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Enumerative combinatorics, graph theory, order theory, posets, matroids, designs and other discrete structures. It also includes algebraic, analytic and probabilistic combinatorics.

12 votes
1 answer
194 views

Ternary sequences satisfying $ x_i + y_i = 1 $ for some $ i $

Consider a set of strings $ {\mathcal S} \subset \{0, 1, 2\}^n $ satisfying the following two conditions: 1.) every string in $ {\mathcal S} $ has exactly $ k $ symbols from $ \{0, 1\} $ (i.e., $ \for …
aleph's user avatar
  • 503
3 votes
1 answer
595 views

Sum of $q$-binomial coefficients

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 tha …
aleph's user avatar
  • 503
2 votes
3 answers
875 views

The number of submodules of $\mathbb{Z}_q^n$

Observe $\mathbb{Z}_q^n = \mathbb{Z}_q \times \cdots \times\mathbb{Z}_q$ as a module over $\mathbb{Z}_q\equiv\mathbb{Z}/q\mathbb{Z}$, for general $q$. I am interested in the following questions: How …
aleph's user avatar
  • 503