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\begin{abstract}
The monic sequence that shifts left under convolution with itself is the 
Catalan numbers with 130+ combinatorial interpretations. Here we 
establish a combinatorial interpretation for the monic sequence that shifts 
left under \emph{composition}: it counts permutations that contain a 3241 pattern only 
as part of a 35241 pattern. We give two recurrences, 
the first allowing relatively fast computation,
the second similar to one for the Catalan numbers. 
Among the $4\times 4!=96$ similarly restricted patterns involving 4 
letters (such as $4\underline{2}31$: a 431 pattern occurs only as part 
of a 4231), four different counting sequences arise: 64 give the Catalan 
numbers, 16 give the Bell numbers, 12 give sequence A051295,
in OEIS, and 4 give a new sequence with an explicit formula.   
\end{abstract}

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