Finite posets and Ferrers shapes (Q5937633)

Given a finite poset \(P\), it has an associated Ferrer's shape \(\lambda(P)\), the number of boxes in the first \(k\) rows (resp. columns) of \(\lambda(P)\) equaling the maximum number of elements in a union of \(k\)-chains (resp. antichains of \(P\)). Given the quite slippery duality-like correspondence between chain theory and anti-chain conditions of Rival-Zaguia for example, it is clear from the definition of \(\lambda(P)\) and the experience of partition theory, where the ordering is by refinement, that it should be possible to obtain a plethora of interesting results which aim to extend not only information about poset structure in general (i.e., without extra conditions on the class other than finiteness) but to better illustrate the delicate balance between comparability and noncomparability pictured in the actual shapes of \(\lambda(P)\). Obviously, from the setting it is clear that results obtained would generalize classical results if the latter can be cast in poset language, as with Young tableaux for ``permutation posets'' for example. The list is quite considerable and touches on many other areas more numerous than a short review needs to list, other than to point out that here we have an excellent introduction to a very important part of general poset theory originating in fundamental work by C. Greene, D. J. Kleitman and many others, the authors of this survey, which allows the beginner to commence virtually ab initio, included.