Transactions of the American Mathematical Society

Author(s): Petter Brändén
Journal: Trans. Amer. Math. Soc. 358 (2006), 3697-3716.
MSC (2000): Primary 05A15, 26C10; Secondary 05A19, 05A05, 20F55
Posted: February 20, 2006
Retrieve article in: PDF

Abstract: We prove that certain linear operators preserve the Pólya frequency property and real-rootedness, and apply our results to settle some conjectures and open problems in combinatorics proposed by Bóna, Brenti and Reiner-Welker.

1.: M. Bóna, Corrigendum: ``Symmetry and unimodality in -stack sortable permutations'', J. Combin. Theory Ser. A 99 (2002), no. 1, 191-194. MR 1911466 (2003g:05007b)
2.: -, A survey of stack-sorting disciplines, Electron. J. Combin. 9(2) (2002). MR 2028290 (2004j:05012)
3.: -, Symmetry and unimodality in -stack sortable permutations, J. Combin. Theory Ser. A 98 (2002), no. 1, 201-209. MR 1897934 (2003g:05007a)
4.: M. Bousquet-Mélou, Multi-statistic enumeration of two-stack sortable permutations, Electron. J. Combin. 5 (1998), no. 1, Research Paper 21, 12 pp. (electronic). MR 1614300 (99b:05001)
5.: P. Brändén, On operators on polynomials preserving real-rootedness and the Neggers-Stanley conjecture, J. Algebraic Comb. 20 (2004), no. 2, 119-130. MR 2104673
6.: F. Brenti, Unimodal, log-concave and Pólya frequency sequences in combinatorics, Mem. Amer. Math. Soc. 81 (1989), no. 413, viii+106. MR 0963833 (90d:05014)
7.: -, Log-concave and unimodal sequences in algebra, combinatorics, and geometry: an update, Jerusalem combinatorics '93, Contemp. Math., vol. 178, Amer. Math. Soc., Providence, RI, 1994, pp. 71-89. MR 1310575 (95j:05026)
8.: -, -Eulerian polynomials arising from Coxeter groups, European J. Combin. 15 (1994), no. 5, 417-441. MR 1292954 (95i:05013)
9.: -, A class of -symmetric functions arising from plethysm, J. Combin. Theory Ser. A 91 (2000), no. 1-2, 137-170. MR 1779778 (2001i:05153)
10.: F. Brenti, G. F. Royle, and D. G. Wagner, Location of zeros of chromatic and related polynomials of graphs, Canad. J. Math. 46 (1994), no. 1, 55-80. MR 1260339 (94k:05077)
11.: L. Comtet, Advanced combinatorics, enlarged ed., D. Reidel Publishing Co., Dordrecht, 1974. MR 0460128 (57:124)
12.: T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977), no. 4, 801-817. MR 0568321 (58:27921)
13.: J. Dedieu, Obreschkoff's theorem revisited: what convex sets are contained in the set of hyperbolic polynomials?, J. Pure Appl. Algebra 81 (1992), no. 3, 269-278. MR 1179101 (93g:12001)
14.: S. Dulucq, S. Gire, and O. Guibert, A combinatorial proof of J. West's conjecture, Discrete Math. 187 (1998), no. 1-3, 71-96. MR 1630680 (99f:05053)
15.: A. Edrei, On the generating functions of totally positive sequences. II, J. Analyse Math. 2 (1952), 104-109. MR 0053175 (14:732e)
16.: D. Foata and M. Schützenberger, Théorie géométrique des polynômes eulériens, Lecture Notes in Mathematics, Vol. 138, Springer-Verlag, Berlin, 1970. MR 0272642 (42:7523)
17.: S.V. Fomin and A.V. Zelevinsky, Y-systems and generalized associahedra, Ann. of Math. 158 (2003), no. 158, 977-1018. MR 2031858 (2004m:17010)
18.: J. Garloff and D. G. Wagner, Hadamard products of stable polynomials are stable, J. Math. Anal. Appl. 202 (1996), no. 3, 797-809. MR 1408355 (97e:30010)
19.: I. P. Goulden and J. West, Raney paths and a combinatorial relationship between rooted nonseparable planar maps and two-stack-sortable permutations, J. Combin. Theory Ser. A 75 (1996), no. 2, 220-242. MR 1401000 (98a:05082)
20.: G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, University Press, Cambridge, 1952, 2d ed. MR 0046395 (13:727e)
21.: B. Jacquard and G. Schaeffer, A bijective census of nonseparable planar maps, J. Combin. Theory Ser. A 83 (1998), no. 1, 1-20. MR 1629428 (99f:05054)
22.: S. Karlin, Total positivity. Vol. I, Stanford University Press, Stanford, Calif., 1968. MR 0230102 (37:5667)
23.: M. Marden, Geometry of polynomials, Second edition. Mathematical Surveys, No. 3, American Mathematical Society, Providence, R.I., 1966. MR 0225972 (37:1562)
24.: N. Obreschkoff, Verteilung und Berechnung der Nullstellen reeller Polynome, VEB Deutscher Verlag der Wissenschaften, Berlin, 1963. MR 0164003 (29:1302)
25.: G. Pólya, Collected papers, Vol. II: Location of zeros, Edited by R. P. Boas, Mathematicians of Our Time, Vol. 8. The MIT Press, Cambridge, Mass.-London, 1974. MR 58:21342
26.: E. D. Rainville, Special functions, The Macmillan Co., New York, 1960. MR 0107725 (21:6447)
27.: V. Reiner, Descents and one-dimensional characters for classical Weyl groups, Discrete Math. 140 (1995), no. 1-3, 129-140. MR 1333715 (96d:05116)
28.: V. Reiner and V. Welker, On the Charney-Davis and the Neggers-Stanley conjectures, http://www.math.umn.edu/~reiner/Papers/papers.html (2002).
29.: J. Schur, Zwei sätze über algebraische gleichungen mit lauter reellen wurzeln, J. Reine Angew. Math. 144 (1914), no. 2, 75-88.
30.: R. P. Stanley, Log-concave and unimodal sequences in algebra, combinatorics, and geometry, Graph theory and its applications: East and West (Jinan, 1986), Ann. New York Acad. Sci., vol. 576, New York Acad. Sci., New York, 1989, pp. 500-535. MR 1110850 (92e:05124)
31.: -, Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62, Cambridge University Press, Cambridge, 1999. MR 1676282 (2000k:05026)
32.: -, Positivity problems and conjectures in algebraic combinatorics, Mathematics: frontiers and perspectives, Amer. Math. Soc., Providence, RI, 2000, pp. 295-319. MR 1754784 (2001f:05001)
33.: J. R. Stembridge, Some permutation representations of Weyl groups associated with the cohomology of toric varieties, Adv. Math. 106 (1994), no. 2, 244-301. MR 1279220 (95f:20011)
34.: R. A. Sulanke, The Narayana distribution, J. Statist. Plann. Inference 101 (2002), no. 1-2, 311-326, Special issue on lattice path combinatorics and applications (Vienna, 1998). MR 1878867
35.: D. G. Wagner, The partition polynomial of a finite set system, J. Combin. Theory Ser. A 56 (1991), no. 1, 138-159. MR 1082848 (92b:05005)
36.: -, Enumeration of functions from posets to chains, European J. Combin. 13 (1992), no. 4, 313-324. MR 1179527 (94c:05008)
37.: -, Total positivity of Hadamard products, J. Math. Anal. Appl. 163 (1992), no. 2, 459-483. MR 1145841 (93f:15020)

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 05A15, 26C10, 05A19, 05A05, 20F55

Petter Brändén
Affiliation: Matematik, Chalmers tekniska högskola och Göteborgs universitet, S-412 96 Göteborg, Sweden
Address at time of publication: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1043
Email: branden@math.chalmers.se, branden@umich.edu

DOI: 10.1090/S0002-9947-06-03856-6
PII: S 0002-9947(06)03856-6
Received by editor(s): March 22, 2004
Received by editor(s) in revised form: September 14, 2004
Posted: February 20, 2006
Additional Notes: This research was financed by the EC's IHRP Programme, within the Research Training Network ``Algebraic Combinatorics in Europe'', grant HPRN-CT-2001-00272, while the author was at Universitá di Roma ``Tor Vergata'', Rome, Italy.
Copyright of article: Copyright 2006, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS