Real-rootedness of variations of Eulerian polynomials

Haglund, James; Zhang, Philip B.

doi:10.1016/j.aam.2019.05.001

Cited by 16 publications

(5 citation statements)

References 34 publications

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“…In particular, if we use the notation p(x) ⋖ q(x) to denote that p and q interlace, after fixing k and n, our conjecture implies that one can produce the sequence of polynomials (1,...,1) , x), and h * (R k,(1,...,1) , x) = h * (Δ k,n , x). It is reasonable to search for recurrences that these polynomials satisfy and use techniques as those used in [20] or [19].…”

Section: Conjecture 512 Letmentioning

confidence: 99%

Lattice points in slices of prisms

Ferroni,

McGinnis

2024

Can. J. Math.-J. Can. Math.

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We conduct a systematic study of the Ehrhart theory of certain slices of rectangular prisms. Our polytopes are generalizations of the hypersimplex and are contained in the larger class of polypositroids introduced by Lam and Postnikov; moreover, they coincide with polymatroids satisfying the strong exchange property up to an affinity. We give a combinatorial formula for all the Ehrhart coefficients in terms of the number of weighted permutations satisfying certain compatibility properties. This result proves that all these polytopes are Ehrhart positive. Additionally, via an extension of a result by Early and Kim, we give a combinatorial interpretation for all the coefficients of the $h^*$ -polynomial. All of our results provide a combinatorial understanding of the Hilbert functions and the h-vectors of all algebras of Veronese type, a problem that had remained elusive up to this point. A variety of applications are discussed, including expressions for the volumes of these slices of prisms as weighted combinations of Eulerian numbers; some extensions of Laplace’s result on the combinatorial interpretation of the volume of the hypersimplex; a multivariate generalization of the flag Eulerian numbers and refinements; and a short proof of the Ehrhart positivity of the independence polytope of all uniform matroids.

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Section: Conjecture 512 Letmentioning

confidence: 99%

Lattice points in slices of prisms

Ferroni,

McGinnis

2024

Can. J. Math.-J. Can. Math.

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“…for k = n. In both cases, multiplication on the left by the given matrix preserves the interlacing of sequences of real-rooted polynomials with nonnegative coefficients (see [20,Theorem 2.4] for a more general statement, in the former case, and Lemma 2.3 in the latter) and thus the first statement follows by induction on n. We now deduce that A n (x) d n,k,j (x) for every n ∈ N and all 0 ≤ k, j ≤ n with k+j ≤ n. We first show that the sequence (d n,k (x)) 0≤k≤n is interlacing for every n ∈ N (as already mentioned, this also follows from [11,Theorem 3.6]). Indeed, by the first statement and Proposition 4.2 (g) we have…”

Section: Barycentric Subdivisionsmentioning

confidence: 99%

“…, n}, were studied implicitly in[11, Section 3.2] and explicitly in [7, Section 4] (we also set d n,k (x) := 1 for n = k = 0). These polynomials interpolate between the Eulerian polynomial d n,0 (x) = A n (x) andd n,n (x) = w∈Sn: Fix(w)=∅x exc(w) , called the nth derangement polynomial; see, for instance, [11, Section 3.2][19,20] and references therein. As part of their study of the derangement transformation [11, Section 3.2], using tools from the theory of multivariate stable polynomials, Brändén and Solus showed that (d n,k (x)) 0≤k≤n is an interlacing sequence of real-rooted polynomials for every n ∈ N (this follows from[11, Theorem 3.6]).…”

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confidence: 99%

Symmetric Decompositions, Triangulations and Real‐rootedness

Athanasiadis

Tzanaki

2021

Mathematika

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Polynomials which afford nonnegative, real-rooted symmetric decompositions have been investigated recently in algebraic, enumerative and geometric combinatorics. Brändén and Solus have given sufficient conditions under which the image of a polynomial under a certain operator associated to barycentric subdivision has such a decomposition. This paper gives a new proof of their result which generalizes to subdivision operators in the setting of uniform triangulations of simplicial complexes, introduced by the first author. Sufficient conditions under which these decompositions are also interlacing are described. Applications yield new classes of polynomials in geometric combinatorics which afford nonnegative, real-rooted symmetric decompositions. Some interesting questions in f -vector theory arise from this work.

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“…In addition, different generalizations of Eulerian polynomials and Eulerian numbers were considered, see Haglund-Zhang [24], Han-Mao-Zeng [25], Rzadkowski-Urlińska [35], Zhu [48], Zhuang [52]. But recently in many new combinatorial enumerations, there bring out more and more combinatorial triangles satisfying certain three-term recurrence similar to recurrence relations (1.8), (1.9) and (1.11).…”

Section: Structure Of This Papermentioning

confidence: 99%

A unified approach to combinatorial triangles: a generalized Eulerian polynomial

Zhu

2020

Preprint

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Motivated by the classical Eulerian number, descent and excedance numbers in the hyperoctahedral groups, an triangular array from staircase tableaux and so on, we study a triangular array [T n,k ] n,k≥0 satisfying the recurrence relation:We derive a functional transformation for its row-generating function T n (x) from the row-generating function A n (x) of another array [A n,k ] n,k satisfying a two-term recurrence relation. Based on this transformation, we can get properties of T n,k and T n (x) including nonnegativity, log-concavity, real rootedness, explicit formula and so on. Then we extend the famous Frobenius formula, the γ positivity decomposition and the David-Barton formula for the classical Eulerian polynomial to those of a generalized Eulerian polynomial. We also get an identity for the generalized Eulerian polynomial with the general derivative polynomial. Finally, we apply our results to an array from the Lambert function, a triangular array from staircase tableaux and the alternating-runs triangle of type B in a unified approach.

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Real-rootedness of variations of Eulerian polynomials

Cited by 16 publications

References 34 publications

Lattice points in slices of prisms

Lattice points in slices of prisms

Symmetric Decompositions, Triangulations and Real‐rootedness

A unified approach to combinatorial triangles: a generalized Eulerian polynomial

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