# 1. Statistic finder

Imagine you have some data for a statistic $f:$ Dyck paths $= \mathcal{D} \longrightarrow \mathbb{Z}$:

\begin{align} [1, 0, 1, 0, 1, 0, 1, 0] &\mapsto 16 \\ [1, 0, 1, 0, 1, 1, 0, 0] &\mapsto 5 \\ [1, 0, 1, 1, 0, 0, 1, 0] &\mapsto 6 \\ [1, 0, 1, 1, 0, 1, 0, 0] &\mapsto 3 \\ [1, 0, 1, 1, 1, 0, 0, 0] &\mapsto 1 \end{align}

We aim to find a statistic $st$ on a collection $\mathcal{S}$ and a map $\phi : \mathcal{D} \longrightarrow \mathcal{S}$ such that for all $D$ in our sample data $$st(\ \phi(D)\ ) = f(D).$$

We go to StatisticFinder/DyckPaths and input the data by assigning to the Dyck path $[1,0,1,0,1,0,1,0]$ the value $16$ etc. We finally hit search for statistic and obtain

   1 After applying to 132 avoiding permutation
2 your input data matches the following statistic in the shown values:
3
4 Identifier St000001
5
6 Values
7 [1,0,1,1,0,0,1,0] => [4,2,3,1] => 6
8 [1,0,1,1,0,1,0,0] => [3,2,4,1] => 3
9 [1,0,1,0,1,0,1,0] => [4,3,2,1] => 16
10 [1,0,1,1,1,0,0,0] => [2,3,4,1] => 1
11 [1,0,1,0,1,1,0,0] => [3,4,2,1] => 5
12
13 Description
14 This is the number of ways to write a permutation as a product of simple transpositions.


   1 After applying to partition
2 your input data matches the following statistic in the shown values:
3
4 Identifier St000003
5
6 Values
7 [1,0,1,0,1,0,1,0] => [3,2,1] => 16
8 [1,0,1,1,1,0,0,0] => [1,1,1] => 1
9 [1,0,1,0,1,1,0,0] => [2,2,1] => 5
10 [1,0,1,1,0,1,0,0] => [2,1,1] => 3
11 [1,0,1,1,0,0,1,0] => [3,1,1] => 6
12
13 Description
14 This is the number of standard Young tableaux of the partition.


These entries were originally inputted as statistics for permutations and partitions, but combinatorial maps from Dyck paths to $132$-avoiding permutations (Mp00025) and to integer partitions (Mp00027) connect these values to known statistics in the database.

The combinatorial collections considered in this project come in finite levels (e.g., the length of a permutation).

The generating function of a statistic $st : \mathcal{S} \longrightarrow \mathbb{Z}$ is the family of (Laurent) polynomials given by $$f_x(q) = \sum_{elt \in \mathcal{S}_x} q^{st(elt)}$$ where $\mathcal{S}_x$ is one of the levels of $\mathcal{S}$.

We would also like to find a statistic $st$ on some set of objects $\mathcal{S}$ and a map $\phi : \mathcal{D} \longrightarrow \mathcal{S}$ having the same generating function.

### 1.1.1. Distribution search by levels

The first and simpler option is to provide all values of a given level set, and to perform a distribution search such as

$$[1,2] \mapsto 0 \quad [2,1] \mapsto 1$$

\begin{align*} [1,2,3] \mapsto 0 \quad [1,3,2] \mapsto 1 \quad [2,1,3] \mapsto 1 \\ [2,3,1] \mapsto 2 \quad [3,1,2] \mapsto 2 \quad [3,2,1] \mapsto 3 \end{align*}

In this case, you will get (among other matching distributions) the following output:

After applying to partition your input data matches the following statistic in the shown values:

   1 Identifier St000004
2
3 Values
4 [1,2] => 0
5 [2,1] => 1
6 [1,2,3] => 0
7 [1,3,2] => 2
8 [2,1,3] => 1
9 [2,3,1] => 2
10 [3,1,2] => 1
11 [3,2,1] => 3
12
13 Description
14 The major index of a permutation.


   1 Identifier St000018
2
3 Values
4 [1,2] => 0
5 [2,1] => 1
6 [1,2,3] => 0
7 [1,3,2] => 1
8 [2,1,3] => 1
9 [2,3,1] => 2
10 [3,1,2] => 2
11 [3,2,1] => 3
12
13 Description
14 The number of inversions of a permutation.


   1 Identifier St000156
2
3 Values
4 [1,2] => 0
5 [2,1] => 1
6 [1,2,3] => 0
7 [1,3,2] => 2
8 [2,1,3] => 1
9 [2,3,1] => 3
10 [3,1,2] => 1
11 [3,2,1] => 2
12
13 Description
14 The Denert index of a permutation.


### 1.1.2. Other distribution searches

Beside this search for statistics with the same distribution on the levels of the sets, you can also do more sophisticated searches by providing values all-at-once. Entering

   1 [1,2]
2 [2,1]
3 ====> 0,1
4 [1,2,3]
5 [1,3,2]
6 [2,1,3]
7 [2,3,1]
8 [3,1,2]
9 [3,2,1]
10 ====> 0,1,1,2,2,3


yields the same search as above. But it is possible to further refine this input.

   1 [1,2] => 0
2 [2,1] => 1
3 [1,2,3] => 0
4 [2,1,3] => 1
5 [2,3,1] => 2
6 [1,3,2]
7 [3,1,2]
8 ====> 1,2
9 [3,2,1] => 3


Now, all values are attached to one permutation, except that the two permutations $[1,3,2]$ and $[3,1,2]$ get the two values $1$ and $2$ together, without specifying which gets which. This results in only finding the major index and the number of inversions, but not the Denert index.

# 2. Map finder

The map finder works analogously to the statistic finder.

Feel free to try to search for your favorite map from Dyck paths to permutations.

# 3. Contributing to FindStat

Everyone with knowledge on combinatorial statistics can contribute by adding a new statistic or editing an existing statistic. Any statistic which is relevant in someone's research is worth being contained in the database. You do not necessarily need to create an account in order to contribute, but we recommend doing so.

## 3.1. Contributing statistics

Every contributed database entry must contain

1. A combinatorial collection. Examples are Permutations and IntegerPartitions.

2. Data for the statistic. We require a statistic to be defined and provided for at least 10 and at most 1200 elements (if possible, you should provide data for 1000 elements). Every entry contains one line and is of the form object_name => value. Here, object_name must be provided in the same way it is used in the StatisticsDatabase.

3. A detailed description of the statistic.
4. Sage or (pseudo-)code for computing the statistic (if applicable).

5. References to literature containing the statistic (if applicable).
6. The author's name.

Every new statistic must first be "verified" by an editor. This verification is not meant to be a mathematical verification, but only a quick check that the provided statistic seems to be meaningful.

## 3.2. Contributing maps

Sorry, this is not yet possible - but it will be soon.

## 3.3. Creating and editing wiki pages

You must have signed up for an account in order to be allowed to create a new wiki page.

To create a new wiki page, simply type the page name into the URL. If possible you should use one of the given templates.

### 3.3.1. References

The macro for references can be used in the following 4 ways:

• Reference(identifier,author,title,publishing information) simply defines a reference
• Example: <<Reference("Nob12","A. Nobody","A proof of the Riemann hypothesis","Some journal '''1''' (2012)")>> renders as [Nob12].

• Reference(identifier,author,title,publishing information) defines a reference with some additional infos
• Example: <<Reference("Yes12","A. Yesbody","No proof of the Riemann hypothesis","Some journal '''2''' (2012)","Theorem 1")>> renders as [Yes12, Theorem 1].

• Reference(identifier) produces another instance of a reference
• Example: <<Reference("Yes12")>> renders as [Yes12].

• Reference(identifier) produces another instance of a reference plus some additional infos
• Example: <<Reference("Yes12","Theorem 3.14")>> renders as [Yes12, Theorem 3.14].

The list of references is produced by calling <<Reference>> without any parameters:

References:

[Nob12]   A. Nobody, A proof of the Riemann hypothesis, Some journal 1 (2012).

[Yes12]   A. Yesbody, No proof of the Riemann hypothesis, Some journal 2 (2012).

Interesting system pages