Indecomposable Permutations, Hypermaps and Labeled Dyck Paths

Hypermaps were introduced as an algebraic tool for the representation of embeddings of graphs on an orientable surface. Recently a bijection was given between hypermaps and indecomposable permutations; this sheds new light on the subject by connecting a hypermap to a simpler object. In this paper, a bijection between indecomposable permutations and labelled Dyck paths is proposed, from which a few enumerative results concerning hypermaps and maps follow. We obtain for instance an inductive formula for the number of hypermaps with n darts, p vertices and q hyper-edges; the latter is also the number of indecomposable permutations of with p cycles and q left-to-right maxima. The distribution of these parameters among all permutations is also considered.Comment: 30 pages 4 Figures. submitte

Sports Injury Prevention in Danbury, CT

Sports Injuries, especially overuse injuries, cause a significant amount of morbidity and public health cost, especially for young athletes. Importantly, many of these injuries can be prevented with proper training and some novel approaches to warm-up and training. To address this concern, I worked with Orthopedic Specialists of CT and the athletic trainers at Danbury High School to come up with some educational materials for patients on effective ways to prevent sports medicine injuries.https://scholarworks.uvm.edu/fmclerk/1353/thumbnail.jp

Parallelogram polyominoes, the sandpile model on a complete bipartite graph, and a q,t-Narayana polynomial

We classify recurrent configurations of the sandpile model on the complete bipartite graph K_{m,n} in which one designated vertex is a sink. We present a bijection from these recurrent configurations to decorated parallelogram polyominoes whose bounding box is a m*n rectangle. Several special types of recurrent configurations and their properties via this bijection are examined. For example, recurrent configurations whose sum of heights is minimal are shown to correspond to polyominoes of least area. Two other classes of recurrent configurations are shown to be related to bicomposition matrices, a matrix analogue of set partitions, and (2+2)-free partially ordered sets. A canonical toppling process for recurrent configurations gives rise to a path within the associated parallelogram polyominoes. This path bounces off the external edges of the polyomino, and is reminiscent of Haglund's well-known bounce statistic for Dyck paths. We define a collection of polynomials that we call q,t-Narayana polynomials, defined to be the generating function of the bistatistic (area,parabounce) on the set of parallelogram polyominoes, akin to the (area,hagbounce) bistatistic defined on Dyck paths in Haglund (2003). In doing so, we have extended a bistatistic of Egge, Haglund, Kremer and Killpatrick (2003) to the set of parallelogram polyominoes. This is one answer to their question concerning extensions to other combinatorial objects. We conjecture the q,t-Narayana polynomials to be symmetric and prove this conjecture for numerous special cases. We also show a relationship between Haglund's (area,hagbounce) statistic on Dyck paths, and our bistatistic (area,parabounce) on a sub-collection of those parallelogram polyominoes living in a (n+1)*n rectangle

How Secure Are Retirement Nest Eggs?

Life's uncertainties can upend the best-laid retirement plans. Health can fail as people grow older, or their spouses can become ill. Older people can lose their jobs, and often have trouble finding new ones. Marriages can end in widowhood or divorce. Health, employment, and marital shocks near retirement can have serious financial repercussions, raising out-of-pocket medical spending, reducing earnings, disrupting retirement saving, and forcing people to dip prematurely into their nest eggs. This brief examines different types of negative events that can strike near retirement. It reports the incidence of widowhood, divorce, job layoffs, disability, and various medical conditions over a 10-year period, and estimates their impact on household wealth. Data come from the Health and Retirement Study (HRS), a nationally representative survey of older Americans conducted by the University of Michigan for the National Institute on Aging. The survey interviewed a large sample of non-institutionalized adults ages 51 to 61 in 1992 and re-interviewed them every other year. The analysis uses data through 2002, the most recent year available. The results show that many people in their 50s and 60s experience negative shocks that threaten retirement security. Job layoffs, divorce, and the onset of work disabilities near retirement substantially erode retirement savings. The findings highlight the limitations of the safety net when things go wrong in late midlife. This Brief was written for the Center for Retirement Research based at Boston College

Health Problems and Job Layoffs Crack Retirement Nest Eggs

As people prepare for retirement, even the best-laid plans can go awry. Health fails. Spouses or other family members become ill. Marriages end in widowhood or divorce. Investments sour. People lose their jobs. Health, employment, and marital shocks can have serious financial consequences -- out-of-pocket health bills, reduced earnings, disrupted retirement saving -- forcing people as young as 50 or 60 to dip into their nest eggs