Continued Fractions and Catalan Problems
نویسندگان
چکیده
منابع مشابه
Continued Fractions and Catalan Problems
We find a generating function expressed as a continued fraction that enumerates ordered trees by the number of vertices at different levels. Several Catalan problems are mapped to an ordered-tree problem and their generating functions also expressed as a continued fraction. Among these problems is the enumeration of (132)-pattern avoiding permutations that have a given number of increasing patt...
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Let N be a positive non-square integer and a(1),a(2),...,a(3) be the partial denominators in the period of length s = s(N) of the continued fraction for radicalN. Also let Sigma(N) = a(3) - a(3-1) + -... +/- a(1), and let h(d) be the class-number of Q( radicald). Hirzebruch (unpublished) recently found the surprising theorem (which is a special case of more general results): If p is a prime ide...
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General T-fractions and M-fractions whose approximants form diagonals in two-point Pad6 tables are subsumed here under the study of Perron-Carath6odory continued fractions (PC-fractions) whose approximants form diagonals in weak two-point Pad6 tables. The correspondence of PCfractions with pairs of formal power series is characterized in terms of Toeplitz determinants. For the subclass of posit...
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Given a continued fraction, we construct a certain function that is discontinuous at every rational number p/q. We call this discontinuity the “gap”. We then try to characterize the gap sizes, and find, to the first order, the size is 1/q2, and that, for higher orders, the gap appears to be perfectly ’randomly’ distributed, in that it is Cauchy-dense on the unit square, and thus, this function ...
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ژورنال
عنوان ژورنال: The Electronic Journal of Combinatorics
سال: 2000
ISSN: 1077-8926
DOI: 10.37236/1523