Efficient Regular Polygon Dissections

Ranking efficient units by regular polygon area in DEA

Counting symmetry classes of dissections of a convex regular polygon

This paper proves explicit formulas for the number of dissections of a convex regular polygon modulo the action of the cyclic and dihedral groups. The formulas are obtained by making use of the Cauchy-Frobenius Lemma as well as bijections between rotationally symmetric dissections and simpler classes of dissections. A number of special cases of these formulas are studied. Consequently, some kno...

Counting polygon dissections in the projective plane

For each value of k ≥ 2, we determine the number pn of ways of dissecting a polygon in the projective plane into n subpolygons with k + 1 sides each. In particular, if k = 2 we recover a result of Edelman and Reiner (1997) on the number of triangulations of the Möbius band having n labelled points on its boundary. We also solve the problem when the polygon is dissected into subpolygons of arbit...

Polygon Dissections and Some Generalizations of Cluster Complexes for the Classical Reflection Groups

We define a simplicial complex ∆ W in terms of polygon dissections for every classical reflection group W and positive integer m. For m = 1, ∆ W is isomorphic to the cluster complex corresponding to W , defined in [8]. We enumerate the faces of ∆ W and show that the entries of its h-vector are given by the generalized Narayana numbers N W (i), defined in [3]. We also prove that for any m ≥ 1 th...

Polygon Dissections and Euler, Fuss, Kirkman, and Cayley Numbers

We give a short and elementary proof of the formulas for classical numbers of polygon dissections. We describe the relationship between the proof, our recent work in knot theory, and Jones’ work on planar algebras. Over one hundred years ago Arthur Cayley considered the following problem: “ The partitions are made by non-intersecting diagonals; the problem which have been successively considere...