Geodesic
Enumeration of generalized Dyck paths based on the height of down-steps modulo \(k\)
Clemens Heuberger1 ; Sarah J. Selkirk1 ; Stephan Wagner2
1University of Klagenfurt
2Uppsala Universitet
The electronic journal of combinatorics, Tome 30 (2023) no. 1
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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 $y = -t$, and consists of steps from the step-set $\{(1, 1), (1, -k)\}$. We enumerate the family of $k_t$-Dyck paths by considering the number of down-steps at a height of $i$ modulo $k$. Given a tuple $(a_1, a_2, \ldots, a_k)$ we find an exact enumeration formula for the number of $k_t$-Dyck paths of length $(k+1)n$ with $a_i$ down-steps at a height of $i$ modulo $k$, $1 \leq i \leq k$. The proofs given are done via bijective means or with generating functions.
DOI : 10.37236/11218
Classification : 05A15, 05E05
Mots-clés : Dyck path, Fuss-Catalan number, Schur-positivity, Lagrange inversion formula

Clemens Heuberger  1   ; Sarah J. Selkirk  1   ; Stephan Wagner  2

1 University of Klagenfurt
2 Uppsala Universitet 
@article{10_37236_11218,
     author = {Clemens Heuberger and Sarah J. Selkirk and Stephan Wagner},
     title = {Enumeration of generalized {Dyck} paths based on the height of down-steps modulo \(k\)},
     journal = {The electronic journal of combinatorics},
     year = {2023},
     volume = {30},
     number = {1},
     doi = {10.37236/11218},
     zbl = {1514.05013},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/11218/}
}
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Clemens Heuberger; Sarah J. Selkirk; Stephan Wagner. Enumeration of generalized Dyck paths based on the height of down-steps modulo \(k\). The electronic journal of combinatorics, Tome 30 (2023) no. 1. doi: 10.37236/11218

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