Skip to main content
SpringerLink
Log in

Certain variants of multipermutohedron ideals

  • Published:
Proceedings - Mathematical Sciences Aims and scope Submit manuscript

Abstract

Multipermutohedron ideals have rich combinatorial properties. An explicit combinatorial formula for the multigraded Betti numbers of a multipermutohedron ideal and their Alexander duals are known. Also, the dimension of the Artinian quotient of an Alexander dual of a multipermutohedron ideal is the number of generalized parking functions. In this paper, monomial ideals which are certain variants of multipermutohedron ideals are studied. Multigraded Betti numbers of these variant monomial ideals and their Alexander duals are obtained. Further, many interesting combinatorial properties of multipermutohedron ideals are extended to these variant monomial ideals.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bayer D, Peeva I and Sturmfels B, Monomial resolutions, Math. Res. Lett. 5 (1998) 31–46

    Article  MathSciNet  MATH  Google Scholar 

  2. Bayer D and Sturmfels B, Cellular resolution of monomial modules, J. für die Reine und Angewandte Mathematik 502 (1998) 123–140

    MathSciNet  MATH  Google Scholar 

  3. Kumar A and Kumar C, Multigraded Betti numbers of multipermutohedron ideals, J. Ramanujan Math. Soc. 28 (1) (2013) 1–18

    MathSciNet  MATH  Google Scholar 

  4. Kumar A and Kumar C, Alexander duals of multipermutohedron ideals, Proc. Indian Acad. Sci. (Math Sci.) 124 (1) (2014) 1–15

    Article  MathSciNet  MATH  Google Scholar 

  5. Manjunath M and Sturmfels B, Monomials, binomials and Riemann-Roch, J. Algebraic Comb. 37 (4) (2013) 737–756

    Article  MathSciNet  MATH  Google Scholar 

  6. Mohammadi F and Shokrieh F, Divisors on graphs, binomial and monomial ideals, and cellular resolutions, Math. Z. 283 (1) (2016) 59–102

    Article  MathSciNet  MATH  Google Scholar 

  7. Miller E and Sturmfels B, Combinatorial commutative algebra, Graduate Texts in Mathematics 227 (2004) (Springer)

  8. Miller E, Alexander duality for monomial ideals and their resolutions, Rejecta Mathematica 1 (1) (2009) 18–57

    Google Scholar 

  9. Pitman J and Stanley R, A polytope related to empirical distributions, plane trees, parking functions, and the associahedron, Discret. Comput. Geom. 27 (2002) 603–634

    Article  MathSciNet  MATH  Google Scholar 

  10. Postnikov A and Shapiro B, Trees, parking functions, syzygies, and deformation of Monomial ideals, Trans. Am. Math. Soc. 356 (2004) 3109–3142

    Article  MathSciNet  MATH  Google Scholar 

  11. Yan C H, On the enumeration of generalized parking functions, in: Proceedings of the 31st Southeastern International Conference on Combinatorics, Graph Theory and Computing (2000) (Boca Raton, FL, 2000) Congressus Numerantium, vol. 147, pp. 201–209

  12. Yan C H, Generalized parking functions, tree inversions, and multicolored graphs, Adv. Appl. Math. 27 (2–3) (2001) 641–670

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

Thanks are due to the anonymous referee for many valuable suggestions that improved the overall presentation of the paper. The first author is thankful to CSIR, Govt. of India for financial support.

Author information

Author notes

  1. AJAY KUMAR

    Present address: DAV University, Jalandhar, Punjab, 144 012, India

Authors and Affiliations

  1. Indian Institute of Science Education and Research, Mohali, Knowledge City, Sector 81, SAS Nagar, Punjab, 140 306, India

    AJAY KUMAR & CHANCHAL KUMAR

Authors
  1. AJAY KUMAR
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. CHANCHAL KUMAR
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to CHANCHAL KUMAR.

Additional information

Communicating Editor: B V Rajarama Bhat

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

KUMAR, A., KUMAR, C. Certain variants of multipermutohedron ideals. Proc Math Sci 126, 479–500 (2016). https://doi.org/10.1007/s12044-016-0313-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12044-016-0313-4

Keywords

2010 Mathematics Subject Classification.

Search

Navigation