9 research outputs found
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 patterns of length k. This extends and illuminates a result of
Robertson, Wilf and Zeilberger for the case k=3.Comment: 9 pages, 1 figur
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On the Average Genus of a Graph
Not all rational numbers are possibilities for the average genus of an individual graph. The smallest such numbers are determined, and varied examples are constructed to demonstrate that a single value of average genus can be shared by arbitrarily many different graphs. It is proved that the number one is a limit point of the set of possible values for average genus and that the complete graph K4 is the only 3-connected graph whose average genus is less than one. Several problems for future study are suggested
The imbedding sum of a graph
Consiglio Nazionale delle Ricerche - Biblioteca Centrale - P.le Aldo Moro, 7 Rome / CNR - Consiglio Nazionale delle RichercheSIGLEITItal
Continued fractions and Catalan problems
Consiglio Nazionale delle Ricerche - Biblioteca Centrale - P.le Aldo Moro, 7 Rome / CNR - Consiglio Nazionale delle RichercheSIGLEITItal