Partition numbers and Gaussian binomial coefficient

Question

• Let $a(n)$ be A000041 i.e. the number of partitions of $n$ (the partition numbers).

• Let $T(n, k)$ be A083906. Here

$$ T(n, k) = [q^k]\sum\limits_{m=0}^{n} \binom{n}{m}_q $$

where $\binom{n}{m}_q$ denotes Gaussian binomial coefficient.

I conjecture that

$$ T(n, k) = [n > k](T(n-1, k) + a(k)) + [n \leqslant k](2T(n-1, k) - T(n-2, k) + T(n-2, k - n + 1)), \\ T(0, k) = [k = 0], T(1, k) = 2[k = 0]. $$

where square bracket denotes Iverson bracket.

I also conjecture that

$$ \sum\limits_{i = 0}^{n}T(n-i, i) = a(n+1) $$

Note that upper limit of the summation can be reduced.

Is there a way to prove it? Is there a way to get a simple closed form for $T(n,k)$ based on the given recursion?

Answer 1

Consider Rogers-Szego polynomials defined by $$ H_n(t)=\sum\limits_{m=0}^{n} \binom{n}{m}_qt^m. $$

In Andrews, "Theory of partitions", exercise 6 in chapter 3 gives three term recurrence satisfied by this polynomial

$$ H_{n}(t)=(1+t)H_{n-1}(t)-(1-q^{n-1})tH_{n-2}(t). $$

Since

$$ T(n,k)=[q^k]H_n(1), $$

one can easily work out the case $n\le k$ of the recurrence for $T(n,k)$ from here.

As to the conjecture $$ \sum\limits_{i = 0}^{n}T(n-i, i) = a(n+1), $$ it can be proved using (3.3.9) from Andrew's book $$ \frac{1}{(q;q)_{m+1}}=\sum\limits_{j=0}^{\infty} q^j\binom{m+j}{m}_q, $$ as follows: $$ \sum\limits_{i = 0}^{n}T(n-i, i) =\sum\limits_{i = 0}^{n}[q^i]\sum\limits_{m= 0}^{n-i}\binom{n-i}{m}_q=\sum\limits_{i = 0}^{n}[q^{n-i}]\sum\limits_{m= 0}^{i}\binom{i}{m}_q\\=[q^{n}]\sum\limits_{i = 0}^{n}q^i\sum\limits_{m= 0}^{i}\binom{i}{m}_q=[q^{n}]\sum\limits_{m = 0}^{n}\sum\limits_{i=m}^{n}q^i\binom{i}{m}_q=[q^{n}]\sum\limits_{m = 0}^{n}q^m\sum\limits_{i=0}^{n-m}q^i\binom{i+m}{m}_q\\=[q^{n}]\sum\limits_{m = 0}^{n}q^m\sum\limits_{i=0}^{\infty}q^i\binom{i+m}{m}_q =[q^{n}]\sum\limits_{m = 0}^{n}q^m\frac{1}{(q;q)_{m+1}}\\=[q^{n+1}]\sum\limits_{m = 0}^{n}\frac{1-(1-q^{m+1})}{(q;q)_{m+1}}=[q^{n+1}]\sum\limits_{m = 0}^{n}\left(\frac{1}{(q;q)_{m+1}}-\frac{1}{(q;q)_{m}}\right) =[q^{n+1}]\left(\frac{1}{(q;q)_{n+1}}-1\right)=a(n+1) $$