Denote by $ \binom{n}{k}_q = \prod_{i=0}^{k-1} \frac{ q^{n-i} - 1 }{ q^{k-i} - 1 } $, $ k = 0, 1, \ldots, n $, the $ q $-binomial (Gaussian) coefficients. These numbers are symmetric, in the sense that $ \binom{n}{k}_q = \binom{n}{n-k}_q $, unimodal, and hence maximized for $ k = \lfloor n/2 \rfloor $ (or $ k = \lceil n/2 \rceil $). I am interested in the sum $$ \sum^{n}_{\substack{ k=0 \\ k \, \equiv \, h \pmod \ell }} \binom{n}{k}_q $$ where $ h $ is a parameter over which the sum is to be maximized. Question: Is it true that, for any fixed $ q $, $ n $, $ \ell $, the choice $ h = \lfloor n/2 \rfloor $ (or, equivalently, $ h = \lceil n/2 \rceil $) maximizes the sum?
Special cases:
- This is known to be true for binomial coefficients ($ q \to 1 $). It follows from the result of Katona (1972) stating that $$ \sum^{n}_{\substack{ k=0 \\ k \, \equiv \, \lfloor n/2 \rfloor \pmod \ell }} \binom{n}{k} $$ is the maximum cardinality of a family of subsets $ \mathcal{F} \subseteq 2^{\{1, \ldots, n\}} $ satisfying the condition that if $ A, B \in \mathcal{F} $ and $ A \subsetneq B $, then $ |B| - |A| \geqslant \ell $. The proof relies on the theory of partial orders. A simpler and direct proof for this case would also be appreciated.
- The case $ \ell = 2 $. I saw the following identity in one of the answers to this question: $$ \sum_{i=0}^n(-1)^i\binom ni_q=\begin{cases}0,&n\, \text{odd}\\ \prod_{j=1}^{n/2}(1-q^{2j-1}),&n\, \text{even}\end{cases} $$ If this is true, it implies the statement I am interested in when $ \ell = 2 $. Does someone know how to prove this identity?
Background: The sum represents the number of subspaces of $ \mathbb{F}_q^n $ whose dimension is congruent to $ h $ modulo $ \ell $. The set of all subspaces of $ \mathbb{F}_q^n $ partially ordered by inclusion satisfies the so-called LYM condition. It follows from this fact and a result of Kleitman (1975) that $$ \max_h \sum^{n}_{\substack{ k=0 \\ k \, \equiv \, h \pmod \ell }} \binom{n}{k}_q $$ is the maximum cardinality of a family of subspaces satisfying the condition that if $ U \subsetneq V $, then $ \dim V - \dim U \geqslant \ell $. I was wondering if the maximizer can be found explicitly.
