I am not an expert in combinatorics, but I need to count (or to approximate) the number of antichains of a poset.

The idea relies on approximating this number by embedding my poset into another one, that I call "rectangular" but actually I don't know if there is already a standard definition.

Let's say that my poset has depth $d$ (t.i. the maximal lenght of a chain) and spread $a$ (t.i., the max of cardinalities among all maximal antichains). Then I embedd this poset into another one, that has $d$ independent chains of lenght $a$, not considering the $0$ and $1$ elements. I claim that the second one contains more antichains than the first one, and that this number is $d \cdot 2^a$.

Is this idea correct? May I have a better upper bound of the number of antichains in a general poset?

Thank you all in advance for any answer!