The 'new' d-complete posets (i.e. those which are not trees, shapes,
or shifted shapes) 'explain' the possession of the jeu de taquin and
hook length properties for many but not all of the other posets with
up to 8 elements which have been found to have these properties by
computer search. At the end of this page we will indicate why one
should expect there to exist many jeu de taquin posets which do not
fit into any class of posets which is defined with purely local
structural conditions, such as the class of d-complete posets. We do
not know what to make of the 'extra' hook length posets.
It is not hard to see that a connected jeu de taquin poset must have a unique maximal element. Connected d-complete posets must also have unique maximal elements.
The disjoint union of hook length posets will obviously be hook length.
*Update* In November 2000, Richard Stanley told us the following counterexample to the converse of this statement: If P1 is the four element chain and P2 is the three element "V" poset, then P1 U P2 is hook length, but P2 is not hook length.
In 1994 we collected data only on connected posets. There are 14,512 connected posets with 8 elements. Only 232 of these have the hook length property (around 1.6%). d-Complete posets account for 181 of these (around 78%). Of these, 127 were known to be members of infinite classes of hook length posets prior to 1998 (trees, shapes, shifted shapes, and double-tailed diamonds). So 54 of the remaining 105 hook length posets with 8 elements now have new d-complete explanations of their possession of the hook length property.
None of the hundreds of small connected hook length posets have more than one maximal element, and so it is natural to conjecture that any connected hook length poset must have only one maximal element.
It seems to be a coincidence that the number of 8 element posets with the hook length property is so close to the number of 8 element posets with the jeu de taquin property (232 versus 236, with 181 of each d-complete). For 7 element posets, the comparable numbers are 75 and 77. For both of these, 69 are d-complete. But only 2 of the "extra" 6 hook length posets appear amongst the "extra" 8 jeu de taquin posets.
This construction indicates that characterizing all posets which have the jdt property could be quite difficult. On the one hand, the purely local d-complete condition implies the global jeu de taquin property. (The proof succeeds only by using the classification theorem heavily - there are only 15 classes of slant irreducible d-complete posets, and within each class the common non-uniform portion of each poset is fairly small and can be analysed in an ad hoc fashion.) On the other hand, for general jeu de taquin posets one must tackle a global problem. For the jeu de taquin posets which partly rely upon having a long "correction" neck on top, one may need to develop a measure of how badly jdt can fail by in the lower portion in order to know how many elements will be needed in the neck. And it is conceivable that there could exist other global mechanisms for compensating for local defects with respect to the jdt property.
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