Colored partitions of a convex polygon by noncrossing diagonals

  title={Colored partitions of a convex polygon by noncrossing diagonals},
  author={Daniel Birmajer and J. Gil and M. Weiner},
  journal={Discret. Math.},
For any positive integers a and b , we enumerate all colored partitions made by noncrossing diagonals of a convex polygon into polygons whose number of sides is congruent to b modulo a . For the number of such partitions made by a fixed number of diagonals, we give both a recurrence relation and an explicit representation in terms of partial Bell polynomials. We use basic properties of these polynomials to efficiently incorporate restrictions on the type of polygons allowed in the partitions. 
From Dyck Paths to Standard Young Tableaux
We present nine bijections between classes of Dyck paths and classes of standard Young tableaux (SYT). In particular, we consider SYT of flag and rectangular shapes, we give Dyck path descriptionsExpand
A family of Bell transformations
A glimpse of the versatility of Bell transformations is shown by discussing the enumeration of several combinatorial configurations, including rational Dyck paths, rooted planar maps, and certain classes of permutations. Expand
The general case of cutting of GML surfaces and bodies
Generalized Mobius-Listing bodies and surfaces are generalizations of the classic Mobius band. The original motivation is that for solutions of boundary value problems the knowledge of the domain isExpand
Encoding and avoiding 2-connected patterns in polygon dissections and outerplanar graphs
It is proved that this is always algebraic, through an explicit combinatorial decomposition depending on $\Delta $, and the decomposition also gives a defining system for $D_{\Delta }(z,\mathbf{0})$, which encodes polygon dissections that restrict these patterns as subgraphs. Expand
Some relatives of the Catalan sequence
A sufficient condition for positive definiteness of the sequence c_n(a_2,\ldots,a_r) is found and several examples from OEIS are checked. Expand
The general case of cutting of Generalized Möbius-Listing surfaces and bodies
The original motivation to study Generalized Mobius-Listing GML surfaces and bodies was the observation that the solution of boundary value problems greatly depends on the domains. Since around 2010Expand
Realization of a Method for Calculating Bell Polynomials Based on Compositae of Generating Functions
In this paper different computational methods for calculating partial and n-th complete Bell polynomials are considered. As one of the new methods, the authors propose to use the method forExpand


Nested sets, set partitions and Kirkman-Cayley dissection numbers
  • G. Gaiffi
  • Computer Science, Mathematics
  • Eur. J. Comb.
  • 2015
A proof by explicit bijections of the famous Kirkman-Cayley formula for the number of dissections of a convex polygon is shown. Expand
XII. On the K-partitions of the R-gon and R-ace
  • T. Kirkman
  • Engineering
  • Philosophical Transactions of the Royal Society of London
  • 1857
I. By the k-partitions of an r-gon, I mean the number of ways in which it can be divided by k — 1 diagonals, of which none crosses another; two ways being different only when no cyclical permutationExpand
Polygon Dissections and Euler, Fuss, Kirkman, and Cayley Numbers
A short and elementary proof of the formulas for classical numbers of polygon dissections is given and the relationship between the proof, recent work in knot theory, and Jones' work on planar algebras is described. Expand
Polygon Dissections and Standard Young Tableaux
  • R. Stanley
  • Computer Science, Mathematics
  • J. Comb. Theory, Ser. A
  • 1996
A simple bijection is given between dissections of a convex (n+2)-gon withddiagonals not intersecting in their interiors and standard Young tableaux of shape (d+1, d+1, 1n?1?d).
Variants of Schroeder Dissections
Some formulae are given for the enumeration of certain types of dissections of the convex (n+2)-gon by non-crossing diagonals. The classical Schroeder and Motzkin numbers are addressed using aExpand
Some Convolution Identities and an Inverse Relation Involving Partial Bell Polynomials
We prove an inverse relation and a family of convolution formulas involving partial Bell polynomials. Known and some presumably new combinatorial identities of convolution type are discussed. OurExpand
Advanced Combinatorics The Art Of Finite And Infinite Expansions
The On-Line Encyclopedia of Integer Sequences
  • N. Sloane
  • Mathematics, Computer Science
  • Electron. J. Comb.
  • 1994
The On-Line Encyclopedia of Integer Sequences (or OEIS) is a database of some 130000 number sequences which serves as a dictionary, to tell the user what is known about a particular sequence and is widely used. Expand
Note on a Geometrical Theorem
Exponential polynomials