Journal of Integer Sequences, Vol. 14 (2011), Article 11.4.2 |

Department of Mathematics

Rose-Hulman Institute of Technology

Terre Haute, IN 47803

USA

Jeffrey Liese

Department of Mathematics

California Polytechnic State University

San Luis Obispo, CA 93407-0403

USA

Jeffrey Remmel

Department of Mathematics

University of California, San Diego

La Jolla, CA 92093-0112

USA

**Abstract:**

Kitaev, Liese, Remmel, and Sagan recently defined generalized factor
order on words comprised of letters from a partially ordered set
by setting
if there is a contiguous subword
of of the same length as such that the -th character of
is greater than or equal to the -th character of for all .
This subword is called an embedding of into . For the case
where is the positive integers with the usual ordering, they
defined the weight of a word
to be
wt, and the corresponding weight generating
function
wt. They then
defined two words and to be Wilf equivalent, denoted
,
if and only if
. They also defined the related
generating function
wt where
is the set of all words such
that the only embedding of into is a suffix of , and showed
that
if and only if
. We continue
this study by giving an explicit formula for if factors
into a weakly increasing word followed by a weakly decreasing word. We
use this formula as an aid to classify Wilf equivalence for all words
of length 3. We also show that coefficients of related generating
functions are well-known sequences in several special cases. Finally,
we discuss a conjecture that if
then and must be
rearrangements, and the stronger conjecture that there also must be a
weight-preserving bijection on words over the positive integers
such that is a rearrangement of for all , and
embeds if and only if embeds .

(Concerned with sequences A000045 A000071 A000073 A000078 A000124 A000126 A000292 A001591 A001949 A007800 A008466 A008937 A014162 A050231 A050232 A050233 A107066 A145112 A145113 A172119.)

Received May 24 2010;
revised version received March 16 2011.
Published in *Journal of Integer Sequences*, March 26 2011.

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