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Your data matches 841 different statistics following compositions of up to 3 maps.
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Matching statistic: St001385
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
St001385: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1
[2]
=> 1
[1,1]
=> 1
[3]
=> 2
[2,1]
=> 1
[1,1,1]
=> 1
[4]
=> 6
[3,1]
=> 2
[2,2]
=> 1
[2,1,1]
=> 1
[1,1,1,1]
=> 1
Description
The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition.
Equivalently, given an integer partition $\lambda$, this is the number of molecular combinatorial species that decompose into a product of atomic species of sizes $\lambda_1,\lambda_2,\dots$. In particular, the value on the partition $(n)$ is the number of atomic species of degree $n$, [2].
Matching statistic: St000048
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000048: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000048: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> []
=> 1
[2]
=> []
=> 1
[1,1]
=> [1]
=> 1
[3]
=> []
=> 1
[2,1]
=> [1]
=> 1
[1,1,1]
=> [1,1]
=> 2
[4]
=> []
=> 1
[3,1]
=> [1]
=> 1
[2,2]
=> [2]
=> 1
[2,1,1]
=> [1,1]
=> 2
[1,1,1,1]
=> [1,1,1]
=> 6
Description
The multinomial of the parts of a partition.
Given an integer partition $\lambda = [\lambda_1,\ldots,\lambda_k]$, this is the multinomial
$$\binom{|\lambda|}{\lambda_1,\ldots,\lambda_k}.$$
For any integer composition $\mu$ that is a rearrangement of $\lambda$, this is the number of ordered set partitions whose list of block sizes is $\mu$.
Matching statistic: St000001
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000001: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000001: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1] => [1] => 1
[2]
=> [1,0,1,0]
=> [1,2] => [1,2] => 1
[1,1]
=> [1,1,0,0]
=> [2,1] => [2,1] => 1
[3]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => [3,2,1] => 2
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => 1
[2,2]
=> [1,1,1,0,0,0]
=> [3,1,2] => [3,1,2] => 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => 2
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,2,3,1] => 6
Description
The number of reduced words for a permutation.
This is the number of ways to write a permutation as a minimal length product of simple transpositions. E.g., there are two reduced words for the permutation $[3,2,1]$, which are $(1,2)(2,3)(1,2) = (2,3)(1,2)(2,3)$.
Matching statistic: St000124
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000124: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000124: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1] => [1] => 1
[2]
=> [1,0,1,0]
=> [1,2] => [1,2] => 1
[1,1]
=> [1,1,0,0]
=> [2,1] => [2,1] => 1
[3]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => [3,1,2] => 2
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => 1
[2,2]
=> [1,1,1,0,0,0]
=> [3,1,2] => [2,3,1] => 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,2,3] => 2
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,1,2,3] => 6
Description
The cardinality of the preimage of the Simion-Schmidt map.
The Simion-Schmidt bijection transforms a [3,1,2]-avoiding permutation into a [3,2,1]-avoiding permutation. More generally, it can be thought of as a map $S$ that turns any permutation into a [3,2,1]-avoiding permutation. This statistic is the size of $S^{-1}(\pi)$ for each permutation $\pi$.
The map $S$ can also be realized using the quotient of the $0$-Hecke Monoid of the symmetric group by the relation $\pi_i \pi_{i+1} \pi_i = \pi_{i+1} \pi_i$, sending each element of the fiber of the quotient to the unique [3,2,1]-avoiding element in that fiber.
Matching statistic: St000278
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000278: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000278: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1,0]
=> []
=> 1
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> []
=> 1
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1]
=> 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2]
=> 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> 2
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 2
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 6
Description
The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions.
This is the multinomial of the multiplicities of the parts, see [1].
This is the same as $m_\lambda(x_1,\dotsc,x_k)$ evaluated at $x_1=\dotsb=x_k=1$,
where $k$ is the number of parts of $\lambda$.
An explicit formula is $\frac{k!}{m_1(\lambda)! m_2(\lambda)! \dotsb m_k(\lambda) !}$
where $m_i(\lambda)$ is the number of parts of $\lambda$ equal to $i$.
Matching statistic: St000810
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000810: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000810: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1,0]
=> []
=> 1
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> []
=> 1
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1]
=> 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2]
=> 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> 2
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 2
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 6
Description
The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to monomial symmetric functions.
For example, $p_{22} = 2m_{22} + m_4$, so the statistic on the partition $22$ is 3.
Matching statistic: St000958
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000958: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000958: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1,0]
=> [1] => 1
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,2] => 1
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [2,1] => 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,3,2] => 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [2,1,3] => 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => 2
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 6
Description
The number of Bruhat factorizations of a permutation.
This is the number of factorizations $\pi = t_1 \cdots t_\ell$ for transpositions $\{ t_i \mid 1 \leq i \leq \ell\}$ such that the number of inversions of $t_1 \cdots t_i$ equals $i$ for all $1 \leq i \leq \ell$.
Matching statistic: St000959
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000959: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000959: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1,0]
=> [1] => 1
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,2] => 1
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [2,1] => 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,3,2] => 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [2,1,3] => 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => 2
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 6
Description
The number of strong Bruhat factorizations of a permutation.
This is, the number of factorizations $\pi = t_1 \cdots t_\ell$ for transpositions $\{ t_i \mid 1 \leq i \leq \ell\}$ such that $t_1 \cdots t_i$ has more inversions than $t_1 \cdots t_{i-1}$ for all $1 \leq i \leq \ell$.
Matching statistic: St001220
(load all 20 compositions to match this statistic)
(load all 20 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001220: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001220: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [2,1] => [1,2] => 1
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => 1
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,4,3,2] => 2
[2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => [1,3,2] => 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [1,5,4,3,2] => 6
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,4,3,2] => 2
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,3,2,4] => 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,3,4,2] => 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 1
Description
The width of a permutation.
Let $w$ be a permutation. The interval $[e,w]$ in the weak order is ranked, and we define $r_i=r_i(w)$ to be the number of elements at rank $i$ in $[e,w]$, where $i \in \{0, \dots, \ell(w)\}$. The ''width'' of
$w$ is the maximum of $\{r_0,r_1,\ldots,r_{\ell(w)}\}$. See [1].
Matching statistic: St001387
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001387: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001387: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> []
=> []
=> []
=> 1
[2]
=> []
=> []
=> []
=> 1
[1,1]
=> [1]
=> [1]
=> [1]
=> 1
[3]
=> []
=> []
=> []
=> 1
[2,1]
=> [1]
=> [1]
=> [1]
=> 1
[1,1,1]
=> [1,1]
=> [2]
=> [1,1]
=> 2
[4]
=> []
=> []
=> []
=> 1
[3,1]
=> [1]
=> [1]
=> [1]
=> 1
[2,2]
=> [2]
=> [1,1]
=> [2]
=> 1
[2,1,1]
=> [1,1]
=> [2]
=> [1,1]
=> 2
[1,1,1,1]
=> [1,1,1]
=> [2,1]
=> [2,1]
=> 6
Description
Number of standard Young tableaux of the skew shape tracing the border of the given partition.
Let $\lambda \vdash n$ be a diagram with the given partition as shape.
Add $n$ additional boxes, one in each column $1,\dotsc,n$, and let this be $\mu$.
The statistic is the number of standard Young tableaux of skew shape $\mu/\lambda$,
which is equal to $\frac{n!}{\prod_{i} (\mu_i - \lambda_i)!}$.
For example, $\lambda=[2,1,1]$ gives $\mu = [4,2,1,1]$.
The first row in the skew shape $\mu/\lambda$ has two boxes, so the number of SYT of
shape $\mu/\lambda$ is then $4!/2 = 12$.
This statistic shows up in the study of skew specialized Macdonald polynomials,
where a type of charge statistic give rise to a $q$-analogue of the above formula.
The following 831 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001532The leading coefficient of the Poincare polynomial of the poset cone. St001779The order of promotion on the set of linear extensions of a poset. St000881The number of short braid edges in the graph of braid moves of a permutation. St000963The 2-shifted major index of a permutation. St001309The number of four-cliques in a graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001856The number of edges in the reduced word graph of a permutation. St001487The number of inner corners of a skew partition. St000769The major index of a composition regarded as a word. St001846The number of elements which do not have a complement in the lattice. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001462The number of factors of a standard tableaux under concatenation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000633The size of the automorphism group of a poset. St000902 The minimal number of repetitions of an integer composition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001399The distinguishing number of a poset. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001470The cyclic holeyness of a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001820The size of the image of the pop stack sorting operator. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001884The number of borders of a binary word. St000297The number of leading ones in a binary word. St000360The number of occurrences of the pattern 32-1. St000392The length of the longest run of ones in a binary word. St000454The largest eigenvalue of a graph if it is integral. St000488The number of cycles of a permutation of length at most 2. St000664The number of right ropes of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000758The length of the longest staircase fitting into an integer composition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000850The number of 1/2-balanced pairs in a poset. St000982The length of the longest constant subword. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001095The number of non-isomorphic posets with precisely one further covering relation. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001130The number of two successive successions in a permutation. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. 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 $eA$. 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. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001330The hat guessing number of a graph. St001423The number of distinct cubes in a binary word. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001520The number of strict 3-descents. St001530The depth of a Dyck path. St001556The number of inversions of the third entry of a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001948The number of augmented double ascents of a permutation. St001964The interval resolution global dimension of a poset. St001372The length of a longest cyclic run of ones of a binary word. St000068The number of minimal elements in a poset. St000527The width of the poset. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000160The multiplicity of the smallest part of a partition. St000181The number of connected components of the Hasse diagram for the poset. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000335The difference of lower and upper interactions. St000456The monochromatic index of a connected graph. St000667The greatest common divisor of the parts of the partition. St000733The row containing the largest entry of a standard tableau. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. 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. 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. St001267The length of the Lyndon factorization of the binary word. St001437The flex of a binary word. St001481The minimal height of a peak of a Dyck path. St001490The number of connected components of a skew partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001498The normalised height of a Nakayama algebra with magnitude 1. St001571The Cartan determinant of the integer partition. St001890The maximum magnitude of the Möbius function of a poset. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000408The number of occurrences of the pattern 4231 in a permutation. St000422The energy of a graph, if it is integral. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000451The length of the longest pattern of the form k 1 2. St000455The second largest eigenvalue of a graph if it is integral. St000534The number of 2-rises of a permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000842The breadth of a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. 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. St001389The number of partitions of the same length below the given integer partition. St000003The number of standard Young tableaux of the partition. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000010The length of the partition. St000026The position of the first return of a Dyck path. St000079The number of alternating sign matrices for a given Dyck path. St000120The number of left tunnels of a Dyck path. St000147The largest part of an integer partition. St000159The number of distinct parts of the integer partition. St000179The product of the hook lengths of the integer partition. St000182The number of permutations whose cycle type is the given integer partition. St000184The size of the centralizer of any permutation of given cycle type. St000321The number of integer partitions of n that are dominated by an integer partition. St000326The position of the first one in a binary word after appending a 1 at the end. St000345The number of refinements of a partition. St000346The number of coarsenings of a partition. St000378The diagonal inversion number of an integer partition. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000531The leading coefficient of the rook polynomial of an integer partition. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000548The number of different non-empty partial sums of an integer partition. St000655The length of the minimal rise of a Dyck path. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000734The last entry in the first row of a standard tableau. St000783The side length of the largest staircase partition fitting into a partition. 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. 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. St000876The number of factors in the Catalan decomposition of a binary word. St000935The number of ordered refinements of an integer partition. St000947The major index east count of a Dyck path. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001161The major index north count of a Dyck path. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001432The order dimension of the partition. St001527The cyclic permutation representation number of an integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001612The number of coloured multisets of cycles such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001780The order of promotion on the set of standard tableaux of given shape. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. 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. St000008The major index of the composition. St000011The number of touch points (or returns) of a Dyck path. St000012The area of a Dyck path. St000014The number of parking functions supported by a Dyck path. St000015The number of peaks of a Dyck path. St000020The rank of the permutation. St000024The number of double up and double down steps of a Dyck path. St000025The number of initial rises of a Dyck path. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000033The number of permutations greater than or equal to the given permutation in (strong) Bruhat order. St000038The product of the heights of the descending steps of a Dyck path. St000040The number of regions of the inversion arrangement of a permutation. St000047The number of standard immaculate tableaux of a given shape. St000051The size of the left subtree of a binary tree. St000053The number of valleys of the Dyck path. St000054The first entry of the permutation. St000056The decomposition (or block) number of a permutation. St000061The number of nodes on the left branch of a binary tree. St000062The length of the longest increasing subsequence of the permutation. St000063The number of linear extensions of a certain poset defined for an integer partition. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000075The orbit size of a standard tableau under promotion. St000083The number of left oriented leafs of a binary tree except the first one. St000084The number of subtrees. St000088The row sums of the character table of the symmetric group. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000108The number of partitions contained in the given partition. St000109The number of elements less than or equal to the given element in Bruhat order. St000110The number of permutations less than or equal to a permutation in left weak order. St000141The maximum drop size of a permutation. St000154The sum of the descent bottoms of a permutation. St000156The Denert index of a permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000201The number of leaf nodes in a binary tree. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000240The number of indices that are not small excedances. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000255The number of reduced Kogan faces with the permutation as type. St000277The number of ribbon shaped standard tableaux. St000288The number of ones in a binary word. St000290The major index of a binary word. St000291The number of descents of a binary word. St000295The length of the border of a binary word. St000296The length of the symmetric border of a binary word. St000305The inverse major index of a permutation. St000306The bounce count of a Dyck path. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000331The number of upper interactions of a Dyck path. St000340The number of non-final maximal constant sub-paths of length greater than one. St000354The number of recoils of a permutation. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000383The last part of an integer composition. St000389The number of runs of ones of odd length in a binary word. St000390The number of runs of ones in a binary word. St000393The number of strictly increasing runs in a binary word. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000396The register function (or Horton-Strahler number) of a binary tree. 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. St000443The number of long tunnels of a Dyck path. St000492The rob statistic of a set partition. St000499The rcb statistic of a set partition. St000501The size of the first part in the decomposition of a permutation. St000504The cardinality of the first block of a set partition. St000505The biggest entry in the block containing the 1. St000507The number of ascents of a standard tableau. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000532The total number of rook placements on a Ferrers board. St000539The number of odd inversions of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000542The number of left-to-right-minima of a permutation. St000545The number of parabolic double cosets with minimal element being the given permutation. St000568The hook number of a binary tree. St000579The number of occurrences of the pattern {{1},{2}} such that 2 is a maximal element. St000617The number of global maxima of a Dyck path. St000619The number of cyclic descents of a permutation. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000626The minimal period of a binary word. St000627The exponent of a binary word. St000630The length of the shortest palindromic decomposition of a binary word. St000631The number of distinct palindromic decompositions of a binary word. St000644The number of graphs with given frequency partition. St000653The last descent of a permutation. St000654The first descent of a permutation. St000657The smallest part of an integer composition. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000676The number of odd rises of a Dyck path. St000677The standardized bi-alternating inversion number of a permutation. St000678The number of up steps after the last double rise of a Dyck path. St000679The pruning number of an ordered tree. St000682The Grundy value of Welter's game on a binary word. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000691The number of changes of a binary word. St000701The protection number of a binary tree. St000702The number of weak deficiencies of a permutation. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000738The first entry in the last row of a standard tableau. St000740The last entry of a permutation. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000746The number of pairs with odd minimum in a perfect matching. St000753The Grundy value for the game of Kayles on a binary word. St000759The smallest missing part in an integer partition. St000764The number of strong records in an integer composition. St000767The number of runs in an integer composition. St000781The number of proper colouring schemes of a Ferrers diagram. St000794The mak of a permutation. St000796The stat' of a permutation. St000797The stat`` of a permutation. St000798The makl of a permutation. St000808The number of up steps of the associated bargraph. St000812The sum of the entries in the column specified by the partition of the change of basis matrix from complete homogeneous symmetric functions to monomial symmetric functions. St000816The number of standard composition tableaux of the composition. 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. 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. St000820The number of compositions obtained by rotating the composition. St000823The number of unsplittable factors of the set partition. St000829The Ulam distance of a permutation to the identity permutation. St000833The comajor index of a permutation. St000840The number of closers smaller than the largest opener in a perfect matching. St000843The decomposition number of a perfect matching. St000847The number of standard Young tableaux whose descent set is the binary word. St000862The number of parts of the shifted shape of a permutation. St000873The aix statistic of a permutation. St000874The position of the last double rise in a Dyck path. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000877The depth of the binary word interpreted as a path. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000885The number of critical steps in the Catalan decomposition of a binary word. St000886The number of permutations with the same antidiagonal sums. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000889The number of alternating sign matrices with the same antidiagonal sums. St000899The maximal number of repetitions of an integer composition. St000900The minimal number of repetitions of a part in an integer composition. St000903The number of different parts of an integer composition. St000904The maximal number of repetitions of an integer composition. St000920The logarithmic height of a Dyck path. St000922The minimal number such that all substrings of this length are unique. St000946The sum of the skew hook positions in a Dyck path. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St000971The smallest closer of a set partition. St000983The length of the longest alternating subword. St000984The number of boxes below precisely one peak. St000988The orbit size of a permutation under Foata's bijection. St000990The first ascent of a permutation. St000991The number of right-to-left minima of a permutation. St000993The multiplicity of the largest part of an integer partition. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001000Number of indecomposable modules with projective 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. 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. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of 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. St001031The height of 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. St001041The depth of the label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001044The number of pairs whose larger element is at most one more than half the size of the perfect matching. St001045The number of leaves in the subtree not containing one in the decreasing labelled binary unordered tree associated with the perfect matching. St001048The number of leaves in the subtree containing 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001052The length of the exterior of a permutation. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001080The minimal length of a factorization of a permutation using the transposition (12) and the cycle (1,. St001081The number of minimal length factorizations of a permutation into star transpositions. St001128The exponens consonantiae of a partition. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001136The largest label with larger sister in the leaf labelled binary unordered tree associated with the perfect matching. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001151The number of blocks with odd minimum. St001162The minimum jump of a permutation. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. 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. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001197The global 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. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
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 $eA$. 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. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in 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. 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. 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. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001298The number of repeated entries in the Lehmer code of a permutation. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001312Number of parabolic noncrossing partitions indexed by the composition. St001313The number of Dyck paths above the lattice path given by a binary word. St001344The neighbouring number of a permutation. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001400The total number of Littlewood-Richardson tableaux of given shape. St001405The number of bonds in a permutation. St001415The length of the longest palindromic prefix of a binary word. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001424The number of distinct squares in a binary word. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001461The number of topologically connected components of the chord diagram of a permutation. St001471The magnitude of a Dyck path. St001480The number of simple summands of the module J^2/J^3. St001482The product of the prefix sums of a permutation. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001485The modular major index of a binary word. St001497The position of the largest weak excedence of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001500The global dimension of magnitude 1 Nakayama algebras. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001517The length of a longest pair of twins in a permutation. St001524The degree of symmetry of a binary word. St001528The number of permutations such that the product with the permutation has the same number of fixed points. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001569The maximal modular displacement of a permutation. St001591The number of graphs with the given composition of multiplicities of Laplacian eigenvalues. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001595The number of standard Young tableaux of the skew partition. St001597The Frobenius rank of a skew partition. St001614The cyclic permutation representation number of a skew partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001652The length of a longest interval of consecutive numbers. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001662The length of the longest factor of consecutive numbers in a permutation. St001667The maximal size of a pair of weak twins for a permutation. St001671Haglund's hag of a permutation. St001675The number of parts equal to the part in the reversed composition. St001732The number of peaks visible from the left. St001733The number of weak left to right maxima of a Dyck path. St001735The number of permutations with the same set of runs. St001741The largest integer such that all patterns of this size are contained in the permutation. St001759The Rajchgot index of a permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St001801Half the number of preimage-image pairs of different parity in a permutation. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001806The upper middle entry of a permutation. St001807The lower middle entry of a permutation. St001808The box weight or horizontal decoration of a Dyck path. St001809The index of the step at the first peak of maximal height in a Dyck path. St001814The number of partitions interlacing the given partition. 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). St001896The number of right descents of a signed permutations. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St001955The number of natural descents for set-valued two row standard Young tableaux. St001965The number of decreasable positions in the corner sum matrix of an alternating sign matrix. St000090The variation of a composition. St000091The descent variation of a composition. St000217The number of occurrences of the pattern 312 in a permutation. St000233The number of nestings of a set partition. St000338The number of pixed points of a permutation. St000358The number of occurrences of the pattern 31-2. St000365The number of double ascents of a permutation. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000650The number of 3-rises of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000779The tier of a permutation. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001705The number of occurrences of the pattern 2413 in a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001866The nesting alignments of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001568The smallest positive integer that does not appear twice in the partition. St000706The product of the factorials of the multiplicities of an integer partition. St000707The product of the factorials of the parts. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001488The number of corners of a skew partition. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000618The number of self-evacuating tableaux of given shape. St000668The least common multiple of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000815The number of semistandard Young tableaux of partition weight of given shape. St000871The number of very big ascents of a permutation. 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. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001563The value of the power-sum symmetric function evaluated at 1. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001645The pebbling number of a connected graph. St001763The Hurwitz number of an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St000022The number of fixed points of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000035The number of left outer peaks of a permutation. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000374The number of exclusive right-to-left minima of a permutation. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000466The Gutman (or modified Schultz) index of a connected graph. St000546The number of global descents of a permutation. St000731The number of double exceedences of a permutation. St000742The number of big ascents of a permutation after prepending zero. St001713The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern. St000284The Plancherel distribution on integer partitions. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000735The last entry on the main diagonal of a standard tableau. St000744The length of the path to the largest entry in a standard Young tableau. St000939The number of characters of the symmetric group whose value on the partition is positive. St000021The number of descents of a permutation. St000023The number of inner peaks of a permutation. St000045The number of linear extensions of a binary tree. St000060The greater neighbor of the maximum. St000082The number of elements smaller than a binary tree in Tamari order. St000100The number of linear extensions of a poset. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000210Minimum over maximum difference of elements in cycles. St000234The number of global ascents of a permutation. St000253The crossing number of a set partition. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000327The number of cover relations in a poset. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000353The number of inner valleys of a permutation. St000385The number of vertices with out-degree 1 in a binary tree. St000402Half the size of the symmetry class of a permutation. St000414The binary logarithm of the number of binary trees with the same underlying unordered tree. St000418The number of Dyck paths that are weakly below a Dyck path. St000420The number of Dyck paths that are weakly above a Dyck path. St000444The length of the maximal rise of a Dyck path. St000486The number of cycles of length at least 3 of a permutation. St000487The length of the shortest cycle of a permutation. St000489The number of cycles of a permutation of length at most 3. 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. St000529The number of permutations whose descent word is the given binary word. St000530The number of permutations with the same descent word as the given permutation. St000543The size of the conjugacy class of a binary word. St000563The number of overlapping pairs of blocks of a set partition. St000570The Edelman-Greene number of a permutation. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000635The number of strictly order preserving maps of a poset into itself. St000640The rank of the largest boolean interval in a poset. St000646The number of big ascents of a permutation. St000663The number of right floats of a permutation. St000690The size of the conjugacy class of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000729The minimal arc length of a set partition. St000732The number of double deficiencies of a permutation. St000741The Colin de Verdière graph invariant. St000762The sum of the positions of the weak records of an integer composition. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000864The number of circled entries of the shifted recording tableau of a permutation. St000872The number of very big descents of a permutation. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000898The number of maximal entries in the last diagonal of the monotone triangle. St000908The length of the shortest maximal antichain in a poset. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St000925The number of topologically connected components of a set partition. St001062The maximal size of a block of a set partition. St001075The minimal size of a block of a set partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. 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. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001246The maximal difference between two consecutive entries of a permutation. St001346The number of parking functions that give the same permutation. St001388The number of non-attacking neighbors of a permutation. St001413Half the length of the longest even length palindromic prefix of a binary word. St001469The holeyness of a permutation. St001531Number of partial orders contained in the poset determined by the Dyck path. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001625The Möbius invariant of a lattice. St001637The number of (upper) dissectors of a poset. St001665The number of pure excedances of a permutation. St001668The number of points of the poset minus the width of the poset. St001686The order of promotion on a Gelfand-Tsetlin pattern. St001712The number of natural descents of a standard Young tableau. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001781The interlacing number of a set partition. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001838The number of nonempty primitive factors of a binary word. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001889The size of the connectivity set of a signed permutation. St001928The number of non-overlapping descents in a permutation. St001959The product of the heights of the peaks of a Dyck path. St000039The number of crossings of a permutation. St000058The order of a permutation. St000105The number of blocks in the set partition. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000219The number of occurrences of the pattern 231 in a permutation. St000221The number of strong fixed points of a permutation. St000236The number of cyclical small weak excedances. St000241The number of cyclical small excedances. St000247The number of singleton blocks of a set partition. St000248The number of anti-singletons of a set partition. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000251The number of nonsingleton blocks of a set partition. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000317The cycle descent number of a permutation. St000325The width of the tree associated to a permutation. St000328The maximum number of child nodes in a tree. St000355The number of occurrences of the pattern 21-3. St000367The number of simsun double descents of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000462The major index minus the number of excedences of a permutation. St000470The number of runs in a permutation. St000485The length of the longest cycle of a permutation. St000496The rcs statistic of a set partition. St000500Eigenvalues of the random-to-random operator acting on the regular representation. St000502The number of successions of a set partitions. St000516The number of stretching pairs of a permutation. St000557The number of occurrences of the pattern {{1},{2},{3}} in a set partition. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000561The number of occurrences of the pattern {{1,2,3}} 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. St000578The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000591The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal. St000592The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal. St000593The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000623The number of occurrences of the pattern 52341 in a permutation. St000649The number of 3-excedences of a permutation. St000666The number of right tethers of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000836The number of descents of distance 2 of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St000943The number of spots the most unlucky car had to go further in a parking function. St000961The shifted major index of a permutation. St000962The 3-shifted major index of a permutation. St000989The number of final rises of a permutation. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001114The number of odd descents of a permutation. St001301The first Betti number of the order complex associated with the poset. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001371The length of the longest Yamanouchi prefix of a binary word. St001381The fertility of a permutation. St001396Number of triples of incomparable elements in a finite poset. St001402The number of separators in a permutation. St001403The number of vertical separators in a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001513The number of nested exceedences of a permutation. St001537The number of cyclic crossings of a permutation. St001545The second Elser number of a connected graph. St001549The number of restricted non-inversions between exceedances. St001550The number of inversions between exceedances where the greater exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001810The number of fixed points of a permutation smaller than its largest moved point. St001847The number of occurrences of the pattern 1432 in a permutation. St001850The number of Hecke atoms of a permutation. St001851The number of Hecke atoms of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St001903The number of fixed points of a parking function. St001905The number of preferred parking spots in a parking function less than the index of the car. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000638The number of up-down runs of a permutation. St000824The sum of the number of descents and the number of recoils of a permutation. St000831The number of indices that are either descents or recoils. St001472The permanent of the Coxeter matrix of the poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001875The number of simple modules with projective dimension at most 1. St000460The hook length of the last cell along the main diagonal of an integer partition. St000632The jump number of the poset. St000782The indicator function of whether a given perfect matching is an L & P matching. St000870The product of the hook lengths of the diagonal cells in an integer partition. 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. St001360The number of covering relations in Young's lattice below a partition. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. 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. St001635The trace of the square of the Coxeter matrix of the incidence algebra of a poset. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000907The number of maximal antichains of minimal length in a poset. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St000717The number of ordinal summands of a poset. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000117The number of centered tunnels of a Dyck path. St000144The pyramid weight of the Dyck path. St000148The number of odd parts of a partition. St000183The side length of the Durfee square of an integer partition. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000228The size of a partition. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St000289The decimal representation of a binary word. St000293The number of inversions of a binary word. St000384The maximal part of the shifted composition of an integer partition. St000395The sum of the heights of the peaks of a Dyck path. St000459The hook length of the base cell of a partition. St000475The number of parts equal to 1 in a partition. St000517The Kreweras number of an integer partition. St000519The largest length of a factor maximising the subword complexity. St000549The number of odd partial sums of an integer partition. St000628The balance of a binary word. St000674The number of hills of a Dyck path. St000705The number of semistandard tableaux on a given integer partition of n with maximal entry n. St000784The maximum of the length and the largest part of the integer partition. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000867The sum of the hook lengths in the first row of an integer partition. St000878The number of ones minus the number of zeros of a binary word. St000897The number of different multiplicities of parts of an integer partition. St000913The number of ways to refine the partition into singletons. St000932The number of occurrences of the pattern UDU in a Dyck path. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St000992The alternating sum of the parts of an integer partition. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. 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. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001127The sum of the squares of the parts of a partition. St001129The product of the squares of the parts of a partition. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. 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. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001243The sum of coefficients in the Schur basis of certain LLT polynomials associated with a Dyck path. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001256Number of simple reflexive modules that are 2-stable reflexive. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001274The number of indecomposable injective modules with projective dimension equal to two. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. 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. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001484The number of singletons of an integer partition. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001523The degree of symmetry of a Dyck path. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001721The degree of a binary word. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St001885The number of binary words with the same proper border set. St001910The height of the middle non-run of a Dyck path. St001966Half the global dimension of the stable Auslander algebra of a sincere Nakayama algebra (with associated Dyck path). St000102The charge of a semistandard tableau. St001960The number of descents of a permutation minus one if its first entry is not one.
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