A Combinatorial Interpretation of the Eigensequence for Composition
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 with itself: 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.
