On the Permanents of Complements of the Direct Sum of identity Matrices

A classic experiment used in testing for ESP abilities has the following general form. A deck of cards, consisting of a, identical cards of type i, 1 I i I r, is shuffled and placed face down (in most such experiments r = 5 and a, = 5 for 1 I i _( 5). The subject then attempts to correctly guess the type of each card as the cards are sequentially removed from the deck. In previous work [2,3], several of the authors have analyzed the effects of allowing various kinds of feedback into this process. For example, after each incorrect guess the subject might be told what the guessed card actually was. Obviously such information, if used appropriately, could significantly increase the number of correct guesses the subject could expect to make during a pass through the deck. Consider the standard deck consisting of 25 cards, 5 cards of each of 5 types. Without any feedback (or ESP ability) the expected number of correct guesses is 5. With complete feedback, a subject can expect to achieve more than 8.64 correct guesses, simply by always guessing the most frequently occurring type in the remaining deck (see [3]). Another very important type of feedback, investigated in [3], was that in which the subject was just told whether each guess is right or wrong (but not the correct identity of an incorrectly guessed card). The optimal strategy for using this kind of partial feedback is extremely complex and, in some cases, counter-intuitive. For example, the optimal strategy can require guessing a type which is not the most likely type in the remaining deck (see [3]). A fundamental quantity in these studies is N( a,, . . . , a,; b, , . . . , b,) which is defined to be the number of arrangements of a deck of u, + . . . +a, = n cards, with uj of type i, such that symbol 1 does not appear in the first b,