×

The \(q\)-log-convexity of the Narayana polynomials of type \(B\). (English) Zbl 1230.05276

Summary: We prove a conjecture of Liu and Wang on the \(q\)-log-convexity of the Narayana polynomials of type \(B\). By using Pieri’s rule and the Jacobi-Trudi identity for Schur functions, we obtain an expansion of a sum of products of elementary symmetric functions in terms of Schur functions with nonnegative coefficients. By the principal specialization, this leads to \(q\)-log-convexity. We also show that the linear transformation with respect to the triangular array of Narayana numbers of type \(B\) is log-convexity preserving.

MSC:

05E05 Symmetric functions and generalizations
33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
05A15 Exact enumeration problems, generating functions

Software:

SF; Maple; ACE
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ardila, F.; Beck, M.; Hosten, S.; Pfeifle, J.; Seashore, K., Root polytopes and growth series of root lattices · Zbl 1261.52010
[2] Benson, M., Growth series of finite extensions of \(Z^n\) are rational, Invent. Math., 73, 251-269 (1983) · Zbl 0498.20022
[3] Brenti, F., Log-concave and unimodal sequences in algebra, combinatorics, and geometry: An update, (Contemp. Math., vol. 178 (1994)), 71-89 · Zbl 0813.05007
[4] Butler, L. M., The \(q\)-log-concavity of \(q\)-binomial coefficients, J. Combin. Theory Ser. A, 54, 54-63 (1990) · Zbl 0718.05007
[5] Butler, L. M.; Flanigan, W. P., A note on log-convexity of \(q\)-Catalan numbers, Ann. Comb., 11, 369-373 (2007) · Zbl 1147.05011
[8] Došlić, T.; Veljan, D., Calculus proofs of some combinatorial inequalities, Math. Inequal. Appl., 6, 197-209 (2003) · Zbl 1029.05009
[9] Fomin, S.; Reading, N., Root systems and generalized associahedra, (Geometric Combinatorics. Geometric Combinatorics, IAS/Park City Math. Ser., vol. 13 (2007), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 63-131 · Zbl 1147.52005
[10] Krattenthaler, C., On the \(q\)-log-concavity of Gaussian binomial coefficients, Monatsh. Math., 107, 333-339 (1989) · Zbl 0713.05001
[11] Leroux, P., Reduced matrices and \(q\)-log-concavity properties of \(q\)-Stirling numbers, J. Combin. Theory Ser. A, 54, 64-84 (1990) · Zbl 0704.05003
[12] Liu, L. L.; Wang, Y., On the log-convexity of combinatorial sequences, Adv. in Appl. Math., 39, 453-476 (2007) · Zbl 1131.05010
[13] Reiner, V., Noncrossing partitions for classical reflection groups, Discrete Math., 177, 195-222 (1992)
[14] Sagan, B. E., Inductive proofs of \(q\)-log concavity, Discrete Math., 99, 298-306 (1992) · Zbl 0764.05096
[15] Simion, R., Combinatorial statistics on type-\(B\) analogues of noncrossing partitions and restricted permutations, Electron. J. Combin., 7 (2000), #R9 · Zbl 0938.05003
[16] Stanley, R. P., Enumerative Combinatorics, vol. 2 (1999), Cambridge University Press: Cambridge University Press New York/Cambridge · Zbl 0928.05001
[17] Stembridge, J. R., A maple package for symmetric functions, J. Symbolic Comput., 20, 755-768 (1995) · Zbl 0849.68068
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.