Hook formulas for skew shapes III. Multivariate and product formulas

We give new product formulas for the number of standard Young tableaux of certain skew shapes and for the principal evaluation of the certain Schubert polynomials. These are proved by utilizing symmetries for evaluations of factorial Schur functions, extensively studied in the first two papers in the series "Hook formulas for skew shapes" [arxiv:1512.08348, arxiv:1610.04744]. We also apply our technology to obtain determinantal and product formulas for the partition function of certain weighted lozenge tilings, and give various probabilistic and asymptotic applications.Comment: 40 pages, 17 figures. This is the third paper in the series "Hook formulas for skew shapes"; v2 added reference to [KO1] (arxiv:1409.1317) where the formula in Corollary 1.1 had previously appeared; v3 Corollary 5.10 added, resembles published versio

A Locality-Oriented Public Welfare Agency: A Case Study of Boundary Maintenance in a Hostile Environment

Boundary maintenance activities are studied in a public welfare agency as a means of establishing the relationship between the nature of these activities and the essential character of a formal organization. Assaults on the agency are observed through a period of social change, in this case an extreme of racial succession among the staff and administration of the agency. Conclusions point to congruence between the character of the organization and its boundary maintenance activity. Skewed or incongruent boundary maintenance produces disorganization and confusion among participants. In the context of racial succession, universalistic patterns are recommended as a solution to the confusion

Building an Interdisciplinary Partnership to Improve Math Skills in the Science Classroom

This presentation is intended for University STEM faculty interested in developing interdisciplinary collaborations to improve the ability of students to do math in their science courses. The development of a partnership between faculty in chemistry and mathematics will be discussed. The presenters’ partnership explored ways to improve rates of chemistry student success by identifying key mathematical stumbling blocks encountered by chemistry students. This collaboration resulted in the formation of a student learning community comprised of a General Chemistry course and Precalculus course. Modifications were made to both courses including the addition of material to the mathematics course and changes in pedagogy in the chemistry course. Attendees will be presented with the framework that was used to develop this partnership and will engage in interdisciplinary discussions with other participants. We hope to promote the development of interdisciplinary teams at other colleges and universities

Matrices with restricted entries and q-analogues of permutations

We study the functions that count matrices of given rank over a finite field with specified positions equal to zero. We show that these matrices are $q$-analogues of permutations with certain restricted values. We obtain a simple closed formula for the number of invertible matrices with zero diagonal, a $q$-analogue of derangements, and a curious relationship between invertible skew-symmetric matrices and invertible symmetric matrices with zero diagonal. In addition, we provide recursions to enumerate matrices and symmetric matrices with zero diagonal by rank, and we frame some of our results in the context of Lie theory. Finally, we provide a brief exposition of polynomiality results for enumeration questions related to those mentioned, and give several open questions.Comment: 29 pages, 2 figures, v2: one additional result, some formulas simplified, and a new reference; v3: corrected typo

Minimal skew semistandard tableaux and the Hillman--Grassl correspondence

Standard tableaux of skew shape are fundamental objects in enumerative and algebraic combinatorics and no product formula for the number is known. In 2014, Naruse gave a formula (NHLF) as a positive sum over excited diagrams of products of hook-lengths. Subsequently, Morales, Pak, and Panova gave a $q$-analogue of this formula in terms of skew semistandard tableaux (SSYT). They also showed, partly algebraically, that the Hillman--Grassl map, restricted to skew semistandard tableaux, is behind their $q$-analogue. We study the problem of circumventing the algebraic part and proving the bijection completely combinatorially, which we do for border strips. For a skew shape, we define minimal semistandard Young tableaux, that are in correspondence with excited diagrams via a new description of the Hillman--Grassl bijection and have an analogue of excited moves. Lastly, we relate the minimal skew SSYT with the terms of the Okounkov-Olshanski formula (OOF) for counting standard tableaux of skew shape. Our construction immediately implies that the summands in the NHLF are less than the summands in the OOF and we characterize the shapes where both formulas have the same number of summands

Hook formulas for skew shapes

International audienceThe celebrated hook-length formula gives a product formula for the number of standard Young tableaux of a straight shape. In 2014, Naruse announced a more general formula for the number of standard Young tableaux of skew shapes as a positive sum over excited diagrams of products of hook-lengths. We give two q-analogues of Naruse's formula for the skew Schur functions and for counting reverse plane partitions of skew shapes. We also apply our results to border strip shapes and their generalizations

Comparative analysis of anisotropic material properties of uniaxial nematics formed by flexible dimers and rod-like monomers

We report temperature dependencies of material properties such as dielectric anisotropy, birefringence, splay (K₁₁), twist (K₂₂), and bend (K₃₃) elastic constants of the uniaxial nematic (N) phase formed by flexible dimers of DTC5C9 and compare their behavior to that of a corresponding monomer MCT5. DTC5C9 forms a twist-bend nematic (Ntb) at temperatures below the N phase. Anisotropic properties of MCT5 are typical of the rod-like mesogens. In particular, birefringence increases as the temperature is reduced, following the classic behavior, described by Haller. The elastic constants also follow the standard behavior, with their ratios being practically temperature-independent. In contrast, DTC5C9 shows a dramatic departure from the standard case. Birefringence changes non-monotonously with temperature, decreasing on approaching the N-Ntb phase transition. K₃₃ decreases strongly to 0.4 pN near the N - Ntb transition, although remains finite. The ratios of the elastic constants in DTC5C9 show a strong temperature dependence that can be associated with the bend-induced changes in the orientational distribution function. The measured elastic properties are consistent with the tendency of the dimeric molecules to adopt bent configurations that give rise to the Ntb phase

Multi-Level Evaluation of a Perinatal Health Program in Rural Southeast Georgia

Problem: Infant mortality has declined steadily in the past decade, however, significant disparities associate with lack of adequate perinatal health services and barriers to access disproportionately impact women residing in rural areas. In Georgia, data suggest significant challenges with respect to birth outcomes, and this problem seems to be exacerbated in rural regions of state. The objective of this presentation is to report on the impact of a regional perinatal health care collaborative implemented in rural southeast Georgia. Method: Analysis of pre-intervention and post-intervention birth outcomes (gestational age, birth weight and infant mortality) served as the focal point programmatic evaluation. Differences in mean gestational age and mean birth weight were analyzed using a t-test (α = 0.05). Proportional differences in low birth weight and infant mortality were assessed using the chi-square test (α = 0.05). Differences were investigated relative to race (white and nonwhite). Results: Analysis of white participants showed no significant difference in any birth outcomes investigated. Furthermore, analysis of non-white PHP participants suggested significant improvements in all birth weight (p \u3c 0.001), gestational age (p = 0.007), low birth weight (p = 0.002), and infant mortality (p = 0.007). Conclusion: The perinatal health program in southeast Georgia demonstrated considerable effectiveness as measured through pre-intervention and post-intervention birth outcomes. The potential for improved health outcomes of high risk pregnant women and infants as a result of adequate perinatal care may also lead to the achievement of Healthy People 2010 within this region