Summary: 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 $composition$: 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.

05A15

eigensequence, 35241OK permutation, restricted pattern, underlined pattern