Q-Pochhammer Symbol - Combinatorial Interpretation

Combinatorial Interpretation

The q-Pochhammer symbol is closely related to the enumerative combinatorics of partitions. The coefficient of in

is the number of partitions of m into at most n parts.

Since, by conjugation of partitions, this is the same as the number of partitions of m into parts of size at most n, by identification of generating series we obtain the identity:

$(a;q)_\infty^{-1} = \sum_{k=0}^\infty \left(\prod_{j=1}^k \frac{1}{1-q^j} \right) a^k = \sum_{k=0}^\infty \frac{a^k}{(q;q)_k}$

as in the above section.

We also have that the coefficient of in

is the number of partitions of m into n or n-1 distinct parts.

By removing a triangular partition with n-1 parts from such a partition, we are left with an arbitrary partition with at most n parts. This gives a weight-preserving bijection between the set of partitions into n or n-1 distinct parts and the set of pairs consisting of a triangular partition having n-1 parts and a partition with at most n parts. By identifying generating series, this leads to the identity:

$(-a;q)_\infty = \prod_{k=0}^\infty (1+aq^k) = \sum_{k=0}^\infty \left(q^{k\choose 2} \prod_{j=1}^k \frac{1}{1-q^j}\right) a^k = \sum_{k=0}^\infty \frac{q^{k\choose 2}}{(q;q)_k} a^k$