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By George E. Andrews, Peter Paule, Axel Riese (auth.), Anton Betten, Axel Kohnert, Reinhard Laue, Alfred Wassermann (eds.)

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Extra resources for Algebraic Combinatorics and Applications: Proceedings of the Euroconference, Algebraic Combinatorics and Applications (ALCOMA), held in Gößweinstein, Germany, September 12–19, 1999

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11 (1959), 223- 224. 15. P. Stanley, Linear homogeneaus diophantine equations and magic labelings of graphs, Duke Math. J. 40 (1973), 607-632. 16. P. Stanley, Combinatorics and Commutative Algebra, Birkhäuser, Boston, 1983. 17. P. Stanley, Enumerative Combinatorics - Volume 1, Wadsworth , Monterey, California, 1986. 18. J . Yee, On Combinatorics of Lecture Hall Partitions, (to appear). Note on the Proper Linear Spaces on 18 Points A. Betten 1 and D. de Mathematisches Seminar der Universität Kiel Ludewig-Meyn-Str.

Compare also [2] where we have explained how a Latin square may be viewed as a linear space: We Iabel the rows and columns and digits by r1, ... , r5 and c1, . . c5 and d1, ... , d6. The Latin Squares of order 6 are the linear spaces of line type (6 3 , 336 ) where the three 6-lines are special: They are {r 1, . , r5}, {c 1, . . , c5} and {d 1, .. , d6}. The 36 3lines correspond to the entries of the Latin Square. Any automorphism either permutes the three 6-lines or not. If it fixes all three, we call the automorphism inner.

However when j = 3, we note b1 = 2 and MacMahon's Partition Analysis V 27 From Theorem 301, Cayley's Theorem follows as a natural corollaryo In Secto 302, we shall apply Partition Analysis to Cayley compositionso This will yield Theorem 301 quite directlyo The short Secto 303 will derive Cayley's Theorem from Theorem 301. y describe some generalizations and relations to other work. 2 Partition Analysis and Cayley Compositions The following is the only strictly Partition Analysis identity that is required: n ~ 1+B (1 - A)(1 - Aß2) 0 >.

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