We find, in the form of a continued fraction, the generating function for the number of (132)-avoiding permutations that have a given number of (123) patterns, and show how to extend this to permutations that have exactly one (132) pattern. We also find some properties of the continued fraction, which is similar to, though more general than, those that were studied by Ramanujan. the electronic ...

In this paper we investigate an extension to Vuillemin's work on continued fraction arithmetic [Vuillemin 87, Vuillemin 88, Vuillemin 90], that permits it to evaluate the standard statistical distribution functions. By this we mean: the normal distribution, the -distribution, the t-distribution, and, in particular, the F-distribution. The underlying representation of non-rational computable rea...

For uniformly chosen random α ∈ [0, 1], it is known the probability the nth digit of the continued-fraction expansion, [α]n converges to the Gauss-Kuzmin distribution P([α]n = k) ≈ log2(1 + 1/k(k + 2)) as n → ∞. In this paper, we show the continued fraction digits of √ d, which are eventually periodic, also converge to the Gauss-Kuzmin distribution as d → ∞ with bounded class number, h(d). The ...

We detail the continued fraction expansion of the square root of the general monic quartic polynomial. We note that each line of the expansion corresponds to addition of the divisor at infinity, and interpret the data yielded by the general expansion. The paper includes a detailed ’reminder exposition’ on continued fractions of quadratic irrationals in function fields. A delightful ‘essay’ [16]...

For integers m ≥ 2, we study divergent continued fractions whose numerators and denominators in each of the m arithmetic progressions modulo m converge. Special cases give, among other things, an infinite sequence of divergence theorems, the first of which is the classical Stern-Stolz theorem. We give a theorem on a class of Poincaré type recurrences which shows that they tend to limits when th...