Two unfortunate properties of pure 𝑓-vectors

Pastine, Adrián; Zanello, Fabrizio

doi:10.1090/S0002-9939-2014-12338-0

Two unfortunate properties of pure $f$-vectors
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by Adrián Pastine and Fabrizio Zanello PDF

Proc. Amer. Math. Soc. 143 (2015), 955-964 Request permission

Abstract:

The set of $f$-vectors of pure simplicial complexes is an important but little understood object in combinatorics and combinatorial commutative algebra. Unfortunately, its explicit characterization appears to be a virtually intractable problem, and its structure is very irregular and complicated. The purpose of this note, where we combine a few different algebraic and combinatorial techniques, is to lend some further evidence to this fact.

We first show that pure (in fact, Cohen-Macaulay) $f$-vectors can be nonunimodal with arbitrarily many peaks, thus improving the corresponding results known for level Hilbert functions and pure $O$-sequences. We provide both an algebraic and a combinatorial argument for this result. Then, answering negatively a question of the second author and collaborators posed in the recent AMS Memoir on pure $O$-sequences, we show that the interval property fails for the set of pure $f$-vectors, even in dimension 2.

References

Mats Boij, Juan C. Migliore, Rosa M. Miró-Roig, Uwe Nagel, and Fabrizio Zanello, On the shape of a pure $O$-sequence, Mem. Amer. Math. Soc. 218 (2012), no. 1024, viii+78. MR 2931681, DOI 10.1090/S0065-9266-2011-00647-7
Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 1251956
P. Byrnes, Ph.D. Thesis, University of Minnesota, in preparation.
Charles J. Colbourn and Jeffrey H. Dinitz (eds.), Handbook of combinatorial designs, 2nd ed., Discrete Mathematics and its Applications (Boca Raton), Chapman & Hall/CRC, Boca Raton, FL, 2007. MR 2246267
Charles J. Colbourn, M. S. Keranen, and D. L. Kreher, $f$-vectors of pure complexes and pure multicomplexes of rank three, Discrete Math. 320 (2014), 26–39. MR 3147204, DOI 10.1016/j.disc.2013.12.001
A. Constantinescu and M. Varbaro, $h$-vectors of matroid complexes, preprint. Available on the arXiv.
Huy Tài Hà, Erik Stokes, and Fabrizio Zanello, Pure $O$-sequences and matroid $h$-vectors, Ann. Comb. 17 (2013), no. 3, 495–508. MR 3090174, DOI 10.1007/s00026-013-0193-6
Tamás Hausel, Quaternionic geometry of matroids, Cent. Eur. J. Math. 3 (2005), no. 1, 26–38. MR 2110782, DOI 10.2478/BF02475653
Takayuki Hibi, What can be said about pure $O$-sequences?, J. Combin. Theory Ser. A 50 (1989), no. 2, 319–322. MR 989204, DOI 10.1016/0097-3165(89)90025-3
June Huh, Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs, J. Amer. Math. Soc. 25 (2012), no. 3, 907–927. MR 2904577, DOI 10.1090/S0894-0347-2012-00731-0
J. Huh, $h$-vectors of matroids and logarithmic concavity, preprint. Available on the arXiv.
June Huh and Eric Katz, Log-concavity of characteristic polynomials and the Bergman fan of matroids, Math. Ann. 354 (2012), no. 3, 1103–1116. MR 2983081, DOI 10.1007/s00208-011-0777-6
T.P. Kirkman, On a problem in combinatorics, Cambridge Dublin Math. J. 2 (1847), 191–204.
M. Lenz, The $f$-vector of a realizable matroid complex is strictly log-concave, Comb. Probab. and Computing, to appear. Available on the arXiv.
C. C. Lindner and C. A. Rodger, Design theory, 2nd ed., Discrete Mathematics and its Applications (Boca Raton), CRC Press, Boca Raton, FL, 2009. MR 2469212
F. S. MacAulay, Some Properties of Enumeration in the Theory of Modular Systems, Proc. London Math. Soc. (2) 26 (1927), 531–555. MR 1576950, DOI 10.1112/plms/s2-26.1.531
Juan Migliore, Uwe Nagel, and Fabrizio Zanello, Pure $O$-sequences: known results, applications, and open problems, Commutative algebra, Springer, New York, 2013, pp. 527–550. MR 3051384, DOI 10.1007/978-1-4614-5292-8_{1}6
Ezra Miller and Bernd Sturmfels, Combinatorial commutative algebra, Graduate Texts in Mathematics, vol. 227, Springer-Verlag, New York, 2005. MR 2110098
Richard P. Stanley, Cohen-Macaulay complexes, Higher combinatorics (Proc. NATO Advanced Study Inst., Berlin, 1976) NATO Adv. Study Inst. Ser. C: Math. Phys. Sci., vol. 31, Reidel, Dordrecht, 1977, pp. 51–62. MR 0572989
Richard P. Stanley, Combinatorics and commutative algebra, 2nd ed., Progress in Mathematics, vol. 41, Birkhäuser Boston, Inc., Boston, MA, 1996. MR 1453579
Richard P. Stanley and Fabrizio Zanello, On the rank function of a differential poset, Electron. J. Combin. 19 (2012), no. 2, Paper 13, 17. MR 2928628, DOI 10.37236/2258
Fabrizio Zanello, A non-unimodal codimension 3 level $h$-vector, J. Algebra 305 (2006), no. 2, 949–956. MR 2266862, DOI 10.1016/j.jalgebra.2006.07.009
Fabrizio Zanello, Interval conjectures for level Hilbert functions, J. Algebra 321 (2009), no. 10, 2705–2715. MR 2512622, DOI 10.1016/j.jalgebra.2007.09.030