You cannot determine a column's description based solely on the serial number; it is possible for two different column part numbers to have the same serial number.
For example, if the same batch of bulk is used to pack a 4.6x150mm column and a 4.6x250mm column on the same day in the same manufacturing facility, and if they are packed during the same production run, the two columns will have the same 10-digit lot number in many cases. Then, if the two columns are packed on the same packing station, and if they have the same sequential number within a given production run, both columns would have the same exact serial number.
Waters does not track columns based on the 14-digit serial number; instead, it tracks columns based on the 10-digit lot number (the first 10 digits of the serial number).
The only way to identify a column description on the basis of serial number is to review a company's purchasing history for specific part numbers and then see if Waters shipped any of those specific part numbers in the specific 10-digit lot number in question.