Geodesic
On a conjecture concerning shuffle-compatible permutation statistics
The electronic journal of combinatorics, Tome 29 (2022) no. 3
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The notion of shuffle-compatible permutation statistics was implicit in Stanley's work on P-partitions and was first explicitly studied by Gessel and Zhuang. The aim of this paper is to prove that the triple ${\rm (udr, pk, des)}$ is shuffle-compatible as conjectured by Gessel and Zhuang, where ${\rm udr}$ denotes the number of up-down runs, ${\rm pk}$ denotes the peak number, and ${\rm des}$ denotes the descent number. This is accomplished by establishing an ${\rm (udr, pk, des)}$-preserving bijection in the spirit of Baker-Jarvis and Sagan's bijective proofs of the shuffle compatibility property of permutation statistics.
DOI : 10.37236/10953
Classification : 05A05, 05C30, 60C05
Mots-clés : descent statistic, shuffle compatibility property 
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     author = {Lihong Yang and Sherry H.F.  Yan},
     title = {On a conjecture concerning shuffle-compatible permutation statistics},
     journal = {The electronic journal of combinatorics},
     year = {2022},
     volume = {29},
     number = {3},
     doi = {10.37236/10953},
     zbl = {1492.05005},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/10953/}
}
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Lihong Yang; Sherry H.F.  Yan. On a conjecture concerning shuffle-compatible permutation statistics. The electronic journal of combinatorics, Tome 29 (2022) no. 3. doi: 10.37236/10953

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