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Your data matches 505 different statistics following compositions of up to 3 maps.
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Matching statistic: St000378
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Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000378: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> 1
([],2)
=> [2]
=> 2
([(0,1)],2)
=> [1]
=> 1
([],3)
=> [3,3]
=> 4
([(1,2)],3)
=> [3]
=> 2
([(0,1),(0,2)],3)
=> [2]
=> 2
([(0,2),(2,1)],3)
=> [1]
=> 1
([(0,2),(1,2)],3)
=> [2]
=> 2
Description
The diagonal inversion number of an integer partition.
The dinv of a partition is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \in \{0,1\}$.
See also exercise 3.19 of [2].
This statistic is equidistributed with the length of the partition, see [3].
Matching statistic: St000644
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000644: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000644: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> 1
([],2)
=> [2]
=> 2
([(0,1)],2)
=> [1]
=> 1
([],3)
=> [3,3]
=> 4
([(1,2)],3)
=> [3]
=> 2
([(0,1),(0,2)],3)
=> [2]
=> 2
([(0,2),(2,1)],3)
=> [1]
=> 1
([(0,2),(1,2)],3)
=> [2]
=> 2
Description
The number of graphs with given frequency partition.
The frequency partition of a graph on $n$ vertices is the partition obtained from its degree sequence by recording and sorting the frequencies of the numbers that occur.
For example, the complete graph on $n$ vertices has frequency partition $(n)$. The path on $n$ vertices has frequency partition $(n-2,2)$, because its degree sequence is $(2,\dots,2,1,1)$. The star graph on $n$ vertices has frequency partition is $(n-1, 1)$, because its degree sequence is $(n-1,1,\dots,1)$.
There are two graphs having frequency partition $(2,1)$: the path and an edge together with an isolated vertex.
Matching statistic: St000814
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000814: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000814: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> 1
([],2)
=> [1,1]
=> 2
([(0,1)],2)
=> [2]
=> 1
([],3)
=> [1,1,1]
=> 4
([(1,2)],3)
=> [2,1]
=> 2
([(0,1),(0,2)],3)
=> [2,1]
=> 2
([(0,2),(2,1)],3)
=> [3]
=> 1
([(0,2),(1,2)],3)
=> [2,1]
=> 2
Description
The 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.
For example, $e_{22} = s_{1111} + s_{211} + s_{22}$, so the statistic on the partition $22$ is $3$.
Matching statistic: St001109
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(load all 3 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> 1
([],2)
=> ([],2)
=> 1
([(0,1)],2)
=> ([(0,1)],2)
=> 2
([],3)
=> ([],3)
=> 1
([(1,2)],3)
=> ([(1,2)],3)
=> 4
([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
Description
The number of proper colourings of a graph with as few colours as possible.
By definition, this is the evaluation of the chromatic polynomial at the first nonnegative integer which is not a zero of the polynomial.
Matching statistic: St001679
Values
([],1)
=> ([],1)
=> 1
([],2)
=> ([],1)
=> 1
([(0,1)],2)
=> ([(0,1)],2)
=> 2
([],3)
=> ([],1)
=> 1
([(1,2)],3)
=> ([(0,1)],2)
=> 2
([(0,1),(0,2)],3)
=> ([(0,1)],2)
=> 2
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 4
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
Description
The number of subsets of a lattice whose meet is the bottom element.
Matching statistic: St000185
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000185: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000185: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> 0 = 1 - 1
([],2)
=> [1,1]
=> 1 = 2 - 1
([(0,1)],2)
=> [2]
=> 0 = 1 - 1
([],3)
=> [1,1,1]
=> 3 = 4 - 1
([(1,2)],3)
=> [2,1]
=> 1 = 2 - 1
([(0,1),(0,2)],3)
=> [2,1]
=> 1 = 2 - 1
([(0,2),(2,1)],3)
=> [3]
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> [2,1]
=> 1 = 2 - 1
Description
The weighted size of a partition.
Let $\lambda = (\lambda_0\geq\lambda_1 \geq \dots\geq\lambda_m)$ be an integer partition. Then the weighted size of $\lambda$ is
$$\sum_{i=0}^m i \cdot \lambda_i.$$
This is also the sum of the leg lengths of the cells in $\lambda$, or
$$
\sum_i \binom{\lambda^{\prime}_i}{2}
$$
where $\lambda^{\prime}$ is the conjugate partition of $\lambda$.
This is the minimal number of inversions a permutation with the given shape can have, see [1, cor.2.2].
This is also the smallest possible sum of the entries of a semistandard tableau (allowing 0 as a part) of shape $\lambda=(\lambda_0,\lambda_1,\ldots,\lambda_m)$, obtained uniquely by placing $i-1$ in all the cells of the $i$th row of $\lambda$, see [2, eq.7.103].
Matching statistic: St001347
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> 0 = 1 - 1
([],2)
=> ([],2)
=> 1 = 2 - 1
([(0,1)],2)
=> ([(0,1)],2)
=> 0 = 1 - 1
([],3)
=> ([],3)
=> 3 = 4 - 1
([(1,2)],3)
=> ([(1,2)],3)
=> 0 = 1 - 1
([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
Description
The number of pairs of vertices of a graph having the same neighbourhood.
Matching statistic: St001961
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St001961: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001961: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> 0 = 1 - 1
([],2)
=> [1,1]
=> 1 = 2 - 1
([(0,1)],2)
=> [2]
=> 0 = 1 - 1
([],3)
=> [1,1,1]
=> 3 = 4 - 1
([(1,2)],3)
=> [2,1]
=> 1 = 2 - 1
([(0,1),(0,2)],3)
=> [2,1]
=> 1 = 2 - 1
([(0,2),(2,1)],3)
=> [3]
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> [2,1]
=> 1 = 2 - 1
Description
The sum of the greatest common divisors of all pairs of parts.
Matching statistic: St000006
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(load all 5 compositions to match this statistic)
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000006: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000006: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> [1,0,1,0]
=> 1
([],2)
=> [2]
=> [1,1,0,0,1,0]
=> 2
([(0,1)],2)
=> [1]
=> [1,0,1,0]
=> 1
([],3)
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 4
([(1,2)],3)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 2
([(0,1),(0,2)],3)
=> [2]
=> [1,1,0,0,1,0]
=> 2
([(0,2),(2,1)],3)
=> [1]
=> [1,0,1,0]
=> 1
([(0,2),(1,2)],3)
=> [2]
=> [1,1,0,0,1,0]
=> 2
Description
The dinv of a Dyck path.
Let $a=(a_1,\ldots,a_n)$ be the area sequence of a Dyck path $D$ (see [[St000012]]).
The dinv statistic of $D$ is
$$ \operatorname{dinv}(D) = \# \big\{ i < j : a_i-a_j \in \{ 0,1 \} \big\}.$$
Equivalently, $\operatorname{dinv}(D)$ is also equal to the number of boxes in the partition above $D$ whose ''arm length'' is one larger or equal to its ''leg length''.
There is a recursive definition of the $(\operatorname{area},\operatorname{dinv})$ pair of statistics, see [2].
Let $a=(0,a_2,\ldots,a_r,0,a_{r+2},\ldots,a_n)$ be the area sequence of the Dyck path $D$ with $a_i > 0$ for $2\leq i\leq r$ (so that the path touches the diagonal for the first time after $r$ steps). Assume that $D$ has $v$ entries where $a_i=0$. Let $D'$ be the path with the area sequence $(0,a_{r+2},\ldots,a_n,a_2-1,a_3-1,\ldots,a_r-1)$, then the statistics are related by
$$(\operatorname{area}(D),\operatorname{dinv}(D)) = (\operatorname{area}(D')+r-1,\operatorname{dinv}(D')+v-1).$$
Matching statistic: St000038
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Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000038: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000038: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> [1,0]
=> 1
([],2)
=> [1,1]
=> [1,1,0,0]
=> 2
([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> 1
([],3)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 4
([(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
([(0,1),(0,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
([(0,2),(2,1)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 1
([(0,2),(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
Description
The product of the heights of the descending steps of a Dyck path.
A Dyck path with 2n letters defines a partition inside an [n] x [n] board. This statistic counts the number of placements of n non-attacking rooks on the board.
By the Gessel-Viennot theory of orthogonal polynomials this corresponds to the 0-moment of the Hermite polynomials.
Summing the values of the statistic over all Dyck paths of fixed size n the number of perfect matchings (2n+1)!! is obtained: up steps are openers, down steps closers and the rooks determine a pairing of openers and closers.
The following 495 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000086The number of subgraphs. St000087The number of induced subgraphs. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000307The number of rowmotion orbits of a poset. St000468The Hosoya index of a graph. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000926The clique-coclique number of a graph. St000935The number of ordered refinements of an integer partition. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001616The number of neutral elements in a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001754The number of tolerances of a finite lattice. St001814The number of partitions interlacing the given partition. St001815The number of order preserving surjections from a poset to a total order. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000005The bounce statistic of a Dyck path. St000009The charge of a standard tableau. St000016The number of attacking pairs of a standard tableau. St000059The inversion number of a standard tableau as defined by Haglund and Stevens. St000081The number of edges of a graph. St000147The largest part of an integer partition. St000169The cocharge of a standard tableau. St000263The Szeged index of a graph. St000265The Wiener index of a graph. St000271The chromatic index of a graph. St000301The number of facets of the stable set polytope of a graph. St000330The (standard) major index of a standard tableau. St000336The leg major index of a standard tableau. St000547The number of even non-empty partial sums of an integer partition. St000869The sum of the hook lengths of an integer partition. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001117The game chromatic index of a graph. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001341The number of edges in the center of a graph. St001345The Hamming dimension of a graph. St001397Number of pairs of incomparable elements in a finite poset. St001619The number of non-isomorphic sublattices of a lattice. St001649The length of a longest trail in a graph. St001664The number of non-isomorphic subposets of a poset. St001666The number of non-isomorphic subposets of a lattice which are lattices. St001697The shifted natural comajor index of a standard Young tableau. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001783The number of odd automorphisms of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001827The number of two-component spanning forests of a graph. St001931The weak major index of an integer composition regarded as a word. St000012The area of a Dyck path. St000013The height of a Dyck path. St000020The rank of the permutation. St000025The number of initial rises of a Dyck path. St000033The number of permutations greater than or equal to the given permutation in (strong) Bruhat order. St000040The number of regions of the inversion arrangement of a permutation. St000109The number of elements less than or equal to the given element in Bruhat order. St000144The pyramid weight of the Dyck path. St000189The number of elements in the poset. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000269The number of acyclic orientations of a graph. St000270The number of forests contained in a graph. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000343The number of spanning subgraphs of a graph. St000346The number of coarsenings of a partition. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000443The number of long tunnels of a Dyck path. St000450The number of edges minus the number of vertices plus 2 of a graph. St000519The largest length of a factor maximising the subword complexity. St000537The cutwidth of a graph. St000545The number of parabolic double cosets with minimal element being the given permutation. St000617The number of global maxima of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000691The number of changes of a binary word. St000922The minimal number such that all substrings of this length are unique. St000972The composition number of a graph. St000984The number of boxes below precisely one peak. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. 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. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001281The normalized isoperimetric 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. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001378The product of the cohook lengths of the integer partition. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001474The evaluation of the Tutte polynomial of the graph at (x,y) equal to (2,-1). St001480The number of simple summands of the module J^2/J^3. St001481The minimal height of a peak of a Dyck path. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. 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. St001645The pebbling number of a connected graph. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. St001717The largest size of an interval in a poset. St001758The number of orbits of promotion on a graph. St001809The index of the step at the first peak of maximal height in a Dyck path. St001869The maximum cut size of a graph. St001924The number of cells in an integer partition whose arm and leg length coincide. St001955The number of natural descents for set-valued two row standard Young tableaux. St000004The major index of a permutation. St000008The major index of the composition. St000010The length of the partition. St000018The number of inversions of a permutation. St000024The number of double up and double down steps of a Dyck path. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000052The number of valleys of a Dyck path not on the x-axis. St000055The inversion sum of a permutation. St000108The number of partitions contained in the given partition. St000133The "bounce" of a permutation. St000154The sum of the descent bottoms of a permutation. St000160The multiplicity of the smallest part of a partition. St000246The number of non-inversions of a permutation. St000295The length of the border of a binary word. St000300The number of independent sets of vertices of a graph. St000304The load of a permutation. St000305The inverse major index of a permutation. St000340The number of non-final maximal constant sub-paths of length greater than one. St000341The non-inversion sum of a permutation. St000347The inversion sum of a binary word. St000348The non-inversion sum of a binary word. St000379The number of Hamiltonian cycles in a graph. St000423The number of occurrences of the pattern 123 or of the pattern 132 in a permutation. St000428The number of occurrences of the pattern 123 or of the pattern 213 in a permutation. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000435The number of occurrences of the pattern 213 or of the pattern 231 in a permutation. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000439The position of the first down step of a Dyck path. St000446The disorder of a permutation. St000532The total number of rook placements on a Ferrers board. St000548The number of different non-empty partial sums of an integer partition. St000676The number of odd rises of a Dyck path. St000692Babson and Steingrímsson's statistic of a permutation. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000867The sum of the hook lengths in the first row of an integer partition. St000921The number of internal inversions of a binary word. St000983The length of the longest alternating subword. St000992The alternating sum of the parts of an integer partition. 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. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. 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. St001079The minimal length of a factorization of a permutation using the permutations (12)(34). St001094The depth index of a set partition. St001161The major index north count of 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. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001172The number of 1-rises at odd height of a Dyck path. St001180Number of indecomposable injective modules with projective dimension at most 1. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. 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. 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. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001311The cyclomatic number of a graph. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001362The normalized Knill dimension of a graph. St001412Number of minimal entries in the Bruhat order matrix of a permutation. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. 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. 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. St001541The Gini index of an integer partition. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001759The Rajchgot index of a permutation. St001874Lusztig's a-function for the symmetric group. St001930The weak major index of a binary word. St001956The comajor index for set-valued two-row standard Young tableaux. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001003The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St000848The balance constant multiplied with the number of linear extensions of a poset. St001624The breadth of a lattice. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000298The order dimension or Dushnik-Miller dimension of a poset. St000331The number of upper interactions of a Dyck path. St000418The number of Dyck paths that are weakly below a Dyck path. St000640The rank of the largest boolean interval in a poset. St000770The major index of an integer partition when read from bottom to top. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. 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. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001268The size of the largest ordinal summand in the poset. St001488The number of corners of a skew partition. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001531Number of partial orders contained in the poset determined by the Dyck path. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001959The product of the heights of the peaks of a Dyck path. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000466The Gutman (or modified Schultz) index of a connected graph. St000632The jump number of the poset. St000741The Colin de Verdière graph invariant. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000120The number of left tunnels of a Dyck path. St000272The treewidth of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000453The number of distinct Laplacian eigenvalues of a graph. St000482The (zero)-forcing number of a graph. St000536The pathwidth of a graph. St000544The cop number of a graph. St000778The metric dimension of a graph. St000785The number of distinct colouring schemes of a graph. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000822The Hadwiger number of the graph. St001029The size of the core of a graph. St001093The detour number of a graph. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension 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. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001261The Castelnuovo-Mumford regularity of a graph. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001270The bandwidth of a graph. St001277The degeneracy of a graph. 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. St001330The hat guessing number of a graph. St001352The number of internal nodes in the modular decomposition of a graph. St001358The largest degree of a regular subgraph of a graph. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001494The Alon-Tarsi number of a graph. St001500The global dimension of magnitude 1 Nakayama algebras. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001580The acyclic chromatic number of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001644The dimension of a graph. St001734The lettericity of a graph. St001792The arboricity of a graph. St001808The box weight or horizontal decoration of a Dyck path. St001883The mutual visibility number of a graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St001962The proper pathwidth of a graph. St000289The decimal representation of a binary word. St000391The sum of the positions of the ones in a binary word. St000454The largest eigenvalue of a graph if it is integral. St000472The sum of the ascent bottoms of a permutation. St000490The intertwining number of a set partition. St000492The rob statistic of a set partition. St000493The los statistic of a set partition. St000494The number of inversions of distance at most 3 of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000498The lcs statistic of a set partition. St000499The rcb statistic of a set partition. St000535The rank-width of a graph. St000558The number of occurrences of the pattern {{1,2}} in a set partition. St000573The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element. St000574The number of occurrences of the pattern {{1},{2}} such that 1 is a minimal and 2 a maximal element. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000578The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton. 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. 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. St000976The sum of the positions of double up-steps of a Dyck path. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. 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. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001120The length of a longest path in a graph. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001323The independence gap of a graph. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001333The cardinality of a minimal edge-isolating set of a graph. St001393The induced matching number of a graph. St001479The number of bridges of a graph. St001512The minimum rank of a graph. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001638The book thickness of a graph. St001642The Prague dimension of a graph. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001721The degree of a binary word. St001736The total number of cycles in a graph. St001743The discrepancy of a graph. St001746The coalition number of a graph. St001826The maximal number of leaves on a vertex of a graph. St000455The second largest eigenvalue of a graph if it is integral. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001570The minimal number of edges to add to make a graph Hamiltonian. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001118The acyclic chromatic index of a graph. St000172The Grundy number of a graph. St000180The number of chains of a poset. St000244The cardinality of the automorphism group of a graph. St000258The burning number of a graph. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000286The number of connected components of the complement of a graph. St000363The number of minimal vertex covers of a graph. St000364The exponent of the automorphism group of a graph. St000452The number of distinct eigenvalues of a graph. St000460The hook length of the last cell along the main diagonal of an integer partition. St000469The distinguishing number of a graph. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000636The hull number of a graph. St000667The greatest common divisor of the parts of the partition. St000722The number of different neighbourhoods in a graph. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000914The sum of the values of the Möbius function of a poset. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St000937The number of positive values of the symmetric group character corresponding to the partition. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001304The number of maximally independent sets of vertices of a graph. St001315The dissociation number of a graph. St001316The domatic number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001342The number of vertices in the center of a graph. St001360The number of covering relations in Young's lattice below a partition. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001389The number of partitions of the same length below the given integer partition. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001527The cyclic permutation representation number of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St001571The Cartan determinant of the integer partition. St001581The achromatic number of a graph. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001670The connected partition number of a graph. St001672The restrained domination number of a graph. St001674The number of vertices of the largest induced star graph in the graph. 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. St001757The number of orbits of toric promotion on a graph. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001802The number of endomorphisms of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001909The number of interval-closed sets of a poset. St001933The largest multiplicity of a part in an integer partition. St001963The tree-depth of a graph. St000273The domination number of a graph. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000287The number of connected components of a graph. St000309The number of vertices with even degree. St000315The number of isolated vertices of a graph. St000444The length of the maximal rise of a Dyck path. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000553The number of blocks of a graph. St000656The number of cuts of a poset. St000674The number of hills of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000759The smallest missing part in an integer partition. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000916The packing number of a graph. St000950Number of tilting modules of the corresponding LNakayama algebra, where a tilting module is a generalised tilting module of projective dimension 1. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St000953The largest degree of an irreducible factor of the Coxeter polynomial of the Dyck path over the rational numbers. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St000981The length of the longest zigzag subpath. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St000997The even-odd crank of an integer partition. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. 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. St001228The vector space dimension of the space of module homomorphisms between J and itself when J denotes the Jacobson radical of the corresponding Nakayama algebra. St001254The vector space dimension of the first extension-group between A/soc(A) and J when A is the corresponding Nakayama algebra with Jacobson radical J. St001286The annihilation number of a graph. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001361The number of lattice paths of the same length that stay weakly above a Dyck path. St001363The Euler characteristic of a graph according to Knill. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001441The number of non-empty connected induced subgraphs of a graph. St001463The number of distinct columns in the nullspace of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St001523The degree of symmetry of a Dyck path. St001691The number of kings in a graph. St001765The number of connected components of the friends and strangers graph. St001828The Euler characteristic of a graph. St001829The common independence number of a graph. St001885The number of binary words with the same proper border set. St001545The second Elser number of a connected graph. St000464The Schultz index of a connected graph. St000474Dyson's crank of a partition. St000477The weight of a partition according to Alladi. St000668The least common multiple of the parts of the partition. St000706The product of the factorials of the multiplicities of an integer partition. 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. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. 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 $eA$. 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 $eA$. 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 $eA$. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001060The distinguishing index of a graph. St000327The number of cover relations in a poset. St000528The height of a poset. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000815The number of semistandard Young tableaux of partition weight of given shape. St000907The number of maximal antichains of minimal length in a poset. St000911The number of maximal antichains of maximal size in a poset. St000912The number of maximal antichains in a poset. St000933The number of multipartitions of sizes given by 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. St001343The dimension of the reduced incidence algebra of a poset. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001813The product of the sizes of the principal order filters in a poset. St000100The number of linear extensions of a poset. St000478Another weight of a partition according to Alladi. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000633The size of the automorphism group of a poset. St000635The number of strictly order preserving maps of a poset into itself. St000681The Grundy value of Chomp on Ferrers diagrams. 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. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St001561The value of the elementary symmetric function evaluated at 1. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St000302The determinant of the distance matrix of a connected graph. St000467The hyper-Wiener index of a connected graph.
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