Geodesic
Intervals in Dyck paths and the wreath conjecture
Jan Petr1 ; Pavel Turek2
1University of Passau
2University of Birmingham
The electronic journal of combinatorics, Tome 32 (2025) no. 4
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Let $\iota_{k}(m,l)$ denote the total number of intervals of length $m$ across all Dyck paths of semilength $k$ such that each interval contains precisely $l$ falls. We give the formula for $\iota_{k}(m,l)$ and show that $\iota_{k}(k,l)=\binom{k}{l}^2$. Motivated by this, we propose two stronger variants of the wreath conjecture due to Baranyai for $n=2k+1$.
DOI : 10.37236/13812

Jan Petr  1   ; Pavel Turek  2

1 University of Passau
2 University of Birmingham 
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     author = {Jan Petr and Pavel Turek},
     title = {Intervals in {Dyck} paths and the wreath conjecture},
     journal = {The electronic journal of combinatorics},
     year = {2025},
     volume = {32},
     number = {4},
     doi = {10.37236/13812},
     zbl = {arXiv:2501.07277},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/13812/}
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Jan Petr; Pavel Turek. Intervals in Dyck paths and the wreath conjecture. The electronic journal of combinatorics, Tome 32 (2025) no. 4. doi: 10.37236/13812

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