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Addison-Wesley, 1989. P. Hilton and J. Pedersen, Catalan numbers, their generalization, and their uses, Math. Intelligencer 13 (1991), 64-75. H. A. Kierstead and W. T. Trotter, Explicit matchings in the middle levels of the Boolean lattice, Order 5 (1988), 163-171. B. Montagh, A simple proof and a generalization of an old result of Chung and Feller, Discrete Math. 87 (1991), 105-108. T. Narayana, Lattice Path Combinatorics with Statistical Applications. Math. Expositions 23, Univ. of Toronto Press, 1979.

W. Cusick, Free Actions on Spaces with Non-zero Euler Characteristic, Topology Appl. 33 (1989), 185-196. A. L. Edmonds, Surface Symmetry I, Michigan Math. J. 29 (1982), 171-183. A. L. Edmonds, Surface Symmetry II, Michigan Math. J. 30 (1983), 143-154. T. W. , New York, 1974. C. M. D. Dissertation, Purdue University, 1989. W. S. , New York, 1967. R. E. Mosher and M. C. Tangora, Cohomology Operations and Applications in Homotopy Theory, Harper & Row, New York, 1968. J. J. Rotman, Theory of Groups: An Introduction, Second Edition, Allyn & Bacon, Boston, 1973.

Graham, Knuth, and Patashnik [7, p. 346] presented a geometric proof of Raney's original result, which upon closer examination also yields MonHigh's generalization. ) Encode the integer arrangement ai' ... ,an as a walk in the plane, starting from the origin and moving (+ 1, +a) from the current position when the ith number is encountered. The ending position is (n,1). Figure 8 shows two periods of the walk for the sequence 2, -1,2, -5,3, -2,1, -2,3. 140 THE BRICKlAYER PROBLEM AND THE STRONG CYCLE LEMMA [February Figure 8.

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Annals of Pure and Applied Logic, Volume 152, Issues 1-3, Pages 1-180 (March 2008) by Elsevier

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