Abstract
We investigate the problem of listing combinations using a special class of operations, prefix shifts. Combinations are represented as bitstrings of 0's and 1's, and prefix shifts are the operations of rotating some prefix of a bitstring by one position to left or right. We give a negative answer to an open problem asked by F. Ruskey and A. Williams (Generating combinations by prefix shifts, In Proc. 11th Annual International Computing and Combinatorics Conference 2005, LNCS 3595, Springer, 2005, pp.570–576), that is whether we can generate combinations by only using three very basic prefix shifts on bitstrings, which are transposition of the first two bits and the rotation of the entire bitstring by one position in either direction (i.e., applying the permutations σ2, σ n and σ n −1 to the indices of the bitstrings).
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This work is supported in part by the National Natural Science Foundation of China under Grant No. 60553001 and the National Grand Fundamental Research 973 Program of China under Grant Nos. 2007CB807900 and 2007CB807901.
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Cheng, Y. Generating Combinations by Three Basic Operations. J. Comput. Sci. Technol. 22, 909–913 (2007). https://doi.org/10.1007/s11390-007-9094-7
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DOI: https://doi.org/10.1007/s11390-007-9094-7