Abstract
The alternating descent statistic on permutations was introduced by Chebikin as a variant of the descent statistic. We show that the alternating descent polynomials on permutations, called alternating Eulerian polynomials, are unimodal via a five-term recurrence relation. We also find a quadratic recursion for the alternating major index q-analog of the alternating Eulerian polynomials. As an interesting application of this quadratic recursion, we show that \((1+q)^{\lfloor n/2\rfloor }\) divides \(\sum _{\pi \in {{\mathfrak {S}}}_n}q^{\mathrm{altmaj}(\pi )}\), where \({{\mathfrak {S}}}_n\) is the set of all permutations of \(\{1,2,\ldots ,n\}\) and \(\mathrm{altmaj}(\pi )\) is the alternating major index of \(\pi \). This leads us to discover a q-analog of \(n!=2^{\ell }m\), m odd, using the statistic of alternating major index. Moreover, we study the \(\gamma \)-vectors of the alternating Eulerian polynomials by using these two recursions and the cd-index. Further intriguing conjectures are formulated, which indicate that the alternating descent statistic deserves more work.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
André, D.: Développement de \(\sec x\) et \(\tan x\). C. R. Math. Acad. Sci. Paris 88, 965–979 (1879)
Andrews, G.E., Gessel, I.M.: Divisibility properties of the \(q\)-tangent numbers. Proc. Amer. Math. Soc. 68, 380–384 (1978)
Athanasiadis, C.A.: Gamma-positivity in combinatorics and geometry. Sém. Lothar. Combin. 77, Article B77i (2018)
Brändén, P.: Unimodality, log-concavity, real-rootedness and beyond. Handbook of Enumerative Combinatorics, CRC Press Book. arXiv:1410.6601
Brittenham, C., Carroll, A.T., Petersen, T.K., Thomas, C.: Unimodality via alternating gamma vectors. Electron. J. Comb. 23(2), #2.40 (2016)
Carlitz, L.: \(q\)-Bernoulli and Eulerian numbers. Trans. Amer. Math. Soc. 76, 332–350 (1954)
Carlitz, L.: A combinatorial property of \(q\)-Eulerian numbers. Amer. Math. Monthly 82, 51–54 (1975)
Chebikin, D.: Variations on descents and inversions in permutations. Electron. J. Combin. 15, #R132 (2008)
Chow, C.-O., Shiu, W.C.: Counting simsun permutations by descents. Ann. Comb. 15, 625–635 (2011)
Comtet, L.: Advanced Combinatorics, Boston (1974)
Foata, D.: Further divisibility properties of the \(q\)-tangent numbers. Proc. Amer. Math. Soc. 81, 143–148 (1981)
Foata, D., Schützenberger, M.-P.: Théorie géométrique des polynômes eulériens. Lecture Notes in Mathematics, vol. 138. Springer, Berlin (1970)
Gessel, I.M., Zhuang, Y.: Counting Permutations by alternating descents. Electron. J. Combin. 21(4), #P4.23 (2014)
Han, G.-N., Jouhet, F., Zeng, J.: Two new triangles of \(q\)-integers via \(q\)-Eulerian polynomials of Type \(A\) and \(B\). Ramanujan J. 31, 115–127 (2013)
Lin, Z.: Proof of Gessel’s \(\gamma \)-positivity conjecture. Electron. J. Combin. 23, #P3.15 (2016)
Lin, Z.: On \(\gamma \)-positive polynomials arising in pattern avoidance. Adv. Appl. Math. 82, 1–22 (2017)
Ma, S.-M., Yeh, Y.-N.: Enumeration of permutations by number of alternating descents. Discrete Math. 339, 1362–1367 (2016)
Ma, S.-M., Ma, J., Yeh, Y.-N.: David–Barton type identities and alternating run polynomials. Adv. Appl. Math. 114, Article 101978 (2020)
OEIS Foundation Inc., The On-Line Encyclopedia of Integer Sequences https://oeis.org (2011)
Park, S.: The \(r\)-multipermutations. J. Combin. Theory Ser. A 67, 44–71 (1994)
Petersen, T.K.: Eulerian numbers. With a foreword by Richard Stanley Birkhäuser Advanced Texts: Basler Lehrbücher. Birkhäuser, New York (2015)
Remmel, J.B.: Generating functions for alternating descents and alternating major index. Ann. Comb. 16, 625–650 (2012)
Sagan, B.E., Tirrell, J.: Lucas atoms. Adv. Math. 374, 107387 (2020)
Stanley, R.P.: A survey of alternating permutations. In: Brualdi, R.A., et al. (eds.) Combinatorics and Graphs, Contemporary Mathematics, vol. 531, pp. 165–196. American Mathematical Society, Providence (2010)
Sundaram, S.: The homology of partitions with an even number of blocks. J. Algebraic Combin. 4, 69–92 (1995)
Acknowledgements
The authors thank Yuan-Hsun Lo for his helpful discussions and the anonymous referee for carefully reading the paper and providing insightful comments and suggestions. The first author is grateful to Prof. Weigen Yan at Jimei University for encouraging him to attend the ICM2014 Satellite Conference: International Conference on Combinatorics and Graphs in Beijing, where this work was initiated.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The first author was supported by the National Natural Science Foundation of China Grant 11871247 and the project of Qilu Young Scholars of Shandong University. The second author was supported by the National Natural Science Foundation of China Grant 12071063. The third author was supported by the National Natural Science Foundation of China Grant 11671037. The fourth author was supported by the National Natural Science Foundation of China Grant 11801424 and a start-up research grant of Wuhan University.
Rights and permissions
About this article
Cite this article
Lin, Z., Ma, SM., Wang, D.G.L. et al. Positivity and divisibility of enumerators of alternating descents. Ramanujan J 58, 203–228 (2022). https://doi.org/10.1007/s11139-021-00460-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11139-021-00460-5