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:

DSM

The Diagnostic and Statistical Manual of Mental Disorders, a manual published by the American Psychiatric Association that categorizes and defines mental disorders.

Overdiagnosis

The diagnosis of a condition or disease more often than it is actually present, leading to unnecessary treatment or intervention.

Diagnostic Labels

Terms used in healthcare and psychology to classify and describe a set of symptoms that are recognized as a particular disorder or disease.

Objectivity

The quality of being unbiased, unprejudiced, and impartial in observation or judgment.