# Automatic Counting of Restricted Dyck Paths via (Numeric and Symbolic) Dynamic Programming

@article{Ekhad2020AutomaticCO, title={Automatic Counting of Restricted Dyck Paths via (Numeric and Symbolic) Dynamic Programming}, author={Shalosh B. Ekhad and Doron Zeilberger}, journal={arXiv: Combinatorics}, year={2020} }

Dyck paths are one of the most important objects in enumerative combinatorics, and there are many papers devoted to counting selected families of Dyck paths. Here we present two approaches for the automatic counting of many such families, using both a "dumb" approach (driven by numeric dynamic programming) that often works in practice, and a "clever" approach, needed for larger problems, driven by "symbolic" dynamic programming. Both approaches are fully automated and implemented in Maple.

## 9 Citations

### Automated counting of restricted Motzkin paths

- Computer ScienceEnumerative Combinatorics and Applications
- 2021

Two fully automated methods for enumerating the paths of families of Motzkin paths with restrictions on peak heights, valley heights, upward- run lengths, downward-run lengths, and flat-runlength are presented.

### Enumerating Restricted Dyck Paths with Context-Free Grammars.

- Mathematics
- 2020

The number of Dyck paths of semilength $n$ is famously $C_n$, the $n$th Catalan number. This fact follows after noticing that every Dyck path can be uniquely parsed according to a context-free…

### Retakh's Motzkin paths and some combinatorial comments

- MathematicsEnumerative Combinatorics and Applications
- 2020

Dyck paths where peaks are only allowed on level 1 and on even-indexed levels, were introduced by Retakh and analysed by Zeilberger, with assistance from Ekhad. We add some combinatorial comments to…

### Counting Standard Young Tableaux With Restricted Runs

- Mathematics
- 2020

The number of Young Tableaux whose shape is a k by n rectangle is famously (nk)! 0! ... (k-1)!/((n+k-1)!(n+k-2)!... n!) implying that for each specific k, that sequence satisfies a linear recurrence…

### A walk in my lattice path garden

- Computer Science
- 2021

Various lattice path models are reviewed. The enumeration is done using generating functions. A few bijective considerations are woven in as well. The kernel method is often used. Computer algebra…

### Automated Counting of Restricted Motzkin Paths

- Computer Science
- 2021

Two fully automated methods for enumerating the paths of families of Motzkin paths with restrictions on peak heights, valley heights, upward- run lengths, downward-run lengths, and flat-runlength are presented.

### #A69 INTEGERS 21 (2021) ENUMERATING RESTRICTED DYCK PATHS WITH CONTEXT FREE GRAMMARS

- Mathematics
- 2021

The number of Dyck paths of semilength n is famously Cn, the nth Catalan number. This fact follows after noticing that every Dyck path can be uniquely parsed according to a context-free grammar. In a…

### Automated Counting of Restricted Motzkin Paths

- Computer Science
- 2021

Two fully automated methods for enumerating the paths of families of Motzkin paths with restrictions on peak heights, valley heights, upward- run lengths, downward-run lengths, and flat-runlength are presented.

### Lattice walks ending on a coordinate hyperplane avoiding backtracking and repeats

- MathematicsEnumerative Combinatorics and Applications
- 2021

We work with lattice walks in Z using step set {±1} that finish with xr+1 = 0. We further impose conditions of avoiding backtracking (i.e. [v,−v]) and avoiding consecutive steps (i.e. [v, v]) each…

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