# Recurrence Relations, Partitions, Generating Functions, Monomials and Noncommutative Algebraic Structure

4. In noncommutative algebra, the term monomial refers to any arrangement of a sequence of variables from a set. For example, in a noncommutative algebraic structure on a set of four variables, {x,y,z,w} , examples of monomials of length

3 are xxx,xyx,xxy,zwy,wzx........

a) Write a generating function for the number of monomials of length, n, in a noncommutative algebraic structure on a set of four variables.

b) Find the number of monomials of length, n, in a noncommutative algebraic structure on a set of four variables.

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#### Solution Preview

Proof:

(a) Let f be the generating function for the number of monomials of length n, in a noncommutative algebraic structure on a set of four variables {x,y,z,w}

When n=1, f(n)=4. There are only 4 cases: ...

#### Solution Summary

Recurrence Relations, Partitions, Generating Functions, Monomials and Noncommutative Algebraic Structure are investigated.