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Download American Mathematical Monthly, volume 105, number 2, by THE MATHEMATICAL ASSOCIATION OF AMERICA PDF

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Additional resources for American Mathematical Monthly, volume 105, number 2, February 1998

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M. S. Wilf, The Editor's Corner: n coins in a fountain, Amer. Math. Monthly 95 (1988), 840-843. G. N. Raney, Functional composition and power series reversion, Trans. Amer. Math. Soc. 94 (1960), 441-451. A. D. Sands, On generalized Catalan mlmbers, Discrete Math. 21 (1978), 219-221. H. S. Snevily, Combinatorics of finite sets. D. Thesis, University of Illinois, 1991. HUNTER S. D. from the University of Illinois under Douglas West. He was a Bateman Research Instructor at Caltech (1991-93) and is' now an Associate Professor of Mathematics at the University of Idaho.

We conclude that the number of q-dominating (k, qk + P )-sequences is (( q+ l)k + p) k P , which equals the . k +qk+p formula claimed. Each q-satisfying sequence of length m has k I's and m - k O's, for some k. By setting m - k = qk + p - 1, we obtain p = m - (q + I)k + 1 and can use the preceding formula to count these sequences. Thus the term for k in the summation is precisely the number of q-satisfying sequences of length m that have k I's . • 1998] THE BRICKLAYER PROBLEM AND THE STRONG CYCLE LEMMA 133 /I\l\ A abc ((abc)de) --+ d e abc (a(bcd)e) abcldel --+ d e abcdlel U abc (ab( cde)) --+ d e abcdell Figure 3.

But no details were given. Dickson waited until March 1913 to write up the paper [D 2 ] containing the division algebra part of his 1906 abstract, but we do not know what Dickson wrote in the original version of this paper. Wedderburn saw the original version and proved a general theorem (in November 1913; see [D 2 , p. 33]), which gave a sufficient condition for Dickson's construction to yield a division algebra. ) Wedderburn's result. From the introduction to Wedderburn's April 1914 paper [W3]' it is clear that in the original version of his paper, Dickson had managed to construct examples of division algebras only of dimension 4 or 9 over the centre.

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