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Number of Ways in Which an n-kilogram Stone can be Broken into Minimum Possible Number of Weights to Weigh All Integral Weights from 1 to n kg(s) on a Bi-panel Scale
Abstract
A seller has an n kilogram stone which he wants to break into minimum
possible number of weights using which on a bi-panel scale he can sell in whole
kilograms up to n kilogram(s) of goods in one weighing. As in tradition, he can
place weights on both the panels but goods on only one panel. Our intention is
to find out the number of all different partitions possible with the minimum
number parts broken from an n kilogram stone so that all integral weights from
1 to n kilograms can be weighed using them. We call such partitions a feasible
partition. By the way, we will prove some rules of feasibility which will lead
us to our main result, the recursive function to find out the number of all
feasible partitions of any positive integer n.
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An M-partition of a positive integer m is a partition with as few parts as possible such that any positive integer less than m has a partition made up of parts taken from that partition of m. This is equivalent to partitioning a weight m so as to be able to weigh any integer weight l < m with as few weights as possible and only one scale pan. We show that the number of parts of an M-partition is a log-linear function of m and the M-partitions of m correspond to lattice points in a polytope. We exhibit a recurrence relation for counting the number of M-partitions of m and, for ``half'' of the positive integers, this recurrence relation will have a generating function. The generating function will be, in some sense, the same as the generating function for counting the number of distinct binary partitions for a given integer.
M-partitions: Optimal Partitions of Weight for One Scale Pan
- E Shea
E. O'Shea, M-partitions: Optimal Partitions of Weight for One Scale Pan, Discrete Math.
289 (2004), 81-93.