Binomial coefficients and central trinomial coefficients play important roles in combinatorics. Let p > 3 be a prime. We show that $T_{p-1}\equiv (p/3)3^{p-1}\left(modp²\right)$, where the central trinomial coefficient Tₙ is the constant term in the expansion of $(1+x+x^{-1})ⁿ$. We also prove three congruences modulo p³ conjectured by Sun, one of which is ${\sum }_{k=0}^{p-1}\left(\genfrac{}{}{0pt}{}{p-1}{k}\right)\left(\genfrac{}{}{0pt}{}{2k}{k}\right)({(-1)}^k-{(-3)}^{-k})\equiv (p/3)(3^{p-1}-1)\left(modp³\right)$. In addition, we get some new combinatorial identities.

We establish q-analogs for four congruences involving central binomial coefficients. The q-identities necessary for this purpose are shown via the q-WZ method.

For any odd prime p we obtain q-analogues of van Hamme’s and Rodriguez-Villegas’ supercongruences involving products of three binomial coefficients such as ${\sum }_{k=0}^{(p-1)/2}{\left[\genfrac{}{}{0pt}{}{2k}{k}\right]}_{q²}^3\left(q^{2k}\right)/\left((-q²;q²){²}_k(-q;q){²}_{2k}²\right)\equiv 0\left(mod\left[p\right]²\right)$ for p≡ 3 (mod 4), ${\sum }_{k=0}^{(p-1)/2}{\left[\genfrac{}{}{0pt}{}{2k}{k}\right]}_{q³}\left({(q;q³)}_k{(q²;q³)}_k{q}^{3k}\right)\left({(q⁶;q⁶)}_k²\right)\equiv 0\left(mod\left[p\right]²\right)$ for p≡ 2 (mod 3), where $\left[p\right]=1+q+\cdots +q^{p-1}$ and $(a;q)ₙ=(1-a)(1-aq)\cdots (1-a{q}^{n-1})$. We also prove q-analogues of the Sun brothers’ generalizations of the above supercongruences. Our proofs are elementary in nature and use the theory of basic hypergeometric series and combinatorial q-binomial identities including a new q-Clausen type summation formula.

In this paper, we examine the class of "deco" polyominoes and the succession rule describing their construction. These polyominoes are enumerated according to their directed height by factorial numbers. By changing some aspects of the "factorial" rule, we obtain some succession rules that describe various "deco" polyomino subclasses. By enumerating the subclasses according to their height and width, we find the following well-known numbers: Stirling numbers of the first and second kind,...