Hook and shifted hook numbers

AbstractIn a 1977 paper by J. Herman and F. Chung, several families of counterexamples to the conjecture that a tableau shape is uniquely determined (up to reflection, i.e. conjugation) by its multiset of hook numbers were presented. They also showed that by extending the definition of hook length a tableau shape is uniquely determined (up to conjugation) by its extended multiset of hook numbers.Here we provide an infinite family of counterexamples to the conjecture that a shifted tableau shape is uniquely determined by its multiset of shifted hook numbers.Regarding this uniqueness question, there is a great contrast between tableau shapes and shifted tableau shapes. Indeed, the first author had conjectured that there was just one example of nonuniqueness (the two shifted shapes consisting of three cells given by the partitions (2,1) and (3)) in the shifted case. It was the view of the second author that this conjecture was quite unlikely. The fact is that in the first five million cases, there is only one other pair of examples and there seems to be reason to believe that the next example of nonuniqueness follows after just about a mole (about 6×1023) of distinguishable cases. This statement is based on our construction of an infinite family of pairs of shifted shapes possessing the property that for each pair the shapes are different but the multiset of hook numbers are the same, and the fact that a computer search revealed no further examples. It is for these reasons that the conjecture, which seemed unlikely (and indeed was false), should be thought all the more remarkable for being so very nearly true. We are lead to push our luck and conjecture once again that no additional examples exist. The result (Theorem 6) of Section 6 is in this direction