Illinois Journal of Mathematics

The unimodality of pure $O$-sequences of type three in three variables

Bernadette Boyle

Full-text: Open access


Since the 1970’s, great interest has been taken in the study of pure $O$-sequences, which are in bijective correspondence to the Hilbert functions of Artinian level monomial algebras. Much progress has been made in classifying these by their shape. It has been shown that all monomial complete intersections, Artinian algebras in two variables and Artinian level monomial algebras with type two in both three and four variables have unimodal Hilbert functions. This paper proves that Artinian level monomial algebras of type three in three variables have unimodal Hilbert functions. We will also discuss the licciness of these algebras.

Article information

Illinois J. Math., Volume 58, Number 3 (2014), 757-778.

Received: 18 June 2014
Revised: 17 December 2014
First available in Project Euclid: 9 September 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 13D40: Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series 13E10: Artinian rings and modules, finite-dimensional algebras
Secondary: 13C40: Linkage, complete intersections and determinantal ideals [See also 14M06, 14M10, 14M12] 13F20: Polynomial rings and ideals; rings of integer-valued polynomials [See also 11C08, 13B25] 05E40: Combinatorial aspects of commutative algebra


Boyle, Bernadette. The unimodality of pure $O$-sequences of type three in three variables. Illinois J. Math. 58 (2014), no. 3, 757--778. doi:10.1215/ijm/1441790389.

Export citation


  • A. Bigatti, A. Geramita and J. Migliore, Geometric consequences of extremal behavior in a theorem of Macaulay, Trans. Amer. Math. Soc. 346 (1994), no. 1, 203–235.
  • M. Boij, J. Migliore, R. Miró-Roig, U. Nagel and F. Zanello, On the shape of a pure $O$-sequence, Mem. Amer. Math. Soc. 218 (2012), no. 2024.
  • B. Boyle, The unimodality of pure $O$-sequences of type two in four variables, to appear in Rocky Mt. J. Math.
  • H. Brenner and A. Kaid, Syzygy bundles on $\mathbb{P}$ and the weak Lefschetz property, Illinois J. Math. 51 (2007), 1299–1308.
  • M. K. Chari, Two decompositions in topological combinatorics with applications to matroid complexes, Trans. Amer. Math. Soc. 349 (1997), no. 10, 3925–3943.
  • D. Cook \bsuffixII and U. Nagel, The weak Lefschetz property, monomial ideals, and lozenges, Illinois J. Math. 55 (2011), no. 1, 377–395.
  • A. V. Geramita, Inverse systems of fat points: Waring's problem, secant varieties and Veronese varieties and parametric spaces of Gorenstein ideals, The Curves Seminar at Queen's, vol. X, Queen's Papers in Pure and Appl. Math., vol. 102, 1996, pp. 3–114.
  • A. V. Geramita, T. Harima, J. Migliore and Y. S. Shin, The Hilbert function of a level algebra, Mem. Amer. Math. Soc. 186 (2007), no. 872.
  • A. V. Geramita and J. Migliore, A generalized liaison addition, J. Algebra 163 (1994), 139–164.
  • T. Harima, J. Migliore, U. Nagel and J. Watanabe, The weak and strong Lefschetz properties for Artinian $K$-algebras, J. Algebra 262 (2003), 99–126.
  • T. Hausel, Quaternionic geometry of matroids, Cent. Eur. J. Math. 3 (2005), no. 1, 26–38.
  • T. Hibi, What can be said about pure $O$-sequences? J. Combin. Theory Ser. A 50 (1989), no. 2, 319–322.
  • C. Huneke and B. Ulrich, Liaison of monomial ideals, Bull. Lond. Math. Soc. 39 (2007), 384–392.
  • A. Iarrobino and V. Kanev, Power sums, Gorenstein algebras, and determinantal loci, Lecture Notes in Mathematics, vol. 1721, Springer, Heidelberg, 1999.
  • P. Kaski, P. Östergård and R. J. Patric, Classification algorithms for codes and designs, Algorithms and Computation in Mathematics, vol. 15, Springer, Berlin, 2006.
  • J. Kleppe, R. Miró-Roig, J. Migliore, U. Nagel and C. Peterson, Gorenstein liaison, complete intersection liaison invariants and unobstructedness, Mem. Amer. Math. Soc. 154 (2001), no. 732.
  • C. Merino, S. D. Noble, M. Ramírez-Ibáñez and R. Villarroel-Flores, On the structure of the $h$-vector of a paving matroid, European J. Combin. 33 (2012), no. 8, 1787–1799.
  • J. Migliore, Introduction to liaison theory and deficiency modules, Progress in Mathematics, vol. 165, Birkhäuser, Boston, 1998.
  • J. Migliore and U. Nagel, Monomial ideals and the Gorenstein liaison class of a complete intersection, Compos. Math. 133 (2002), 25–36.
  • U. Nagel, Even liaison classes generated by Gorenstein linkage, J. Algebra 209 (1998), 543–584.
  • S. Oh, Generalized permutohedra, $h$-vectors of cotransversal matroids and pure $O$-sequences, Electron. J. Combin. 20 (2013), no. 3, \bnumberP14.
  • L. Reid, L. Roberts and M. Roitman, On complete intersections and their Hilbert functions, Canad. Math. Bull. 34 (1991), no. 4, 525–535.
  • J. Schweig, On the $h$-vector of a lattice path matroid, Electron. J. Combin. 17 (2010), no. 1, \bnumberN3.
  • R. Stanley, Cohen–Macaulay complexes, Higher combinatorics (M. Aigner, ed.), Reidel, Dordrecht, 1977, pp. 51–62.
  • R. Stanley, Weyl groups, the hard Lefschetz theorem, and the Sperner property, SIAM J. Algebr. Discrete Methods 1 (1980), 168–184.
  • E. Stokes, The $h$-vectors of 1-dimensional matroid complexes and a conjecture of Stanley, preprint; available at
  • J. Watanabe, The Dilworth number of Artinian rings and finite posets with rank function, Commutative algebra and combinatorics, Advanced Studies in Pure Math., vol. 11, Kinokuniya Co. North Holland, Amsterdam, 1987.
  • F. Zanello, A non-unimodal codimesion three level $h$-vector, J. Algebra 305 (2006), no. 2, 949–956.