On Operators on Polynomials Preserving Real-Rootedness and the Neggers-Stanley Conjecture
Petter Brändén
DOI: 10.1023/B:JACO.0000047295.93525.df
Abstract
We refine a technique used in a paper by Schur on real-rooted polynomials. This amounts to an extension of a theorem of Wagner on Hadamard products of Pólya frequency sequences. We also apply our results to polynomials for which the Neggers-Stanley Conjecture is known to hold. More precisely, we settle interlacing properties for E-polynomials of series-parallel posets and column-strict labelled Ferrers posets.
Pages: 119–130
Keywords: neggers-Stanley conjecture; real-rooted polynomials; Sturm sequence
Full Text: PDF
References
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2. J. Dedieu, “Obreschkoff's theorem revisited: What convex sets are contained in the set of hyperbolic polynomials?” J. Pure Appl. Algebra 81(3) (1992), 269-278.
3. C.D. Godsil, Algebraic Combinatorics, Chapman and Hall Mathematics Series. Chapman & Hall, New York, 1993.
4. G.H. Hardy, J.E. Littlewood, and G. Pólya. Inequalities, 2nd ed., Cambridge, at the University Press, 1952.
5. O.J. Heilmann and E.H. Lieb, “Theory of monomer-dimer systems,” Comm. Math. Phys. 25 (1972), 190-232.
6. B.J. Levin, Distribution of Zeros of Entire Functions, American Mathematical Society, Providence, R.I., 1964.
7. M. Marden, Geometry of Polynomials, 2nd edition, Mathematical Surveys, No.
3. American Mathematical Society, Providence, R.I., 1966.
8. A. Nijenhuis, “On permanents and the zeros of rook polynomials,” J. Combinatorial Theory Ser. A 21(2) (1976), 240-244.
9. N. Obreschkoff, Verteilung und Berechnung der Nullstellen reeller Polynome, VEB Deutscher Verlag der Wissenschaften, Berlin,
1963. BR \ddot AND ÉN
10. V. Reiner and V. Welker, “On the Charney-Davis and the Neggers-Stanley conjectures,” http://www.math. umn.edu/\~reiner/Papers/papers.html, 2002.
11. J. Schur, “Zwei s\ddot atze \ddot uber algebraische gleichungen mit lauter reellen wurzeln,” J. Reine Angew. Math. 144(2) (1914), 75-88.
12. R. Simion, “A multi-indexed Sturm sequence of polynomials and unimodality of certain combinatorial sequences,” J. Combin. Theory Ser. A 36(1) (1984), 15-22.
13. R.P. Stanley, Enumerative Combinatorics. vol. 2, volume 62 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1999.
14. D.G. Wagner, “Enumeration of functions from posets to chains,” European J. Combin. 13(4) (1992), 313-324.
15. D.G. Wagner, “Total positivity of hadamard products,” J. Math. Anal. Appl. 163(2) (1992), 459-483.