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On the second number of Plutarch. (English) Zbl 0917.01011

From the introduction of this article: “In this MONTHLY 104, 344–350 (1997; Zbl 0873.01002), Stanley discusses the following statement of Plutarch: “Chrysippus says that the number of compound propositions that can be made from only ten simple propositions exceeds a million. (Hipparcus, to be sure, refuted this by showing that on the affirmative side there are 103,049 compound statements, and on the negative side 310,952).” In particular, he describes Hough’s discovery: 103,049 appears to be the tenth Schröder number \(s(10)\). He also writes that the number 310,952 (\(\ldots\)) remains an enigma. We propose here an interpretation of the number 310,954 similar to Hough’s one. This might suggest a “misprint”, a miscalculation in the original statement, or the elimination of two cases for some reason connected to Stoic logic.” – The authors obtain their result by a number of bracketings on the string \(NO\; x_1 x_2 \ldots x_{10}\), applying recursion relations and generating functions.

MSC:

01A20 History of Greek and Roman mathematics
05A15 Exact enumeration problems, generating functions

Citations:

Zbl 0873.01002
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