The toric ℎ-vectors of partially ordered sets

Bayer, Margaret; Ehrenborg, Richard

doi:10.1090/S0002-9947-00-02657-X

The toric $h$-vectors of partially ordered sets
HTML articles powered by AMS MathViewer

by Margaret M. Bayer and Richard Ehrenborg PDF

Trans. Amer. Math. Soc. 352 (2000), 4515-4531 Request permission

Abstract:

An explicit formula for the toric $h$-vector of an Eulerian poset in terms of the $\mathbf {cd}$-index is developed using coalgebra techniques. The same techniques produce a formula in terms of the flag $h$-vector. For this, another proof based on Fine’s algorithm and lattice-path counts is given. As a consequence, it is shown that the Kalai relation on dual posets, $g_{n/2}(P)=g_{n/2}(P^*)$, is the only equation relating the $h$-vectors of posets and their duals. A result on the $h$-vectors of oriented matroids is given. A simple formula for the $\mathbf {cd}$-index in terms of the flag $h$-vector is derived.

References

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. MR 1322061
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. MR 777703, DOI 10.1090/conm/034/777703
Margaret M. Bayer and Andrew Klapper, A new index for polytopes, Discrete Comput. Geom. 6 (1991), no. 1, 33–47. MR 1073071, DOI 10.1007/BF02574672
Louis J. Billera, Richard Ehrenborg, and Margaret Readdy, The $c$-$2d$-index of oriented matroids, J. Combin. Theory Ser. A 80 (1997), no. 1, 79–105. MR 1472106, DOI 10.1006/jcta.1997.2797
Louis J. Billera, Richard Ehrenborg, and Margaret Readdy, The $cd$-index of zonotopes and arrangements, Mathematical essays in honor of Gian-Carlo Rota (Cambridge, MA, 1996) Progr. Math., vol. 161, Birkhäuser Boston, Boston, MA, 1998, pp. 23–40. MR 1627367
Louis J. Billera and Carl W. Lee, A proof of the sufficiency of McMullen’s conditions for $f$-vectors of simplicial convex polytopes, J. Combin. Theory Ser. A 31 (1981), no. 3, 237–255. MR 635368, DOI 10.1016/0097-3165(81)90058-3
Francisco Brenti, Twisted incidence algebras and Kazhdan-Lusztig-Stanley functions, Adv. Math. 148 (1999), 44–74.
Richard Ehrenborg and Margaret Readdy, Coproducts and the $cd$-index, J. Algebraic Combin. 8 (1998), no. 3, 273–299. MR 1651249, DOI 10.1023/A:1008614816374
S. A. Joni and G.-C. Rota, Coalgebras and bialgebras in combinatorics, Stud. Appl. Math. 61 (1979), no. 2, 93–139. MR 544721, DOI 10.1002/sapm197961293
Joseph P. S. Kung (ed.), Gian-Carlo Rota on combinatorics, Contemporary Mathematicians, Birkhäuser Boston, Inc., Boston, MA, 1995. Introductory papers and commentaries. MR 1392961
P. McMullen, The maximum numbers of faces of a convex polytope, Mathematika 17 (1970), 179–184. MR 283691, DOI 10.1112/S0025579300002850
D. M. Y. Sommerville, The relations connecting the angle-sums and volume of a polytope in space of $n$ dimensions, Proc. Roy. Soc. London Ser. A 115 (1927), 103–119.
Richard P. Stanley, The number of faces of a simplicial convex polytope, Adv. in Math. 35 (1980), no. 3, 236–238. MR 563925, DOI 10.1016/0001-8708(80)90050-X
Radu Bǎdescu, On a problem of Goursat, Gaz. Mat. 44 (1939), 571–577. MR 0000087
Richard P. Stanley, Flag $f$-vectors and the $cd$-index, Math. Z. 216 (1994), no. 3, 483–499. MR 1283084, DOI 10.1007/BF02572336
Dennis Stanton and Dennis White, Constructive combinatorics, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1986. MR 843332, DOI 10.1007/978-1-4612-4968-9