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Your data matches 734 different statistics following compositions of up to 3 maps.
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Matching statistic: St000897
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
St000897: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1 = 2 - 1
[2]
=> 1 = 2 - 1
[1,1]
=> 1 = 2 - 1
[3]
=> 1 = 2 - 1
[2,1]
=> 1 = 2 - 1
[1,1,1]
=> 1 = 2 - 1
[4]
=> 1 = 2 - 1
[3,1]
=> 1 = 2 - 1
[2,2]
=> 1 = 2 - 1
[2,1,1]
=> 2 = 3 - 1
[1,1,1,1]
=> 1 = 2 - 1
[5]
=> 1 = 2 - 1
[4,1]
=> 1 = 2 - 1
[3,2]
=> 1 = 2 - 1
[3,1,1]
=> 2 = 3 - 1
[2,2,1]
=> 2 = 3 - 1
[2,1,1,1]
=> 2 = 3 - 1
[1,1,1,1,1]
=> 1 = 2 - 1
Description
The number of different multiplicities of parts of an integer partition.
Matching statistic: St000183
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00308: Integer partitions —Bulgarian solitaire⟶ Integer partitions
St000183: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000183: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1]
=> 1 = 2 - 1
[2]
=> [1,1]
=> 1 = 2 - 1
[1,1]
=> [2]
=> 1 = 2 - 1
[3]
=> [2,1]
=> 1 = 2 - 1
[2,1]
=> [2,1]
=> 1 = 2 - 1
[1,1,1]
=> [3]
=> 1 = 2 - 1
[4]
=> [3,1]
=> 1 = 2 - 1
[3,1]
=> [2,2]
=> 2 = 3 - 1
[2,2]
=> [2,1,1]
=> 1 = 2 - 1
[2,1,1]
=> [3,1]
=> 1 = 2 - 1
[1,1,1,1]
=> [4]
=> 1 = 2 - 1
[5]
=> [4,1]
=> 1 = 2 - 1
[4,1]
=> [3,2]
=> 2 = 3 - 1
[3,2]
=> [2,2,1]
=> 2 = 3 - 1
[3,1,1]
=> [3,2]
=> 2 = 3 - 1
[2,2,1]
=> [3,1,1]
=> 1 = 2 - 1
[2,1,1,1]
=> [4,1]
=> 1 = 2 - 1
[1,1,1,1,1]
=> [5]
=> 1 = 2 - 1
Description
The side length of the Durfee square of an integer partition.
Given a partition $\lambda = (\lambda_1,\ldots,\lambda_n)$, the Durfee square is the largest partition $(s^s)$ whose diagram fits inside the diagram of $\lambda$. In symbols, $s = \max\{ i \mid \lambda_i \geq i \}$.
This is also known as the Frobenius rank.
Matching statistic: St000697
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000697: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000697: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> []
=> 0 = 2 - 2
[2]
=> []
=> 0 = 2 - 2
[1,1]
=> [1]
=> 0 = 2 - 2
[3]
=> []
=> 0 = 2 - 2
[2,1]
=> [1]
=> 0 = 2 - 2
[1,1,1]
=> [1,1]
=> 0 = 2 - 2
[4]
=> []
=> 0 = 2 - 2
[3,1]
=> [1]
=> 0 = 2 - 2
[2,2]
=> [2]
=> 0 = 2 - 2
[2,1,1]
=> [1,1]
=> 0 = 2 - 2
[1,1,1,1]
=> [1,1,1]
=> 1 = 3 - 2
[5]
=> []
=> 0 = 2 - 2
[4,1]
=> [1]
=> 0 = 2 - 2
[3,2]
=> [2]
=> 0 = 2 - 2
[3,1,1]
=> [1,1]
=> 0 = 2 - 2
[2,2,1]
=> [2,1]
=> 1 = 3 - 2
[2,1,1,1]
=> [1,1,1]
=> 1 = 3 - 2
[1,1,1,1,1]
=> [1,1,1,1]
=> 1 = 3 - 2
Description
The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core.
For any positive integer $k$, one associates a $k$-core to a partition by repeatedly removing all rim hooks of size $k$.
This statistic counts the $3$-rim hooks that are removed in this process to obtain a $3$-core.
Matching statistic: St001175
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00308: Integer partitions —Bulgarian solitaire⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001175: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1]
=> 0 = 2 - 2
[2]
=> [1,1]
=> 0 = 2 - 2
[1,1]
=> [2]
=> 0 = 2 - 2
[3]
=> [2,1]
=> 0 = 2 - 2
[2,1]
=> [2,1]
=> 0 = 2 - 2
[1,1,1]
=> [3]
=> 0 = 2 - 2
[4]
=> [3,1]
=> 0 = 2 - 2
[3,1]
=> [2,2]
=> 1 = 3 - 2
[2,2]
=> [2,1,1]
=> 0 = 2 - 2
[2,1,1]
=> [3,1]
=> 0 = 2 - 2
[1,1,1,1]
=> [4]
=> 0 = 2 - 2
[5]
=> [4,1]
=> 0 = 2 - 2
[4,1]
=> [3,2]
=> 1 = 3 - 2
[3,2]
=> [2,2,1]
=> 1 = 3 - 2
[3,1,1]
=> [3,2]
=> 1 = 3 - 2
[2,2,1]
=> [3,1,1]
=> 0 = 2 - 2
[2,1,1,1]
=> [4,1]
=> 0 = 2 - 2
[1,1,1,1,1]
=> [5]
=> 0 = 2 - 2
Description
The size of a partition minus the hook length of the base cell.
This is, the number of boxes in the diagram of a partition that are neither in the first row nor in the first column.
Matching statistic: St000903
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St000903: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
St000903: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 10 => [1,2] => 2
[2]
=> 100 => [1,3] => 2
[1,1]
=> 110 => [1,1,2] => 2
[3]
=> 1000 => [1,4] => 2
[2,1]
=> 1010 => [1,2,2] => 2
[1,1,1]
=> 1110 => [1,1,1,2] => 2
[4]
=> 10000 => [1,5] => 2
[3,1]
=> 10010 => [1,3,2] => 3
[2,2]
=> 1100 => [1,1,3] => 2
[2,1,1]
=> 10110 => [1,2,1,2] => 2
[1,1,1,1]
=> 11110 => [1,1,1,1,2] => 2
[5]
=> 100000 => [1,6] => 2
[4,1]
=> 100010 => [1,4,2] => 3
[3,2]
=> 10100 => [1,2,3] => 3
[3,1,1]
=> 100110 => [1,3,1,2] => 3
[2,2,1]
=> 11010 => [1,1,2,2] => 2
[2,1,1,1]
=> 101110 => [1,2,1,1,2] => 2
[1,1,1,1,1]
=> 111110 => [1,1,1,1,1,2] => 2
Description
The number of different parts of an integer composition.
Matching statistic: St000124
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000124: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000124: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1] => 1 = 2 - 1
[2]
=> [1,0,1,0]
=> [2,1] => 1 = 2 - 1
[1,1]
=> [1,1,0,0]
=> [1,2] => 1 = 2 - 1
[3]
=> [1,0,1,0,1,0]
=> [2,1,3] => 1 = 2 - 1
[2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 1 = 2 - 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,3,2] => 1 = 2 - 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 1 = 2 - 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 2 = 3 - 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 1 = 2 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 1 = 2 - 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => 1 = 2 - 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => 1 = 2 - 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => 2 = 3 - 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1 = 2 - 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => 2 = 3 - 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 2 = 3 - 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => 1 = 2 - 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => 1 = 2 - 1
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: St000243
(load all 63 compositions to match this statistic)
(load all 63 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000243: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000243: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,2] => 1 = 2 - 1
[2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1 = 2 - 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1 = 2 - 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1 = 2 - 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1 = 2 - 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1 = 2 - 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1 = 2 - 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2 = 3 - 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1 = 2 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1 = 2 - 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 1 = 2 - 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => 1 = 2 - 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => 2 = 3 - 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1 = 2 - 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2 = 3 - 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1 = 2 - 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => 2 = 3 - 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => 1 = 2 - 1
Description
The number of cyclic valleys and cyclic peaks of a permutation.
This is given by the number of indices $i$ such that $\pi_{i-1} > \pi_i < \pi_{i+1}$ with indices considered cyclically. Equivalently, this is the number of indices $i$ such that $\pi_{i-1} < \pi_i > \pi_{i+1}$ with indices considered cyclically.
Matching statistic: St000291
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00261: Binary words —Burrows-Wheeler⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00261: Binary words —Burrows-Wheeler⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 10 => 10 => 1 = 2 - 1
[2]
=> 100 => 100 => 1 = 2 - 1
[1,1]
=> 110 => 110 => 1 = 2 - 1
[3]
=> 1000 => 1000 => 1 = 2 - 1
[2,1]
=> 1010 => 1100 => 1 = 2 - 1
[1,1,1]
=> 1110 => 1110 => 1 = 2 - 1
[4]
=> 10000 => 10000 => 1 = 2 - 1
[3,1]
=> 10010 => 11000 => 1 = 2 - 1
[2,2]
=> 1100 => 1010 => 2 = 3 - 1
[2,1,1]
=> 10110 => 11100 => 1 = 2 - 1
[1,1,1,1]
=> 11110 => 11110 => 1 = 2 - 1
[5]
=> 100000 => 100000 => 1 = 2 - 1
[4,1]
=> 100010 => 101000 => 2 = 3 - 1
[3,2]
=> 10100 => 11000 => 1 = 2 - 1
[3,1,1]
=> 100110 => 110010 => 2 = 3 - 1
[2,2,1]
=> 11010 => 11100 => 1 = 2 - 1
[2,1,1,1]
=> 101110 => 111010 => 2 = 3 - 1
[1,1,1,1,1]
=> 111110 => 111110 => 1 = 2 - 1
Description
The number of descents of a binary word.
Matching statistic: St000321
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00308: Integer partitions —Bulgarian solitaire⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000321: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000321: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1]
=> []
=> 1 = 2 - 1
[2]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,1]
=> [2]
=> []
=> 1 = 2 - 1
[3]
=> [2,1]
=> [1]
=> 1 = 2 - 1
[2,1]
=> [2,1]
=> [1]
=> 1 = 2 - 1
[1,1,1]
=> [3]
=> []
=> 1 = 2 - 1
[4]
=> [3,1]
=> [1]
=> 1 = 2 - 1
[3,1]
=> [2,2]
=> [2]
=> 2 = 3 - 1
[2,2]
=> [2,1,1]
=> [1,1]
=> 1 = 2 - 1
[2,1,1]
=> [3,1]
=> [1]
=> 1 = 2 - 1
[1,1,1,1]
=> [4]
=> []
=> 1 = 2 - 1
[5]
=> [4,1]
=> [1]
=> 1 = 2 - 1
[4,1]
=> [3,2]
=> [2]
=> 2 = 3 - 1
[3,2]
=> [2,2,1]
=> [2,1]
=> 2 = 3 - 1
[3,1,1]
=> [3,2]
=> [2]
=> 2 = 3 - 1
[2,2,1]
=> [3,1,1]
=> [1,1]
=> 1 = 2 - 1
[2,1,1,1]
=> [4,1]
=> [1]
=> 1 = 2 - 1
[1,1,1,1,1]
=> [5]
=> []
=> 1 = 2 - 1
Description
The number of integer partitions of n that are dominated by an integer partition.
A partition $\lambda = (\lambda_1,\ldots,\lambda_n) \vdash n$ dominates a partition $\mu = (\mu_1,\ldots,\mu_n) \vdash n$ if $\sum_{i=1}^k (\lambda_i - \mu_i) \geq 0$ for all $k$.
Matching statistic: St000333
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000333: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000333: 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 = 2 - 1
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 1 = 2 - 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1 = 2 - 1
[2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 1 = 2 - 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1 = 2 - 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1 = 2 - 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 1 = 2 - 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 1 = 2 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 2 = 3 - 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1 = 2 - 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 1 = 2 - 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 1 = 2 - 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 2 = 3 - 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 1 = 2 - 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 2 = 3 - 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 2 = 3 - 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 1 = 2 - 1
Description
The dez statistic, the number of descents of a permutation after replacing fixed points by zeros.
This descent set is denoted by $\operatorname{ZDer}(\sigma)$ in [1].
The following 724 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000345The number of refinements of a partition. St000390The number of runs of ones in a binary word. St000659The number of rises of length at least 2 of a Dyck path. St000761The number of ascents in an integer composition. St000834The number of right outer peaks of a permutation. St000905The number of different multiplicities of parts of an integer composition. St000920The logarithmic height of a Dyck path. St000935The number of ordered refinements of an integer partition. 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. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. 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 $K[x]/(x^n)$. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001597The Frobenius rank of a skew partition. St001673The degree of asymmetry of an integer composition. St001722The number of minimal chains with small intervals between a binary word and the top element. St001732The number of peaks visible from the left. St000017The number of inversions of a standard tableau. St000057The Shynar inversion number of a standard tableau. St000142The number of even parts of a partition. St000217The number of occurrences of the pattern 312 in a permutation. St000292The number of ascents of a binary word. St000317The cycle descent number of a permutation. St000358The number of occurrences of the pattern 31-2. St000386The number of factors DDU in a Dyck path. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000480The number of lower covers of a partition in dominance order. St000648The number of 2-excedences of a permutation. St000660The number of rises of length at least 3 of a Dyck path. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000871The number of very big ascents of a permutation. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001083The number of boxed occurrences of 132 in a permutation. St001092The number of distinct even parts of a partition. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001394The genus of a permutation. St001470The cyclic holeyness of a permutation. St001596The number of two-by-two squares inside a skew partition. St001712The number of natural descents of a standard Young tableau. St001727The number of invisible inversions of a permutation. St000015The number of peaks of a Dyck path. St000159The number of distinct parts of the integer partition. St000201The number of leaf nodes in a binary tree. St000258The burning number of a graph. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000783The side length of the largest staircase partition fitting into a partition. St000824The sum of the number of descents and the number of recoils of a permutation. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. 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 $eA$. 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:
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 $eA$. St001432The order dimension of the partition. St001471The magnitude of a Dyck path. St001486The number of corners of the ribbon associated with an integer composition. St001530The depth of a Dyck path. St000001The number of reduced words for a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000035The number of left outer peaks of a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000053The number of valleys of the Dyck path. St000078The number of alternating sign matrices whose left key is the permutation. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000155The number of exceedances (also excedences) of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000255The number of reduced Kogan faces with the permutation as type. St000277The number of ribbon shaped standard tableaux. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000306The bounce count of a Dyck path. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000325The width of the tree associated to a permutation. 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. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000346The number of coarsenings of a partition. St000354The number of recoils of a permutation. St000396The register function (or Horton-Strahler number) of a binary tree. St000451The length of the longest pattern of the form k 1 2. St000470The number of runs in a permutation. St000547The number of even non-empty partial sums of an integer partition. St000570The Edelman-Greene number of a permutation. St000617The number of global maxima of a Dyck path. St000628The balance of a binary word. St000662The staircase size of the code of a permutation. St000679The pruning number of an ordered tree. St000703The number of deficiencies of a permutation. 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. St000764The number of strong records in an integer composition. St000765The number of weak records in an integer composition. St000767The number of runs in an integer composition. St000781The number of proper colouring schemes of a Ferrers diagram. St000785The number of distinct colouring schemes of a graph. St000805The number of peaks of the associated bargraph. 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. St000816The number of standard composition tableaux of the composition. St000820The number of compositions obtained by rotating the composition. St000829The Ulam distance of a permutation to the identity permutation. St000862The number of parts of the shifted shape of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000883The number of longest increasing subsequences of a permutation. St000886The number of permutations with the same antidiagonal sums. St000889The number of alternating sign matrices with the same antidiagonal sums. St000919The number of maximal left branches of a binary tree. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000999Number of indecomposable projective module with injective dimension equal to the global dimension 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. St001028Number of simple modules with injective dimension equal to the dominant dimension 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. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001142The projective dimension of the socle of the regular module as a bimodule 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. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001182Number of indecomposable injective modules with codominant dimension at least two 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 St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001199The dominant 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 $eA$. 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. St001220The width of a permutation. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. 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. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001340The cardinality of a minimal non-edge isolating set of a graph. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001487The number of inner corners of a skew partition. St001489The maximum of the number of descents and the number of inverse descents. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. 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. St001591The number of graphs with the given composition of multiplicities of Laplacian eigenvalues. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001665The number of pure excedances of a permutation. St001729The number of visible descents of a permutation. St001735The number of permutations with the same set of runs. St001737The number of descents of type 2 in a permutation. St001741The largest integer such that all patterns of this size are contained in the permutation. St001778The largest greatest common divisor of an element and its image in a permutation. 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. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St001874Lusztig's a-function for the symmetric group. St001884The number of borders of a binary word. St001928The number of non-overlapping descents in a permutation. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000002The number of occurrences of the pattern 123 in a permutation. St000021The number of descents of a permutation. St000023The number of inner peaks of a permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000039The number of crossings of a permutation. St000052The number of valleys of a Dyck path not on the x-axis. St000089The absolute variation of a composition. St000091The descent variation of a composition. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. 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. St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000131The number of occurrences of the contiguous pattern [.,[[[[.,.],.],.],. St000141The maximum drop size of a permutation. St000149The number of cells of the partition whose leg is zero and arm is odd. St000150The floored half-sum of the multiplicities of a partition. St000185The weighted size of a partition. St000204The number of internal nodes of a binary tree. St000218The number of occurrences of the pattern 213 in a permutation. St000220The number of occurrences of the pattern 132 in a permutation. St000222The number of alignments in the permutation. St000223The number of nestings in the permutation. St000233The number of nestings of a set partition. St000238The number of indices that are not small weak excedances. St000242The number of indices that are not cyclical small weak excedances. St000252The number of nodes of degree 3 of a binary tree. St000256The number of parts from which one can substract 2 and still get an integer partition. St000257The number of distinct parts of a partition that occur at least twice. St000316The number of non-left-to-right-maxima of a permutation. St000353The number of inner valleys of a permutation. St000355The number of occurrences of the pattern 21-3. St000356The number of occurrences of the pattern 13-2. St000357The number of occurrences of the pattern 12-3. St000360The number of occurrences of the pattern 32-1. St000365The number of double ascents of a permutation. St000366The number of double descents of a permutation. St000367The number of simsun double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000374The number of exclusive right-to-left minima of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000376The bounce deficit of a Dyck path. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000407The number of occurrences of the pattern 2143 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. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000481The number of upper covers of a partition in dominance order. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000486The number of cycles of length at least 3 of a permutation. St000516The number of stretching pairs of a permutation. St000534The number of 2-rises of a permutation. St000538The number of even inversions of a permutation. St000647The number of big descents of a permutation. St000649The number of 3-excedences of a permutation. St000650The number of 3-rises of a permutation. St000661The number of rises of length 3 of a Dyck path. St000664The number of right ropes of a permutation. St000665The number of rafts of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000711The number of big exceedences of a permutation. St000731The number of double exceedences of a permutation. St000732The number of double deficiencies of a permutation. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000766The number of inversions of an integer composition. St000768The number of peaks in an integer composition. St000769The major index of a composition regarded as a word. St000779The tier of a permutation. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000807The sum of the heights of the valleys of the associated bargraph. St000836The number of descents of distance 2 of a permutation. St000872The number of very big descents of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000879The number of long braid edges in the graph of braid moves of a permutation. St000884The number of isolated descents of a permutation. St000962The 3-shifted major index of a permutation. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001061The number of indices that are both descents and recoils of a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 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. St001091The number of parts in an integer partition whose next smaller part has the same size. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001114The number of odd descents of a permutation. St001115The number of even descents of a permutation. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001214The aft of an integer partition. 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. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. 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. St001234The number of indecomposable three dimensional modules with projective dimension one. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001411The number of patterns 321 or 3412 in a permutation. 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. St001469The holeyness of a permutation. St001513The number of nested exceedences of a permutation. St001535The number of cyclic alignments of a permutation. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001584The area statistic between a Dyck path and its bounce path. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001705The number of occurrences of the pattern 2413 in a permutation. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. 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. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). 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. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001856The number of edges in the reduced word graph of a permutation. St001871The number of triconnected components of a graph. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001931The weak major index of an integer composition regarded as a word. St001932The number of pairs of singleton blocks in the noncrossing set partition corresponding to a Dyck path, that can be merged to create another noncrossing set partition. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000881The number of short braid edges in the graph of braid moves of a permutation. St000931The number of occurrences of the pattern UUU in a Dyck path. St001520The number of strict 3-descents. St000842The breadth of a permutation. St000568The hook number of a binary tree. St000619The number of cyclic descents of a permutation. St000624The normalized sum of the minimal distances to a greater element. St000627The exponent of a binary word. St000652The maximal difference between successive positions of a permutation. St000847The number of standard Young tableaux whose descent set is the binary word. St000988The orbit size of a permutation under Foata's bijection. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001052The length of the exterior of a permutation. St001162The minimum jump of a permutation. St001246The maximal difference between two consecutive entries of a permutation. St001344The neighbouring number of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000369The dinv deficit of a Dyck path. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000497The lcb statistic of a set partition. St000539The number of odd inversions of a permutation. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000562The number of internal points of a set partition. St000572The dimension exponent of a set partition. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000586The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal. St000595The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000598The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, 3 is maximal, (2,3) 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. 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. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000612The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) are consecutive in a block. St000614The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000646The number of big ascents of a permutation. St000710The number of big deficiencies of a permutation. St000747A variant of the major index of a set partition. St000751The number of occurrences of either of the pattern 2143 or 2143 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. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000963The 2-shifted major index of a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001139The number of occurrences of hills of size 2 in a Dyck path. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001552The number of inversions between excedances and fixed points of a permutation. St001556The number of inversions of the third entry of a permutation. St001731The factorization defect of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001866The nesting alignments of a signed permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001964The interval resolution global dimension of a poset. St000542The number of left-to-right-minima of a permutation. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001390The number of bumps occurring when Schensted-inserting the letter 1 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. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St000007The number of saliances of the permutation. St000061The number of nodes on the left branch of a binary tree. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000991The number of right-to-left minima of a permutation. 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 $eA$. St001201The grade of the simple module $S_0$ in the special CNakayama 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. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001734The lettericity of a graph. St000352The Elizalde-Pak rank of a permutation. St000654The first descent of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. 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 $c_{n−1}$. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. 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. St001035The convexity degree of the parallelogram polyomino associated with the 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. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001569The maximal modular displacement of a permutation. St001668The number of points of the poset minus the width of the poset. St000237The number of small exceedances. St000406The number of occurrences of the pattern 3241 in a permutation. St000461The rix statistic of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000750The number of occurrences of the pattern 4213 in a permutation. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001301The first Betti number of the order complex associated with the poset. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001557The number of inversions of the second entry of a permutation. St000982The length of the longest constant subword. St000219The number of occurrences of the pattern 231 in a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001095The number of non-isomorphic posets with precisely one further covering relation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001846The number of elements which do not have a complement in the lattice. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St001490The number of connected components of a skew partition. St000297The number of leading ones in a binary word. St000629The defect of a binary word. St000753The Grundy value for the game of Kayles on a binary word. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St001039The maximal height of a column in the parallelogram polyomino associated with a 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. St000013The height of a Dyck path. St000026The position of the first return of a Dyck path. St000054The first entry of the permutation. St000064The number of one-box pattern of a permutation. St000111The sum of the descent tops (or Genocchi descents) of a permutation. St000144The pyramid weight of the Dyck path. 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. St000203The number of external nodes of a binary tree. St000213The number of weak exceedances (also weak excedences) of a permutation. St000326The position of the first one in a binary word after appending a 1 at the end. St000439The position of the first down step of a Dyck path. St000443The number of long tunnels of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000630The length of the shortest palindromic decomposition of a binary word. St000638The number of up-down runs of a permutation. St000676The number of odd rises of a Dyck path. St000702The number of weak deficiencies of a permutation. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000740The last entry of a permutation. St000746The number of pairs with odd minimum in a perfect matching. St000839The largest opener of a set partition. St000876The number of factors in the Catalan decomposition of a binary word. St000878The number of ones minus the number of zeros of a binary word. St000893The number of distinct diagonal sums of an alternating sign matrix. St000922The minimal number such that all substrings of this length are unique. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001049The smallest label in the subtree not containing 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001180Number of indecomposable injective modules with projective dimension at most 1. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. 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. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. 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. St001461The number of topologically connected components of the chord diagram of a permutation. St001497The position of the largest weak excedence of a permutation. 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. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001256Number of simple reflexive modules that are 2-stable reflexive. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001896The number of right descents of a signed permutations. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001396Number of triples of incomparable elements in a finite poset. St001578The minimal number of edges to add or remove to make a graph a line graph. St001948The number of augmented double ascents of a permutation. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001371The length of the longest Yamanouchi prefix of a binary word. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000392The length of the longest run of ones in a binary word. St000877The depth of the binary word interpreted as a path. St000885The number of critical steps in the Catalan decomposition of a binary word. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001462The number of factors of a standard tableaux under concatenation. St001720The minimal length of a chain of small intervals in a lattice. St000056The decomposition (or block) number of a permutation. St000260The radius of a connected graph. St000295The length of the border of a binary word. St000296The length of the symmetric border of a binary word. St000456The monochromatic index of a connected graph. St000657The smallest part of an integer composition. St000694The number of affine bounded permutations that project to a given permutation. St000788The number of nesting-similar perfect matchings of a perfect matching. St000891The number of distinct diagonal sums of a permutation matrix. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001041The depth of the label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001260The permanent of an alternating sign matrix. St001267The length of the Lyndon factorization of the binary word. St001413Half the length of the longest even length palindromic prefix of a binary word. St001437The flex of a binary word. St001481The minimal height of a peak of a Dyck path. St001566The length of the longest arithmetic progression in a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001590The crossing number of a perfect matching. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001768The number of reduced words of a signed permutation. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001889The size of the connectivity set of a signed permutation. St001946The number of descents in a parking function. St000022The number of fixed points of a permutation. St000153The number of adjacent cycles of a permutation. 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. St000221The number of strong fixed points of a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000405The number of occurrences of the pattern 1324 in a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000666The number of right tethers of a permutation. St000787The number of flips required to make a perfect matching noncrossing. St000894The trace of an alternating sign matrix. St000943The number of spots the most unlucky car had to go further in a parking function. St001131The number of trivial trees on the path to label one in the decreasing labelled binary unordered tree associated with the perfect matching. 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. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001381The fertility of a permutation. St001429The number of negative entries in a signed permutation. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001524The degree of symmetry of a binary word. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001845The number of join irreducibles minus the rank of a lattice. St001850The number of Hecke atoms of a permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St000417The size of the automorphism group of the ordered tree. St001058The breadth of the ordered tree. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St000527The width of the poset. St001616The number of neutral elements in a lattice. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St000455The second largest eigenvalue of a graph if it is integral. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St001516The number of cyclic bonds of a permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000068The number of minimal elements in a poset. 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. St000058The order of a permutation. St000084The number of subtrees. St000105The number of blocks in the set partition. St000328The maximum number of child nodes in a tree. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000504The cardinality of the first block of a set partition. St000759The smallest missing part in an integer partition. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000823The number of unsplittable factors of the set partition. St000953The largest degree of an irreducible factor of the Coxeter polynomial of the Dyck path over the rational numbers. St000983The length of the longest alternating subword. 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. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001060The distinguishing index of a graph. St001062The maximal size of a block of a set partition. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001075The minimal size of a block of a set partition. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St001500The global dimension of magnitude 1 Nakayama algebras. St001733The number of weak left to right maxima of a Dyck path. St001814The number of partitions interlacing the given partition. St000154The sum of the descent bottoms of a permutation. St000210Minimum over maximum difference of elements in cycles. St000253The crossing number of a set partition. St000729The minimal arc length of a set partition. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001806The upper middle entry of a permutation. St000234The number of global ascents of a permutation. St000247The number of singleton blocks of a set partition. St000462The major index minus the number of excedences of a permutation. St000496The rcs statistic of a set partition. 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. St000563The number of overlapping pairs of blocks of a set partition. St000573The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton 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. 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. 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. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000989The number of final rises of a permutation. St001130The number of two successive successions in a permutation. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001402The number of separators in a permutation. St001403The number of vertical separators in a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001781The interlacing number of a set partition. St001847The number of occurrences of the pattern 1432 in a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001903The number of fixed points of a parking function. St001624The breadth of a lattice. St000741The Colin de Verdière graph invariant. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000546The number of global descents of a permutation. St001875The number of simple modules with projective dimension at most 1. St000782The indicator function of whether a given perfect matching is an L & P matching. St000251The number of nonsingleton blocks of a set partition. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000574The number of occurrences of the pattern {{1},{2}} such that 1 is a minimal and 2 a maximal element. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001545The second Elser number of a connected graph. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St000495The number of inversions of distance at most 2 of a permutation. St000633The size of the automorphism group of a poset. St000831The number of indices that are either descents or recoils. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000925The number of topologically connected components of a set partition. St000990The first ascent of a permutation. St001399The distinguishing number of a poset. St000338The number of pixed points of a permutation. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000500Eigenvalues of the random-to-random operator acting on the regular representation. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. 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. St000850The number of 1/2-balanced pairs in a poset. St000961The shifted major index of a permutation. St001851The number of Hecke atoms of a signed permutation. St001472The permanent of the Coxeter matrix of the poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000640The rank of the largest boolean interval in 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. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000524The number of posets with the same order polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000632The jump number of the poset. St000717The number of ordinal summands of a poset. 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. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St000264The girth of a graph, which is not a tree. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000477The weight of a partition according to Alladi. St000668The least common multiple of the parts of the partition. St000675The number of centered multitunnels of a Dyck path. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000762The sum of the positions of the weak records of an integer composition. St000770The major index of an integer partition when read from bottom to top. St000806The semiperimeter of the associated bargraph. St000815The number of semistandard Young tableaux of partition weight of given shape. 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. St001118The acyclic chromatic index of a graph. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000102The charge of a semistandard tableau.
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