The Catalan Numbers C_n Count the Number of Strings of N

question 68

Short Answer

The Catalan numbers C_n count the number of strings of n +’s and n −’s with the following property: as each string is read from left to right, the number of +’s encountered is always at least as large as the number
of −’s.
(a) Verify this by listing these strings of lengths 2, 4, and 6 and showing that there are C₁ , C₂ , and C₃ of
these, respectively.
(b) Explain how counting these strings is the same as counting the number of ways to correctly parenthesize
strings of variables

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Definitions:

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Lessen Dissonance

The act of reducing conflict or inconsistency, especially in the context of cognitive dissonance in psychology, where individuals strive for internal consistency between their beliefs and actions.

Frequent-Flyer Miles

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A decision-making process in which options are evaluated in a way that one attribute cannot make up for a disadvantage in another.