Abstract. Π0 1 classes are important to the logical analysis of many parts of mathematics. The Π0 1 classes form a lattice. As with the lattice of computably enumerable sets, it is natural to explore the relationship between this lattice and the Turing degrees. We focus on an analog of maximality, or more precisely, hyperhypersimplicity, namely the notion of a thin class. We prove a number of results relating automorphisms, invariance, and thin classes. Our main results are an analog of Martin’s work on hyperhypersimple sets and high degrees, using thin classes and anc degrees, and an analog of Soare’s work demonstrating that maximal sets form an orbit. In particular, we show that the collection of perfect thin classes (a notion which is definable in the lattice of Π0 1 classes) forms an orbit in the lattice of Π01 classes; and a degree is anc iff it contains a perfect thin class. Hence the class of anc degrees is an invariant class for the lattice of Π0 1 classes. We remark that the automorphism result is proven via a ∆0 3 automorphism, and demonstrate that this complexity is necessary. 1.

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Garsia and Haiman (J. Algebraic. Combin. 5 (1996), 191 − 244) conjectured that a certain sum Cn(q,t) of rational functions in q,t reduces to a polynomial in q,t with nonnegative integral coefficients. Haglund later discovered (Adv. Math., in press), and with Garsia proved (Proc. Nat. Acad. Sci. 98 (2001), 4313 − 4316) the refined conjecture Cn(q,t) = ∑ q area t bounce. Here the sum is over all Catalan lattice paths and area and bounce have simple descriptions in terms of the path. In this article we give an extension of (area, bounce) to Schröder lattice paths, and introduce polynomials defined by summing q area t bounce over certain sets of Schröder paths. We derive recurrences and special values for these polynomials, and conjecture they are symmetric in q,t. We also describe a much stronger conjecture involving rational functions in q,t and the ∇ operator from the theory of Macdonald symmetric functions.

Generalizing the notion of placing rooks on a Ferrers board leads to a new class of combinatorial models and a new class of rook polynomials. Connections are established with absolute Stirling numbers and permutations, Bessel polynomials, matchings, multiset permutations, hypergeometric functions, Abel polynomials and forests, and polynomial sequences of binomial type. Factorization and reciprocity theorems are proved and a q-analogue is given.