Nesrine Benyahia Tani
Zahra Yahi
Sadek Bouroubi

Ordered and non-ordered non-congruent convex quadrilaterals inscribed in a regular n -gon

The paper is published: Rostocker Mathematisches Kolloquium, Rostock. Math. Kolloq. 68, 71 - 79 (2013)

MSC: 05A15   Exact enumeration problems, generating functions [See also 33Cxx, 33Dxx]
  05C05   Trees
  05C30   Enumeration in graph theory

Abstract:   Using several arguments, some authors showed that the number of non-congruent triangles inscribed in a regular \mbox{$n$-gon} equals $\{n^{2}/12\}$, where $\left\{ x\right\}$ is the nearest integer to $x$. In this paper, we revisit the same problem, but study the number of ordered and non-ordered non-congruent convex quadrilaterals, for which we give simple closed formulas using Partition Theory. The paper is complemented by a study of two further kinds of quadrilaterals called proper and improper non-congruent convex quadrilaterals, which allows to give a formula that connects the number of triangles and ordered quadrilaterals. This formula can be considered as a new combinatorial interpretation of a certain identity in Partition Theory.

Keywords:   Congruent triangles, Congruent quadrilaterals, Ordered quadrilaterals, proper quadrilaterals, Integer partitions.
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