This first **FindStatFact** is about the connection of** two classes of statistics on Dyck paths**, namely statistics that are **Narayana distributed** and those that have the same distribution as the** area statistic**. Some quick background, which is taken from www.findstat.org/DyckPaths:

- A
**Dyck path**of size is a a lattice path in from to consisting of . They are counted by the -th**Catalan number**. - The
**Narayana number**then counts Dyck paths with exactly peaks (www.findstat.org/St000015/), or Dyck paths with exactly**left tunnels**(www.findstat.org/St000120/), or various other refinements of Dyck paths. - The
**area**of a Dyck path is given by the number of complete boxes below and completely above the diagonal (www.findstat.org/St000012/). Other statistics that are equidistributed with the area are for example the**bounce**and the**dinv**statistics (www.findstat.org/St000005/ and www.findstat.org/St000006/).

If you now **search the FindStat database for the number of left tunnels** by clicking “*run the finder on the first 200 of these values*” on this statistic at www.findstat.org/St000120/, you will find that starting with a** Dyck path** ,

- turning into a
**-avoiding permutation**(description),- Let . Define a pair of standard Young tableaux of the same shape with at most two rows, such that records the ‘s in in and ‘s in , and records the ‘s in in and ‘s in . is then the unique permutation RSK-corresponding to . Observe that it is -avoiding since and have at most rows.

- turning into a
**binary search tree**(description), and finally- The binary tree is obtained from by inserting the one-line notation of into an empty binary tree.

- turning again into a
**Dyck path**(description),- Map into a Dyck path recursively by sending to followed by the left subtree of followed by followed by the right subtree of .

**maps** the** number of left tunnels **of** to the area **of . The first map to -avoiding permutations is known to preserve several statistics including the various tunnel statistics, the more surprising part is that magically, the **area statistics** comes into play…

**We find it quite surprising that such a map naturally exists!**

**Did YOU know of such a map?****Is there a direct/easy description of this map?****Was this map studied before?**

Below, we list all Dyck paths of size , , and , together with their images:

1010 => ([[1],[2]],[[1],[2]]) => 21 => [[.,.],.] => 1100 => 1 1100 => ([[1,2]],[[1,2]]) => 12 => [.,[.,.]] => 1010 => 0 101010 => ([[1,3],[2]],[[1,3],[2]]) => 213 => [[.,.],[.,.]] => 110010 => 1 101100 => ([[1,3],[2]],[[1,2],[3]]) => 231 => [[.,.],[.,.]] => 110010 => 1 110010 => ([[1,2],[3]],[[1,3],[2]]) => 312 => [[.,[.,.]],.] => 110100 => 2 110100 => ([[1,2],[3]],[[1,2],[3]]) => 132 => [.,[[.,.],.]] => 101100 => 1 111000 => ([[1,2,3]],[[1,2,3]]) => 123 => [.,[.,[.,.]]] => 101010 => 0 10101010 => ([[1,3],[2,4]],[[1,3],[2,4]]) => 2143 => [[.,.],[[.,.],.]] => 11001100 => 2 10101100 => ([[1,3],[2,4]],[[1,2],[3,4]]) => 2413 => [[.,.],[[.,.],.]] => 11001100 => 2 10110010 => ([[1,3,4],[2]],[[1,3,4],[2]]) => 2134 => [[.,.],[.,[.,.]]] => 11001010 => 1 10110100 => ([[1,3,4],[2]],[[1,2,4],[3]]) => 2314 => [[.,.],[.,[.,.]]] => 11001010 => 1 10111000 => ([[1,3,4],[2]],[[1,2,3],[4]]) => 2341 => [[.,.],[.,[.,.]]] => 11001010 => 1 11001010 => ([[1,2],[3,4]],[[1,3],[2,4]]) => 3142 => [[.,[.,.]],[.,.]] => 11010010 => 2 11001100 => ([[1,2],[3,4]],[[1,2],[3,4]]) => 3412 => [[.,[.,.]],[.,.]] => 11010010 => 2 11010010 => ([[1,2,4],[3]],[[1,3,4],[2]]) => 3124 => [[.,[.,.]],[.,.]] => 11010010 => 2 11010100 => ([[1,2,4],[3]],[[1,2,4],[3]]) => 1324 => [.,[[.,.],[.,.]]] => 10110010 => 1 11011000 => ([[1,2,4],[3]],[[1,2,3],[4]]) => 1342 => [.,[[.,.],[.,.]]] => 10110010 => 1 11100010 => ([[1,2,3],[4]],[[1,3,4],[2]]) => 4123 => [[.,[.,[.,.]]],.] => 11010100 => 3 11100100 => ([[1,2,3],[4]],[[1,2,4],[3]]) => 1423 => [.,[[.,[.,.]],.]] => 10110100 => 2 11101000 => ([[1,2,3],[4]],[[1,2,3],[4]]) => 1243 => [.,[.,[[.,.],.]]] => 10101100 => 1 11110000 => ([[1,2,3,4]],[[1,2,3,4]]) => 1234 => [.,[.,[.,[.,.]]]] => 10101010 => 0

Hm, looks like the number on the far RHS is the length of the longest sequence of zeros between two ones, on the LHS, EXCEPT for

this entry: 10101010 => ([[1,3],[2,4]],[[1,3],[2,4]]) => 2143 => [[.,.],[[.,.],.]] => 11001100 => 2 ….

I just redid the calculation, it seems your observation was just a coincidence for small values… Thanks for checking anyway!