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      Non-crossing chords of a polygon with forbidden positions

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          Abstract

          In this paper we investigate non-crossing chords of simple polygons in the plane systematically. We first develop the Euler characteristic of a family of line-segments, and subsequently study the structure of the diagonals and epigonals of a polygon. A special phenomenon is that the Euler characteristic of a set of diagonals (or epigonals) characterizes the geometric property of polygons, such as convexity. In particular, a positive answer is given to an open problem proposed by Shephard. The main contributions of the present paper extend such research to non-crossing diagonals and epigonals with forbidden positions. We find that the Euler characteristic of diagonals (or epigonals) with forbidden positions determine the types of polygon in surprising ways. Incidentally, some kinds of generalized Catalan's number naturally arise.

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          Most cited references4

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          The Associahedron and Triangulations of the n-gon

          Carl W Lee (1989)
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            Polygon Dissections and Euler, Fuss, Kirkman, and Cayley Numbers

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              Divided differences of inverse functions and partitions of a convex polygon

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                Author and article information

                Journal
                2016-11-09
                Article
                1611.03166
                e407c691-ddfe-407e-9a35-cefaa18e2b6d

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

                History
                Custom metadata
                51E12, 05B25, 51D20, 51E30
                19 pages
                math.CO

                Combinatorics
                Combinatorics

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