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ArcPy contains a ListFields method, which:

Lists the fields in a feature class, shapefile, or table in a specified dataset.

The resulting object has a count method, but I can't see how to return the field count from this (shouldn't this return an integer?).

Aside from iterating through the fields and incrementing a counter, how can I return the field count?

import arcpy
fields = arcpy.ListFields(<table>)
count = fields.count

[Dbg]>>> count
<built-in method count of list object at 0x0EDEABE8>
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To count the number of items in any list (including a list of fields) you can use the Python len function.

For example:

print len(fields)
count = len(fields)

or

print len(arcpy.ListFields(<table>))
count = len(arcpy.ListFields(<table>))

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