Inorganic chemistry of carbon
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There is a rich variety of carbon chemistry that does not fall within the realm of organic chemistry and is thus called inorganic carbon chemistry.
Perhaps best well known are the oxides of carbon, carbon dioxide and carbon monoxide. Other types include (but are not limited to) inorganic salts and complexes of the carbon-containing polyatomic ions, cyanide, cyanate, thiocyanate, carbonate, and carbide.
The known inorganic chemistry of the allotropes of carbon (diamond, graphite, and the fullerenes) blossomed with the discovery of buckminsterfullerene in the late twentieth century as additional fullerenes and their various derivatives were discovered. One such class of derivatives is inclusion compounds, in which an ion is enclosed by the all-carbon shell of the fullerene. This inclusion is denoted by the "@" symbol. For example, an ion consisting of a lithium ion trapped within buckminsterfullerene would be denoted C60@Li+. As with any other ionic compound, this complex ion could in principle pair with a counterion to form a salt.
For a fairly full listing of inorganic carbon compounds on Wikipedia see Inorganic_compounds_by_element#Carbon.