Parity Theorems for Combinatorial Statistics

A q-generalization Gn(q) of a combinatorial sequence Gn which reduces to that sequence when q = 1 is obtained by q-counting a statistic defined on a sequence of finite discrete structures enumerated by Gn. In what follows, we evaluate Gn(−1) for statistics on several classes of discrete structures, giving both algebraic and combinatorial proofs. For the latter, we define appropriate sign-reversing involutions on the associated structures. We shall call the actual algebraic result of such an evaluation at q = −1 a parity theorem (for the statistic on the associated class of discrete structures). Among the structures we study are permutations, binary sequences, Laguerre configurations, derangements, Catalan words, and finite set partitions. As a consequence of our results, we obtain bijective proofs of congruences involving Stirling, Catalan, and Bell numbers. In addition, we modify the ideas used to construct the aforementioned sign-reversing involutions to furnish bijective proofs of combinatorial identities involving sums with alternating signs

Foreword

In order to get a proper measure of modern Estate Planning I think it may be useful to consider, very briefly, some aspects of its history. The family trust was born into our jurisprudence in an environment which had been moulded in that solid and immovable pyramid called feudalism. In that social order there was nothing of imaginative elasticity. Lateral allegiances or entanglements were as little known as lateral movements. All lines of authority moved from the top; all discharges of duties were rendered vertically to the liege lord next above. As with human relationships so also with property. The turreted and fortified provincial seat of that day, like as not surrounded by a deep water moat, provided its own economy; save for its contribution of natural produce, and its quota of fines, recoveries and fighting power to the liege lord, it knew not the outside world. In those days the word value had no significance; the world of wide application was the legal term use. The money changer had yet to make his place in society--so far backward, indeed, had men fallen from the ordered days of Rome that the limits of trade movements, of government authority and of human intercourse were actually and truly a matter of visible horizons

National Autism Indicators Report: Vocational Rehabilitation 2016

Employment is about more than simply earning a paycheck - it influences quality of life, independence, and wellness. Historically, employment outcomes for adults with autism are poor. The U.S. Vocational Rehabilitation system (VR) is designed to provide support to states for implementation of services to assist people with disabilities to prepare for, find, and keep employment. VR data allow us to examine some outcomes for those with autism compared to their peers.To make a difference, research must reach those who need it. The National Autism Indicators Report series presents our research findings in a clearly communicated, open-access, online format to speed the delivery of information to decision-makers while maintaining very high standards of scientific credibility

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