Symmetric Decompositions and Real-Rootedness

Brändén, Petter; Solus, Liam

doi:10.1093/imrn/rnz059

Cited by 36 publications

(51 citation statements)

References 44 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Remark 5.1 (A second proof). The real-rootedness of the a and b polynomials in the symmetric decomposition of d n,r (z) with respect to n was also recently observed, independently, by the second author and P. Brändén [14]. There, the authors actually prove the stronger statement that the a and b polynomials alway interlace.…”

Section: Applicationsmentioning

confidence: 52%

Derangements, Ehrhart theory, and local h-polynomials

Gustafsson¹,

Solus²

2020

Advances in Mathematics

Self Cite

View full text Add to dashboard Cite

The Eulerian polynomials and derangement polynomials are two well-studied generating functions that frequently arise in combinatorics, algebra, and geometry. When one makes an appearance, the other often does so as well, and their corresponding generalizations are similarly linked. This is this case in the theory of subdivisions of simplicial complexes, where the Eulerian polynomial is an h-polynomial and the derangement polynomial is its local h-polynomial. Separately, in Ehrhart theory the Eulerian polynomials are generalized by the h * -polynomials of s-lecture hall simplices. Here, we show that derangement polynomials are analogously generalized by the box polynomials, or local h * -polynomials, of the s-lecture hall simplices, and that these polynomials are all real-rooted. We then connect the two theories by showing that the local h-polynomials of common subdivisions in algebra and topology are realized as local h * -polynomials of s-lecture hall simplices. We use this connection to address some open questions on real-rootedness and unimodality of generating polynomials, some from each side of the story.

show abstract

Section: Applicationsmentioning

confidence: 52%

Derangements, Ehrhart theory, and local h-polynomials

Gustafsson¹,

Solus²

2020

Advances in Mathematics

Self Cite

View full text Add to dashboard Cite

show abstract

“…As pointed out by Brändén and Solus [3], the polynomial p(x) is alternatingly increasing if and only if the pair of polynomials in its symmetric decomposition are both unimodal and have nonnegative coefficients. Thus the bi-γ-positivity of p(x) implies that p(x) is alternatingly increasing.…”

Section: The 1/k-derangement Polynomials Of Type Bmentioning

confidence: 96%

The $1/k$-Eulerian Polynomials of Type $B$

Yeh

et al. 2020

Electron. J. Combin.

View full text Add to dashboard Cite

In this paper, we define the $1/k$-Eulerian polynomials of type $B$. Properties of these polynomials, including combinatorial interpretations, recurrence relations and $\gamma$-positivity are studied. In particular, we show that the $1/k$-Eulerian polynomials of type $B$ are $\gamma$-positive when $k>0$. Moreover, we define the $1/k$-derangement polynomials of type $B$, denoted $d_n^B(x;k)$. We show that the polynomials $d_n^B(x;k)$ are bi-$\gamma$-positive when $k\geq 1/2$. In particular, we get a symmetric decomposition of the polynomials $d_n^B(x;1/2)$ in terms of the classical derangement polynomials.

show abstract

“…where d + n,r (z) and d − n,r (z) are γ-positive polynomials with centers of symmetry n 2 and n+1 2 , respectively. Such a decomposition is called the symmetric decomposition of polynomials by Brändén and Solus [9]. Recently, Gustafsson and Solus [17] proved that both d + n,r (z) and d − n,r (z) have only real roots, and Brändén and Solus [9] further proved that d + n,r (z) ≪ d − n,r (z).…”

Section: Binomial Eulerian Polynomials For Colored Permutationsmentioning

confidence: 99%

“…Such a decomposition is called the symmetric decomposition of polynomials by Brändén and Solus [9]. Recently, Gustafsson and Solus [17] proved that both d + n,r (z) and d − n,r (z) have only real roots, and Brändén and Solus [9] further proved that d + n,r (z) ≪ d − n,r (z). Recently, Athanasiadis [5] introduced a generalization A n,r (z) of A n (z) to the wreath product group Z r ≀ S n and further studied their symmetric function generalizations.…”

Section: Binomial Eulerian Polynomials For Colored Permutationsmentioning

confidence: 99%

Real-rootedness of variations of Eulerian polynomials

Haglund

Zhang

2019

Advances in Applied Mathematics

View full text Add to dashboard Cite

The binomial Eulerian polynomials, introduced by Postnikov, Reiner, and Williams, are γ-positive polynomials and can be interpreted as h-polynomials of certain flag simplicial polytopes. Recently, Athanasiadis studied analogs of these polynomials for colored permutations. In this paper, we generalize them to s-inversion sequences and prove that these new polynomials have only real roots by the method of interlacing polynomials. Three applications of this result are presented. The first one is to prove the real-rootedness of binomial Eulerian polynomials, which confirms a conjecture of Ma, Ma, and Yeh. The second one is to prove that the symmetric decomposition of binomial Eulerian polynomials for colored permutations is real-rooted. Thirdly, our polynomials for certain s-inversion sequences are shown to admit a similar geometric interpretation related to edgewise subdivisions of simplexes.

show abstract

Symmetric Decompositions and Real-Rootedness

Cited by 36 publications

References 44 publications

Derangements, Ehrhart theory, and local h-polynomials

Derangements, Ehrhart theory, and local h-polynomials

The $1/k$-Eulerian Polynomials of Type $B$

Real-rootedness of variations of Eulerian polynomials

Contact Info

Product

Resources

About