Remarks and conjectures regarding combinatorics of discrete partial functions

We discuss combinatorics of discrete relations, (total) functions, and partial functions. For two sets A and B , we present formulas for the calculations of the number of relations from A to B that are not: (i) functions, (ii) one-to-one functions, or (iii) onto functions. W e provide formulas to calculate the number of functions from A to B that are not: (i) one-to-one or (ii) onto. Also, we determine the number of partial and proper partial functions from A to B . Moreover, we state some conjectures and pose questions for the reader. This paper generates numerous integer sequences, some of which can be found in The On-Line Encyclopedia of Integer Sequences (OEIS) .