PF SEQUENCES IN COMBINATORICS 5

s0 t*=0

= (E^W^yXlXP,^ ) +

e1+1(P)Wl))Z1+1)

io to

= £(Q,«2)(*)£(P,u*)(2)(l + *)• (4)

Hence, using (4), we get

* 1 * W M W = * ( l - * )

l m

"

) l

E c n » « M ( ; 4 7 )

= W{PiUfl)(z)W{QlU2)(z),

as desired. •

We will see later how Proposition 1.3.1 and equation (5) actually tell us that also Con-

jectures 1.1 and 2.1 are preserved by the ordinal sum operation.

Let us now turn our attention to the disjoint union operation. Let (PyuJi) and (Q,u?2)

be two labeled posets. If we denote by p and Q the partial orderings on P and Q then

recall that the disjoint union P + Q is the poset that has P (± J Q as underlying set partially

ordered by letting x y if either x,y £ P and x p t/, or #, y € Q and # Q t/. We define a

labeling of P + Q by o! -f LU2 = u^ 0 CJ2. Again note that u?x -f w2 is natural if and only if

both UJI and u;2 are natural. Now, by the definition of the order polynomial and by a direct

counting argument we easily see that

n(P + Q,^ + u2] z) = tyP^u zWiQ,^] z).

Using this and equation (2) it is possible to show that

(1 - z):

WiP + QM+unz)^* {

where p = \P\ and q = \Q\. The proof of this formula will be given in section 4.7 (see

Theorem 4.7.4) in a more general setting. However, the preceding formula is complicated

enough to make it difficult to understand whether it preserves the Poset Conjecture or not.

On the other hand, we have found no counterexamples, so that the question is still open.

This matter will be further discussed in greater detail in section 4.7.

We finish this section by presenting one general class of posets and labelings, for which the

Poset Conjecture is true. The following result first appeared (though expressed in different

form) in [56].

Theorem 1.3.2 Let P be a disjoint union of chains. Then the polynomial W(P; z) has only

real zeros.

\dx)

W(P,Lj1;x)W{P,uji;^x^

(l-x) P+i

lr=?