Mutual Interlacing and Eulerian-like Polynomials for Weyl Groups

We use the method of mutual interlacing to prove two conjectures on the real-rootedness of Eulerian-like polynomials: Brenti's conjecture on $q$-Eulerian polynomials for Weyl groups of type $D$, and Dilks, Petersen, and Stembridge's conjecture on affine Eulerian polynomials for irreducible finite Weyl groups. For the former, we obtain a refinement of Brenti's $q$-Eulerian polynomials of type $D$, and then show that these refined Eulerian polynomials satisfy certain recurrence relation. By using the Routh--Hurwitz theory and the recurrence relation, we prove that these polynomials form a mutually interlacing sequence for any positive $q$, and hence prove Brenti's conjecture. For $q=1$, our result reduces to the real-rootedness of the Eulerian polynomials of type $D$, which were originally conjectured by Brenti and recently proved by Savage and Visontai. For the latter, we introduce a family of polynomials based on Savage and Visontai's refinement of Eulerian polynomials of type $D$. We show that these new polynomials satisfy the same recurrence relation as Savage and Visontai's refined Eulerian polynomials. As a result, we get the real-rootedness of the affine Eulerian polynomials of type $D$. Combining the previous results for other types, we completely prove Dilks, Petersen, and Stembridge's conjecture, which states that, for every irreducible finite Weyl group, the affine descent polynomial has only real zeros.Comment: 28 page

Avoiding vincular patterns on alternating words

A word $w=w_1w_2\cdots w_n$ is alternating if either $w_1w_3\cdots$ (when the word is up-down) or $w_1>w_2w_4<\cdots$ (when the word is down-up). The study of alternating words avoiding classical permutation patterns was initiated by the authors in~\cite{GKZ}, where, in particular, it was shown that 123-avoiding up-down words of even length are counted by the Narayana numbers. However, not much was understood on the structure of 123-avoiding up-down words. In this paper, we fill in this gap by introducing the notion of a cut-pair that allows us to subdivide the set of words in question into equivalence classes. We provide a combinatorial argument to show that the number of equivalence classes is given by the Catalan numbers, which induces an alternative (combinatorial) proof of the corresponding result in~\cite{GKZ}. Further, we extend the enumerative results in~\cite{GKZ} to the case of alternating words avoiding a vincular pattern of length 3. We show that it is sufficient to enumerate up-down words of even length avoiding the consecutive pattern $\underline{132}$ and up-down words of odd length avoiding the consecutive pattern $\underline{312}$ to answer all of our enumerative questions. The former of the two key cases is enumerated by the Stirling numbers of the second kind.Comment: 25 pages; To appear in Discrete Mathematic

Pattern-avoiding alternating words

A word $w=w_1w_2\cdots w_n$ is alternating if either $w_1w_3\cdots$ (when the word is up-down) or $w_1>w_2w_4<\cdots$ (when the word is down-up). In this paper, we initiate the study of (pattern-avoiding) alternating words. We enumerate up-down (equivalently, down-up) words via finding a bijection with order ideals of a certain poset. Further, we show that the number of 123-avoiding up-down words of even length is given by the Narayana numbers, which is also the case, shown by us bijectively, with 132-avoiding up-down words of even length. We also give formulas for enumerating all other cases of avoidance of a permutation pattern of length 3 on alternating words

Circumpolar hoods and clouds and their relation to the Martian H2O cycle

Water exists in both vapor and solid phases in the Martian atmosphere. The polar hoods are shrouds of condensate clouds which obscure both polar regions at times during their respective fall and winter seasons. The hemispheric asymmetries in the polar hood clouds are reviewed and their significance to the seasonal water cycle is discussed. Comparisons of images acquired using short wavelength filters, which provide maximum cloud contrast, and long wavelength filters, which are sensitive to surface features including the surface cap, provide correlations between cloud formation and large scale planetary dynamics

American Native Beef Cooperative

The American Native Beef project involved a failed attempt to establish cow and bull slaughter operation in Southeast Oklahoma. The effort was initially organized as a new generation cooperative and raised over US$2.5M from area producers who retained their funds in escrow for over 5 years despite numerous opportunities to withdraw their investment. The business model was restructured several times to attract equity capital from outside investors. The case provides insights into the linkages between business strategy and business structure. It also raises the question as to whether the project could have been successful under the original business model.Agribusiness, Livestock Production/Industries,

Producers Cooperative Oil Mill: A Case Study in Strategic Planning for a Cooperative

This study examines the strategic issues and decisions of Producers Cooperative Oil Mill, a regional oilseed processing cooperative located in Oklahoma City, Oklahoma. The process of strategic planning and strategy implementation in agribusiness has been the topic of numerous research studies and case studies. While cooperative firms have been the subject of case studies focusing on strategic issues, the unique aspects of strategic decisions in a cooperative firm have not been highlighted. The formulation and implementation of strategy in cooperative organizations is impacted by their business structure. This case study provides unique examples of how strategy formulation and implementation is impacted by the unique structure and values of a cooperative firm. The article is formatted such that it can be used in educational programs as three inter-related mini-case studies. Questions for students are included.cooperative, strategic planning, oilseed processing, Agribusiness, Teaching/Communication/Extension/Profession,