Abstract
We present new functional equations connecting the counting series of plane and planar (in the sense of Harary and Palmer) dissections. Simple rigorous expressions for counting symmetric $r$-dissections of polygons and planar $S$-dissections are obtained.
Highlights
Enumeration of triangulations of a regular (n+2)-gon (or ways to dissect a convex (n+2)gon into n triangles by drawing n −1 diagonals, no two of which intersect in their interior) is one of the most well known problems in enumerative combinatorics
In this article we will be especially interested in another equivalent combinatorial interpretation of these numbers, and namely, the number of trivalent plane trees with n + 2 vertices of degree 1, one of those being selected as a root
The correspondence between the dissections of an (n + 2)-gon with a distinguished outer edge into triangles and such trees is shown in the electronic journal of combinatorics 22(1) (2015), #P1.17 (a)
Summary
Enumeration of triangulations of a regular (n+2)-gon (or ways to dissect a convex (n+2)gon into n triangles by drawing n −1 diagonals, no two of which intersect in their interior) is one of the most well known problems in enumerative combinatorics. Despite the impressive progress in solving the problems described above, the explicit formulas for the numbers of all unrooted dissections of polygons in space into r-gons have not been obtained yet This is primarily due to the complexity of accounting for reflection symmetry in such problems. Brown generalized the solution of this problem to the case of a regular (m + 3)-gon triangulation without distinguished edge, i.e. accounting for all possible symmetries He showed in the article [19] that his approach can be effectively used to solve the counting problem of non-isomorphic dissections into quadrangles of a regular (2p + 4)-gon with n internal points. In the present paper we use the ideas expressed by Brown in the articles [17, 19] to prove the formulas (14) and (20) which link the number of the unrooted dissections of polygons in plane and in space
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