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

A Combinatorial Interpretation of the Eigensequence for Composition

David Callan
Department of Statistics
University of Wisconsin-Madison
1300 University Ave.
Madison, WI 53706-1532


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.

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(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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