Abstract
We give a counting formula for the set of rectangular increasing tableaux in terms of generalized Narayana numbers. We define small m–Schröder paths and give a bijection between the set of increasing rectangular tableaux and small m–Schröder paths, generalizing a result of Pechenik [4]. Using K–jeu de taquin promotion, we give a cyclic sieving phenomenon for the set of increasing hook tableaux.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bennett M., Madill B., Stokke A.: Jeu-de-taquin promotion and a cyclic sieving phenomenon for semistandard hook tableaux. Discrete Math. 319, 62–67 (2014)
Frame, J.S., Robinson, G. de B., Thrall, R.M.: The hook graphs of the symmetric groups. Canadian J. Math. 6, 316–324 (1954)
Goulden I.P., Jackson D.M.: Combinatorial Enumeration. Wiley Interscience, New York (1983)
Pechenik O.: Cyclic sieving of increasing tableaux and small Schröder paths. J. Combin. Theory Ser. A 125, 357–378 (2014)
Reiner V., Stanton D., White D.: The cyclic sieving phenomenon. J. Combin. Theory Ser. A 108(1), 17–50 (2004)
Rhoades B.: Cyclic sieving, promotion, and representation theory. J. Combin. Theory Ser. A 117(1), 38–76 (2010)
Rhoades, B.: A skein action of the symmetric group on noncrossing partitions. Preprint. arXiv:1501.04680 (2015)
Sagan, B.: The cyclic sieving phenomenon: a survey. In: Chapman, R. (ed.) Surveys in Combinatorics 2011. London Math. Soc. Lecture Note Series, Vol. 392, pp. 183–234. Cambridge University Press, Cambridge (2011)
Stanley, R.P.: Promotion and evacuation. Electron. J. Combin. 16(2), #R9 (2009)
Sulanke, R.A.: Generalizing Narayana and Schröder numbers to higher dimensions. Electron. J. Combin. 11, #R54 (2004)
Sulanke R.A.: Three dimensional Narayana and Schr¨oder numbers. Theoret. Comput. Sci. 346(2–3), 455–468 (2005)
Thomas H., Yong A.: A jeu de taquin theory for increasing tableaux with applications to K-theoretic calculus. Algebra Number Theory 3(2), 121–148 (2009)
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada.
Rights and permissions
About this article
Cite this article
Pressey, T., Stokke, A. & Visentin, T. Increasing Tableaux, Narayana Numbers and an Instance of the Cyclic Sieving Phenomenon. Ann. Comb. 20, 609–621 (2016). https://doi.org/10.1007/s00026-016-0320-2
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00026-016-0320-2