A Combinatorial Interpretation of the Eigensequence for Composition
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.
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.
Full version: pdf,
(Concerned with sequences
Received August 4 2005;
revised version received December 19 2005.
Published in Journal of Integer Sequences January 3 2006.
Journal of Integer Sequences home page