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:

Gang Wars

Conflicts and violent struggles between organized groups or gangs, often related to territorial disputes, criminal activities, or ideological differences.

Murders

The unlawful premeditated killing of one human being by another.

Nonviolent Drug Offenses

Legal violations involving the possession, use, or distribution of drugs that do not involve violence or the threat of violence.

War on Drugs

A campaign of prohibition and foreign military aid and military intervention undertaken by the U.S. government with the aim of reducing the illegal drug trade.