Geodesic
Generalized 312-avoiding GS-permutations and Lehmer's transformation
Prikladnaya Diskretnaya Matematika. Supplement, no. 10 (2017), pp. 7-9

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Lehmer's transformation of the GS-permutations introduced by I. Gessel and R. Stanley is considered. It is proved that the iteration of Lehmer's transformation of all GS-permutations of order $r\geq1$ leads to the set of all 312-avoiding GS-permutations of order $r$ and thus gives new characterization of these permutations. It is shown that the statistics $\mathrm{rise}$ and $\mathrm{imal}$ on the set of the 312-avoiding GS-permutations of order $r$ have the same distribution. A simple relation connecting the inverses of the generating function of the Narayana polynomials of order $r$ and the exponential generating function of Euler's polynomials of order $r$ is found.
Mots-clés : GS-permutations, Lehmer's transformation, Euler's polynomials
Keywords: 312-avoiding GS-permutations, statistics $\mathrm{rise}$ and $\mathrm{imal}$, Narayana polynomials, generating function, inverse function. 
@article{PDMA_2017_10_a0,
     author = {L. N. Bondarenko and M. L. Sharapova},
     title = {Generalized 312-avoiding {GS-permutations} and {Lehmer's} transformation},
     journal = {Prikladnaya Diskretnaya Matematika. Supplement},
     pages = {7--9},
     publisher = {mathdoc},
     number = {10},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDMA_2017_10_a0/}
}
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L. N. Bondarenko; M. L. Sharapova. Generalized 312-avoiding GS-permutations and Lehmer's transformation. Prikladnaya Diskretnaya Matematika. Supplement, no. 10 (2017), pp. 7-9. http://geodesic.mathdoc.fr/item/PDMA_2017_10_a0/