Enumerative Combinatorics

 
Researchers

Rogério Reis, Nelma Moreira

• The new combinatorial number h(n), introduced by Frank Harary, can be seen as the longest set [n] with a sum free partition. The number h(n) correspond to the same concept of Shur numbers (s(n)) weakening its sum free condition. We proof that for all n, h(n) is always defined (using Ramsey theorem) and computation of values of h(n), for n=1,2,3,4. The last value was obtained using an optimised algorithm that can be used also to compute the known values of s. We obtained a lower bound for h(5). Both numbers h and s can be seen as two players game and for each we tried to determine the longest game. This work was co-authored with Frank Harary and Peter Blanchard.

  [1]Peter Blanchard, Frank Harary and Rogério Reis, Partitions into sum-free sets. INTEGERS: The Electronic Journal of Combinatorial Number Theory, 6 , A7, 2006.

• A family of regular languages representing partitions of [n] was studied, motivated by the configurations of the above games. In particular explicit formulas for their densities and a relationship with well-known integer sequences were established [2].

  [2] Nelma Moreira and Rogério Reis. On the density of languages representing finite set partitions. Journal of Integer Sequences. Vol 8, 05.2.8.2005.

• We studied some problems concerning polyominoes. A string representation for polyominoes that characterises its contours was obtained with which it is easy to express notions such as convexity and the glueing of two polyominoes. An algorithm that determines the size of a polyomino in linear time was designed.