Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.4 |

Department of Statistics

University of Wisconsin-Madison

1300 University Ave.

Madison, WI 53706-1532

USA

**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 similarly restricted patterns involving 4
letters (such as
: 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.

(Concerned with sequences A000108 A000110 A030266 A051295 A084938 and A110447 .)

Received August 4 2005;
revised version received December 19 2005.
Published in *Journal of Integer Sequences* January 3 2006.

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