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On the density of languages representing finite set partitions

By Nelma Moreira and Rogério Reis


We present a family of regular languages representing partitions of a set of n elements in less or equal c parts. The density of those languages is given by partial sums of Stirling numbers of second kind for which we obtain explicit formulas. We also determine the limit frequency of those languages. This work was motivated by computational representations of the configurations of some numerical games. 1 The languages Lc Consider a game where natural numbers are to be placed, by increasing order, in a fixed number of columns, subject to some specific constraints. In these games column order is irrelevant. Numbering the columns, game configurations can be seen as sequences of column numbers where the successive integers are placed. For instance, the string 11213 stands for a configuration where 1, 2, 4 were placed in the first column, 3 was placed in the second and 5 was placed in the third. Because column order is irrelevant, and to have a unique representation for each configuration, it is not allowed to place an integer in the 1 Work partially funded by Fundação para a Ciência e Tecnologia (FCT) and Program POSI. 1 kth column if the (k − 1)th is still empty, for any k> 1. Blanchard and al. [BHR04] and Reis and al. [RMP04] used this kind of representation to study the possible configurations of sum-free games. Given c columns, let Nc = {1,..., c}. We are interested in studying the set of game configurations as strings in (Nc) ⋆ , i.e., in the set of finite sequences of elements of Nc. Game configurations can be characterised by the following language Lc ⊂ (Nc) ⋆: Lc = {a1a2 · · · ak ∈ (Nc) ⋆ | ∀i ∈ Nk, ai ≤ max{a1,..., ai−1} + 1}. For c = 4, there are only 15 strings in L4 of length 4, instead of the total possible 256 i

Year: 2005
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