A Direct Combinatorial Proof of a Positivity Result

"A simple result deserves a simple proof" I Objects lying in four different boxes are rearranged in such a way that the number of objects where the bottom is a rearrangement of the top. The weight W (T) of the multiset permutation T is defined to be (-I) ~-where U, is the number of columns of the form I: 111122333444 , x # y. For example, if n-= 441223133411 then U, = 7 and w (n-) = (-~) ~ =-1. If % is a set, we write w (%) =I,,, w (u) , and if % and Y are sets, we denote as usual ~ ~ Y = { (u , v) ; U E % , U E Y). (a + b-u) ! (c + d-u) ! ~ ! ~ I W(&(a, b, c, d)) = 1 i a ! b ! c ! d !)-1) ()12 r U-r Us in each box stays the same. Askey, Ismail, and Koornwinder proved that the cardinality of the set of rearrangements for which the number of objects changing boxes is even-exceeds the cardinality of the set of rearrangements for which that number is odd. We give a simple counting proof of this fact. Members of four different clubs, each wearing a hat with an insignia of his club, hang their hats on entering the hall. When they leave there is a power failure and the departing pests scramble for hats in the dark. Assuming the hats were picked at an entirely random fashion, would you bet that the number of guests wearing hats with wrong insignias is even? Askey, Ismail, and Koornwinder [ I , p. 2851 proved that the answer is always yes, no matter how many members belong to each club. Their proof was "analytical" in that it employed an inequality of Koornwinder [ 7 ] concerning integrals of products of Laguerre polynomials. Ismail and Tamhankar [5], and independently Gillis and Kleeman [4], gave elementary proofs of this result. However, both of these employed the rather deep "Master Theorem" of McMahon and the relatively sophisticated notion of "generating function". We are going to give a direct counting argument which should be understood by the proverbial bright grade school student. Although our proof is formally from scratch, it does employ the elegant methods of Foata [2], [3]. As a matter of fact it was conceived while we were reading through …