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Why is the acylium ion stable? Doesn't the inductive effect of oxygen destabilize it? Compared to an alkyl cation, is an acylium ion more stable or less stable?

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The acylium ion is stabilized by the positive mesomeric effect of the lone pair of electrons on the oxygen atom onto the empty p-orbital of the $\ce{C+}$. This helps achieve octet completion of the carbon atom, and thus imparts stability to the carbocation.

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(Image by Jü CC BY-SA 4.0, from Wikimedia Commons)

Resonance effects mostly dominate the weak inductive effect, so here the destabilizing negative inductive effect of oxygen atom has only a negligible role.

An alkyl carbocation has no such positive stabilizing mesomeric effect, rather only hyperconjugative effects, which are not able to provide as much stability as is possible in the acylium ion.

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In addition to what Gaurang already wrote, I would like to add another, not quite as popular, interpretation.

A case can be made to treat the $\ce{CO}$ moiety as a ligand to the (then positively charged) alkyl group $\ce{R}.$ The set of (major) resonance structures gains another member: $$\ce{ R-C+=O <-> R-C#O+ <-> R+\bond{<-}^-C#O+ }$$

This should of course highlight the linear coordination of the carbon, and the importance of the two π bonds. Furthermore depending on $\ce{R}$ there will be additional stabilisation via hyperconjugation, and/ or conjugation.
A bent structure (as it is often depicted in the literature) of acylium ions is not stable, and does only exist as a transition or excited state.

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