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Macaulay 3 (earlier version Macaulay 2) is a software package for research in commutative algebra and algebraic geometry and related fields.
The Manual or the Language overview are good reference points. There is also a book.
The program is available on our server Abel and is started up with "Macaulay" and end it with "quit" at the Macaulay prompt. Macaulay 2 is started up with the command "M2". The program is free and installs on Gnu/Linux machines, Solaris or Windows machines.
An example shows how to compute with matrices over a polynomial ring over a finite field R. An example shows how to work with polynomials. Some combinatorial functions like partitions are built in. Look at some examples in the manual.

Simplicity, Clarity, Generality B.W. Kernighan, R. Pike, in "The Practice of Programming".