Below we give a complete list of all the coherent systems with n components and k minimal path sets for some n and k.
A system is a map s:{0,1}x…x{0,1}→{0,1} where xi=1 (=0) if the ith component works (does not work) and s(x1,…,xn)=1 (=0) if the system works (does not work).
A system is coherent if s is increasing (in a wide sense) in all the variables and it is strictly increasing in each variable at least in a point.
They are given by their minimal path sets.
A path set is a set P such that if all the components in P work, then the system works.
A path set is minimal if its does not contain other path sets.
If the minimal path sets of the system are P1,…,Pk, then:
s(x1,…,xn)=max_i (min_{j∈Pi} xj).
List of coherent systems
n=5
The 180 systems with 5 components were obtained in:
Jorge Navarro and Rafael Rubio (2010). Computations of coherent systems with five components. Communications in Statistics Simulation and Computation 39 (1), 68-84. JCR2010 102/110. Citas JCR hasta 2014: 23.
n=6
The 16145 systems with 6 components are in: Systems
n=7
There is a unique coherent system with 7 components and 1 minimal path set:
1: s(x1,…,x7)=min(x1,…,x7), P1={1,…,7} (series).
There are 966 coherent systems with 7 components and 2 minimal path sets. They are given in the following file:
Systems with 7 components and 2 minimal path sets
Systems with 7 components and 2 minimal path sets and their minimal signatures
There are 76482 coherent systems with 7 components and 3 minimal path sets. They are given in the following file:
Systems with 7 components and 3 minimal path sets
Systems with 7 components and 3 minimal path sets and their minimal signatures
There are 1868650 coherent systems with 7 components and 4 minimal path sets. The file is too big. Please, contact with the author at: jorgenav@um.es.
n=8
There is a unique coherent system with 8 components and 1 minimal path set:
1: s(x1,…,x8)=min(x1,…,x8), P1={1,…,8} (series).
There are 3025 coherent systems with 8 components and 2 minimal path sets. They are given in the following file:
Systems with 8 components and 2 minimal path sets and their minimal signatures
There are 638736 coherent systems with 8 components and 3 minimal path sets. They are given in the following file:
Systems with 8 components and 2 minimal path sets and their minimal signatures
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This work was partially supported by Ministerio de Ciencia y Tecnología under grant MTM2009-08311.