Carbocation Stability: Attracting Inductive vs Mesomeric Effects

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duchuy
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Homework Statement
determine the stability of these following carbocations
Relevant Equations
x
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Hi,
Sorry in advance if the vocab is incorrect, I'm translating this from french.
In the correction, it's written that the 4th carbocation is the less stable due to the attracting inductif effect of the carbonyle group.
But isn't this carbocation stabilised by the mesomer effect of the C=O bond? And if it isn't stabilised by the mesomer effect can you please explain to me why?
And in case it is, how do we determine the stability between the 1st and 4th carbocation?
Thank you so much!
 

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The carbonyl group is polarised due to the resonance C=O ↔ C+-O-. The carbocation is destabilised by being next to a δ+ carbon.
Resonance with the carbonyl group is possible, but it would give a structure C=C-O+, which is less stable because it has the + charge on the more electronegative O atom.
So 4 is definitely less stable than 1. Whether it's less stable than 2 or 3 I'm not so sure.
 
#1 is an allyl cation. A good organic textbook will have an explanation of allyl group stability. The other two are secondary and tertiary alkyl carbocations, and a good textbook should also explain their order of stability.