×

zbMATH — the first resource for mathematics

Stirling numbers in braid matroid Kazhdan-Lusztig polynomials. (English) Zbl 1402.05031
Summary: Restricted Whitney numbers of the first kind appear in the combinatorial recursion for the matroid Kazhdan-Lusztig polynomials. In the special case of braid matroids (the matroid associated to the partition lattice, the complete graph, the type A Coxeter arrangement and the symmetric group) these restricted Whitney numbers are Stirling numbers of the first kind. We use this observation to obtain a formula for the coefficients of the Kazhdan-Lusztig polynomials for braid matroids in terms of sums of products of Stirling numbers of the first kind. This results in new identities between Stirling numbers of the first kind and Stirling numbers of the second kind, as well as a non-recursive formula for the braid matroid Kazhdan-Lusztig polynomials.

MSC:
05B35 Combinatorial aspects of matroids and geometric lattices
05A15 Exact enumeration problems, generating functions
05E10 Combinatorial aspects of representation theory
52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
11B73 Bell and Stirling numbers
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Adamchik, Victor, On Stirling numbers and Euler sums, J. Comput. Appl. Math., 79, 1, 119-130, (1997), MR1437973 · Zbl 0877.39001
[2] Björner, Anders; Brenti, Francesco, Combinatorics of Coxeter Groups, Grad. Texts in Math., vol. 231, (2005), Springer: Springer New York, MR2133266 · Zbl 1110.05001
[3] Brenti, Francesco, P-kernels, IC bases and Kazhdan-Lusztig polynomials, J. Algebra, 259, 2, 613-627, (2003), MR1955535 · Zbl 1022.20015
[4] Elias, Ben; Proudfoot, Nicholas; Wakefield, Max, The Kazhdan-Lusztig polynomial of a matroid, Adv. Math., 299, 36-70, (2016), MR3519463 · Zbl 1341.05250
[5] Gedeon, Katie; Proudfoot, Nicholas; Young, Benjamin, Kazhdan-Lusztig polynomials of matroids: a survey of results and conjectures, Sém. Lothar. Combin., 78B, (2017), Art. 80, 12. MR3678662 · Zbl 1384.05172
[6] Kazhdan, David; Lusztig, George, Representations of Coxeter groups and Hecke algebras, Invent. Math., 53, 2, 165-184, (1979), MR560412 · Zbl 0499.20035
[7] Kazhdan, David; Lusztig, George, Schubert varieties and Poincaré duality, (Geometry of the Laplace Operator, Proc. Sympos. Pure Math.. Geometry of the Laplace Operator, Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979, (1980)), 185-203, MR573434 · Zbl 0461.14015
[8] Proudfoot, Nicholas, The algebraic geometry of Kazhdan-Lusztig-Stanley polynomials, (2018), to appear in EMS Surv.
[9] Orlik, Peter; Terao, Hiroaki, Arrangements of Hyperplanes, Grundlehren Math. Wiss., vol. 300, (1992), Springer-Verlag: Springer-Verlag Berlin, MR1217488 · Zbl 0757.55001
[10] Oxley, James, Matroid Theory, Oxf. Grad. Texts Math., vol. 21, (2011), Oxford University Press: Oxford University Press Oxford, MR2849819 · Zbl 1254.05002
[11] Proudfoot, Nicholas; Xu, Yuan; Young, Benjamin, The Z-polynomial of a matroid, Electron. J. Combin., 25, 1, (2018), Paper 1.26, 21 pp. MR3785005 · Zbl 1380.05022
[12] Proudfoot, Nicholas; Young, Ben, Configuration spaces, \(\operatorname{FS}^{\operatorname{op}}\)-modules, and Kazhdan-Lusztig polynomials of braid matroids, New York J. Math., 23, 813-832, (2017), MR3690232 · Zbl 06747482
[13] Stanley, Richard P., Subdivisions and local h-vectors, J. Amer. Math. Soc., 5, 4, 805-851, (1992), MR1157293 · Zbl 0768.05100
[14] Stanley, Richard P., Enumerative Combinatorics, vol. 1, Cambridge Stud. Adv. Math., vol. 49, (2012), Cambridge University Press: Cambridge University Press Cambridge, MR2868112 · Zbl 1247.05003
[15] Wakefield, Max, A flag Whitney number formula for matroid Kazhdan-Lusztig polynomials, Electron. J. Combin., 25, 1, (2018), Paper 1.22, 14 pp. MR3785001 · Zbl 1380.05023
[16] Welsh, D. J.A., Matroid Theory, London Math. Soc. Monogr. Ser., vol. 8, (1976), Academic Press [Harcourt Brace Jovanovich, Publishers]: Academic Press [Harcourt Brace Jovanovich, Publishers] London-New York, MR0427112 · Zbl 0343.05002
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.