The primitive Eulerian polynomial. (English) Zbl 07923207
Summary: We introduce the primitive Eulerian polynomial \(P_{\mathcal{A}}(z)\) of a central hyperplane arrangement \(\mathcal{A}\). It is a reparametrization of its cocharacteristic polynomial. Previous work by the first author implicitly shows that for simplicial arrangements, \(P_{\mathcal{A}}(z)\) has nonnegative coefficients. For reflection arrangements of types A and B, the same work interprets the coefficients of \(P_{\mathcal{A}}(z)\) using the (flag) excedance statistic on (signed) permutations.
The main result of this article is to provide an interpretation of the coefficients of \(P_{\mathcal{A}}(z)\) for all simplicial arrangements using only the geometry and combinatorics of \(\mathcal{A}\). This new interpretation sheds more light to the case of reflection arrangements and, for the first time, gives combinatorial significance to the coefficients of the primitive Eulerian polynomial of the reflection arrangement of type D, for which no well-behaved excedance statistic is known. In type B, we establish a link between the primitive Eulerian polynomial and the \(1/2\)-Eulerian polynomial of C. D. Savage and G. Viswanathan [Electron. J. Comb. 19, No. 1, Research Paper P9, 21 p. (2012; Zbl 1243.05022)]. We present some results and conjectures regarding the real-rootedness of \(P_{\mathcal{A}}(z)\).
The main result of this article is to provide an interpretation of the coefficients of \(P_{\mathcal{A}}(z)\) for all simplicial arrangements using only the geometry and combinatorics of \(\mathcal{A}\). This new interpretation sheds more light to the case of reflection arrangements and, for the first time, gives combinatorial significance to the coefficients of the primitive Eulerian polynomial of the reflection arrangement of type D, for which no well-behaved excedance statistic is known. In type B, we establish a link between the primitive Eulerian polynomial and the \(1/2\)-Eulerian polynomial of C. D. Savage and G. Viswanathan [Electron. J. Comb. 19, No. 1, Research Paper P9, 21 p. (2012; Zbl 1243.05022)]. We present some results and conjectures regarding the real-rootedness of \(P_{\mathcal{A}}(z)\).
MSC:
| 52C35 | Arrangements of points, flats, hyperplanes (aspects of discrete geometry) |
| 05A05 | Permutations, words, matrices |
Citations:
Zbl 1243.05022References:
| [1] | Ron M. Adin, Francesco Brenti, and Yuval Roichman. Descent numbers and major indices for the hyperoctahedral group. Adv. in Appl. Math., 27(2-3):210-224, 2001. Special issue in honor of Dominique Foata’s 65th birthday (Philadelphia, PA, 2000). doi:10.1006/aama.2001.0731. · Zbl 0995.05008 · doi:10.1006/aama.2001.0731 |
| [2] | Marcelo Aguiar and Swapneel Mahajan. Topics in hyperplane arrangements, vol-ume 226 of Mathematical Surveys and Monographs. American Mathematical So-ciety, Providence, RI, 2017. doi:10.1090/surv/226. · Zbl 1388.14146 · doi:10.1090/surv/226 |
| [3] | Ron M. Adin and Yuval Roichman. The flag major index and group actions on polynomial rings. European J. Combin., 22(4):431-446, 2001. doi:10.1006/ eujc.2000.0469. · Zbl 1058.20031 · doi:10.1006/eujc.2000.0469 |
| [4] | Jose Bastidas. The polytope algebra of generalized permutahedra. Algebr. Comb., 4(5):909-946, 2021. doi:10.5802/alco.185. · Zbl 1480.05141 · doi:10.5802/alco.185 |
| [5] | Anders Björner and Francesco Brenti. Combinatorics of Coxeter groups, volume 231 of Graduate Texts in Mathematics. Springer, New York, 2005. doi:10.1007/ 3-540-27596-7. · Zbl 1110.05001 · doi:10.1007/3-540-27596-7 |
| [6] | Anders Björner, Paul H. Edelman, and Günter M. Ziegler. Hyperplane arrange-ments with a lattice of regions. Discrete Comput. Geom., 5(3):263-288, 1990. doi:10.1007/BF02187790. · Zbl 0698.51010 · doi:10.1007/BF02187790 |
| [7] | Eli Bagno and David Garber. On the excedance number of colored permutation groups. Sém. Lothar. Combin., 53:Art. B53f, 17, 2004. URL: https://www.mat. univie.ac.at/ slc/wpapers/s53bagngarb.html. · Zbl 1084.05001 |
| [8] | Robert Gary Bland. Complementary orthogonal subspaces of N-dimensional Eu-clidean space and orientability of matroids. PhD thesis, Cornell University, 1974. URL: https://www.proquest.com/docview/302689515. |
| [9] | BLVS + 99] Anders Björner, Michel Las Vergnas, Bernd Sturmfels, Neil White, and Günter M. Ziegler. Oriented matroids, volume 46 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, second edition, 1999. doi: 10.1017/CBO9780511586507. · Zbl 0944.52006 · doi:10.1017/CBO9780511586507 |
| [10] | Petter Brändén. On operators on polynomials preserving real-rootedness and the Neggers-Stanley conjecture. J. Algebraic Combin., 20(2):119-130, 2004. doi: 10.1023/B:JACO.0000047295.93525.df. · Zbl 1056.05006 · doi:10.1023/B:JACO.0000047295.93525.df |
| [11] | Francesco Brenti. q-Eulerian polynomials arising from Coxeter groups. European J. Combin., 15(5):417-441, 1994. doi:10.1006/eujc.1994.1046. · Zbl 0809.05012 · doi:10.1006/eujc.1994.1046 |
| [12] | Kenneth S. Brown. Semigroups, rings, and Markov chains. J. Theoret. Probab., 13(3):871-938, 2000. doi:10.1023/A:1007822931408. · Zbl 0980.60014 · doi:10.1023/A:1007822931408 |
| [13] | Thomas Brady and Colum Watt. K(π, 1)’s for Artin groups of finite type. In Proceedings of the Conference on Geometric and Combinatorial Group The-ory, Part I (Haifa, 2000), volume 94, pages 225-250, 2002. doi:10.1023/A: 1020902610809. · Zbl 1053.20034 · doi:10.1023/A:1020902610809 |
| [14] | Anders Björner and Michelle L. Wachs. Geometrically constructed bases for ho-mology of partition lattices of type A, B and D. Electron. J. Combin., 11(2):Re-search Paper 3, 26, 2004. doi:10.37236/1860. · Zbl 1064.05151 · doi:10.37236/1860 |
| [15] | Michael Cuntz and István Heckenberger. Finite Weyl groupoids. J. Reine Angew. Math., 702:77-108, 2015. doi:10.1515/crelle-2013-0033. · Zbl 1358.20037 · doi:10.1515/crelle-2013-0033 |
| [16] | Soojin Cho and Kyoungsuk Park. A combinatorial proof of a symmetry of (t, q)-Eulerian numbers of type B and type D. Ann. Comb., 22(1):99-134, 2018. doi: 10.1007/s00026-018-0372-6. · Zbl 1384.05004 · doi:10.1007/s00026-018-0372-6 |
| [17] | Paul H. Edelman. A partial order on the regions of R n dissected by hyperplanes. Trans. Amer. Math. Soc., 283(2):617-631, 1984. doi:10.2307/1999150. · Zbl 0555.06003 · doi:10.2307/1999150 |
| [18] | Leonhard Euler. Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum. 1755. |
| [19] | Dominique Foata and Guo-Niu Han. Signed words and permutations. V. A sextuple distribution. · Zbl 1215.05012 |
| [20] | Ramanujan J., 19(1):29-52, 2009. doi:10.1007/ s11139-007-9059-z. · Zbl 1215.05012 · doi:10.1007/s11139-007-9059-z |
| [21] | Michael Fire. Statistics on wreath products. 2004. arXiv:math/0409421. |
| [22] | Dominique Foata. Étude algébrique de certains problèmes d’analyse combinatoire et du calcul des probabilités. Publ. Inst. Statist. Univ. Paris, 14:81-241, 1965. · Zbl 0133.41304 |
| [23] | Dominique Foata. Eulerian polynomials: from Euler’s time to the present. In The legacy of Alladi Ramakrishnan in the mathematica sciences, pages 253-273. · Zbl 1322.01007 |
| [24] | Springer, New York, 2010. doi:10.1007/978-1-4419-6263-8_15. · Zbl 1322.01007 · doi:10.1007/978-1-4419-6263-8_15 |
| [25] | Georg Frobenius. Über die Bernoullischen Zahlen und die Eulerschen Polynome. Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften, pages 809-847, 1910. · Zbl 1264.11013 |
| [26] | David Geis. On simplicial arrangements in P 3 (R) with splitting polynomial. 2019. arXiv:1902.11185. |
| [27] | Meinolf Geck and Götz Pfeiffer. Characters of finite Coxeter groups and Iwahori-Hecke algebras, volume 21 of London Mathematical Society Monographs. New Series. The Clarendon Press, Oxford University Press, New York, 2000. · Zbl 0996.20004 |
| [28] | Curtis Greene and Thomas Zaslavsky. On the interpretation of Whitney num-bers throug arrangements of hyperplanes, zonotopes, non-Rado partitions, and orientations of graphs. Trans. Amer. Math. Soc., 280(1):97-126, 1983. doi: 10.2307/1999604. · Zbl 0539.05024 · doi:10.2307/1999604 |
| [29] | Michel Jambu and Hiroaki Terao. Free arrangements of hyperplanes and supersolv-able lattices. Adv. in Math., 52(3):248-258, 1984. doi:10.1016/0001-8708(84) 90024-0. · Zbl 0575.05016 · doi:10.1016/0001-8708(84)90024-0 |
| [30] | Adalbert Kerber. Representations of permutation groups. I. Lecture Notes in Math-ematics, Vol. 240. Springer-Verlag, Berlin-New York, 1971. · Zbl 0232.20014 |
| [31] | Michel Las Vergnas. Matroïdes orientables. C. R. Acad. Sci. Paris Sér. A-B, 280:Ai, A61-A64, 1975. · Zbl 0304.05013 |
| [32] | Swapneel Arvind Mahajan. Shuffles, shellings and projections. PhD thesis, Cornell University, 2002. URL: https://www.proquest.com/docview/251796807. |
| [33] | Arnaldo Mandel. Topology of oriented matroids. PhD thesis, University of Water-loo (Canada), 1982. URL: https://www.proquest.com/docview/303277038. |
| [34] | Peter McMullen. The polytope algebra. Adv. Math., 78(1):76-130, 1989. doi: 10.1016/0001-8708(89)90029-7. · Zbl 0686.52005 · doi:10.1016/0001-8708(89)90029-7 |
| [35] | Peter McMullen. On simple polytopes. Invent. Math., 113(2):419-444, 1993. doi: 10.1007/BF01244313. · Zbl 0803.52007 · doi:10.1007/BF01244313 |
| [36] | Shi-Mei Ma and Toufik Mansour. The 1/k-Eulerian polynomials and k-Stirling permutations. Discrete Math., 338(8):1468-1472, 2015. doi:10.1016/j.disc. 2015.03.015. · Zbl 1310.05009 · doi:10.1016/j.disc.2015.03.015 |
| [37] | Shi-Mei Ma and Yeong-Nan Yeh. Eulerian polynomials, Stirling permutations of the second kind and perfect matchings. Electron. J. Combin., 24(4):Paper No. 4.27, 18, 2017. doi:10.37236/7288. · Zbl 1373.05005 · doi:10.37236/7288 |
| [38] | Isabella Novik, Alexander Postnikov, and Bernd Sturmfels. Syzygies of ori-ented matroids. Duke Math. J., 111(2):287-317, 2002. doi:10.1215/ S0012-7094-02-11124-7. · Zbl 1022.13002 · doi:10.1215/S0012-7094-02-11124-7 |
| [39] | The OEIS Foundation Inc. The On-Line Encyclopedia of Integer Sequences. https://oeis.org. |
| [40] | T. Kyle Petersen. Eulerian numbers. Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuse Advanced Texts: Basel Textbooks]. Birkhäuser/Springer, New York, 2015. With a foreword by Richard Stanley. doi:10.1007/ 978-1-4939-3091-3. · Zbl 1337.05001 · doi:10.1007/978-1-4939-3091-3 |
| [41] | Nathan Reading. Lattice congruences, fans and Hopf algebras. J. Combin. Theory Ser. A, 110(2):237-273, 2005. doi:10.1016/j.jcta.2004.11.001. · Zbl 1133.20027 · doi:10.1016/j.jcta.2004.11.001 |
| [42] | Victor Reiner. Quotients of coxeter complexes and P-partitions. PhD thesis, Mas-sachusetts Institute of Technology, 1990. URL: https://www-users.cse.umn. edu/ reiner/Papers/MyThesis.pdf. · Zbl 0751.06002 |
| [43] | The Sage Developers. SageMath, the Sage Mathematics Software System (Version 9.2.0), 2021. https://www.sagemath.org. |
| [44] | Carla D. Savage and Michael J. Schuster. Ehrhart series of lecture hall polytopes and Eulerian polynomials for inversion sequences. J. Combin. Theory Ser. A, 119(4):850-870, 2012. doi:10.1016/j.jcta.2011.12.005. · Zbl 1237.05017 · doi:10.1016/j.jcta.2011.12.005 |
| [45] | G. C. Shephard and J. A. Todd. Finite unitary reflection groups. Canad. J. Math., 6:274-304, 1954. doi:10.4153/cjm-1954-028-3. · Zbl 0055.14305 · doi:10.4153/cjm-1954-028-3 |
| [46] | Richard P. Stanley. An introduction to hyperplane arrangements. In Geometric com-binatorics, volume 13 of IAS/Park City Math. Ser., pages 389-496. Amer. Math. Soc., Providence, RI, 2007. doi:10.1090/pcms/013/08. · Zbl 1136.52009 · doi:10.1090/pcms/013/08 |
| [47] | Einar Steingrimsson. Permutations statistics of indexed and poset permutations. PhD thesis, Massachusetts Institute of Technology, 1992. URL: https://www. proquest.com/docview/304017819. · Zbl 0790.05002 |
| [48] | John R. Stembridge. Coxeter cones and their h-vectors. Adv. Math., 217(5):1935-1961, 2008. doi:10.1016/j.aim.2007.09.002. · Zbl 1201.52010 · doi:10.1016/j.aim.2007.09.002 |
| [49] | Carla D. Savage and Gopal Viswanathan. The 1/k-Eulerian polynomials. Electron. J. Combin., 19(1):Paper 9, 21, 2012. doi:10.37236/16. · Zbl 1243.05022 · doi:10.37236/16 |
| [50] | Carla D. Savage and Mirkó Visontai. The s-Eulerian polynomials have only real roots. Trans. Amer. Math. Soc., 367(2):1441-1466, 2015. doi:10.1090/ S0002-9947-2014-06256-9. · Zbl 1316.05002 · doi:10.1090/S0002-9947-2014-06256-9 |
| [51] | Jacques Tits. Buildings of spherical type and finite BN-pairs. Lecture Notes in Mathematics, Vol. 386. Springer-Verlag, Berlin-New York, 1974. · Zbl 0295.20047 |
| [52] | Eliana Tolosa Villareal. Positive signed sum systems and odd-lecture hall poly-topes. Master’s thesis, San Francisco State University, 2021. |
| [53] | Thomas Zaslavsky. Facing up to arrangements: face-count formulas for partition of space by hyperplanes. Mem. Amer. Math. Soc., 1(issue 1, 154):vii+102, 1975. doi:10.1090/memo/0154. · Zbl 0296.50010 · doi:10.1090/memo/0154 |
| [54] | Günter M. Ziegler. Lectures on polytopes, volume 152 of Graduate Texts in Math-ematics. Springer-Verlag, New York, 1995. doi:10.1007/978-1-4613-8431-1. · Zbl 0823.52002 · doi:10.1007/978-1-4613-8431-1 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
