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2 results

  • COMBINATORIAL EXPANSIONS OF KAZHDAN–LUSZTIG POLYNOMIALS

  • FRANCESCO BRENTI
  • Journal:
    Journal of the London Mathematical Society / Volume 55 / Issue 3 / June 1997
    Published online by Cambridge University Press:
    01 June 1997, pp. 448-472
    Print publication:
    June 1997

  • We introduce two related families of polynomials, easily computable by simple recursions into which any Kazhdan–Lusztig (and inverse Kazhdan–Lusztig) polynomial of any Coxeter group can be expanded linearly, and we give combinatorial interpretations to the coefficients in these expansions. This yields a combinatorial rule for computing the Kazhdan–Lusztig polynomials in terms of paths in a directed graph, and a completely combinatorial reformulation of the nonnegativity conjecture [15, p. 166].

  • Location of Zeros of Chromatic and Related Polynomials of Graphs

  • Francesco Brenti, Gordon F. Royle, David G. Wagner
  • Journal:
    Canadian Journal of Mathematics / Volume 46 / Issue 1 / 01 February 1994
    Published online by Cambridge University Press:
    20 November 2018, pp. 55-80
    Print publication:
    01 February 1994
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  • We consider the location of zeros of four related classes of polynomials, one of which is the class of chromatic polynomials of graphs. All of these polynomials are generating functions of combinatorial interest. Extensive calculations indicate that these polynomials often have only real zeros, and we give a variety of theoretical results which begin to explain this phenomenon. In the course of the investigation we prove a number of interesting combinatorial identities and also give some new sufficient conditions for a polynomial to have only real zeros.