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This paper gives an elementary and self-contained proof of Conway's Basic Theorem on rational tangles. This theorem states that two rational tangles are topologically equivalent if and only if they have the same associated rational fraction. Our proof divides into a geometric half that relates the arithmetic of continued fractions to the topology of tangles and an algebraic part that defines the fraction of any tangle via the bracket model for the Jones polynomial. We present an application to molecular biology. ᮊ 1997 Academic Press

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We introduce a new tool, the factorial polynomials, to study rook equivalence of Ferrers boards. We provide a set of invariants for rook equivalence as well as a very simple algorithm for deciding rook equivalence of Ferrers boards. We then count the number of Ferrers boards rook equivalent to a given Ferrers board.

dedicated to the memory of gian-carlo rotaGeneralizing 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. Academic Press

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