Down-step statistics in generalized Dyck paths

@article{Asinowski2020DownstepSI,
  title={Down-step statistics in generalized Dyck paths},
  author={Andrei Asinowski and Benjamin Hackl and Sarah J. Selkirk},
  journal={Discret. Math. Theor. Comput. Sci.},
  year={2020},
  volume={24}
}
The number of down-steps between pairs of up-steps in $k_t$-Dyck paths, a generalization of Dyck paths consisting of steps $\{(1, k), (1, -1)\}$ such that the path stays (weakly) above the line $y=-t$, is studied. Results are proved bijectively and by means of generating functions, and lead to several interesting identities as well as links to other combinatorial structures. In particular, there is a connection between $k_t$-Dyck paths and perforation patterns for punctured convolutional codes… 
[PDF] Semantic Reader
3 Citations
Methods Citations
1

3 Citations

Raised $k$-Dyck paths

Raised k -Dyck paths are a generalization of k -Dyck paths that may both begin and end at a nonzero height. In this paper, we develop closed formulas for the number of raised k -Dyck paths from (0 ,…

Enumeration of Generalized Dyck Paths Based on the Height of Down-Steps Modulo $k$

For fixed non-negative integers $k$, $t$, and $n$, with $t < k$, a $k_t$-Dyck path of length $(k+1)n$ is a lattice path that starts at $(0, 0)$, ends at $((k+1)n, 0)$, stays weakly above the line $y…

MIN-turns and MAX-turns in k-Dyck paths: a pure generating function approach

k-Dyck paths differ from ordinary Dyck paths by using an up-step of length k. We analyze at which level the path is after the s-th up-step and before the (s + 1)st up-step. In honour of Rainer Kemp…

27 References

Ascents in Non-Negative Lattice Paths

Non-negative {\L}ukasiewicz paths are special two-dimensional lattice paths never passing below their starting altitude which have only one single special type of down step. They are well-known and…

Returns and Hills on Generalized Dyck Paths

This paper finds that the probability that a randomly chosen ternary path has an even number of hills approaches 125/169 as the length of the path approaches infinity.

On k-Dyck paths with a negative boundary

Paths that consist of up-steps of one unit and down-steps of $k$ units, being bounded below by a horizontal line $-t$, behave like $t+1$ ordered tuples of $k$-Dyck paths, provided that $t\le k$. We…

Catalan Numbers, Their Generalization, and Their Uses

Probably the most prominent among the special integers that arise in combinatorial contexts are the binomial coefficients (~). These have many uses and, often, fascinating interpretations [9]. We…

Counting walks with large steps in an orthant

In the past fifteen years, the enumeration of lattice walks with steps taken in a prescribed set S and confined to a given cone, especially the first quadrant of the plane, has been intensely…

Generalized Dyck tilings

Bijections for a class of labeled plane trees

The On-Line Encyclopedia of Integer Sequences

  • N. Sloane
  • Computer Science
    Electron. J. Comb.
  • 1994
The On-Line Encyclopedia of Integer Sequences (or OEIS) is a database of some 130000 number sequences which serves as a dictionary, to tell the user what is known about a particular sequence and is widely used.

Basic analytic combinatorics of directed lattice paths

On the enumeration of perforation patterns for punctured convolutional codes, Poster presentation at the 4th international conference on

  • Formal Power Series and Algebraic Combinatorics,
  • 1992