Positivity of permutation pattern character polynomials

Positivity results for Stanley’s character polynomials

In Stanley [16], the author introduces expressions for the normalized characters of the symmetric group and states some positivity conjectures for these expressions. Here, we give an affirmative partial answer to Stanley’s positivity conjectures about the expressions using results on Kerov polynomials. In particular, we use new positivity results in Goulden and Rattan [7]. We shall see that the...

Combinatorial interpretation and positivity of Kerov’s character polynomials

Kerov’s polynomials give irreducible character values of the symmetric group in term of the free cumulants of the associated Young diagram. Using a combinatorial approach with maps, we prove in this article a positivity result on their coefficients, which extends a conjecture of S. Kerov. Résumé. Les polynômes de Kerov expriment les valeurs des caractères irréductibles du groupe symmétrique en ...

Positivity Of Trigonometri Polynomials

Deciding positivity of real polynomials

We describe an algorithm for deciding whether or not a real polynomial is positive semideenite. The question is reduced to deciding whether or not a certain zero-dimensional system of polynomial equations has a real root.

Positivity of Legendrian Thom polynomials

We study Legendrian singularities arising in complex contact geometry. We define a one-parameter family of bases in the ring of Legendrian characteristic classes such that any Legendrian Thom polynomial has nonnegative coefficients when expanded in these bases. The method uses a suitable Lagrange Grassmann bundle on the product of projective spaces. This is an extension of a nonnegativity resul...