This work explores the possibility of producing some specific power laws (with exponential cut-off) out from a partitional dynamics. The data obtained from the partitions of integers in the interval 1-60 have been used to find the coefficients of a power law and its exponential cut-off, and also of a single hyperbolic form after renormalization. There is also the speculation that this type of power law, so easily derived from arithmetic minimalist operations, may underlie the communication exchanges of living cells and the structural games of self-constructing agents endowed with relative structural freedom.