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Whitney Numbers for Poset Cones

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Abstract

Hyperplane arrangements dissect \(\mathbb {R}^{n}\) into connected components called chambers, and a well-known theorem of Zaslavsky counts chambers as a sum of nonnegative integers called Whitney numbers of the first kind. His theorem generalizes to count chambers within any cone defined as the intersection of a collection of halfspaces from the arrangement, leading to a notion of Whitney numbers for each cone. This paper focuses on cones within the braid arrangement, consisting of the reflecting hyperplanes xi = xj inside \(\mathbb {R}^{n}\) for the symmetric group, thought of as the type An− 1 reflection group. Here,

  • cones correspond to posets,

  • chambers within the cone correspond to linear extensions of the poset,

  • the Whitney numbers of the cone interestingly refine the number of linear extensions of the poset.

We interpret this refinement for all posets as counting linear extensions according to a statistic that generalizes the number of left-to-right maxima of a permutation. When the poset is a disjoint union of chains, we interpret this refinement differently, using Foata’s theory of cycle decomposition for multiset permutations, leading to a simple generating function compiling these Whitney numbers.

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References

  1. Aguiar, M., Mahajan, S.: Topics in hyperplane arrangements. American Mathematica Society (2017)

  2. Bandelt, H.-J., Chepoi, V., Knauer, K.: COMs: complexes of oriented matroids. J. Combin. Theory Ser. A 156, 195–237 (2018)

    Article  MathSciNet  Google Scholar 

  3. Beck, M., Sanyal, R.: Combinatorial reciprocity theorems: An invitation to enumerative geometric combinatorics Graduate Studies in Mathematics 195, American Mathematical Society (2018)

  4. Björner, A., Wachs, M.L.: Generalized quotients in Coxeter groups. Trans. Amer. Math. Soc. 308, 1–37 (1988)

    Article  MathSciNet  Google Scholar 

  5. Björner, A., Wachs, M.L.: q-hook length formulas for forests. J. Combin Theory Ser. A 52, 165–187 (1989)

    Article  MathSciNet  Google Scholar 

  6. Björner, A., Wachs, M.L.: Permutation statistics and linear extensions of posets. J. Combin. Theory Ser. A 58, 85–114 (1991)

    Article  MathSciNet  Google Scholar 

  7. Brändén, P.: q-Narayana numbers and the flag h-vector of J(2 ×n). Discrete Math. 281(1-3), 67–81 (2004)

    Article  MathSciNet  Google Scholar 

  8. Brown, K.: Semigroups, rings, and markov chains. J. Theor. Probab. 13(3), 342–351 (2000)

    Article  MathSciNet  Google Scholar 

  9. Foata, D.: Etude algébrique de certains problèmes d’analyse combinatoire et du calcul des probabilités. PhD thesis, Publ. Inst. Statist. Univ. Paris (1965)

  10. Foata, D., Cartier, P.: Problèmes Combinatoires De Commutation et réarrangements. Springer Verlag, Lecture Notes in Mathematics (1969)

  11. Garoufalidis, S., Lê, T.T.Q., Zeilberger, D.: The Quantum MacMahon Master Theorem. J. Algebra (2003)

  12. Gessel, I. M.: Multipartite P-partitions and inner products of skew Schur functions. Combinatorics and algebra (Boulder, Colo., 1983), 289–317, Contemp. Math. 34, Amer. Math. Soc., Providence, RI (1984)

  13. Gente, R.: The Varchenko Matrix for Cones. PhD thesis, Universität Marburg (2013)

  14. Humphreys, J.: Reflection Groups and Coxeter Groups Cambridge Studies in Advanced Mathematics, vol. 29. Cambridge University Press, Cambridge (1990)

    Book  Google Scholar 

  15. Knuth, D.E.: The art of computer programming. Addison-Wesley (2015)

  16. Konvalinka, Matjaz, Pak, Igor: Non-commutative Extensions of macMahon’s Master Theorem. Adv. Math. 216(1), 29–61 (2007)

    Article  MathSciNet  Google Scholar 

  17. Lehrer, G., Solomon, L.: On the action of the symmetric group on the cohomology of the complement of its reflecting hyperplanes. J. Algebra 104(2), 410–424 (1986)

    Article  MathSciNet  Google Scholar 

  18. Orlik, P., Solomon, L.: Unitary reflection groups and cohomology. Invent. Math. 59(1), 77–94 (1980)

    Article  MathSciNet  Google Scholar 

  19. Orlik, P., Terao, H.: Arrangements of hyperplanes, volume 300 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (1992)

    Google Scholar 

  20. Postnikov, A., Reiner, V., Williams, L.: Faces of generalized permutohedra. Doc. Math. 13, 207–273 (2008)

    MathSciNet  MATH  Google Scholar 

  21. Reiner, V.: Quotients of Coxeter complexes and P-partitions. Mem. Amer. Math. Soc. 460, 95 (1992)

    MathSciNet  MATH  Google Scholar 

  22. Reiner, V., Saliola, F., Welker, V.: Spectra of symmetrized shuffling operators. (English summary). Mem. Amer. Math. Soc. 1072, 228 (2014)

    MATH  Google Scholar 

  23. Reiner, V., Welker, V.: On the charney-Davis and neggers-Stanley conjectures. J. Comb. Theory Ser. A 109(2), 247–280 (2005)

    Article  MathSciNet  Google Scholar 

  24. Rényi, A.: Théorie des éléments saillants d?une suite d?observations. Ann. Fac. Sci. Univ. Clermont-Ferrand 8, 7–13 (1962)

    MathSciNet  Google Scholar 

  25. Saito, K.: Principal Γ-cone for a tree. Adv. Math. 212(2), 645–668 (2007)

    Article  MathSciNet  Google Scholar 

  26. Stanley, R.: Enumerative Combinatorics, vol. 2. Cambridge University Press, Cambridge (1999)

    Book  Google Scholar 

  27. Stanley, R.: An introduction to hyperplane arrangements. Geometric Combinatorics IAS/Park City Mathematics Series, pp. 389–496 (2007)

  28. Stanley, R.: Enumerative Combinatorics, 2nd edn., vol. 1. Cambridge University Press, New York (2012)

    Google Scholar 

  29. Steinberg, R.: Differential equations invariant under finite reflection groups. Trans. Amer. Math. Soc. 112, 392–400 (1964)

    Article  MathSciNet  Google Scholar 

  30. Stembridge, J.: Coxeter cones and their h-vectors. Adv. Math. 217 (5), 1935–1961 (2008)

    Article  MathSciNet  Google Scholar 

  31. Zaslavsky, T.: Facing up to arrangements: face-count formulas for partitions of space by hyperplanes. Mem. Amer. Math. Soc. 1(1), 154 (1975)

    MathSciNet  MATH  Google Scholar 

  32. Zaslavsky, T.: A combinatorial analysis of topological dissections. Adv. Math. 25(3), 267–285 (1977)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors gratefully acknowledge Dennis Stanton for conversations about MacMahon’s Master Theorem, as well as Anders Björner, Jesus DeLoera, Theo Douvropoulos, Michael Falk, Ira Gessel, Benjamin Steinberg, Volkmar Welker, Chi-Ho Yuen for enlightening discussions and references. They thank Philip Zhang for asking them questions about real-rootedness at the 2019 Mid-Atlantic Algebra, Geometry, and Combinatorics Workshop. Finally, the authors thank an anonymous referee for helpful comments.

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Authors and Affiliations

  1. School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

    Galen Dorpalen-Barry & Victor Reiner

  2. Department of Mathematics, Sungkyunkwan University, Suwon, Gyeonggi-do, 16419, South Korea

    Jang Soo Kim

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  1. Galen Dorpalen-Barry
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  3. Victor Reiner
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Correspondence to Galen Dorpalen-Barry.

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The first and third authors were partially supported by NSF grant DMS-1601961. The second author was supported by NRF grants #2019R1F1A1059081 and #2016R1A5A1008055.

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Dorpalen-Barry, G., Kim, J. & Reiner, V. Whitney Numbers for Poset Cones. Order 38, 283–322 (2021). https://doi.org/10.1007/s11083-020-09541-4

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