Hipparchus, Plutarch, Schröder, and Hough

American Mathematical Monthly

Volumn 104, Issue 4, 1997, Pages 344-350

  Stanley, Richard P ^a  

^a MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0031528908     PISSN: 00029890     EISSN: None     Source Type: Journal    
DOI: 10.2307/2974582     Document Type: Article

References (36)

1. K.-R. Biermann and J. Mau, Überprüfung einer frühen Anwendung der Kombinatorik in der Logik, J. Symbolic Logic 23 (1958), 129-132.
2. N. L. Biggs, The roots of combinatorics, Historia Mathematica 6 (1979), 109-136.
3. J. Bonin, L. W. Shapiro, and R. Simion, Some g-analogues of the Schröder numbers arising from combinatorial statistics on lattice paths, J. Stat. Planning and Inference 34 (1993), 35-55.
4. A. Cayley, On the analytical form called trees, Part II, Philos. Mag. (4) 18 (1859), 374-378.
5. L. Comtet, Calcul pratique des coefficients de Taylor d'une fonction algébrique, Enseignement Math. 10 (1964), 267-270.
6. L. Comtet, Advanced Combinatorics, Reidel, Dordrecht/Boston, 1974.
7. A. Erdélyi and I. M. H. Etherington, Some problems of non-associative combinations (2), Edinburgh Math. Notes 32 (1940), 7-12.
8. I. M. H. Etherington, Some problems of non-associative combinations (1), Edinburgh Math. Notes 32 (1940), 1-6.
9. L. Euler, Summarium, Novi Commentarii academiae scientiarum Petropolitanae 7 (1758/59), 13-15. Reprinted in Opera Omnia (1) 26 (1953), xvi-xviii.
10. L. Euler, Summarium, Novi Commentarii academiae scientiarum Petropolitanae 7 (1758/59), 13-15. Reprinted in Opera Omnia (1) 26 (1953), xvi-xviii.
11. D. Foata and D. Zeilberger, A classic proof of a recurrence for a very classical sequence, preprint. Available from the website http://www.math.temple.edu/ ~ zeilberg/mamarim/mamarimhtml/classic.html.
12. M. Gardner, Catalan numbers, Mathematical Games, Scientific American 234 (June, 1976), pp. 120-125, and bibliography on p. 132. Reprinted (with an Addendum) in Chapter 20 of Time Travel and Other Mathematical Bewilderments, W. H. Freeman, New York, 1988.
13. M. Gardner, Catalan numbers, Mathematical Games, Scientific American 234 (June, 1976), pp. 120-125, and bibliography on p. 132. Reprinted (with an Addendum) in Chapter 20 of Time Travel and Other Mathematical Bewilderments, W. H. Freeman, New York, 1988.
14. T. L. Heath, A History of Greek Mathematics, Oxford University Press, Oxford; 1921; reprinted by Dover, New York, 1981.
15. T. L. Heath, A Manual of Greek Mathematics, Oxford University Press, Oxford, 1931; reprinted by Dover, New York, 1963.
16. P. Hilton and J. Pedersen, Catalan numbers, their generalizations, and their uses, Math. Intelligencer 13 (Spring, 1991), 64-75.
17. M. Klazar, On abab-frec and abba-free set partitions, Europ. J. Combinatorics 17 (1996), 53-68.
18. W. Kneale and M. Kneale, The Development of Logic, Oxford University Press, Oxford, 1962, 1971.
19. M. Lothaire, Combinatorics on Words, Encyclopedia of Mathematics and Its Applications, vol. 17, Addison-Wesley, Reading, Massachusetts, 1983.
20. J. Łukasiewicz, Selected Works (L. Borkowski, ed.), North-Holland, Amsterdam, 1970.
21. O. Neugebauer, A History of Ancient Mathematical Astronomy, vol. 1, Springer-Verlag, Berlin/Heidelberg/New York, 1975.
22. Plutarch, Moralia, vol. IX (introduction by E. L. Minar, Jr.), Loeb Classical Library, Harvard University Press, Cambridge, Massachusetts, 1961.
23. Plutarch, The Rise and Fall of Athens: Nine Greek Lives, translated by I. Scott-Kilvert, Penguin, London, 1960.
24. G. N. Raney, Functional composition patterns and power series reversion, Trans. Amer. Math. Soc. 94 (1960), 441-451.
25. D. G. Rogers and L. W. Shapiro, Deques, trees and lattice paths, in Combinatorial Mathematics VIII (K. L. McAvaney, ed.), Lecture Notes in Mathematics, no. 884, Springer-Verlag, Berlin, pp. 293-303.
26. A. Rome, Procédés anciens de calcul des combinaisons, Ann. Soc. sci. Bruxelles, ser. A 50 (1930), 97-104.
27. E. Schröder, Vier combinatorische Probleme, Z. für Math. Physik 15 (1870), 361-376.
28. A. de Segner, Enumeratio modorum, quibus figurae planae rectilineae per diagonales dividuntur in triangula, Novi Commentarii academiae scientiarum Petropolitanae 7 (1758/59), 203-209.
29. L. W. Shapiro and A. B. Stephens, Bootstrap percolation, the Schröder numbers, and the n-kings problem, SIAM J. Discrete Math. 4 (1991), 275-280.
30. N. J. A. Sloane and S. Plouffe, The Encyclopedia of Integer Sequences, Academic Press, San Diego, 1995.
31. R. Stanley, Differentiably finite power series, European J. Combinatorics 1 (1980), 175-188.
32. R. Stanley, Enumerative Combinatorics, vol. 1, Wadsworth & Brooks/Cole, Belmont, CA, 1986; second printing, Cambridge University Press, 1996.
33. R. Stanley, Enumerative Combinatorics, vol. 2, Cambridge University Press, Cambridge, in preparation.
34. G. J. Toomer, Hipparchus, in Dictionary of Scientific Biography (C. C. Gillipsie, editor in chief), vol. XV, supplement I, Scribner's, New York, 1978, pp. 207-224.
35. B. L. van der Waerden, Geometry and Algebra in Ancient Civilizations, Springer-Verlag, Berlin, 1983.
36. J. West, Generating trees and the Catalan and Schröder numbers, Discrete Math. 146} (1995), 247-262.