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Combinatorics
Published in Sriraman Sridharan, R. Balakrishnan, Foundations of Discrete Mathematics with Algorithms and Programming, 2019
Sriraman Sridharan, R. Balakrishnan
The total number of unordered partitions of an n-set into mutually disjoint subsets is the n-th Bell number, denoted by Bn $ B_n $ . Since nk $ \left\{ n\atop k\right\} $ is the number of unordered partitions of an n-set into k mutually disjoint non-empty subsets, we have the following proposition.
Discrete Mathematics
Published in Dan Zwillinger, CRC Standard Mathematical Tables and Formulas, 2018
The Bell number Bn denotes the number of partitions of a set with n elements. For example: the 5 ways to partition the 3-element set {a,b,c} $ \{ a, b, c\} $ are: {(a),(b),(c)}, $ \{ (a), (b), (c)\} , $ {(a), (b,c{(b), $ (b, c \{ (b), $ (a,c{(c), $ (a, c \{ (c), $ (a, b and {(a, b, c The Bell numbers may be written in terms of the Stirling subset numbers: Bn=∑nm. $ B_{n} = \mathop \sum \limits_{{}}^{{}} \left\{ {\begin{array}{*{20}l} n \hfill \\ m \hfill \\ \end{array} } \right\}. $
A recursive LMI-based algorithm for efficient vertex reduction in LPV systems
Published in International Journal of Control, 2022
Adrián Sanjuan, Damiano Rotondo, Fatiha Nejjari, Ramon Sarrate
The number of possible partitions of a set, which corresponds to the Bell number, grows exponentially with the cardinality of the set itself. This fact prevents the development of an optimal search algorithm due to the computational complexity of the problem. To overcome such complexity, an algorithm (Algorithm 3) that introduces some constraints to the search has been developed and assessed in an academic example. As a result, this algorithm finds a combinable partition with the minimal (or close to the minimal) number of elements much more efficiently than through a brute-force search.