2000 character limit reached
On a q-analogue of the Zeta polynomial of posets
Published 19 Feb 2024 in math.CO | (2402.11979v2)
Abstract: We introduce a q-analogue of the classical Zeta polynomial of finite partially ordered sets, as a polynomial in one variable x with coefficients depending on the indeterminate q. We prove some properties of this polynomial invariant, including its behaviour with respect to duality, product and disjoint union. The leading term is a q-analogue of the number of maximal chains, but not always with non-negative coefficients. The value at q=0 turns out to be essentially the characteristic polynomial.
- Shellability of noncrossing partition lattices. Proc. Amer. Math. Soc., 135(4):939–949, 2007.
- Chains in shard intersection lattices and parabolic support posets. J. Comb., 9(2):309–325, 2018.
- Posets related to the connectivity set of Coxeter groups. J. Algebra, 303(2):831–846, 2006.
- David Bessis. The dual braid monoid. Ann. Sci. École Norm. Sup. (4), 36(5):647–683, 2003.
- Linear inequalities for flags in graded partially ordered sets. J. Combin. Theory Ser. A, 89(1):77–104, 2000.
- Anders Björner. Shellable and Cohen-Macaulay partially ordered sets. Trans. Amer. Math. Soc., 260(1):159–183, 1980.
- K(π,1)𝐾𝜋1K(\pi,1)italic_K ( italic_π , 1 )’s for Artin groups of finite type. In Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part I (Haifa, 2000), volume 94, pages 225–250, 2002.
- Non-crossing partition lattices in finite real reflection groups. Trans. Amer. Math. Soc., 360(4):1983–2005, 2008.
- Frédéric Chapoton. q𝑞qitalic_q-analogues of Ehrhart polynomials. Proc. Edinb. Math. Soc. (2), 59(2):339–358, 2016.
- The SageMath Developers. sagemath/sage: 10.0, 2023.
- Ira M. Gessel. A historical survey of P𝑃Pitalic_P-partitions. In The mathematical legacy of Richard P. Stanley, pages 169–188. Amer. Math. Soc., Providence, RI, 2016.
- Jon McCammond. Noncrossing partitions in surprising locations. Amer. Math. Monthly, 113(7):598–610, 2006.
- T. Kyle Petersen. On the shard intersection order of a Coxeter group. SIAM J. Discrete Math., 27(4):1880–1912, 2013.
- Nathan Reading. Noncrossing partitions and the shard intersection order. In 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009), volume AK of Discrete Math. Theor. Comput. Sci. Proc., pages 745–756. Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2009.
- Nathan Reading. Noncrossing partitions and the shard intersection order. J. Algebraic Combin., 33(4):483–530, 2011.
- Richard P. Stanley. Ordered structures and partitions. American Mathematical Society, Providence, R.I., 1972. Memoirs of the American Mathematical Society, No. 119.
- Richard P. Stanley. Two poset polytopes. Discrete Comput. Geom., 1:9–23, 1986.
- Richard P. Stanley. Subdivisions and local hℎhitalic_h-vectors. J. Am. Math. Soc., 5(4):805–851, 1992.
- Richard P. Stanley. Enumerative combinatorics. Volume 1, volume 49 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, second edition, 2012.
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.