Jean Fromentin

Schur and weak Schur partitions

We enumerate (in lexicographic order) all the terminal Schur $k$-partitions, i.e., such that it is not possible to extend the partition by placing the next successive integer while keeping the sum-free property. To reduce the size of the data we regroup such partitions into cluster. The for each cluster we determine the mean and the maximal partition length :

Type Colors $k$ Terminal partitions Cluster size Correlation Files
Strong 3 138 10 0.66 data png
Strong 4 92 292 017 1 000 000 -0.17 data png
Weak 3 8 066 100 0.88 data png
Weak 4 536 994 391 720 1 000 000 000 0.69 data png

Source code is availbale on GitHub.

Reference

[1] Investigating Monte-Carlo methods on the weak Schur Problem with S. Eliahou, C. Fonlupt, V. Marion-Poly, D. Robilliard and F. Teytaud in Evolutionary Computation in Combinatorial Optimization, vol. 7832 of Lectures Notes in Comput. Sci., 191-201 (2013) HAL