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no. 5
Matroidal mixed Eulerian numbers
Katz, Eric1; Kutler, Max1
1 The Ohio State University Department of Mathematics 231 West 18th Avenue Columbus OH 43210-1174
Algebraic Combinatorics, Volume 7 (2024) no. 5, pp. 1479-1506.

We make a systematic study of matroidal mixed Eulerian numbers which are certain intersection numbers in the matroid Chow ring generalizing the mixed Eulerian numbers introduced by Postnikov. These numbers are shown to be valuative and obey a log-concavity relation. We establish recursion formulas and use them to relate matroidal mixed Eulerian numbers to the characteristic and Tutte polynomials, reproving results of Huh–Katz and Berget–Spink–Tseng. Generalizing Postnikov, we show that these numbers are equal to certain weighted counts of binary trees. Lastly, we study these numbers for perfect matroid designs, proving that they generalize the remixed Eulerian numbers of Nadeau–Tewari.

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DOI: 10.5802/alco.382
Classification: 05B35, 05A15, 05E99
Keywords: Eulerian numbers, Chow rings of matroids, log concavity, Tutte polynomials

Katz, Eric 1; Kutler, Max 1

1 The Ohio State University Department of Mathematics 231 West 18th Avenue Columbus OH 43210-1174
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Katz, Eric; Kutler, Max. Matroidal mixed Eulerian numbers. Algebraic Combinatorics, Volume 7 (2024) no. 5, pp. 1479-1506. doi : 10.5802/alco.382. https://alco.centre-mersenne.org/articles/10.5802/alco.382/

[1] Adiprasito, Karim; Huh, June; Katz, Eric Hodge theory for combinatorial geometries, Ann. of Math. (2), Volume 188 (2018) no. 2, pp. 381-452 | DOI | MR | Zbl

[2] Berget, Andrew; Eur, Christopher; Spink, Hunter; Tseng, Dennis Tautological classes of matroids, Invent. Math., Volume 233 (2023) no. 2, pp. 951-1039 | DOI | MR | Zbl

[3] Berget, Andrew; Spink, Hunter; Tseng, Dennis Log-concavity of matroid h-vectors and mixed Eulerian numbers, Duke Math. J., Volume 172 (2023) no. 18, pp. 3475-3520 | DOI | MR | Zbl

[4] Bjorner, Anders The homology and shellability of matroids and geometric lattices, Matroid applications (Encyclopedia Math. Appl.), Volume 40, Cambridge Univ. Press, Cambridge, 1992, pp. 226-283 | DOI | MR | Zbl

[5] Braden, Tom; Huh, June; Matherne, Jacob P.; Proudfoot, Nicholas; Wang, Botong A semi-small decomposition of the Chow ring of a matroid, Adv. Math., Volume 409 (2022), Paper no. 108646, 49 pages | DOI | MR | Zbl

[6] Brion, Michel Piecewise polynomial functions, convex polytopes and enumerative geometry, Parameter spaces (Warsaw, 1994) (Banach Center Publ.), Volume 36, Polish Acad. Sci. Inst. Math., Warsaw, 1996, pp. 25-44 | MR | Zbl

[7] Brylawski, Thomas; Oxley, James The Tutte polynomial and its applications, Matroid applications (Encyclopedia Math. Appl.), Volume 40, Cambridge Univ. Press, Cambridge, 1992, pp. 123-225 | DOI | MR | Zbl

[8] Cox, David A.; Little, John B.; Schenck, Henry K. Toric varieties, Graduate Studies in Mathematics, 124, American Mathematical Society, Providence, RI, 2011, xxiv+841 pages | DOI | MR

[9] Dawson, Jeremy E. A collection of sets related to the Tutte polynomial of a matroid, Graph theory, Singapore 1983 (Lecture Notes in Math.), Volume 1073, Springer, Berlin, 1984, pp. 193-204 | DOI | MR | Zbl

[10] Derksen, Harm; Fink, Alex Valuative invariants for polymatroids, Adv. Math., Volume 225 (2010) no. 4, pp. 1840-1892 | DOI | MR

[11] Deza, M. Perfect matroid designs, Matroid applications (Encyclopedia Math. Appl.), Volume 40, Cambridge Univ. Press, Cambridge, 1992, pp. 54-72 | DOI | MR | Zbl

[12] Deza, M.; Singhi, N. M. Some properties of perfect matroid designs, Ann. Discrete Math., Volume 6 (1980), pp. 57-76 | DOI | MR | Zbl

[13] Edidin, Dan; Graham, William Equivariant intersection theory, Invent. Math., Volume 131 (1998) no. 3, pp. 595-634 | DOI | MR

[14] Eur, Christopher The Geometry of Divisors on Matroids, Ph. D. Thesis, University of California, Berkeley (2020)

[15] Feichtner, Eva Maria; Yuzvinsky, Sergey Chow rings of toric varieties defined by atomic lattices, Invent. Math., Volume 155 (2004) no. 3, pp. 515-536 | DOI | MR | Zbl

[16] Fulton, William; Sturmfels, Bernd Intersection theory on toric varieties, Topology, Volume 36 (1997) no. 2, pp. 335-353 | DOI | MR | Zbl

[17] Horiguchi, Tatsuya Mixed Eulerian numbers and Peterson Schubert calculus, Int. Math. Res. Not. IMRN (2024) no. 2, pp. 1422-1471 | DOI | MR | Zbl

[18] Huh, June Tropical geometry of matroids, Current developments in mathematics 2016, Int. Press, Somerville, MA, 2018, pp. 1-46 | MR | Zbl

[19] Huh, June; Katz, Eric Log-concavity of characteristic polynomials and the Bergman fan of matroids, Math. Ann., Volume 354 (2012) no. 3, pp. 1103-1116 | DOI | MR | Zbl

[20] Katz, Eric; Payne, Sam Piecewise polynomials, Minkowski weights, and localization on toric varieties, Algebra Number Theory, Volume 2 (2008) no. 2, pp. 135-155 | DOI | MR | Zbl

[21] Nadeau, Philippe; Tewari, Vasu A q-deformation of an algebra of Klyachko and Macdonald’s reduced word formula, 2021 | arXiv

[22] Nadeau, Philippe; Tewari, Vasu Remixed Eulerian numbers, Forum Math. Sigma, Volume 11 (2023), Paper no. e65, 26 pages | DOI | MR | Zbl

[23] Postnikov, Alexander Permutohedra, associahedra, and beyond, Int. Math. Res. Not. IMRN (2009) no. 6, pp. 1026-1106 | DOI | MR | Zbl

[24] Stanley, Richard P. Enumerative combinatorics. Volume 1, Cambridge Studies in Advanced Mathematics, 49, Cambridge University Press, Cambridge, 2012, xiv+626 pages | MR

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