By Anthony Mendes, Jeffrey Remmel

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**Example text**

J keeping track of how many 0’s, . . , j’s were in r by sorting r in decreasing order. Finally, let a be the sequence with part i equal to k if and only if there is an decrease of size k in the sequence c after place i. † For example, suppose that r = 0 1 2 3 0 1 0 0 2 3 1 3. The table below shows τr−1 , τr ,c, and a: r τr−1 τr c a 1 2 3 0 1 2 12 8 5 12 10 4 3 3 3 0 0 1 = = = = = 4 5 6 7 8 3 0 1 0 0 3 11 7 10 9 9 3 11 6 2 2 2 1 1 1 0 1 0 0 1 9 10 11 12 2 3 1 3 4 2 6 1 8 7 5 1 0 0 0 0 0 0 0 0 −1 From this construction, the x weight in a fixed point is xdes(τr ) and the y −1 weight in a fixed point is y ris(τr ) .

4. Published uses of brick tabloids in permutation enumeration Although there have been other connections from the ring of symmetric functions to the permutation enumeration of the symmetric group, Brenti established a direct connection with the homomorphism ξ f1 [13, 14]. 3, Beck and Remmel provided the ideas which we used to prove the theorems in [4, 6]. In addition to these works, there have been other publications which further investigate the methods introduced in the previous section (as is the case with this monograph, all authors of these publications have had direct ties to Remmel).

Suppose the last brick in T is of length j. The weight on the last brick given by ν1 is xj−1 /(x − y)j−1 . This enables us to replace our choice of either x or −y in the nonterminal cells of T with x. If the last brick in T is longer than j cells, the weight on the last brick given by ν1 is xj−1 (−y)/(x − y)j . This enables us to replace the last j choices for x or −y with one −y followed by j − 1 x’s. One object which may be formed in this manner is found below. 31). 3 where brick are scanned from left to right for the first occurrence of either a −y or two consecutive bricks with a decrease in the integer labeling between them.