What is the right terminology / methods for this standard signal processing task? Say we have a signal s = [1,2,2,1,3,3,2,2,1,1] or similar (only 1,2,3 can occur). I want to look for a particular pattern, e.g. [1,3,3,2], which can occur anywhere (0 or many times) in the signal. I would like to perform the computation as a convolution with some filter f. 

I've convinced myself that f = [100,1000,1000,10] will do the job, i.e. sending the pattern p = [1,3,3,2] to f = 10.^fliplr(p), in matlab syntax. That is, the convolution s*f will take the value 1*10 + 3*1000 + 3*1000 + 2*100 = 6210 if and only if we have the pattern p in the right place, at least as long as the pattern we are looking for is short enough. 

For a long enough pattern, e.g. [1,1,1,1,1,1,1,1,1,1,2] (ten 1s followed by a 2),  the corresponding filter f = [100,10,..,10] would give a "false positive" for the pattern [2,2,2,2,2,2,2,2,2,2,1], since the convolution would give the values 10*10 + 2*100 = 300 for the first, and 2*10*10 + 1*100 = 300 for the second. (That I needed 10 1s to "trick" the filter is obviously related to the fact that my filter "encodings" of the 1s and 2s differ by a factor 10. Changing the encoding to e.g. {1,2,3} ->  {10,1000,100000} should let me handle correspondingly longer patterns).

I have a strong feeling that this is a very standard signal processing task. What is the standard terminology and methods concerning this problem? (I.e. what key words should I google.)
The thing you are looking for is the matched filter, which is essentially the (optimal) linear approach to look for a known pattern in a signal (which may be impacted by additive zero mean noise).

The impulse response (i.e., the sequence you are convolving with) of the matched filter is just the pattern reversed, i.e. $f = [2, 3, 3, 1]$ will do the job. If the input signal is normalized in RMS amplitude, the matthed filter will give maximum  output if and only if the pattern occurs.
The thing you are looking for is the matched filter, which is essentially the (optimal) linear approach to look for a known pattern in a signal (which may be impacted by additive zero mean noise).

The impulse response (i.e., the sequence you are convolving with) of the matched filter is just the pattern reversed, i.e. $f = [2, 3, 3, 1]$ will do the job. If the input signal is normalized in RMS amplitude, the matthed filter will give maximum  output if and only if the pattern occurs.