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

.