Minimal Permutations and 2-Regular Skew Tableaux

Bouvel and Pergola introduced the notion of minimal permutations in the study of the whole genome duplication-random loss model for genome rearrangements. Let $\mathcal{F}_d(n)$ denote the set of minimal permutations of length $n$ with $d$ descents, and let $f_d(n)= |\mathcal{F}_d(n)|$. They derived that $f_{n-2}(n)=2^{n}-(n-1)n-2$ and $f_n(2n)=C_n$, where $C_n$ is the $n$-th Catalan number. Mansour and Yan proved that $f_{n+1}(2n+1)=2^{n-2}nC_{n+1}$. In this paper, we consider the problem of counting minimal permutations in $\mathcal{F}_d(n)$ with a prescribed set of ascents. We show that such structures are in one-to-one correspondence with a class of skew Young tableaux, which we call $2$-regular skew tableaux. Using the determinantal formula for the number of skew Young tableaux of a given shape, we find an explicit formula for $f_{n-3}(n)$. Furthermore, by using the Knuth equivalence, we give a combinatorial interpretation of a formula for a refinement of the number $f_{n+1}(2n+1)$.Comment: 19 page

Modeling the impact of racial and ethnic disparities on COVID-19 epidemic dynamics.

BACKGROUND: The impact of variable infection risk by race and ethnicity on the dynamics of SARS-CoV-2 spread is largely unknown. METHODS: Here, we fit structured compartmental models to seroprevalence data from New York State and analyze how herd immunity thresholds (HITs), final sizes, and epidemic risk change across groups. RESULTS: A simple model where interactions occur proportionally to contact rates reduced the HIT, but more realistic models of preferential mixing within groups increased the threshold toward the value observed in homogeneous populations. Across all models, the burden of infection fell disproportionately on minority populations: in a model fit to Long Island serosurvey and census data, 81% of Hispanics or Latinos were infected when the HIT was reached compared to 34% of non-Hispanic whites. CONCLUSIONS: Our findings, which are meant to be illustrative and not best estimates, demonstrate how racial and ethnic disparities can impact epidemic trajectories and result in unequal distributions of SARS-CoV-2 infection. FUNDING: K.C.M. was supported by National Science Foundation GRFP grant DGE1745303. Y.H.G. and M.L. were funded by the Morris-Singer Foundation. M.L. was supported by SeroNet cooperative agreement U01 CA261277

An integrated optical CO2 sensor. Phase 0: Design and fabrication of critical elements

Significant progress has been made toward all of the goals for the first phase of the project short of actual fabrication of a light path. Two alternative approaches to fabricating gold mirrors using the basic LIGA process were developed, one using electroplated solid gold mirrors and the second using gold plated over a nickel base. A new method of fabrication, the transfer mask process, was developed and demonstrated. Analysis of the projected surface roughness and beam divergence effects was completed. With gold surface with low surface roughness scattering losses are expected to be insignificant. Beam divergence due to diffraction will require a modification of the original design, but should be eliminated by fabricating mirrors 1000 mu m in height by 1000 mu m in width and using a source with an initial beam radius greater than 300 mu m. This may eliminate any need for focusing optics. Since the modified design does not affect the mask layout, ordering of the mask and fabrication of the test structures can begin immediately at the start of Phase 1

Development and Field Testing of a MALL for Filipino with a Reusable Framework for Mobile-Based Drills

This paper describes the development and field testing of Ibigkas! Filipino, a mobile game that exercises learners’ fluency in identifying synonyms (kasingkahulugan) and antonyms (kasalungat) in the Filipino language. Twenty-four students from Grades 4, 5, and 6 were invited to play and answer comprehension tests to determine whether the game helped them improve their understanding of the content. Self-report questionnaires assessed the extent to which they enjoyed it. Additionally, three teachers were invited to a focus group discussion (FGD) to gather their insights about the game and how they may use it in their classes. Self-report feedback from students showed the game was fun, interesting, and sufficiently challenging. A significant increase in the post-test comprehension scores of the Grade 4 participants was found. This shows the potential of the game to make learning fun while helping realize learning goals. Teachers indicated they can use the game to supplement their Filipino classes and that the students will be receptive to the idea of utilizing a game for learning

Development and Field Testing of a Narrative-Centered Digital Game for English Comprehension

This paper describes the development and field testing of Learning Likha: Rangers to the Rescue, a narrative-centered, mobile-based digital game for practicing English comprehension. Twenty-seven (27) student participants from Grades 4, 5, and 6 were invited to play the game and answer a comprehension test to determine their level of understanding of the game’s contents. Self-report questionnaires were also used to assess the extent to which they enjoyed playing the game. Three (3) teachers were likewise invited for a focus group discussion (FGD) to gather their insights about the game and how they may use it in their classes. Student’s self-reported feedback indicated they found the game fun, interesting, and sufficiently challenging. Post-test comprehension scores were generally good. Younger participants scored lower than their older peers but the differences were found to be not significant. Teachers indicated the game has the potential to be used as a supplement for their classes and that their students would enjoy playing it