Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-25T07:45:19.953Z Has data issue: false hasContentIssue false
  • English
  • Français

ON $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\gamma $-VECTORS AND THE DERIVATIVES OF THE TANGENT AND SECANT FUNCTIONS

Part of: Combinatorics

Published online by Cambridge University Press:  10 April 2014

SHI-MEI MA
SHI-MEI MA*
Affiliation:
School of Mathematics and Statistics, Northeastern University at Qinhuangdao, Hebei 066004, PR China email [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we show that the $\gamma $-vectors of Coxeter complexes (of types A and B) and associahedrons (of types A and B) can be obtained by using derivative polynomials of the tangent and secant functions. We provide a unified grammatical approach to generate these $\gamma $-vectors and the coefficient arrays of Narayana polynomials, Legendre polynomials and Chebyshev polynomials of both kinds.

Keywords

derivative polynomialsContext-free grammarsNarayana polynomialsLegendre polynomials

MSC classification

Secondary: 05A05: Permutations, words, matrices 05A15: Exact enumeration problems, generating functions
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 90 , Issue 2 , October 2014 , pp. 177 - 185
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

References

Boyadzhiev, K. N., ‘Derivative polynomials for tanh, tan, sech and sec in explicit form’, Fibonacci Quart. 45 (2007), 291–303.Google Scholar
Chen, W. Y. C., ‘Context-free grammars, differential operators and formal power series’, Theoret. Comput. Sci. 117 (1993), 113–129.Google Scholar
Chow, C.-O., ‘Counting involutory, unimodal, and alternating signed permutations’, Discrete Math. 36 (2006), 2222–2228.Google Scholar
Chow, C.-O., ‘On certain combinatorial expansions of the Eulerian polynomials’, Adv. in Appl. Math. 41 (2008), 133–157.Google Scholar
Foata, D. and Schützenberger, M. P., Théorie géometrique des polynômes eulériens, Lecture Notes in Mathematics, 138 (Springer, Berlin, 1970).CrossRefGoogle Scholar
Franssens, G. R., ‘Functions with derivatives given by polynomials in the function itself or a related function’, Anal. Math. 33 (2007), 17–36.CrossRefGoogle Scholar
Gal, S. R., ‘Real root conjecture fails for five- and higher-dimensional spheres’, Discrete Comput. Geom. 34 (2005), 269–284.Google Scholar
Hoffman, M. E., ‘Derivative polynomials for tangent and secant’, Amer. Math. Monthly 102 (1995), 23–30.Google Scholar
Hoffman, M. E., ‘Derivative polynomials, Euler polynomials, and associated integer sequences’, Electron. J. Combin. 6 (1999), R21.Google Scholar
Ma, S.-M., ‘Derivative polynomials and enumeration of permutations by number of interior and left peaks’, Discrete Math. 312 (2012), 405–412.Google Scholar
Ma, S.-M., ‘An explicit formula for the number of permutations with a given number of alternating runs’, J. Combin. Theory Ser. A 119 (2012), 1660–1664.CrossRefGoogle Scholar
Ma, S.-M., ‘A family of two-variable derivative polynomials for tangent and secant’, Electron. J. Combin. 20(1) (2013), P11.Google Scholar
Ma, S.-M., ‘Some combinatorial arrays generated by context-free grammars’, European J. Combin. 34 (2013), 1081–1091.Google Scholar
Mansour, T. and Sun, Y., ‘Identities involving Narayana polynomials and Catalan numbers’, Discrete Math. 309 (2009), 4079–4088.CrossRefGoogle Scholar
Marberg, E., ‘Actions and identities on set partitions’, Electron. J. Combin. 19 (2012), P28.Google Scholar
Nevo, E. and Petersen, T. K., ‘On γ-vectors satisfying the Kruskal-Katona inequalities’, Discrete Comput. Geom. 45 (2011), 503–521.Google Scholar
Postnikov, A., Reiner, V. and Williams, L., ‘Faces of generalized permutohedra’, Documenta Math. 13 (2008), 207–273.Google Scholar
Shapiro, L. W., Woan, W. J. and Getu, S., ‘Runs, slides and moments’, SIAM J. Algebraic Discrete Methods 4 (1983), 459–466.Google Scholar
Simion, R., ‘A type-B associahedron’, Adv. in Appl. Math. 30 (2003), 2–25.Google Scholar
Sloane, N. J. A., The On-Line Encyclopedia of Integer Sequences, published electronically at http://oeis.org, 2010.Google Scholar
Stanley, R. P., ‘A survey of alternating permutations’, in: Combinatorics and Graphs, Contemporary Mathematics, 531 (eds. Brualdi, R. A. et al. ) (American Mathematical Society, Providence, RI, 2010), 165–196.Google Scholar