This question is about a possible loophole in a role of Greene and Kleitman that Zarathustra Brady let me know.
The article in question is "Longer chains in the network of entire partitions sorted by wholesale" (available online here). In that document, they calculate the length of the longest chain in the order of dominance in the partitions and, in general, give an algorithm to find the longest chain in any interval of order of domination.
In order of domination we have a coverage relationship $ lambda gtrdot mu $ If and only if $ mu $ is obtained from $ lambda $ moving a single box in a row $ i $ row $ i + 1 $or moving a single frame in the column $ i + 1 $ to the column $ i $. In the first case, Greene and Kleitman say that $ lambda gtrdot mu $ is a Step h (Because, perhaps confusingly for modern readers, the box moved a unit horizontally according to its non-standard scheme of drawing partitions with vertical parts, see Figure 2), and in the second case they say that $ lambda gtrdot mu $ is a V step (because the box moved a unit vertically according to its representation). Keep in mind, as the authors point out, that it is possible that $ lambda gtrdot mu $ it is both a step H and a step V (and in fact this is the source of the possible lagoon!).
Greene and Kleitman say a chain $ lambda ^ 0> lambda ^ 1> cdots> lambda ^ L $ is a H string yes every step $ lambda ^ i> lambda ^ i + 1} = lambda ^ i gtrdot lambda ^ i + 1} $ it's a step H, and similarly say that the chain is a V string yes every step $ lambda ^ i> lambda ^ i + 1} = lambda ^ i gtrdot lambda ^ i + 1} $ It's a step in V. Also, they say $ lambda ^ 0> lambda ^ 1> cdots> lambda ^ L $ is a HV chain if there is any index $ i $ such that $ lambda ^ 0> cdots> lambda ^ i $ it's an H chain and $ lambda ^ i> cdots> lambda ^ L $ It is a V chain.
In a crucial motto of the article, Lemma 3, they affirm that if $ lambda = lambda ^ 0> lambda ^ 1> cdots> lambda ^ L = mu $ is any chain in order of domination, then there is some HV chain between $ lambda $ and $ mu $ in length at least $ L $. The argument they give is: we can assume that each step in the chain is a hedging relationship; we assure that it is true for the chains $ lambda_0> lambda_1> lambda_2 $ of length 2; then, by repeatedly applying this case of length 2 we can convert any length string $ L $ to an HV chain of at least length $ L $.
But this last point about the repeated application of the case of length 2 seems suspicious, for the following reason. Suppose we have a chain of length 3 $ lambda_0> lambda_1> lambda_2> lambda_3 $ such that $ lambda_0> lambda_1 $ it's a V step that is not an H step, $ lambda_1> lambda_2 $ it's a step V and H, and $ lambda_2> lambda_3 $ it is a step H that is not a step V. (This situation may arise: $ (5,4,3,2)> (4,4,4,2)> (4,4,3,3)> (4,4,3,2,1) $.) Then the problem is that, from the perspective of substrings of length 2, things look good: $ lambda_0> lambda_1> lambda_2 $ it is a V chain, so it is in particular an HV chain; Similary $ lambda_1> lambda_2> lambda_3 $ it is an H chain, so in particular it is an HV chain. But $ lambda_0> lambda_1> lambda_2> lambda_3 $ It is obviously not an HV chain.
Question: Is this a real oversight in the role of Greene-Kleitman? If so, is Lemma 3 true, and can the test be repaired?