How to use FindStat

# 1. Statistic finder

An example for how to use the statistic finder.

## 1.1. Background

Imagine you have some data for a function $f:$ Dyck paths $= \mathcal{D} \longrightarrow \mathbb{Z}$ (such a function is a statistic on Dyck paths):

$[1, 0, 1, 0, 1, 0, 1, 0] \mapsto 16$

$[1, 0, 1, 0, 1, 1, 0, 0] \mapsto 5$

$[1, 0, 1, 1, 0, 0, 1, 0] \mapsto 6$

$[1, 0, 1, 1, 0, 1, 0, 0] \mapsto 3$

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

We would like to find a statistic $st$ on some set of objects $\mathcal{S}$ and a map $\phi : \mathcal{D} \longrightarrow \mathcal{S}$ which would yield these same values. That is to say we want to find a statistic $st : \mathcal{S} \longrightarrow \mathbb{Z}$ and $\phi:\mathcal{D} \longrightarrow \mathcal{S}$ such that $st(\phi(D)) = f(D)$ for all $D$ in our sample data.

## 1.2. Finding statistics

To begin, we select Dyck paths under the StatisticFinder tab. Next check the box for size 4, this corresponds to the fact that we have Dyck words of size 4. We input the data, assigning to the Dyck path $[1,0,1,0,1,0,1,0]$ the value $16$ etc. Last click submit data at the bottom of the page and the magic begins by searching the database of known statistics:

After applying to 132 avoiding permutation
your input data matches the following statistic in the shown values:

Identifier St000001

Values
[1,0,1,1,0,0,1,0] => [4,2,3,1] => 6
[1,0,1,1,0,1,0,0] => [3,2,4,1] => 3
[1,0,1,0,1,0,1,0] => [4,3,2,1] => 16
[1,0,1,1,1,0,0,0] => [2,3,4,1] => 1
[1,0,1,0,1,1,0,0] => [3,4,2,1] => 5

Description
This is the number of ways to write a permutation as a product of simple transpositions.

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

Identifier St000003

Values
[1,0,1,0,1,0,1,0] => [3,2,1] => 16
[1,0,1,1,1,0,0,0] => [1,1,1] => 1
[1,0,1,0,1,1,0,0] => [2,2,1] => 5
[1,0,1,1,0,1,0,0] => [2,1,1] => 3
[1,0,1,1,0,0,1,0] => [3,1,1] => 6

Description
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 and to partitions connect these values to known statistics in the database.

# 2. Statistic distribution search

## 2.1. Background

The combinatorial collections considered in this project come in levels (e.g., the length of a Dyck path in our example). Now, the generating function of a statistic $st : \mathcal{S} \longrightarrow \mathbb{Z}$ on a collection with respect to this decomposition into levels is the family of (Laurent) polynomials in one variable $q$ given by $f_k(q) = \sum_{obj \in \mathcal{S}_k} q^{st(obj)}$ where $\mathcal{S}_k$ 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 only same generating function rather than checking for the exact same values.

## 2.2. Finding statistics with the same generating function

The first and simpler option is to provide all values of a given level set, and to check the distribution checkbox to run the distribution search.

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

\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:

Identifier St000004

Values
[1,2] => 0
[2,1] => 1
[1,2,3] => 0
[1,3,2] => 2
[2,1,3] => 1
[2,3,1] => 2
[3,1,2] => 1
[3,2,1] => 3

Description
The major index of a permutation.

Identifier St000018

Values
[1,2] => 0
[2,1] => 1
[1,2,3] => 0
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 2
[3,1,2] => 2
[3,2,1] => 3

Description
The number of inversions of a permutation.

Identifier St000156

Values
[1,2] => 0
[2,1] => 1
[1,2,3] => 0
[1,3,2] => 2
[2,1,3] => 1
[2,3,1] => 3
[3,1,2] => 1
[3,2,1] => 2

Description
The Denert index of a permutation.

## 2.3. 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 using the free box. Entering

[1,2]
[2,1]
====> 0,1
[1,2,3]
[1,3,2]
[2,1,3]
[2,3,1]
[3,1,2]
[3,2,1]
====> 0,1,1,2,2,3

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

[1,2] => 0
[2,1] => 1
[1,2,3] => 0
[2,1,3] => 1
[2,3,1] => 2
[1,3,2]
[3,1,2]
====> 1,2
[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.

# 3. Statistics database

You can also browse the statistics database. The database is searched when the StatisticFinder is running, and all entries of the database which agree on the data you provided are shown.

# 4. Contributing to FindStat

## 4.1. Who can contribute?

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.

## 4.3. Wiki pages for combinatorial collections/statistics/maps

Every combinatorial collection/statistic/map should ideally have an associated wiki page. See the section on wiki pages for more information.

## 4.4. Adding statistics to the FindStat database

Every database entry must contain the following:

1. A combinatorial collection. Examples are Permutations and IntegerPartitions. If you don't know, you probably shouldn't be entering data. If you do not see the collection you want, you can make a request on our Requests page.

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 StatisticFinder, please recheck there if you are uncertain how to provide data. Moreover, the more data is provided, the more likely it will be that it will be meaningful to someone else.

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.

# 5. Adding and editing wiki pages

## 5.1. How do I create a new wiki page?

You must have signed up for an account in order to be allowed to create a new wiki page. Only users with full editor rights can delete (obsolete) pages.

To create a new wiki page, simply type the page name into the URL. Alternatively, link the new page name from another wiki page, and then follow the link. Every new page should be linked from old pages.

## 5.2. What should a wiki page contain?

To create a new wiki page you should use one of the given templates. We provide templates for combinatorial collections, maps and statistics.

• All combinatorial objects should have top-level pages (e.g. Permutations).

• All combinatorial statistics correspond to some combinatorial object. Its wiki page should thus be on the second level, below the combinatorial object. For an example, see Permutations/Descents-Major.

• All combinatorial statistics should be linked from the Statistics section in the corresponding combinatorial object, see e.g. Permutations#Statistics.

• Combinatorial maps are treated in the same way as combinatorial statistics, see e.g. Permutations#Maps.

## 5.4. 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).

# 6. Combinatorial maps

## 6.1. Browsing combinatorial maps

See the pdf of the graph of all combinatorial maps.