Density of $f$-ideals and $f$-ideals in mixed small degrees
DOI:
https://doi.org/10.7146/math.scand.a-129244Abstract
A squarefree monomial ideal is called an $f$-ideal if its Stanley–Reisner and facet simplicial complexes have the same $f$-vector. We show that $f$-ideals generated in a fixed degree have asymptotic density zero when the number of variables goes to infinity. We also provide novel algorithms to construct $f$-ideals generated in small degrees.
References
Abbasi, G. Q., Ahmad, S., Anwar, I., and Baig, W. A., $f$-ideals of degree $2$, Algebra Colloq. 19 (2012), no. 1, 921–926. https://doi.org/10.1142/S1005386712000788
Anwar, I., Mahmood, H., Binyamin, M. A., and Zafar, M. K., On the characterization of $f$-ideals, Comm. Algebra 42 (2014), no. 9, 3736–3741. https://doi.org/10.1080/00927872.2013.792092
Budd, S., and Van Tuyl, A., Newton complementary dual of $f$-ideals, Canad. Math. Bull. 62 (2019), no. 2, 231–241. https://doi.org/10.4153/s0008439518000024
Faridi, S., The facet ideal of a simplicial complex, Manuscripta Math. 109 (2002), no. 2, 159–174. https://doi.org/10.1007/s00229-002-0293-9
bibitem tswu2 Guo, J., and Wu, T., On the $(n,d)^th$ $f$-ideals, J. Korean Math. Soc. 52 (2015), no. 4, 685–697. https://doi.org/10.4134/JKMS.2015.52.4.685
Guo, J., Wu, T., and Liu, Q., $f$-ideals and $f$-graphs, Comm. Algebra 45 (2017), no. 8, 3207–3220.
Liu, A-M., Guo, J., and Wu, T., The Cohen–Macaulay property of $f$-ideals, texttt arxiv:2010.04317v2.
Mahmood, H., Anwar, I., and Zafar, M. K., A construction of Cohen–Macaulay $f$-graphs, J. Algebra Appl. 13 (2014), no. 6, 1450012, 7 pp. https://doi.org/10.1142/S0219498814500121
Mahmood, H., Anwar, I., Binyamin, M. A., and Yasmeen, S., On the connectedness of $f$-simplicial complexes, J. Algebra Appl. 16 (2017), no. 1, 1750017, 9 pp. https://doi.org/10.1142/S0219498817500177
Mahmood, H., Ur Rehman, F., and Binyamin, M. A., A note on $f$-graphs J. Algebra Appl. 19 (2020), no. 10, 2050193, 9 pp. https://doi.org/10.1142/S0219498820501935
Mahmood, H., Ur Rehman, F., Nguyen, T. T., and Binyamin, M. A., Quasi $f$-ideals, texttt arxiv:2009.04773v1.
Mahmood, H., Ur Rehman, F., Nguyen, T. T., and Binyamin, M. A., Quasi $f$-simplicial complexes and quasi $f$-graphs, texttt arxiv:2009.03641v1.
Stanley, R. P., Combinatorics and commutative algebra, Second edition. Progress in Mathematics, 41. Birkhäuser Boston, Inc., Boston, MA, 1996.
