356The number of occurrences of the pattern 13-2. St000359The number of occurrences of the pattern 23-1. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000384The maximal part of the shifted composition of an integer partition. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000431The number of occurrences of the pattern 213 or of the pattern 321 in a permutation. St000441The number of successions of a permutation. St000454The largest eigenvalue of a graph if it is integral. St000459The hook length of the base cell of a partition. St000475The number of parts equal to 1 in a partition. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000519The largest length of a factor maximising the subword complexity. St000546The number of global descents of a permutation. St000548The number of different non-empty partial sums of an integer partition. St000647The number of big descents of a permutation. St000651The maximal size of a rise in a permutation. St000665The number of rafts of a permutation. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000692Babson and Steingrímsson's statistic of a permutation. St000711The number of big exceedences of a permutation. St000731The number of double exceedences of a permutation. St000741The Colin de Verdière graph invariant. St000766The number of inversions of an integer composition. St000778The metric dimension of a graph. St000784The maximum of the length and the largest part of the integer partition. St000867The sum of the hook lengths in the first row of an integer partition. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001079The minimal length of a factorization of a permutation using the permutations (12)(34). St001083The number of boxed occurrences of 132 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001127The sum of the squares of the parts of a partition. 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. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001263The index of the maximal parabolic seaweed algebra associated with the composition. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001298The number of repeated entries in the Lehmer code of a permutation. St001323The independence gap of a graph. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001345The Hamming dimension of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001391The disjunction number of a graph. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001489The maximum of the number of descents and the number of inverse descents. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001513The number of nested exceedences of a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001565The number of arithmetic progressions of length 2 in 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. St001578The minimal number of edges to add or remove to make a graph a line graph. St001584The area statistic between a Dyck path and its bounce path. St001647The number of edges that can be added without increasing the clique number. St001648The number of edges that can be added without increasing the chromatic number. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001689The number of celebrities in a graph. St001692The number of vertices with higher degree than the average degree in a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001712The number of natural descents of a standard Young tableau. St001726The number of visible inversions of a permutation. St001727The number of invisible inversions of a permutation. St001729The number of visible descents of a permutation. St001742The difference of the maximal and the minimal degree in a graph. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001777The number of weak descents in an integer composition. St001810The number of fixed points of a permutation smaller than its largest moved point. St001842The major index of a set partition. St001862The number of crossings of a signed permutation. St001869The maximum cut size 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). St001874Lusztig's a-function for the symmetric group. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001928The number of non-overlapping descents in a permutation. St001949The rigidity index of a graph. St001960The number of descents of a permutation minus one if its first entry is not one. St000166The depth minus 1 of an ordered tree. St000094The depth of an ordered tree. St000168The number of internal nodes of an ordered tree. St000521The number of distinct subtrees of an ordered tree. St000526The number of posets with combinatorially isomorphic order polytopes. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001668The number of points of the poset minus the width of the poset. St000058The order of a permutation. St000335The difference of lower and upper interactions. St000381The largest part of an integer composition. St000392The length of the longest run of ones in a binary word. St000442The maximal area to the right of an up step of a Dyck path. St000528The height of a poset. St000628The balance of a binary word. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000808The number of up steps of the associated bargraph. St000912The number of maximal antichains in a poset. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000982The length of the longest constant subword. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective 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. St001285The number of primes in the column sums of the two line notation of a permutation. St001343The dimension of the reduced incidence algebra of a poset. St001372The length of a longest cyclic run of ones of a binary word. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001589The nesting number of a perfect matching. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001717The largest size of an interval in a poset. St000080The rank of the poset. St000211The rank of the set partition. St000238The number of indices that are not small weak excedances. St000332The positive inversions of an alternating sign matrix. St000668The least common multiple of the parts of the partition. St000670The reversal length of a permutation. St000708The product of the parts of an integer partition. St000710The number of big deficiencies of a permutation. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001152The number of pairs with even minimum in a perfect matching. St001189The number of simple modules with dominant and codominant dimension equal to zero 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. 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. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse 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. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001592The maximal number of simple paths between any two different vertices of a graph. St001651The Frankl number of a lattice. St001760The number of prefix or suffix reversals needed to sort a permutation. St000011The number of touch points (or returns) of a Dyck path. St000054The first entry of the permutation. St000056The decomposition (or block) number of a permutation. St000060The greater neighbor of the maximum. St000083The number of left oriented leafs of a binary tree except the first one. St000084The number of subtrees. St000099The number of valleys of a permutation, including the boundary. St000110The number of permutations less than or equal to a permutation in left weak order. St000153The number of adjacent cycles of a permutation. St000201The number of leaf nodes in a binary tree. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000216The absolute length of a permutation. St000240The number of indices that are not small excedances. St000254The nesting number of a set partition. St000299The number of nonisomorphic vertex-induced subtrees. St000327The number of cover relations in a poset. St000389The number of runs of ones of odd length in a binary word. St000444The length of the maximal rise of a Dyck path. St000453The number of distinct Laplacian eigenvalues of a graph. St000502The number of successions of a set partitions. St000507The number of ascents of a standard tableau. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000553The number of blocks of a graph. St000619The number of cyclic descents of a permutation. St000652The maximal difference between successive positions of a permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000744The length of the path to the largest entry in a standard Young tableau. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. 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. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000843The decomposition number of a perfect matching. St000877The depth of the binary word interpreted as a path. St000886The number of permutations with the same antidiagonal sums. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St000925The number of topologically connected components of a set partition. St000932The number of occurrences of the pattern UDU in a Dyck path. St000933The number of multipartitions of sizes given by an integer partition. St000942The number of critical left to right maxima of the parking functions. St000956The maximal displacement of a permutation. St000971The smallest closer of a set partition. St000983The length of the longest alternating subword. St000988The orbit size of a permutation under Foata's bijection. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. 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. St001052The length of the exterior of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001093The detour number of a graph. St001114The number of odd descents of a permutation. St001128The exponens consonantiae of a partition. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001151The number of blocks with odd minimum. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ 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. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001220The width of a permutation. 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. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001246The maximal difference between two consecutive entries of a permutation. St001286The annihilation number of a graph. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001462The number of factors of a standard tableaux under concatenation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St001555The order of a signed permutation. St001590The crossing number of a perfect matching. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001637The number of (upper) dissectors of a poset. St001674The number of vertices of the largest induced star graph in the graph. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001737The number of descents of type 2 in a permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001806The upper middle entry of a permutation. St001807The lower middle entry of a permutation. St001809The index of the step at the first peak of maximal height in a Dyck path. St000019The cardinality of the support of a permutation. St000023The number of inner peaks of a permutation. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000067The inversion number of the alternating sign matrix. St000144The pyramid weight of the Dyck path. St000174The flush statistic of a semistandard tableau. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000209Maximum difference of elements in cycles. St000224The sorting index of a permutation. St000234The number of global ascents of a permutation. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000290The major index of a binary word. St000305The inverse major index of a permutation. St000336The leg major index of a standard tableau. St000366The number of double descents of a permutation. St000377The dinv defect of an integer partition. St000386The number of factors DDU in a Dyck path. St000439The position of the first down step of a Dyck path. St000446The disorder of a permutation. St000461The rix statistic of a permutation. St000462The major index minus the number of excedences of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000538The number of even inversions of a permutation. St000539The number of odd inversions of a permutation. St000565The major index of a set partition. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000646The number of big ascents of a permutation. St000656The number of cuts of a poset. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000663The number of right floats of a permutation. St000680The Grundy value for Hackendot on posets. St000691The number of changes of a binary word. St000717The number of ordinal summands of a poset. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000732The number of double deficiencies of a permutation. St000787The number of flips required to make a perfect matching noncrossing. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000824The sum of the number of descents and the number of recoils of a permutation. St000834The number of right outer peaks of a permutation. St000836The number of descents of distance 2 of a permutation. St000837The number of ascents of distance 2 of a permutation. St000840The number of closers smaller than the largest opener in a perfect matching. St000871The number of very big ascents of a permutation. St000873The aix statistic of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000906The length of the shortest maximal chain in a poset. St000931The number of occurrences of the pattern UUU in a Dyck path. St000934The 2-degree of an integer partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000961The shifted major index of a permutation. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St000984The number of boxes below precisely one peak. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. 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. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001047The maximal number of arcs crossing a given arc of a perfect matching. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001082The number of boxed occurrences of 123 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001115The number of even descents of a permutation. St001153The number of blocks with even minimum in a set partition. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001274The number of indecomposable injective modules with projective dimension equal to two. St001394The genus of a permutation. St001423The number of distinct cubes in a binary word. St001427The number of descents of a signed permutation. St001469The holeyness of a permutation. St001470The cyclic holeyness of a permutation. St001480The number of simple summands of the module J^2/J^3. St001485The modular major index of a binary word. 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. St001512The minimum rank of a graph. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001671Haglund's hag of a permutation. St001684The reduced word complexity of a permutation. St001728The number of invisible descents of a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001769The reflection length of a signed permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001823The Stasinski-Voll length of a signed permutation. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001864The number of excedances of a signed permutation. St001896The number of right descents of a signed permutations. St001905The number of preferred parking spots in a parking function less than the index of the car. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001935The number of ascents in a parking function. St001946The number of descents in a parking function. St001948The number of augmented double ascents of a permutation. St001965The number of decreasable positions in the corner sum matrix of an alternating sign matrix. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000643The size of the largest orbit of antichains under Panyushev complementation. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. 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. St000485The length of the longest cycle of a permutation. St000489The number of cycles of a permutation of length at most 3. St001062The maximal size of a block of a set partition. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001959The product of the heights of the peaks of a Dyck path. St000730The maximal arc length of a set partition. St001080The minimal length of a factorization of a permutation using the transposition (12) and the cycle (1,. St000061The number of nodes on the left branch of a binary tree. St000326The position of the first one in a binary word after appending a 1 at the end. St000504The cardinality of the first block of a set partition. St000675The number of centered multitunnels of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000823The number of unsplittable factors of the set partition. 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. St000898The number of maximal entries in the last diagonal of the monotone triangle. St000937The number of positive values of the symmetric group character corresponding to the partition. St001346The number of parking functions that give the same permutation. St001500The global dimension of magnitude 1 Nakayama algebras. St000289The decimal representation of a binary word. St000297The number of leading ones in a binary word. St000391The sum of the positions of the ones in a binary word. St000472The sum of the ascent bottoms of a permutation. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000494The number of inversions of distance at most 3 of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000503The maximal difference between two elements in a common block. St000653The last descent of a permutation. St000728The dimension of a set partition. St000753The Grundy value for the game of Kayles on a binary word. St000792The Grundy value for the game of ruler on a binary word. 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. St000874The position of the last double rise in a Dyck path. St000957The number of Bruhat lower covers of a permutation. St000989The number of final rises of a permutation. 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. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St000100The number of linear extensions of a poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001779The order of promotion on the set of linear extensions of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St000455The second largest eigenvalue of a graph if it is integral. 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 $eA$. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001118The acyclic chromatic index of a graph. 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 $eA$. St001380The number of monomer-dimer tilings of a Ferrers diagram. St000184The size of the centralizer of any permutation of given cycle type. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000293The number of inversions of a binary word. St000395The sum of the heights of the peaks of a Dyck path. St000531The leading coefficient of the rook polynomial of an integer partition. St000631The number of distinct palindromic decompositions of a binary word. St000759The smallest missing part in an integer partition. St000922The minimal number such that all substrings of this length are unique. St001018Sum of projective dimension of the indecomposable injective modules of 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. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001360The number of covering relations in Young's lattice below a partition. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001488The number of corners of a skew partition. 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. St001568The smallest positive integer that does not appear twice in the partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001612The number of coloured multisets of cycles such that the multiplicities of colours are given by a partition. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001814The number of partitions interlacing the given partition. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001885The number of binary words with the same proper border set. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000260The radius of a connected graph. St001964The interval resolution global dimension of a poset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001722The number of minimal chains with small intervals between a binary word and the top element. St000259The diameter of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000707The product of the factorials of the parts. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000815The number of semistandard Young tableaux of partition weight of given shape. St001060The distinguishing index of a graph. St001545The second Elser number of a connected graph. St000418The number of Dyck paths that are weakly below a Dyck path. St000420The number of Dyck paths that are weakly above a Dyck path. St000456The monochromatic index of a connected graph. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000736The last entry in the first row of a semistandard tableau. St000770The major index of an integer partition when read from bottom to top. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part 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. 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 $K[x]/(x^n)$. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001531Number of partial orders contained in the poset determined by the Dyck path. St001569The maximal modular displacement of a permutation. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000177The number of free tiles in the pattern. St000178Number of free entries. St001520The number of strict 3-descents. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order.