The short toric polynomial. (English) Zbl 1376.06006
Summary: We introduce the short toric polynomial associated to a graded Eulerian poset. This polynomial contains the same information as the two toric polynomials introduced by Stanley, but allows different algebraic manipulations. The intertwined recurrence defining Stanley’s toric polynomials may be replaced by a single recurrence, in which the degree of the discarded terms is independent of the rank. A short toric variant of the formula by Bayer and Ehrenborg, expressing the toric \(h\)-vector in terms of the \(cd\)-index, may be stated in a rank-independent form, and it may be shown using weighted lattice path enumeration and the reflection principle. We use our techniques to derive a formula expressing the toric \(h\)-vector of a dual simplicial Eulerian poset in terms of its \(f\)-vector. This formula implies Gessel’s formula for the toric \(h\)-vector of a cube, and may be used to prove that the nonnegativity of the toric \(h\)-vector of a simple polytope is a consequence of the Generalized Lower Bound Theorem holding for simplicial polytopes.
MSC:
| 06A07 | Combinatorics of partially ordered sets |
| 05A15 | Exact enumeration problems, generating functions |
| 06A11 | Algebraic aspects of posets |
| 52B05 | Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.) |
Keywords:
Eulerian poset; toric \(h\)-vector; Narayana numbers; reflection principle; Morgan-Voyce polynomialSoftware:
OEISReferences:
| [1] | Margaret M. Bayer, Face numbers and subdivisions of convex polytopes, Polytopes: abstract, convex and computational (Scarborough, ON, 1993) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 440, Kluwer Acad. Publ., Dordrecht, 1994, pp. 155 – 171. · Zbl 0809.52012 |
| [2] | Margaret M. Bayer and Louis J. Billera, Counting faces and chains in polytopes and posets, Combinatorics and algebra (Boulder, Colo., 1983) Contemp. Math., vol. 34, Amer. Math. Soc., Providence, RI, 1984, pp. 207 – 252. , https://doi.org/10.1090/conm/034/777703 Margaret M. Bayer and Louis J. Billera, Generalized Dehn-Sommerville relations for polytopes, spheres and Eulerian partially ordered sets, Invent. Math. 79 (1985), no. 1, 143 – 157. · Zbl 0561.52007 · doi:10.1007/BF01388660 |
| [3] | Margaret M. Bayer and Richard Ehrenborg, The toric \?-vectors of partially ordered sets, Trans. Amer. Math. Soc. 352 (2000), no. 10, 4515 – 4531. · Zbl 0971.52012 |
| [4] | Margaret M. Bayer and Gábor Hetyei, Flag vectors of Eulerian partially ordered sets, European J. Combin. 22 (2001), no. 1, 5 – 26. · Zbl 0971.06004 · doi:10.1006/eujc.2000.0414 |
| [5] | Margaret M. Bayer and Gábor Hetyei, Generalizations of Eulerian partially ordered sets, flag numbers, and the Möbius function, Discrete Math. 256 (2002), no. 3, 577 – 593 (English, with English and French summaries). LaCIM 2000 Conference on Combinatorics, Computer Science and Applications (Montreal, QC). · Zbl 1020.06001 · doi:10.1016/S0012-365X(02)00336-9 |
| [6] | Margaret M. Bayer and Andrew Klapper, A new index for polytopes, Discrete Comput. Geom. 6 (1991), no. 1, 33 – 47. · Zbl 0761.52009 · doi:10.1007/BF02574672 |
| [7] | L. J. Billera and Francesco Brenti, Quasisymmetric functions and Kazhdan-Lusztig polynomials, preprint 2007, to appear in Israel Journal of Mathematics, arXiv:0710.3965v2 [math.CO]. · Zbl 1269.20030 |
| [8] | Louis J. Billera and Gábor Hetyei, Decompositions of partially ordered sets, Order 17 (2000), no. 2, 141 – 166. · Zbl 0982.06002 · doi:10.1023/A:1006420120193 |
| [9] | Clara Chan, Plane trees and \?-vectors of shellable cubical complexes, SIAM J. Discrete Math. 4 (1991), no. 4, 568 – 574. · Zbl 0737.05040 · doi:10.1137/0404049 |
| [10] | G. Hetyei, On the \?\?-variation polynomials of André and Simsun permutations, Discrete Comput. Geom. 16 (1996), no. 3, 259 – 275. · Zbl 0862.05002 · doi:10.1007/BF02711512 |
| [11] | G. Hetyei, A second look at the toric \( h\)-polynomial of a cubical complex, preprint 2010, arXiv:1002.3601v1 [math.CO], to appear in Ann. Combin. · Zbl 1256.05260 |
| [12] | Kalle Karu, Hard Lefschetz theorem for nonrational polytopes, Invent. Math. 157 (2004), no. 2, 419 – 447. · Zbl 1077.14071 · doi:10.1007/s00222-004-0358-3 |
| [13] | Kalle Karu, The \?\?-index of fans and posets, Compos. Math. 142 (2006), no. 3, 701 – 718. · Zbl 1103.14029 · doi:10.1112/S0010437X06001928 |
| [14] | C. Krattenthaler, The enumeration of lattice paths with respect to their number of turns, Advances in combinatorial methods and applications to probability and statistics, Stat. Ind. Technol., Birkhäuser Boston, Boston, MA, 1997, pp. 29 – 58. · Zbl 0882.05004 |
| [15] | C. Krattenthaler, personal communication. |
| [16] | C. Lee, Sweeping the \( cd\)-Index and the Toric \( H\)-Vector, preprint 2009, http://www.ms.uky. edu/\~lee/cd.pdf |
| [17] | A. M. Morgan-Voyce, Ladder Network Analysis Using Fibonacci Numbers, IRE Trans. Circuit Th. CT-6, 321-322, Sep. 1959. |
| [18] | N.J.A. Sloane, “On-Line Encyclopedia of Integer Sequences,” http://www.research.att.com/ñjas/sequences · Zbl 1274.11001 |
| [19] | Richard P. Stanley, Flag \?-vectors and the \?\?-index, Math. Z. 216 (1994), no. 3, 483 – 499. · Zbl 0805.06003 · doi:10.1007/BF02572336 |
| [20] | Richard Stanley, Generalized \?-vectors, intersection cohomology of toric varieties, and related results, Commutative algebra and combinatorics (Kyoto, 1985) Adv. Stud. Pure Math., vol. 11, North-Holland, Amsterdam, 1987, pp. 187 – 213. · Zbl 0652.52007 |
| [21] | Richard P. Stanley, The number of faces of simplicial polytopes and spheres, Discrete geometry and convexity (New York, 1982) Ann. New York Acad. Sci., vol. 440, New York Acad. Sci., New York, 1985, pp. 212 – 223. · doi:10.1111/j.1749-6632.1985.tb14556.x |
| [22] | Richard P. Stanley, Enumerative combinatorics. Vol. 1, Cambridge Studies in Advanced Mathematics, vol. 49, Cambridge University Press, Cambridge, 1997. With a foreword by Gian-Carlo Rota; Corrected reprint of the 1986 original. · Zbl 0889.05001 |
| [23] | Richard P. Stanley, Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62, Cambridge University Press, Cambridge, 1999. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. · Zbl 0928.05001 |
| [24] | M. N. S. Swamy, Properties of the Polynomials Defined by Morgan-Voyce, Fibonacci Quart. 4 (1966), 73-81. · Zbl 0133.32304 |
| [25] | M. N. S. Swamy, Further properties of Morgan-Voyce polynomials, Fibonacci Quart. 6 (1968), no. 2, 167 – 175. · Zbl 0155.08204 |
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