- FindStat

Your data matches 811 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001215
(load all 29 compositions to match this statistic)

Mapping

St001215: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0]
 => 1
[1,1,0,0]
 => 0
[1,0,1,1,0,0]
 => 1
[1,1,0,0,1,0]
 => 1
[1,1,0,1,0,0]
 => 2
[1,1,1,0,0,0]
 => 0
[1,0,1,1,0,0,1,0]
 => 2
[1,0,1,1,1,0,0,0]
 => 1
[1,1,0,0,1,1,0,0]
 => 1
[1,1,0,1,1,0,0,0]
 => 2
[1,1,1,0,0,0,1,0]
 => 1
[1,1,1,0,0,1,0,0]
 => 2
[1,1,1,1,0,0,0,0]
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => 1
[1,1,1,0,0,0,1,1,0,0]
 => 1
[1,1,1,1,0,0,0,0,1,0]
 => 1
[1,1,1,1,1,0,0,0,0,0]
 => 0
[1,1,1,1,1,1,0,0,0,0,0,0]
 => 0

Description

Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. Then the statistic gives the vector space dimension of the second Ext-group between X and the regular module. For the first 196 values, the statistic also gives the number of indecomposable non-projective modules

$X$ such that

$\tau(X)$ has codominant dimension equal to one and projective dimension equal to one.

Matching statistic: St001216
(load all 13 compositions to match this statistic)

Mapping

St001216: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0]
 => 1
[1,1,0,0]
 => 0
[1,0,1,1,0,0]
 => 1
[1,1,0,0,1,0]
 => 1
[1,1,0,1,0,0]
 => 2
[1,1,1,0,0,0]
 => 0
[1,0,1,1,0,0,1,0]
 => 2
[1,0,1,1,1,0,0,0]
 => 1
[1,1,0,0,1,1,0,0]
 => 1
[1,1,0,1,1,0,0,0]
 => 2
[1,1,1,0,0,0,1,0]
 => 1
[1,1,1,0,0,1,0,0]
 => 2
[1,1,1,1,0,0,0,0]
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => 1
[1,1,1,0,0,0,1,1,0,0]
 => 1
[1,1,1,1,0,0,0,0,1,0]
 => 1
[1,1,1,1,1,0,0,0,0,0]
 => 0
[1,1,1,1,1,1,0,0,0,0,0,0]
 => 0

Description

The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module.

Matching statistic: St001225
(load all 32 compositions to match this statistic)

Mapping

St001225: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0]
 => 1
[1,1,0,0]
 => 0
[1,0,1,1,0,0]
 => 1
[1,1,0,0,1,0]
 => 1
[1,1,0,1,0,0]
 => 2
[1,1,1,0,0,0]
 => 0
[1,0,1,1,0,0,1,0]
 => 2
[1,0,1,1,1,0,0,0]
 => 1
[1,1,0,0,1,1,0,0]
 => 1
[1,1,0,1,1,0,0,0]
 => 2
[1,1,1,0,0,0,1,0]
 => 1
[1,1,1,0,0,1,0,0]
 => 2
[1,1,1,1,0,0,0,0]
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => 1
[1,1,1,0,0,0,1,1,0,0]
 => 1
[1,1,1,1,0,0,0,0,1,0]
 => 1
[1,1,1,1,1,0,0,0,0,0]
 => 0
[1,1,1,1,1,1,0,0,0,0,0,0]
 => 0

Description

The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra.

Matching statistic: St001274
(load all 13 compositions to match this statistic)

Mapping

St001274: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0]
 => 1
[1,1,0,0]
 => 0
[1,0,1,1,0,0]
 => 1
[1,1,0,0,1,0]
 => 1
[1,1,0,1,0,0]
 => 2
[1,1,1,0,0,0]
 => 0
[1,0,1,1,0,0,1,0]
 => 2
[1,0,1,1,1,0,0,0]
 => 1
[1,1,0,0,1,1,0,0]
 => 1
[1,1,0,1,1,0,0,0]
 => 2
[1,1,1,0,0,0,1,0]
 => 1
[1,1,1,0,0,1,0,0]
 => 2
[1,1,1,1,0,0,0,0]
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => 1
[1,1,1,0,0,0,1,1,0,0]
 => 1
[1,1,1,1,0,0,0,0,1,0]
 => 1
[1,1,1,1,1,0,0,0,0,0]
 => 0
[1,1,1,1,1,1,0,0,0,0,0,0]
 => 0

Description

The number of indecomposable injective modules with projective dimension equal to two.

Matching statistic: St001278
(load all 32 compositions to match this statistic)

Mapping

St001278: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0]
 => 1
[1,1,0,0]
 => 0
[1,0,1,1,0,0]
 => 1
[1,1,0,0,1,0]
 => 1
[1,1,0,1,0,0]
 => 2
[1,1,1,0,0,0]
 => 0
[1,0,1,1,0,0,1,0]
 => 2
[1,0,1,1,1,0,0,0]
 => 1
[1,1,0,0,1,1,0,0]
 => 1
[1,1,0,1,1,0,0,0]
 => 2
[1,1,1,0,0,0,1,0]
 => 1
[1,1,1,0,0,1,0,0]
 => 2
[1,1,1,1,0,0,0,0]
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => 1
[1,1,1,0,0,0,1,1,0,0]
 => 1
[1,1,1,1,0,0,0,0,1,0]
 => 1
[1,1,1,1,1,0,0,0,0,0]
 => 0
[1,1,1,1,1,1,0,0,0,0,0,0]
 => 0

Description

The number of indecomposable modules that are fixed by

$\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. The statistic is also equal to the number of non-projective torsionless indecomposable modules in the corresponding Nakayama algebra. See theorem 5.8. in the reference for a motivation.

Matching statistic: St000012
(load all 8 compositions to match this statistic)

Mapping

Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St000012: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0]
 => [1,1,0,0]
 => 1
[1,1,0,0]
 => [1,0,1,0]
 => 0
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1
[1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 2
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 0
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 1
[1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 2
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0

Description

The area of a Dyck path. This is the number of complete squares in the integer lattice which are below the path and above the x-axis. The 'half-squares' directly above the axis do not contribute to this statistic. 1. Dyck paths are bijection with '''area sequences'''

$(a_1,\ldots,a_n)$ such that

$a_1 = 0, a_{k+1} \leq a_k + 1$ . 2. The generating function

$\mathbf{D}_n(q) = \sum_{D \in \mathfrak{D}_n} q^{\operatorname{area}(D)}$ satisfy the recurrence

$\mathbf{D}_{n+1}(q) = \sum q^k \mathbf{D}_k(q) \mathbf{D}_{n-k}(q).$ 3. The area is equidistributed with [[St000005]] and [[St000006]]. Pairs of these statistics play an important role in the theory of

$q,t$ -Catalan numbers.

Matching statistic: St000018
(load all 12 compositions to match this statistic)

Mapping

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000018: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0]
 => [2,1] => 1
[1,1,0,0]
 => [1,2] => 0
[1,0,1,1,0,0]
 => [2,1,3] => 1
[1,1,0,0,1,0]
 => [1,3,2] => 1
[1,1,0,1,0,0]
 => [3,1,2] => 2
[1,1,1,0,0,0]
 => [1,2,3] => 0
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => 2
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => 1
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => 2
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => 1
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => 2
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 0
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 0
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,2,3,4,5,6] => 0

Description

The number of inversions of a permutation. This equals the minimal number of simple transpositions

$(i,i+1)$ needed to write

$\pi$ . Thus, it is also the Coxeter length of

$\pi$ .

Matching statistic: St000019
(load all 16 compositions to match this statistic)

Mapping

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000019: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0]
 => [2,1] => 1
[1,1,0,0]
 => [1,2] => 0
[1,0,1,1,0,0]
 => [2,1,3] => 1
[1,1,0,0,1,0]
 => [1,3,2] => 1
[1,1,0,1,0,0]
 => [3,1,2] => 2
[1,1,1,0,0,0]
 => [1,2,3] => 0
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => 2
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => 1
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => 2
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => 1
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => 2
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 0
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 0
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,2,3,4,5,6] => 0

Description

The cardinality of the support of a permutation. A permutation

$\sigma$ may be written as a product

$\sigma = s_{i_1}\dots s_{i_k}$ with

$k$ minimal, where

$s_i = (i,i+1)$ denotes the simple transposition swapping the entries in positions

$i$ and

$i+1$ . The set of indices

$\{i_1,\dots,i_k\}$ is the '''support''' of

$\sigma$ and independent of the chosen way to write

$\sigma$ as such a product. See [2], Definition 1 and Proposition 10. The '''connectivity set''' of

$\sigma$ of length

$n$ is the set of indices

$1 \leq i < n$ such that

$\sigma(k) < i$ for all

$k < i$ . Thus, the connectivity set is the complement of the support.

Matching statistic: St000024
(load all 59 compositions to match this statistic)

Mapping

Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0]
 => [1,1,0,0]
 => 1
[1,1,0,0]
 => [1,0,1,0]
 => 0
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1
[1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 2
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 0
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 1
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2
[1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0

Description

The number of double up and double down steps of a Dyck path. In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.

Matching statistic: St000029
(load all 16 compositions to match this statistic)

Mapping

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000029: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0]
 => [2,1] => 1
[1,1,0,0]
 => [1,2] => 0
[1,0,1,1,0,0]
 => [2,1,3] => 1
[1,1,0,0,1,0]
 => [1,3,2] => 1
[1,1,0,1,0,0]
 => [3,1,2] => 2
[1,1,1,0,0,0]
 => [1,2,3] => 0
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => 2
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => 1
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => 2
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => 1
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => 2
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 0
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 0
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,2,3,4,5,6] => 0

Description

The depth of a permutation. This is given by

$\operatorname{dp}(\sigma) = \sum_{\sigma_i>i} (\sigma_i-i) = |\{ i \leq j : \sigma_i > j\}|.$ The depth is half of the total displacement [4], Problem 5.1.1.28, or Spearman’s disarray [3]

$\sum_i |\sigma_i-i|$ . Permutations with depth at most

$1$ are called ''almost-increasing'' in [5].

The following 801 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000030The sum of the descent differences of a permutations. St000216The absolute length of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000340The number of non-final maximal constant sub-paths of length greater than one. St000374The number of exclusive right-to-left minima of a permutation. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000494The number of inversions of distance at most 3 of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000670The reversal length of a permutation. St000703The number of deficiencies of a permutation. St000795The mad of a permutation. St000809The reduced reflection length of the permutation. St000831The number of indices that are either descents or recoils. St000957The number of Bruhat lower covers of a permutation. St000984The number of boxes below precisely one peak. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. 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. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001480The number of simple summands of the module J^2/J^3. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St000443The number of long tunnels of a Dyck path. St001007Number of simple modules with projective dimension 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. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. 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. St000004The major index of a permutation. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000010The length of the partition. St000021The number of descents of a permutation. St000041The number of nestings of a perfect matching. St000052The number of valleys of a Dyck path not on the x-axis. St000053The number of valleys of the Dyck path. St000057The Shynar inversion number of a standard tableau. St000065The number of entries equal to -1 in an alternating sign matrix. St000067The inversion number of the alternating sign matrix. St000076The rank of the alternating sign matrix in the alternating sign matrix poset. St000081The number of edges of a graph. St000083The number of left oriented leafs of a binary tree except the first one. St000120The number of left tunnels of a Dyck path. St000141The maximum drop size of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000156The Denert index of a permutation. St000161The sum of the sizes of the right subtrees of a binary tree. St000168The number of internal nodes of an ordered tree. St000211The rank of the set partition. St000214The number of adjacencies of a permutation. St000218The number of occurrences of the pattern 213 in a permutation. St000220The number of occurrences of the pattern 132 in a permutation. St000223The number of nestings in the permutation. St000224The sorting index of a permutation. St000234The number of global ascents of a permutation. St000237The number of small exceedances. St000238The number of indices that are not small weak excedances. St000245The number of ascents of a permutation. St000246The number of non-inversions of a permutation. St000305The inverse major index of a permutation. St000306The bounce count of a Dyck path. St000331The number of upper interactions of a Dyck path. St000332The positive inversions of an alternating sign matrix. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000354The number of recoils of a permutation. St000355The number of occurrences of the pattern 21-3. St000356The number of occurrences of the pattern 13-2. St000358The number of occurrences of the pattern 31-2. St000359The number of occurrences of the pattern 23-1. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000441The number of successions of a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000446The disorder of a permutation. St000463The number of admissible inversions of a permutation. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000497The lcb statistic of a set partition. St000502The number of successions of a set partitions. St000539The number of odd inversions of a permutation. St000572The dimension exponent of a set partition. St000632The jump number of the poset. St000647The number of big descents of a permutation. St000653The last descent of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000692Babson and Steingrímsson's statistic of a permutation. St000728The dimension of a set partition. St000731The number of double exceedences of a permutation. St000732The number of double deficiencies of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000794The mak of a permutation. St000797The stat`` of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000833The comajor index of a permutation. St000849The number of 1/3-balanced pairs in a poset. St000874The position of the last double rise in a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000996The number of exclusive left-to-right maxima of a permutation. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001061The number of indices that are both descents and recoils of a permutation. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001192The maximal dimension of

$Ext_A^2(S,A)$ for a simple module

$S$ over the corresponding Nakayama algebra

$A$ . St001194The injective dimension of

$A/AfA$ in the corresponding Nakayama algebra

$A$ when

$Af$ is the minimal faithful projective-injective left

$A$ -module St001197The global dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001205The number of non-simple indecomposable projective-injective modules of the algebra

$eAe$ in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001298The number of repeated entries in the Lehmer code of a permutation. St001397Number of pairs of incomparable elements in a finite poset. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001428The number of B-inversions of a signed permutation. St001479The number of bridges of a graph. St001489The maximum of the number of descents and the number of inverse descents. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001512The minimum rank of a graph. St001726The number of visible inversions of a permutation. St001727The number of invisible inversions of a permutation. St001729The number of visible descents of a permutation. St001760The number of prefix or suffix reversals needed to sort a permutation. St001843The Z-index of a set partition. St001869The maximum cut size of a graph. St000011The number of touch points (or returns) of a Dyck path. St000013The height of a Dyck path. St000015The number of peaks of a Dyck path. St000025The number of initial rises of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000056The decomposition (or block) number of a permutation. St000086The number of subgraphs. St000153The number of adjacent cycles of a permutation. St000240The number of indices that are not small excedances. St000325The width of the tree associated to a permutation. St000345The number of refinements of a partition. St000444The length of the maximal rise of a Dyck path. St000470The number of runs in a permutation. St000638The number of up-down runs of a permutation. St000676The number of odd rises of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000702The number of weak deficiencies of a permutation. St000722The number of different neighbourhoods in a graph. St000740The last entry of a permutation. St000839The largest opener of a set partition. St000935The number of ordered refinements of an integer partition. St000991The number of right-to-left minima of a permutation. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. 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. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n-1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a Dyck path as follows: St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001389The number of partitions of the same length below the given integer partition. St001461The number of topologically connected components of the chord diagram of a permutation. St001530The depth of a Dyck path. St001725The harmonious chromatic number of a graph. St001733The number of weak left to right maxima of a Dyck path. St001809The index of the step at the first peak of maximal height in a Dyck path. St000058The order of a permutation. St000439The position of the first down step of a Dyck path. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path)

$[c_0,c_1,...,c_{n-1}]$ by adding

$c_0$ to

$c_{n-1}$ . St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000002The number of occurrences of the pattern 123 in a permutation. St000008The major index of the composition. St000022The number of fixed points of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000035The number of left outer peaks of a permutation. St000042The number of crossings of a perfect matching. St000051The size of the left subtree of a binary tree. St000080The rank of the poset. St000089The absolute variation of a composition. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000133The "bounce" of a permutation. St000147The largest part of an integer partition. St000157The number of descents of a standard tableau. St000159The number of distinct parts of the integer partition. St000169The cocharge of a standard tableau. St000204The number of internal nodes of a binary tree. St000209Maximum difference of elements in cycles. St000233The number of nestings of a set partition. St000288The number of ones in a binary word. St000289The decimal representation of a binary word. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000293The number of inversions of a binary word. St000304The load of a permutation. St000330The (standard) major index of a standard tableau. St000362The size of a minimal vertex cover of a graph. St000366The number of double descents of a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length

$3$ . St000377The dinv defect of an integer partition. St000378The diagonal inversion number of an integer partition. St000391The sum of the positions of the ones in a binary word. St000392The length of the longest run of ones in a binary word. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000445The number of rises of length 1 of a Dyck path. St000462The major index minus the number of excedences of a permutation. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000491The number of inversions of a set partition. St000496The rcs statistic of a set partition. St000503The maximal difference between two elements in a common block. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000534The number of 2-rises of a permutation. St000538The number of even 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. St000546The number of global descents of a permutation. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000586The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. 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. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000607The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000646The number of big ascents of a permutation. St000648The number of 2-excedences of a permutation. St000651The maximal size of a rise in a permutation. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000662The staircase size of the code of a permutation. 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. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000693The modular (standard) major index of a standard tableau. St000711The number of big exceedences of a permutation. St000730The maximal arc length of a set partition. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000769The major index of a composition regarded as a word. St000783The side length of the largest staircase partition fitting into a partition. St000787The number of flips required to make a perfect matching noncrossing. St000792The Grundy value for the game of ruler on a binary word. St000796The stat' of a permutation. St000798The makl of a permutation. St000834The number of right outer peaks of a permutation. St000836The number of descents of distance 2 of a permutation. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000864The number of circled entries of the shifted recording tableau of a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000868The aid statistic in the sense of Shareshian-Wachs. St000877The depth of the binary word interpreted as a path. St000884The number of isolated descents of a permutation. St000919The number of maximal left branches of a binary tree. St000931The number of occurrences of the pattern UUU in a Dyck path. St000947The major index east count of a Dyck path. St000956The maximal displacement of a permutation. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St000989The number of final rises of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001077The prefix exchange distance of a permutation. St001083The number of boxed occurrences of 132 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001115The number of even descents of a permutation. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001139The number of occurrences of hills of size 2 in a Dyck path. St001153The number of blocks with even minimum in a set partition. St001161The major index north count of a Dyck path. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001176The size of a partition minus its first part. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001214The aft of an integer partition. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001280The number of parts of an integer partition that are at least two. 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. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001362The normalized Knill dimension of a graph. St001372The length of a longest cyclic run of ones of a binary word. St001375The pancake length of a permutation. St001377The major index minus the number of inversions of a permutation. St001391The disjunction number of a graph. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001411The number of patterns 321 or 3412 in a permutation. St001427The number of descents of a signed permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001484The number of singletons of an integer partition. St001552The number of inversions between excedances and fixed points of a permutation. St001584The area statistic between a Dyck path and its bounce path. St001596The number of two-by-two squares inside a skew partition. St001631The number of simple modules

$S$ with

$dim Ext^1(S,A)=1$ in the incidence algebra

$A$ of the poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001665The number of pure excedances of a permutation. St001671Haglund's hag of a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001697The shifted natural comajor index of a standard Young tableau. St001712The number of natural descents of a standard Young tableau. St001718The number of non-empty open intervals in a poset. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001777The number of weak descents in an integer composition. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St001801Half the number of preimage-image pairs of different parity in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001812The biclique partition number of a graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001928The number of non-overlapping descents in a permutation. St000007The number of saliances of the permutation. St000054The first entry of the permutation. St000061The number of nodes on the left branch of a binary tree. St000062The length of the longest increasing subsequence of the permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000069The number of maximal elements of a poset. St000071The number of maximal chains in a poset. St000084The number of subtrees. St000093The cardinality of a maximal independent set of vertices of a graph. St000105The number of blocks in the set partition. St000110The number of permutations less than or equal to a permutation in left weak order. St000164The number of short pairs. St000166The depth minus 1 of an ordered tree. St000167The number of leaves of an ordered tree. 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. St000273The domination number of a graph. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St000287The number of connected components of a graph. St000297The number of leading ones in a binary word. St000299The number of nonisomorphic vertex-induced subtrees. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000326The position of the first one in a binary word after appending a 1 at the end. St000328The maximum number of child nodes in a tree. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000390The number of runs of ones in a binary word. St000451The length of the longest pattern of the form k 1 2. St000452The number of distinct eigenvalues of a graph. St000479The Ramsey number of a graph. St000485The length of the longest cycle of a permutation. St000504The cardinality of the first block of a set partition. St000507The number of ascents of a standard tableau. St000527The width of the poset. St000528The height of a poset. St000542The number of left-to-right-minima of a permutation. St000544The cop number of a graph. St000619The number of cyclic descents of a permutation. St000636The hull number of a graph. St000654The first descent of a permutation. St000675The number of centered multitunnels of a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000717The number of ordinal summands of a poset. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000738The first entry in the last row of a standard tableau. St000759The smallest missing part in an integer partition. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000808The number of up steps of the associated bargraph. St000843The decomposition number of a perfect matching. St000844The size of the largest block in the direct sum decomposition of a permutation. St000906The length of the shortest maximal chain in a poset. St000912The number of maximal antichains in a poset. St000916The packing number of a graph. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St000925The number of topologically connected components of a set partition. St000971The smallest closer of a set partition. St000982The length of the longest constant subword. St000990The first ascent of a permutation. St001050The number of terminal closers of a set partition. St001058The breadth of the ordered tree. St001062The maximal size of a block of a set partition. St001081The number of minimal length factorizations of a permutation into star transpositions. St001093The detour number of a graph. St001108The 2-dynamic chromatic number of a graph. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001201The grade of the simple module

$S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001235The global 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. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001268The size of the largest ordinal summand in the poset. St001315The dissociation 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. 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. St001342The number of vertices in the center of a graph. St001343The dimension of the reduced incidence algebra of a poset. St001346The number of parking functions that give the same permutation. St001363The Euler characteristic of a graph according to Knill. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001415The length of the longest palindromic prefix of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001471The magnitude of a Dyck path. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. 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. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001672The restrained domination number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001746The coalition number of a graph. St001779The order of promotion on the set of linear extensions of a poset. St001828The Euler characteristic of a graph. St001829The common independence number of a graph. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000094The depth of an ordered tree. St000203The number of external nodes of a binary tree. St000521The number of distinct subtrees of an ordered tree. St000643The size of the largest orbit of antichains under Panyushev complementation. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St001180Number of indecomposable injective modules with projective dimension at most 1. St001279The sum of the parts of an integer partition that are at least two. St001458The rank of the adjacency matrix of a graph. St000876The number of factors in the Catalan decomposition of a binary word. 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. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the 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. St000217The number of occurrences of the pattern 312 in a permutation. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000039The number of crossings of a permutation. St000219The number of occurrences of the pattern 231 in a permutation. St000317The cycle descent number of a permutation. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000710The number of big deficiencies of a permutation. St001152The number of pairs with even minimum in a perfect matching. St001188The number of simple modules

$S$ with grade

$\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra

$A$ corresponding to the Dyck path. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. 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). St000673The number of non-fixed points of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000095The number of triangles of a graph. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000222The number of alignments in the permutation. St000242The number of indices that are not cyclical small weak excedances. St000327The number of cover relations in a poset. St000365The number of double ascents of a permutation. St000367The number of simsun double descents of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. 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. St000799The number of occurrences of the vincular pattern |213 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. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000837The number of ascents of distance 2 of a permutation. St000840The number of closers smaller than the largest opener in a perfect matching. St000879The number of long braid edges in the graph of braid moves of a permutation. St000895The number of ones on the main diagonal of an alternating sign matrix. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001082The number of boxed occurrences of 123 in a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001234The number of indecomposable three dimensional modules with projective dimension one. St001388The number of non-attacking neighbors of a permutation. St001394The genus of a permutation. St001396Number of triples of incomparable elements in a finite poset. St001498The normalised height of a Nakayama algebra with magnitude 1. St001511The minimal number of transpositions needed to sort a permutation in either direction. St001668The number of points of the poset minus the width of the poset. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001684The reduced word complexity of a permutation. St001874Lusztig's a-function for the symmetric group. St001948The number of augmented double ascents of a permutation. St000060The greater neighbor of the maximum. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000450The number of edges minus the number of vertices plus 2 of a graph. St000746The number of pairs with odd minimum in a perfect matching. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001589The nesting number of a perfect matching. St000235The number of indices that are not cyclical small weak excedances. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001183The maximum of

$projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001769The reflection length of a signed permutation. St001861The number of Bruhat lower covers of a permutation. St001894The depth of a signed permutation. St001862The number of crossings of a signed permutation. St001864The number of excedances of a signed permutation. St001896The number of right descents of a signed permutations. St000896The number of zeros on the main diagonal of an alternating sign matrix. St000353The number of inner valleys of a permutation. St000741The Colin de Verdière graph invariant. St000871The number of very big ascents of a permutation. St001645The pebbling number of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000454The largest eigenvalue of a graph if it is integral. St000466The Gutman (or modified Schultz) index of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000891The number of distinct diagonal sums of a permutation matrix. St000506The number of standard desarrangement tableaux of shape equal to the given partition. 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

$eA$ . St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000296The length of the symmetric border of a binary word. St000393The number of strictly increasing runs in a binary word. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000627The exponent of a binary word. St000631The number of distinct palindromic decompositions of a binary word. 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. St000885The number of critical steps in the Catalan decomposition of a binary word. St000922The minimal number such that all substrings of this length are unique. St001267The length of the Lyndon factorization of the binary word. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001432The order dimension of the partition. St001437The flex of a binary word. St001571The Cartan determinant of the integer partition. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001884The number of borders of a binary word. St001892The flag excedance statistic of a signed permutation. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St000160The multiplicity of the smallest part of a partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000183The side length of the Durfee square of an integer partition. St000225Difference between largest and smallest parts in a partition. St000294The number of distinct factors of a binary word. St000295The length of the border of a binary word. St000518The number of distinct subsequences in a binary word. St000519The largest length of a factor maximising the subword complexity. St000548The number of different non-empty partial sums of an integer partition. St000897The number of different multiplicities of parts of an integer partition. St001198The number of simple modules in the algebra

$eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001206The maximal dimension of an indecomposable projective

$eAe$ -module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ . St001413Half the length of the longest even length palindromic prefix of a binary word. St001424The number of distinct squares in a binary word. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001524The degree of symmetry of a binary word. St001587Half of the largest even part of an integer partition. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001889The size of the connectivity set of a signed permutation. St001930The weak major index of a binary word. St001961The sum of the greatest common divisors of all pairs of parts. St000455The second largest eigenvalue of a graph if it is integral. St001513The number of nested exceedences of a permutation. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001811The Castelnuovo-Mumford regularity of a permutation. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St000092The number of outer peaks of a permutation. St000023The number of inner peaks of a permutation. St000136The dinv of a parking function. St000194The number of primary dinversion pairs of a labelled dyck path corresponding to a parking function. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length

$3$ . St000516The number of stretching pairs of a permutation. St000706The product of the factorials of the multiplicities of an integer partition. St000779The tier of a permutation. St000848The balance constant multiplied with the number of linear extensions of a poset. St001207The Lowey length of the algebra

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001433The flag major index of a signed permutation. St001537The number of cyclic crossings of a permutation. St001557The number of inversions of the second entry of a permutation. St001569The maximal modular displacement of a permutation. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001821The sorting index of a signed permutation. St001823The Stasinski-Voll length of a signed permutation. St001935The number of ascents in a parking function. St001946The number of descents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one. St000099The number of valleys of a permutation, including the boundary. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St001060The distinguishing index of a graph. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001462The number of factors of a standard tableaux under concatenation. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001555The order of a signed permutation. 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. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St000993The multiplicity of the largest part of an integer partition. St000929The constant term of the character polynomial of an integer partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000806The semiperimeter of the associated bargraph. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001281The normalized isoperimetric number of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St000379The number of Hamiltonian cycles in a graph. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000941The number of characters of the symmetric group whose value on the partition is even. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001820The size of the image of the pop stack sorting operator. St001720The minimal length of a chain of small intervals in a lattice. St000456The monochromatic index of a connected graph. St001128The exponens consonantiae of a partition. St000464The Schultz index of a connected graph. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001545The second Elser number of a connected graph. St000762The sum of the positions of the weak records of an integer composition. St001488The number of corners of a skew partition. St001330The hat guessing number of a graph. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000907The number of maximal antichains of minimal length in a poset. St000472The sum of the ascent bottoms of a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St000043The number of crossings plus two-nestings of a perfect matching. St000154The sum of the descent bottoms of a permutation. St000173The segment statistic of a semistandard tableau. St000174The flush statistic of a semistandard tableau. St000360The number of occurrences of the pattern 32-1. St000486The number of cycles of length at least 3 of a permutation. St000562The number of internal points of a set partition. St000565The major index of a set partition. St000624The normalized sum of the minimal distances to a greater element. St000663The number of right floats of a permutation. St000873The aix statistic of a permutation. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001118The acyclic chromatic index of a graph. St001403The number of vertical separators in a permutation. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001469The holeyness of a permutation. St001470The cyclic holeyness of a permutation. St001549The number of restricted non-inversions between exceedances. St001550The number of inversions between exceedances where the greater exceedance is linked. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001728The number of invisible descents of a permutation. St001781The interlacing number of a set partition. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001866The nesting alignments of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001903The number of fixed points of a parking function. St000091The descent variation of a composition. St000100The number of linear extensions of a poset. St000254The nesting number of a set partition. St000307The number of rowmotion orbits of a poset. St000570The Edelman-Greene number of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000942The number of critical left to right maxima of the parking functions. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001114The number of odd descents of a permutation. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001282The number of graphs with the same chromatic polynomial. 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. St001354The number of series nodes in the modular decomposition of a graph. St001525The number of symmetric hooks on the diagonal of a partition. St001561The value of the elementary symmetric function evaluated at 1. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001737The number of descents of type 2 in a permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001863The number of weak excedances of a signed permutation. St001893The flag descent of a signed permutation. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000824The sum of the number of descents and the number of recoils of a permutation. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St000264The girth of a graph, which is not a tree. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000422The energy of a graph, if it is integral. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001964The interval resolution global dimension of a poset. St000460The hook length of the last cell along the main diagonal of an integer partition. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000940The number of characters of the symmetric group whose value on the partition is zero. St001175The size of a partition minus the hook length of the base cell. St000618The number of self-evacuating tableaux of given shape. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000707The product of the factorials of the parts. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001568The smallest positive integer that does not appear twice in the partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000781The number of proper colouring schemes of a Ferrers diagram. St000928The sum of the coefficients of the character polynomial of an integer partition. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001593This is the number of standard Young tableaux of the given shifted shape. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. 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. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. 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. St000699The toughness times the least common multiple of 1,. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000284The Plancherel distribution on integer partitions. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000509The diagonal index (content) of a partition. St000782The indicator function of whether a given perfect matching is an L & P matching. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000881The number of short braid edges in the graph of braid moves of a permutation. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000477The weight of a partition according to Alladi. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. 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. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St000936The number of even values of the symmetric group character corresponding to the partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000997The even-odd crank of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. 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. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001651The Frankl number of a lattice. St001875The number of simple modules with projective dimension at most 1. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001623The number of doubly irreducible elements of a lattice. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition.