Linear extension sums as valuations on cones
Adrien Boussicault
, Valentin Féray
, Alain Lascoux
and Victor Reiner
DOI: 10.1007/s10801-011-0316-2
Abstract
The geometric and algebraic theory of valuations on cones is applied to understand identities involving summing certain rational functions over the set of linear extensions of a poset.
Pages: 573–610
Keywords: keywords poset; rational function identities; valuation of cones; lattice points; affine semigroup ring; Hilbert series; total residue; root system; weight lattice
Full Text: PDF
References
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17. Knutson, A., Miller, E.: Gröbner geometry of Schubert polynomials. Ann. Math. 161, 1245-1318 (2005)
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2. Barvinok, A.I., Pommersheim, J.E.: An algorithmic theory of lattice points in polyhedra. In: New Perspectives in Algebraic Combinatorics, Berkeley, CA, 1996-1997. Math. Sci. Res. Inst. Publ., vol. 38, pp. 91-147. Cambridge University Press, Cambridge (1999)
3. Berline, N., Vergne, M.: Local Euler-Maclaurin formula for polytopes. Mosc. Math. J. 7(3), 355-386 (2007). Also see p. 573
4. Björner, A.: Topological methods. In: Handbook of Combinatorics, vols. 1, 2, pp. 1819-1872. Amsterdam, Elsevier (1995)
5. Björner, A., Wachs, M.L.: Permutation statistics and linear extensions of posets. J. Comb. Theory, Ser. A 58, 85-114 (1991)
6. Björner, A., Las Vergnas, M., Sturmfels, B., White, N., Ziegler, G.M.: Oriented Matroids, 2nd edn. Encyclopedia of Mathematics and Its Applications, vol.
46. Cambridge University Press, Cambridge (1999)
7. Boussicault, A.: Operations on Posets and Rational Identities of Type A. International Conference on Formal Power Series and Algebraic Combinatorics, vol. 19 (2007)
8. Boussicault, A.: Action du groupe symétrique sur certaines fractions rationnelles suivi de Puissances paires du Vandermonde. Ph.D. thesis (2009). Available at
9. Boussicault, A., Féray, V.: Application of graph combinatorics to rational identities of type A. Electron. J. Comb. 16(1), R145 (2009)
10. Brion, M., Vergne, M.: Arrangements of hyperplanes I. Rational functions and Jeffrey-Kirwan residue. Ann. Sci. Ec. Norm. Super. 32(5), 715-741 (1999)
11. Chapoton, F., Hivert, F., Novelli, J.-C., Thibon, J.-Y.: An operational calculus for the Mould operad. Int. Math. Res. Not. IMRN 9, Art. ID rnn018 (2008). 22 pp.
12. Fomin, S.V., Kirillov, A.N.: The Yang-Baxter equation, symmetric functions, and Schubert polynomials. Discrete Math. 153(1-3), 123-143 (1996)
13. Geissinger, L.: The face structure of a poset polytope. In: Proceedings of the Third Caribbean Conference on Combinatorics and Computing. University West Indies, Barbados (1981)
14. Gessel, I.M.: Multipartite P -partitions and inner products of skew Schur functions. In: Combinatorics and Algebra (Boulder, Colo., 1983). Contemp. Math., vol. 34, pp. 289-317 Am. Math. Soc., Providence (1984)
15. Greene, C.: A rational-function identity related to the Murnaghan-Nakayama formula for the characters of Sn. J. Algebr. Comb. 1(3), 235-255 (1992)
16. Ilyuta, G.: Calculus of linear extensions and Newton interpolation.
17. Knutson, A., Miller, E.: Gröbner geometry of Schubert polynomials. Ann. Math. 161, 1245-1318 (2005)
18. Lascoux, A.: Symmetric Functions and Combinatorial Operators on Polynomials. CBMS Regional Conference Series in Mathematics, vol.
99. Am. Math. Soc., Providence (2003)
19. Macdonald, I.G.: Notes on Schubert polynomials, Publications du LACIM, Univ. du Québec a Montréal (1991)
20. Littlewood, D.E.: The Theory of Group Characters, 2nd edn. AMS, Providence (1950)
21. Miller, E., Sturmfels, B.: Combinatorial Commutative Algebra. Graduate Texts in Mathematics, vol.
227. Springer, New York (2005)
22. Postnikov, A., Reiner, V., Williams, L.: Faces of generalized permutohedra. Doc. Math. 13, 207-273 (2008). [math.CO] J Algebr Comb (2012) 35:573-610
23. Stanley, R.P.: Enumerative Combinatorics, vols. 1,
2. Cambridge Studies in Advanced Mathematics, vols. 49,
62. Cambridge University Press, Cambridge (1997)
24. Stanley, R.P.: Two poset polytopes. Discrete Comput. Geom. 1, 9-23 (1986)
25. Sturmfels, B.: Gröbner Bases and Convex Polytopes. University Lecture Series, vol.
8. Am. Math.
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