Abstract
The s-lecture hall polytopes P s are a class of integer polytopes defined by Savage and Schuster which are closely related to the lecture hall partitions of Eriksson and Bousquet-Mélou. We define a half-open parallelopiped Par s associated with P s and give a simple description of its integer points. We use this description to recover earlier results of Savage et al. on the δ-vector (or h*-vector) and to obtain the connections to s-ascents and s-descents, as well as some generalizations of these results.
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References
Bousquet-Mélou M., Eriksson K.:: Lecture hall partitions. Ramanujan J. 1(1), 101–111 (1997)
Corteel, S., Lee, S., Savage, C.D.: Enumeration of sequences constrained by the ratio of consecutive parts. Sém. Lothar. Combin. 54A, Art. B54Aa (2005)
Corteel S., Savage C.D.: Anti-lecture hall compositions. Discrete Math. 263, 275–280 (2003)
Ehrhart, E.: Sur les polyèdres rationnels homothétiques à n dimensions. C. R. Acad. Sci. Paris 254, 616–618 (1962)
Pensyl, T.W., Savage, C.D.: Rational lecture hall polytopes and inflated Eulerian polynomials. Ramanujan J. 31(1-2), 97–114 (2013)
Pensyl, T.W., Savage, C.D.: Lecture hall partitions and the wreath products \({C_{k} \wr S_{n}}\). Integers 12B, #A10 (2012/13)
Savage C.D., Schuster M.J.: Ehrhart series of lecture hall polytopes and Eulerian polynomials for inversion sequences. J. Combin. Theory Ser. A 119(4), 850–870 (2012)
Savage, C.D., Viswanathan, G.: The 1/k-Eulerian polynomials. Electron. J. Combin. 19, #P9 (2012)
Stanley R.P.: Decompositions of rational convex polytopes. Ann. Discrete Math. 6, 333–342 (1980)
Stanley, R.P.: Enumerative Combinatorics. Vol. 1, second ed. Cambridge Stud. Adv. Math., Vol. 49. Cambridge University Press, Cambridge (2012)
Stanley, R.P.: Enumerative Combinatorics. Vol. 2. Cambridge Stud. Adv. Math., Vol. 62. Cambridge University Press, Cambridge (1999)
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Fu Liu is partially supported by the Hellman Fellowship from UC Davis.
Richard Stanley is partially supported by NSF grant DMS-1068625.
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Liu, F., Stanley, R.P. The Lecture Hall Parallelepiped. Ann. Comb. 18, 473–488 (2014). https://doi.org/10.1007/s00026-014-0235-8
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DOI: https://doi.org/10.1007/s00026-014-0235-8