By THE MATHEMATICAL ASSOCIATION OF AMERICA
Read Online or Download American Mathematical Monthly, volume 105, number 2, February 1998 PDF
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Extra resources for American Mathematical Monthly, volume 105, number 2, February 1998
Sample text
TABLE 2. R. Free Actions on T n + 1 by Groups of Order 2n, n ~ 7. r. r. on T nr + 1 if and only if n is odd. 2. Ifr is even, then is odd, then Upper Bound ZZn o. Ifr The proofs are straightforward (see [6] for details), but one observation here is that such results may never have been speculated without a computer to do the dirty work. Technology has become an indispensable tool in our continuing search to discover beautiful connections in science, mathematics, and nature. ACKNOWLEDGMENT. This paper would not have been possible without the assistance of Larry W.
Kierstead and Trotter [9] strengthened the Cycle Lemma on (k, k + 1)-arrangements, showing that each 0 plays a special role. For 1 ~ i ~ k + 1, they proved that every (k, k + 1)-arrangement has a unique linearization ending in a 0 such that exactly i of the prefixes ending at O's have more O's than l's. They noted that this result is implicit in the work of Feller [6] and Narayana [11], and they used it to construct new explicit perfect matchings in the bipartite graph of the inclusion relation on the k-sets and k + I-sets of a 2k + I-element set.
4 with r = l. We now have the sequence 1 -) 7T 1T n+1 -) 7T 1U 3 -) G -) l. 7 gives us a means for determining all such sequences (if any) for a given G by employing a computer to repeatedly perform the steps in the following algorithm: • • • • Construct all possible homomorphisms a: 7T 1U 3 -) G. Determine which of these are epimorphisms. r. free action. r. free action. Before we begin our search, we quickly address the preliminary question: Are any of the free actions discovered using the preceding algorithm equivalent in a natural way?
