A refinement of the Shuffle Conjecture with cars of two sizes and t=1/qt=1/q

The original Shuffle Conjecture of Haglund et al. has a symmetric function side and a combinatorial side. The symmetric function side may be simply expressed as $$ where \nabla is the Macdonald polynomial eigen-operator of Bergeron and Garsia and $h_\mu$ is the homogeneous basis indexed by $\mu=(\mu_1,\mu_2,...,\mu_k)$ partitions of n. The combinatorial side q,t-enumerates a family of Parking Functions whose reading word is a shuffle of k successive segments of 1,2,3,...,n of respective lengths $\mu_1,\mu_2,...,\mu_k$. It can be shown that for t=1/q the symmetric function side reduces to a product of q-binomial coefficients and powers of q. This reduction suggests a surprising combinatorial refinement of the general Shuffle Conjecture. Here we prove this refinement for k=2 and t=1/q. The resulting formula gives a q-analogue of the well studied Narayana numbers.Comment: 17 pages, 11 figure

Indigenous Learning: Weaving the Fabric of Our Histories for Success

Six public housing community leaders were interviewed for this study. Four themes for recognizing indigenous learning were identified. Themes explained leadership style, validation of and motivation for leadership, and personal impact. The findings give rise to considerations of how adult educators can identify and begin to understand indigenous learning systems

Psychobiological perspectives on attachment: implications for health over the lifespan

Journal ArticleOne of the most robust findings to emerge from health psychology over the past 30 years is that individuals in enduring, committed romantic relationships have longer, healthier, and happier lives than unmarried individuals (Kitigawa & Hauser, 1973; Ryff, Singer, Wing, & Love, 2001; Stack & Eshleman, 1998)

Analyzing Experiences with Context-Based Physical Activity Coaching (CB-PAC) in Caregivers of Children with Visual Impairments: A Pilot Study

Caregivers have faced many barriers in providing equitable physical activity (PA) opportunities for their children with visual impairments (VI) (Armstrong et al., 2018; Columna et al., 2017; Sterbova & Kudlacek, 2014). This pilot study focused on two female caregivers of children with VI to understand their experiences and perceptions related to providing PA for their children along with a context-based coaching program to improve their child’s participation in PA. The study utilized a mixed methods design and concluded that support, motivation, strategies for participation, and equipment were important areas to address for caregivers to engage their children in PA.https://soar.usa.edu/otdcapstonessummer2023/1025/thumbnail.jp