Actions on permutations and unimodality of descent polynomials

European Journal of Combinatorics

Volumn 29, Issue 2, 2008, Pages 514-531

  Brändén, Petter ^a  

^a UNIVERSITY OF MICHIGAN   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 38549181579     PISSN: 01956698     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.ejc.2006.12.010     Document Type: Article

References (50)

1. *-vectors, Eulerian polynomials and stable polytopes of graphs. Electron. J. Combin. 11 2 (2004-2006) 13 pages. R6, Electronic
2. Babson E., and Steingrímsson E. Generalized permutation patterns and a classification of the Mahonian statistic. Sém. Lothar. Combin. B44b (2000) 18 pages
3. Björner A., and Farley J.D. Chain polynomials of distributive lattices are 75% unimodal. Electron. J. Combin. 12 (2005) 7 pages. Note 4, Electronic
4. Bóna M. Symmetry and unimodality in t-stack sortable permutations. J. Combin. Theory Ser. A 98 1 (2002) 201-209
5. Bóna M. Corrigendum: Symmetry and unimodality in t-stack sortable permutations. J. Combin. Theory Ser. A 99 1 (2002) 191-194
6. Bóna M. A survey of stack-sorting disciplines. Electron. J. Combin. 9 2 (2002-2003) 16 pages. Article 1, Electronic
7. Bousquet-Melou M. Sorted and/or sortable permutations. Discrete Math. 225 (2000) 25-50
8. Brändén P. On operators on polynomials preserving real-rootedness and the Neggers-Stanley conjecture. J. Algebraic Combin. 20 2 (2004) 119-130
9. Brändén P. Sign-graded posets, unimodality of W-polynomials and the Charney-Davis conjecture. Electron. J. Combin. 11 2 (2004-2006) 15 pages. R9, Electronic
10. Brändén P. Counterexamples to the Neggers-Stanley conjecture. Electron. Res. Announc. Amer. Math. Soc. 10 (2004) 155-158 Electronic
11. Brändén P. On linear transformations preserving the Pólya frequency property. Trans. Amer. Math. Soc. 358 (2006) 3697-3716
12. Brändén P., Claesson A., and Steingrímsson E. Catalan continued fractions and increasing subsequences in permutations. Discrete Math. 258 (2002) 275-287
13. Brenti F. Unimodal, log-concave and Pólya frequency sequences in combinatorics. Mem. Amer. Math. Soc. 81 (1989)
14. Carlitz L., and Scoville R. Generalized Eulerian numbers: Combinatorial applications. J. Reine Angew. Math. 265 (1974) 110-137
15. Charney R., and Davis M. The Euler characteristic of a nonpositively curved, piecewise Euclidean manifold. Pacific J. Math. 171 1 (1995) 117-137
16. C.-O. Chow, On certain combinatorial expansions of the Eulerian polynomials. Preprint
17. Corteel S. Crossings and alignments of permutations. Adv. Appl. Math. 38 (2007) 149-163
18. S. Corteel, L. Williams, Tableaux combinatorics for the asymmetric exclusion process, Adv. Appl. Math. (in press)
19. Dukes W.M.B. Permutation statistics on involutions. European J. Combin. 28 (2006) 186-198
20. Foata D. Groupes de réarrangements et nombres d'Euler. C. R. Acad. Sci. Paris Sr. A-B 275 (1972) 1147-1150
21. Foata D., and Han G.N. Arbres minimax et polynomes d'André. Adv. Appl. Math. 27 (2001) 367-389
22. Foata D., and Schützenberger M.P. Théorie géométrique des polynômes eulériens. Lecture Notes in Mathematics vol. 138 (1970), Springer-Verlag, Berlin, New York
23. Foata D., and Strehl V. Rearrangements of the symmetric group and enumerative properties of the tangent and secant numbers. Math. Z. 137 (1974) 257-264
24. Foata D., and Strehl V. Euler numbers and variations of permutations. Colloquio Internazionale sulle Teorie Combinatorie (Roma, 1973), Tomo I. Atti dei Convegni Lincei vol. 17 (1976), Accad. Naz. Lincei, Rome 119-131
25. Gal S.R. Real root conjecture fails for five- and higher-dimensional spheres. Discrete Comput. Geom. 34 2 (2005) 269-284
26. Gasharov V. On the Neggers-Stanley conjecture and the Eulerian polynomials. J. Combin. Theory Ser. A 82 (1998) 134-146
27. I.M. Gessel, 2005 (Private communication)
28. Guo V.J.W., and Zeng J. The Eulerian distribution on involutions is indeed unimodal. J. Combin. Theory Ser. A 113 (2006) 1061-1071
29. Harper L.H. Stirling behavior is asymptotically normal. Ann. Math. Statist. 38 (1967) 410-414
30. Hetyei G., and Reiner E. Permutation trees and variation statistics. European J. Combin. 19 (1998) 847-866
31. Knuth D.E. The art of computer programming. Fundamental Algorithms vol. 1 (1973), Addison-Wesley, Reading, MA
32. Kreweras G. Joint distribution of three descriptive parameters of bridges. Lecture Notes in Mathematics vol. 1234 (1986), Springer, Berlin
33. Marden M. The Geometry of the zeros of a polynomial in a complex variable. Math. Surveys vol. 3 (1949), Amer. Math. Soc., New York, NY 183 p
34. Neggers J. Representations of finite partially ordered sets. J. Combin. Inform. System Sci. 3 (1978) 113-133
35. Reiner V., and Welker V. On the Charney-Davis and Neggers-Stanley conjectures. J. Combin. Theory Ser. A 109 2 (2005) 247-280
36. A. Postnikov, Total positivity, Grassmannians and networks, math.CO/0609764. Preprint
37. A. Postnikov, V. Reiner, L. Williams, Faces of generalized permutohedra, math.CO/0609184. Preprint
38. Sagan B.E. Unimodality and the reflection principle. Ars Combin. 48 (1998) 65-72
39. Simion R. Combinatorial statistics on non-crossing partitions. J. Combin. Theory Ser. A 66 (1994) 270-301
40. Simion R., and Stanton D. Octabasic Laguerre polynomials and permutation statistics. J. Comput. Appl. Math. 68 (1996) 297-329
41. Simion R., and Ullman D. On the structure of the lattice of noncrossing partitions. Discrete Math. 98 (1991) 193-206
42. Steingrímsson E., and Williams L. Permutation tableaux and permutation patterns. J. Combin. Theory Ser. A 114 (2007) 211-234
43. Stembridge J.R. Enriched P-partitions. Trans. Amer. Math. Soc. 349 2 (1997) 763-788
44. Stembridge J.R. Counterexamples to the poset conjectures of Neggers, Stanley, and Stembridge. Trans. Amer. Math. Soc. 359 (2007) 1115-1128
45. Shapiro L.W., Woan W.J., and Getu S. Runs, slides and moments. SIAM J. Algebr. Discr. Methods 4 4 (1983) 459-466
46. Stanley R.P. Ordered structures and partitions. Mem. Amer. Math. Soc. 119 (1972)
47. Strehl V. Symmetric Eulerian distribution for involutions. Sém. Lothar. Combin. 1 (1981)
48. Wagner D.G. Enumeration of functions from posets to chains. European J. Combin. 13 (1992) 313-324
49. Wagner D.G. Total positivity of Hadamard products. J. Math. Anal. Appl. 163 2 (1992) 459-483
50. Williams L. Enumeration of totally positive Grassmann cells. Adv. Math. 190 (2005) 319-342