|
The toric  -vectors of partially ordered sets
Authors:
Margaret M. Bayer and Richard Ehrenborg
Journal:
Trans. Amer. Math. Soc. 352 (2000), 4515-4531
MSC (2000):
Primary 06A07; Secondary 52B05
Posted:
June 13, 2000
MathSciNet review:
1779486
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: An explicit formula for the toric  -vector of an Eulerian poset in terms of the  -index is developed using coalgebra techniques. The same techniques produce a formula in terms of the flag  -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,  , is the only equation relating the  -vectors of posets and their duals. A result on the  -vectors of oriented matroids is given. A simple formula for the  -index in terms of the flag  -vector is derived.
- 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. MR 1322061
(96a:52012)
- 2.
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. MR 774533
(86f:52010b), http://dx.doi.org/10.1007/BF01388660
- 3.
Margaret
M. Bayer and Andrew
Klapper, A new index for polytopes, Discrete Comput. Geom.
6 (1991), no. 1, 33–47. MR 1073071
(91k:52024), http://dx.doi.org/10.1007/BF02574672
- 4.
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 (98h:05051), http://dx.doi.org/10.1006/jcta.1997.2797
- 5.
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
(99h:52011)
- 6.
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
(82m:52006), http://dx.doi.org/10.1016/0097-3165(81)90058-3
- 7.
Francisco Brenti, Twisted incidence algebras and Kazhdan-Lusztig-Stanley functions, Adv. Math. 148 (1999), 44-74.
- 8.
Richard
Ehrenborg and Margaret
Readdy, Coproducts and the 𝑐𝑑-index, J.
Algebraic Combin. 8 (1998), no. 3, 273–299. MR 1651249
(2000b:52009), http://dx.doi.org/10.1023/A:1008614816374
- 9.
S.
A. Joni and G.-C.
Rota, Coalgebras and bialgebras in combinatorics, Stud. Appl.
Math. 61 (1979), no. 2, 93–139. MR 544721
(81c:05002)
- 10.
Joseph
P. S. Kung (ed.), Gian-Carlo Rota on combinatorics,
Contemporary Mathematicians, Birkhäuser Boston Inc., Boston, MA, 1995.
Introductory papers and commentaries. MR 1392961
(99b:01027)
- 11.
P.
McMullen, The maximum numbers of faces of a convex polytope,
Mathematika 17 (1970), 179–184. MR 0283691
(44 #921)
- 12.
D. M. Y. Sommerville, The relations connecting the angle-sums and volume of a polytope in space of
 dimensions, Proc. Roy. Soc. London Ser. A 115 (1927), 103-119.
- 13.
Richard
P. Stanley, The number of faces of a simplicial convex
polytope, Adv. in Math. 35 (1980), no. 3,
236–238. MR
563925 (81f:52014), http://dx.doi.org/10.1016/0001-8708(80)90050-X
- 14.
Richard
P. Stanley, Enumerative combinatorics. Vol. I, The Wadsworth
& Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced
Books & Software, Monterey, CA, 1986. With a foreword by Gian-Carlo
Rota. MR
847717 (87j:05003)
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. MR 1442260
(98a:05001)
- 15.
Richard
P. Stanley, Flag 𝑓-vectors and the
𝑐𝑑-index, Math. Z. 216 (1994),
no. 3, 483–499. MR 1283084
(96b:06006), http://dx.doi.org/10.1007/BF02572336
- 16.
Dennis
Stanton and Dennis
White, Constructive combinatorics, Undergraduate Texts in
Mathematics, Springer-Verlag, New York, 1986. MR 843332
(88a:05001)
- 1.
- Margaret M. Bayer, Face numbers and subdivisions of convex polytopes, Polytopes: Abstract, Convex, and Computational (Dordrecht-Boston) (T. Bisztriczky, P. McMullen, R. Schneider, and A. Ivic Weiss, eds.), NATO Advanced Science Institute Series, vol. C 440, Kluwer Academic Publishers, 1994, pp. 155-171. MR 96a:52012
- 2.
- Margaret M. Bayer and Louis J. Billera, Generalized Dehn-Sommerville relations for polytopes, spheres and Eulerian partially ordered sets, Invent. Math. 79 (1985), 143-157. MR 86f:52010b
- 3.
- Margaret M. Bayer and Andrew Klapper, A new index for polytopes, Discrete Comput. Geom. 6 (1991), 33-47. MR 91k:52024
- 4.
- Louis J. Billera, Richard Ehrenborg, and Margaret Readdy, The
 -index of oriented matroids, J. Combin. Theory Ser. A 80 (1997), 79-105. MR 98h:05051
- 5.
- -, The cd-index of zonotopes and arrangements, Mathematical essays in honor of Gian-Carlo Rota (Bruce E. Sagan and Richard P. Stanley, eds.), Birkhäuser Boston, Boston, 1998, pp. 23-40. MR 99h:52011
- 6.
- Louis J. Billera and Carl W. Lee, A proof of the sufficiency of McMullen's conditions for
 -vectors of simplicial polytopes, J. Combin. Theory Ser. A 31 (1981), 237-255. MR 82m:52006
- 7.
- Francisco Brenti, Twisted incidence algebras and Kazhdan-Lusztig-Stanley functions, Adv. Math. 148 (1999), 44-74.
- 8.
- Richard Ehrenborg and Margaret Readdy, Coproducts and the
 -index, J. Algebraic Combin. 8 (1998), 273-299. MR 2000b:52009
- 9.
- S. A. Joni and G.-C. Rota, Coalgebras and bialgebras in combinatorics, Stud. Appl. Math. 61 (1979), 93-139. MR 81c:05002
- 10.
- Joseph P. S. Kung (ed.), Gian-Carlo Rota on combinatorics. Introductory papers and commentaries, Birkhäuser Boston, Boston, 1995. MR 99b:01027
- 11.
- Peter McMullen, The maximum number of faces of a convex polytope, Mathematika 17 (1970), 179-184. MR 44:921
- 12.
- D. M. Y. Sommerville, The relations connecting the angle-sums and volume of a polytope in space of dimensions, Proc. Roy. Soc. London Ser. A 115 (1927), 103-119.
- 13.
- Richard P. Stanley, The number of faces of simplicial convex polytopes, Adv. Math. 35 (1980), 236-238. MR 81f:52014
- 14.
- -, Enumerative combinatorics, Vol. I, Wadsworth and Brooks/Cole, Pacific Grove, 1986. MR 87j:05003; Corrected reprint of 1986 original, Cambridge Univ. Press, 1997. MR 98a:05001
- 15.
- -, Flag
 -vectors and the  -index, Math. Z. 216 (1994), 483-499. MR 96b:06006
- 16.
- Dennis Stanton and Dennis White, Constructive combinatorics, Springer-Verlag, New York, 1986. MR 88a:05001
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC (2000):
06A07,
52B05
Retrieve articles in all journals
with MSC (2000):
06A07,
52B05
Additional Information
Margaret M. Bayer
Affiliation:
Department of Mathematics, University of Kansas, Lawrence, Kansas 66045
Email:
bayer@math.ukans.edu
Richard Ehrenborg
Affiliation:
School of Mathematics, Institute for Advanced Study, Olden Lane, Princeton, New Jersey 08540
Address at time of publication:
Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506
Email:
jrge@ms.uky.edu
DOI:
http://dx.doi.org/10.1090/S0002-9947-00-02657-X
PII:
S 0002-9947(00)02657-X
Keywords:
Partially ordered set,
$h$-vector,
$\mathbf{c}\mathbf{d}$-index,
coalgebra
Received by editor(s):
February 15, 1998
Posted:
June 13, 2000
Additional Notes:
The first author was supported in part at MSRI by NSF grant #DMS 9022140
This work was begun when the second author was an H. C. Wang Assistant Professor at Cornell University and was completed at IAS under the partial support of NSF grant #DMS 97-29992 and NEC Research Institute, Inc.
Article copyright:
© Copyright 2000 American Mathematical Society
|
