Trees

Abstract

Contents 1. Introduction 2. Probabilistic consequences of real-rootedness 3. Unimodality and γ-nonnegativity 3.1. An action on permutations 3.2. γ-nonnegativity of h-polynomials 3.3. Barycentric subdivisions 3.4. Unimodality of h * -polynomials 4. Log-concavity and matroids 5. Infinite log-concavity 6. The Neggers-Stanley conjecture 7. Preserving real-rootedness 7.1. The subdivision operator 8. Common interleavers 8.1. s-Eulerian polynomials 8.2. Eulerian polynomials for finite Coxeter groups 9. Multivariate techniques 9.1. Stable polynomials and matroids 9.2. Strong Rayleigh measures 9.3. The symmetric exclusion process 9.4. The Grace-Walsh-Szegő theorem, and the proof of Theorem 5.1 10. Historical notes References

Permutations of Type $B$ with Fixed Number of Descents and Minus Signs

“…, n} the polynomial j∈S P n,j (x) has only real and simple roots. Combining (47) with Example 7.8.8 in [3] we will note (Theorem 20) that in fact every linear combination c 0 P n,0 (x) + c 1 P n,1 (x) + . .…”

“…, with the initial conditions: p n,0 (x) = P A n (x) for n ≥ 0 and p n,n (x) = xP A n (x) for n ≥ 1. By (32) the polynomial p n,j (x) coincides with A n+1,j+1 (x) considered by Brändén [3], Example 7.8.8. He noted that…”

“…From [3], Example 7.8.8 and (47) we have the following property of the polynomials p n,j (x) and P n,j (x):…”

“…Conger proved that the polynomials p n,j (x) := n k=0 b(n, k, j)x k have only real roots (Theorem 5 in [5]). Brändén [3] showed something stronger: for every n ≥ 1 the sequence of polynomials {p n,j (x)} n j=0 is interlacing, in particular for every c 0 , c 1 , . .…”

Simplices for numeral systems

“…A sequence F m := (f i ) m i=1 of real-rooted polynomials is called (strictly) interlacing if f i (strictly) interlaces f j for all 1 ≤ i < j ≤ m. Let F + m denote the space of all interlacing sequences F m for which f i has only nonnegative coefficients for all 1 ≤ i ≤ n. In [5], Brändén characterized when a matrix G = (G i,j (z)) m i,i=1 of polynomials maps F + m to F + m . We say that such a map preserves interlacing.…”