The Catalan Numbers $C _ { n }$ Count the Number of Strings of N +’S and Number

question 89

Essay

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:

Dependent Samples T-Test

A statistical test used to compare the means of two related groups.

Regression

A statistical method used to model and analyze relationships between variables.

Obtained Value

The actual value or result achieved in an experiment or study, typically used in statistical analysis to compare against theoretical values.

Critical Value

The threshold value that the test statistic must exceed in order for the null hypothesis to be rejected in a hypothesis test.