Your data matches 416 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000307
(load all 3 compositions to match this statistic)
Mapping
St000307: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => 1 = 0 + 1
([],2)
 => 2 = 1 + 1
([(0,1)],2)
 => 1 = 0 + 1
([],3)
 => 4 = 3 + 1
([(1,2)],3)
 => 1 = 0 + 1
([(0,1),(0,2)],3)
 => 2 = 1 + 1
([(0,2),(2,1)],3)
 => 1 = 0 + 1
([(0,2),(1,2)],3)
 => 2 = 1 + 1
Description
The number of rowmotion orbits of a poset. Rowmotion is an operation on order ideals in a poset $P$. It sends an order ideal $I$ to the order ideal generated by the minimal antichain of $P \setminus I$.
Matching statistic: St001175
(load all 3 compositions to match this statistic)
Mapping
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [2]
 => 0
([],2)
 => [2,2]
 => 1
([(0,1)],2)
 => [3]
 => 0
([],3)
 => [2,2,2,2]
 => 3
([(1,2)],3)
 => [6]
 => 0
([(0,1),(0,2)],3)
 => [3,2]
 => 1
([(0,2),(2,1)],3)
 => [4]
 => 0
([(0,2),(1,2)],3)
 => [3,2]
 => 1
Description
The size of a partition minus the hook length of the base cell. This is, the number of boxes in the diagram of a partition that are neither in the first row nor in the first column.
Matching statistic: St001341
(load all 6 compositions to match this statistic)
Mapping
Mp00198: Posets —incomparability graph⟶ Graphs
St001341: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],1)
 => 0
([],2)
 => ([(0,1)],2)
 => 1
([(0,1)],2)
 => ([],2)
 => 0
([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
([(0,1),(0,2)],3)
 => ([(1,2)],3)
 => 1
([(0,2),(2,1)],3)
 => ([],3)
 => 0
([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => 1
Description
The number of edges in the center of a graph. The center of a graph is the set of vertices whose maximal distance to any other vertex is minimal. In particular, if the graph is disconnected, all vertices are in the certer.
Matching statistic: St001783
(load all 5 compositions to match this statistic)
Mapping
Mp00198: Posets —incomparability graph⟶ Graphs
St001783: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],1)
 => 0
([],2)
 => ([(0,1)],2)
 => 1
([(0,1)],2)
 => ([],2)
 => 0
([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
([(0,1),(0,2)],3)
 => ([(1,2)],3)
 => 1
([(0,2),(2,1)],3)
 => ([],3)
 => 0
([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => 1
Description
The number of odd automorphisms of a graph. Let $D$ be an arbitrary orientation of a graph $G$. Then an automorphism of $G$ is odd, if it reverses the orientation of an odd number of edges of $D$. The graphs on $n$ vertices without any odd automorphisms are equinumerous with the number of non-isomorphic $n$-team tournaments, see [2]. The odd automorphisms of the complete graphs are precisely the even permutations.
Matching statistic: St000010
(load all 2 compositions to match this statistic)
Mapping
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [2]
 => 1 = 0 + 1
([],2)
 => [2,2]
 => 2 = 1 + 1
([(0,1)],2)
 => [3]
 => 1 = 0 + 1
([],3)
 => [2,2,2,2]
 => 4 = 3 + 1
([(1,2)],3)
 => [6]
 => 1 = 0 + 1
([(0,1),(0,2)],3)
 => [3,2]
 => 2 = 1 + 1
([(0,2),(2,1)],3)
 => [4]
 => 1 = 0 + 1
([(0,2),(1,2)],3)
 => [3,2]
 => 2 = 1 + 1
Description
The length of the partition.
Matching statistic: St000146
(load all 2 compositions to match this statistic)
Mapping
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St000146: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [2]
 => 1 = 0 + 1
([],2)
 => [2,2]
 => 2 = 1 + 1
([(0,1)],2)
 => [3]
 => 1 = 0 + 1
([],3)
 => [2,2,2,2]
 => 4 = 3 + 1
([(1,2)],3)
 => [6]
 => 1 = 0 + 1
([(0,1),(0,2)],3)
 => [3,2]
 => 2 = 1 + 1
([(0,2),(2,1)],3)
 => [4]
 => 1 = 0 + 1
([(0,2),(1,2)],3)
 => [3,2]
 => 2 = 1 + 1
Description
The Andrews-Garvan crank of a partition. If $\pi$ is a partition, let $l(\pi)$ be its length (number of parts), $\omega(\pi)$ be the number of parts equal to 1, and $\mu(\pi)$ be the number of parts larger than $\omega(\pi)$. The crank is then defined by $$ c(\pi) = \begin{cases} l(\pi) &\text{if \(\omega(\pi)=0\)}\\ \mu(\pi) - \omega(\pi) &\text{otherwise}. \end{cases} $$ This statistic was defined in [1] to explain Ramanujan's partition congruence $$p(11n+6) \equiv 0 \pmod{11}$$ in the same way as the Dyson rank ([[St000145]]) explains the congruences $$p(5n+4) \equiv 0 \pmod{5}$$ and $$p(7n+5) \equiv 0 \pmod{7}.$$
Matching statistic: St000473
(load all 2 compositions to match this statistic)
Mapping
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St000473: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [2]
 => 1 = 0 + 1
([],2)
 => [2,2]
 => 2 = 1 + 1
([(0,1)],2)
 => [3]
 => 1 = 0 + 1
([],3)
 => [2,2,2,2]
 => 4 = 3 + 1
([(1,2)],3)
 => [6]
 => 1 = 0 + 1
([(0,1),(0,2)],3)
 => [3,2]
 => 2 = 1 + 1
([(0,2),(2,1)],3)
 => [4]
 => 1 = 0 + 1
([(0,2),(1,2)],3)
 => [3,2]
 => 2 = 1 + 1
Description
The number of parts of a partition that are strictly bigger than the number of ones. This is part of the definition of Dyson's crank of a partition, see [[St000474]].
Matching statistic: St001280
(load all 4 compositions to match this statistic)
Mapping
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [2]
 => 1 = 0 + 1
([],2)
 => [2,2]
 => 2 = 1 + 1
([(0,1)],2)
 => [3]
 => 1 = 0 + 1
([],3)
 => [2,2,2,2]
 => 4 = 3 + 1
([(1,2)],3)
 => [6]
 => 1 = 0 + 1
([(0,1),(0,2)],3)
 => [3,2]
 => 2 = 1 + 1
([(0,2),(2,1)],3)
 => [4]
 => 1 = 0 + 1
([(0,2),(1,2)],3)
 => [3,2]
 => 2 = 1 + 1
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St001710
(load all 3 compositions to match this statistic)
Mapping
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St001710: 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]
 => 2 = 1 + 1
([(0,1)],2)
 => [1]
 => 1 = 0 + 1
([],3)
 => [3,3]
 => 4 = 3 + 1
([(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,1),(0,2)],3)
 => [2]
 => 2 = 1 + 1
([(0,2),(2,1)],3)
 => [1]
 => 1 = 0 + 1
([(0,2),(1,2)],3)
 => [2]
 => 2 = 1 + 1
Description
The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. Let $\alpha$ be any permutation of cycle type $\lambda$. This statistic is the number of permutations $\pi$ such that $$ \alpha\pi\alpha^{-1} = \pi^{-1}.$$
Matching statistic: St000017
(load all 2 compositions to match this statistic)
Mapping
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000017: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [2]
 => [[1,2]]
 => 0
([],2)
 => [2,2]
 => [[1,2],[3,4]]
 => 1
([(0,1)],2)
 => [3]
 => [[1,2,3]]
 => 0
([],3)
 => [2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => 3
([(1,2)],3)
 => [6]
 => [[1,2,3,4,5,6]]
 => 0
([(0,1),(0,2)],3)
 => [3,2]
 => [[1,2,3],[4,5]]
 => 1
([(0,2),(2,1)],3)
 => [4]
 => [[1,2,3,4]]
 => 0
([(0,2),(1,2)],3)
 => [3,2]
 => [[1,2,3],[4,5]]
 => 1
Description
The number of inversions of a standard tableau. Let $T$ be a tableau. An inversion is an attacking pair $(c,d)$ of the shape of $T$ (see [[St000016]] for a definition of this) such that the entry of $c$ in $T$ is greater than the entry of $d$.
The following 406 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000142The number of even parts of a partition. St000157The number of descents of a standard tableau. St000160The multiplicity of the smallest part of a partition. St000185The weighted size of a partition. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St000519The largest length of a factor maximising the subword complexity. St000547The number of even non-empty partial sums of an integer partition. St000548The number of different non-empty partial sums of an integer partition. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000992The alternating sum of the parts of an integer partition. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001057The Grundy value of the game of creating an independent set in a graph. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001274The number of indecomposable injective modules with projective dimension equal to two. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001827The number of two-component spanning forests of a graph. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001932The number of pairs of singleton blocks in the noncrossing set partition corresponding to a Dyck path, that can be merged to create another noncrossing set partition. St001961The sum of the greatest common divisors of all pairs of parts. St000143The largest repeated part of a partition. St000147The largest part of an integer partition. St000288The number of ones in a binary word. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000378The diagonal inversion number of an integer partition. St000393The number of strictly increasing runs in a binary word. St000668The least common multiple of the parts of the partition. St000733The row containing the largest entry of a standard tableau. 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. St000935The number of ordered refinements of an integer partition. St001267The length of the Lyndon factorization of the binary word. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001437The flex of a binary word. St001587Half of the largest even part of an integer partition. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St000052The number of valleys of a Dyck path not on the x-axis. St000053The number of valleys of the Dyck path. St000059The inversion number of a standard tableau as defined by Haglund and Stevens. St000081The number of edges of a graph. St000148The number of odd parts of a partition. St000169The cocharge of a standard tableau. St000263The Szeged index of a graph. St000265The Wiener index of a graph. St000297The number of leading ones in a binary word. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000330The (standard) major index of a standard tableau. St000331The number of upper interactions of a Dyck path. St000336The leg major index of a standard tableau. St000379The number of Hamiltonian cycles in a graph. St000445The number of rises of length 1 of a Dyck path. St000475The number of parts equal to 1 in a partition. St000869The sum of the hook lengths of an integer partition. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001117The game chromatic index of a graph. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. 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. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001584The area statistic between a Dyck path and its bounce path. St001649The length of a longest trail in a graph. St001697The shifted natural comajor index of a standard Young tableau. St001712The number of natural descents of a standard Young tableau. St001910The height of the middle non-run of a Dyck path. St001931The weak major index of an integer composition regarded as a word. St001955The number of natural descents for set-valued two row standard Young tableaux. St001956The comajor index for set-valued two-row standard Young tableaux. St000015The number of peaks of a Dyck path. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000038The product of the heights of the descending steps of a Dyck path. St000086The number of subgraphs. St000117The number of centered tunnels of a Dyck path. St000335The difference of lower and upper interactions. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000468The Hosoya index of a graph. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000507The number of ascents of a standard tableau. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000655The length of the minimal rise of a Dyck path. St000676The number of odd rises of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000734The last entry in the first row of a standard tableau. 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. 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. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001389The number of partitions of the same length below the given integer partition. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001462The number of factors of a standard tableaux under concatenation. St001498The normalised height of a Nakayama algebra with magnitude 1. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001523The degree of symmetry of a Dyck path. St001814The number of partitions interlacing the given partition. St000439The position of the first down step of a Dyck path. St001180Number of indecomposable injective modules with projective dimension at most 1. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. 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. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St000946The sum of the skew hook positions in a Dyck path. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St000850The number of 1/2-balanced pairs in a poset. St001621The number of atoms of a lattice. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001397Number of pairs of incomparable elements in a finite poset. St001596The number of two-by-two squares inside a skew partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000633The size of the automorphism group of a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001597The Frobenius rank of a skew partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. 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. St001071The beta invariant of the graph. 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. St001120The length of a longest path in a graph. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001354The number of series nodes in the modular decomposition of a graph. St001479The number of bridges of a graph. St001512The minimum rank of a graph. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001691The number of kings in a graph. St001736The total number of cycles in a graph. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001816Eigenvalues of the top-to-random operator acting on a simple module. St001869The maximum cut size of a graph. St000075The orbit size of a standard tableau under promotion. St000120The number of left tunnels of a Dyck path. St000299The number of nonisomorphic vertex-induced subtrees. St000418The number of Dyck paths that are weakly below a Dyck path. St000453The number of distinct Laplacian eigenvalues of a graph. St000708The product of the parts of an integer partition. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St001093The detour number of a graph. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. 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 $eA$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001302The number of minimally dominating sets of vertices of a graph. 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. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001531Number of partial orders contained in the poset determined by the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001592The maximal number of simple paths between any two different vertices 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). St001959The product of the heights of the peaks of a Dyck path. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000455The second largest eigenvalue of a graph if it is integral. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000454The largest eigenvalue of a graph if it is integral. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001624The breadth of a lattice. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000309The number of vertices with even degree. St000315The number of isolated vertices of a graph. St000387The matching number of a graph. St000478Another weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000535The rank-width of a graph. St000567The sum of the products of all pairs of parts. St000640The rank of the largest boolean interval in a poset. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000699The toughness times the least common multiple of 1,. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000929The constant term of the character polynomial of an integer partition. St000934The 2-degree of an integer partition. St000941The number of characters of the symmetric group whose value on the partition is even. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001095The number of non-isomorphic posets with precisely one further covering relation. 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. 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. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001249Sum of the odd parts of a partition. St001271The competition number of a graph. 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. St001333The cardinality of a minimal edge-isolating set of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001353The number of prime nodes in the modular decomposition of a graph. St001356The number of vertices in prime modules of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001393The induced matching number of a graph. St001478The number of nowhere zero 4-flows of a graph. St001525The number of symmetric hooks on the diagonal of a partition. St001541The Gini index of an integer partition. St001561The value of the elementary symmetric function evaluated at 1. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001668The number of points of the poset minus the width of the poset. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000741The Colin de Verdière graph invariant. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000096The number of spanning trees of a graph. St000137The Grundy value of an integer partition. St000274The number of perfect matchings of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000344The number of strongly connected outdegree sequences of a graph. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000442The maximal area to the right of an up step of a Dyck path. St000466The Gutman (or modified Schultz) index of a connected graph. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000661The number of rises of length 3 of a Dyck path. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000693The modular (standard) major index of a standard tableau. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000874The position of the last double rise in a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000947The major index east count of a Dyck path. St000948The chromatic discriminant of a graph. St000976The sum of the positions of double up-steps of a 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. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001139The number of occurrences of hills of size 2 in a Dyck path. St001176The size of a partition minus its first part. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001281The normalized isoperimetric number of a graph. St001383The BG-rank of an integer partition. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001395The number of strictly unfriendly partitions of a graph. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001657The number of twos in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001826The maximal number of leaves on a vertex of a graph. St001645The pebbling number of a connected graph. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000456The monochromatic index of a connected graph. St000618The number of self-evacuating tableaux of given shape. St000936The number of even values of the symmetric group character corresponding to the partition. St000993The multiplicity of the largest part of an integer partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. 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. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001933The largest multiplicity of a part in an integer partition. St001330The hat guessing number of a graph. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000667The greatest common divisor of the parts of the partition. St000770The major index of an integer partition when read from bottom to top. St001118The acyclic chromatic index of a graph. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001527The cyclic permutation representation number of an integer partition. St001564The value of the forgotten symmetric functions when all variables set to 1. St001571The Cartan determinant of the integer partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001651The Frankl number of a lattice. St000068The number of minimal elements in a poset. St000080The rank of the poset. St000100The number of linear extensions of a poset. St000181The number of connected components of the Hasse diagram for the poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000527The width of the poset. St000632The jump number of the poset. St000635The number of strictly order preserving maps of a poset into itself. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element 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. St000908The length of the shortest maximal antichain in a poset. St000909The number of maximal chains of maximal size 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. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001510The number of self-evacuating linear extensions of a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001718The number of non-empty open intervals in a poset. St001779The order of promotion on the set of linear extensions of a poset. St001890The maximum magnitude of the Möbius function of a poset. St001902The number of potential covers of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001964The interval resolution global dimension of a poset. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. 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. St000225Difference between largest and smallest parts in a partition. St000284The Plancherel distribution on integer partitions. St000302The determinant of the distance matrix of a connected graph. St000467The hyper-Wiener index of a connected graph. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000928The sum of the coefficients of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000944The 3-degree of an integer partition. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001128The exponens consonantiae of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the 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. St001432The order dimension of the partition. St001568The smallest positive integer that does not appear twice in the partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St001586The number of odd parts smaller than the largest even part in an integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type.