A combinatorial interpretation of the eigensequence for composition

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 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 only occurs 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.