... and the second follows from (2)
and the bounds for R
5
(3) given in S. Radziszowski’s survey paper [6]. For earlier
work on lower bounds for S(k) see H. Fredricksen [5] and A. Beutelspacher and
W. ... D
k
(n)tobethemaximumofD(P )overall
symmetric sum-free partitions P of [1,n]intok sets. Notethatwemusthave
D
k
(n) ≤ S(k − 1) + 1.
Symmetric Sum-Free Partitions...
... 2)(n − 4) = 2
a
3
b
for some a, b ∈ N. This is only possible for n =6andn = 8, and in these cases
we have λ =(3, 2, 1) and λ =(3, 2, 1
3
), respectively. If k =2,thenh
µ
=8,andso
π
1
π
2
π
3
= ... p-weight for all primes p, if and only if one of the following occurs:
λ =(n) , (1
n
) or (2
2
) .
(2) λ is of maximal p-weight for all primes p>2, if and only if λ is one of the...
... that the
lower bound for the rows of B is reached for some partitions α with [α] ∈ [A].
Let us assume that the biggest length of a column in A is 1. Then A decomposes into
disconnected rows and Υ ... conditions for positivity of [A
1
] − [A
2
].
4 The Cover Partition
In this section we use Remark 3.5 and [Gut1, Theorem 4.2] to determine for the ordinary
and the Schubert prod...
... C
i,j
C
k,l
is reduced and in
W
S
if and on l y if a t least one of the
following hold:
1. i < k and j < l.
2. k + l > n and i < k and j ≤ l.
3. i + j > n and i ≤ k and j ≤ l.
Proof. ... the
rank generating f unction for P for all types except type A.
This verification occurs in Theorem 28 for type B, Theorem 41 fo r type C, and Theo-
rem 52 for type D....
... C
i,j
C
k,l
is reduced and in
W
S
if and on l y if a t least one of the
following hold:
1. i < k and j < l.
2. k + l > n and i < k and j ≤ l.
3. i + j > n and i ≤ k and j ≤ l.
Proof. ... < j < 2 n − i − 1,
(10)
and for i = 0 and j = 1 we have
s
0
Σ
1
(1) = s
0
s
1
= s
1
s
0
= Σ
1
(1)s
0
.
Similar commutation rules for s
i
Σ
0
(j) for all 0 ≤ i ≤...
... coloured with the colours (1), (2) and (3) is 20 and the
number of bi-chromatic blocks coloured with the colours (2) and (3) is at least 7 and so
|T
X
1
∪X
2
∪X
3
|≥27 and this is absurd.
Let us consider ... completes theorem 3 and gives important information about the chro-
matic spectrum of BSQSs(16).
Theorem 4 The upper chromatic number for all BSQSs(16)is¯χ =3; the lower c...
... [10] that
M(k)=k + 2 for k =2, 3 and there is some numerical evidence that this holds for all
k ≥ 1. Note that, unlike for the numbers T(k), the fact that M(k) are finite for all k ≥ 1,
follows ... t
1
< and
q
1
∈ .Thusf(x + t
1
)=a and if at the same time f(y + t
1
)=a, i.e., y + t
1
= e
q
2
for another rational number q
2
= q
1
,pickt
2
so that t
1
<t
2
< and x + t...
... defined by
E(
x
i
) = x
i
for all i ∈ N and linear extension. This transformation is important in the theory of (P, Ω)-
partitions (see Brenti [4] for details). Br and en [3, Lemma 4.4] showed ... (10) and the fact that A
n
(x) has only real zeros,
Wilf can show that R
n
(x) has only real zeros for n ≥ 2 (see B´ona [1, Theorem 1.41] and
Stanley [12] for details). Very recen...
... qusaisymmetric functions and Π is the algebra of peak qua-
sisymmetric functions.
For the interested reader, the duality between NC a nd Q was established through
[5, 6, 10], and between A
E
and ... thank Christine Bessenro dt, Louis Billera and Hugh Thomas
for helpful conversations, Andrew Rechnitzer for programming assistance, and the referee
for helpful comments. John Ste...
... 2.3, we get a formula for the special graded Betti number of a
t-connected sum of two simplicial complexes for t 3.
Corollary 3.4. Let ∆
1
and ∆
2
be simplicial complexes on V
1
and V
2
respective ... 3.3. In H
n
, the
vertex v
i
is connected to all the other vertices for i t, and v
j
and v
j
′
are not connected
to each other for all t + 1 j, j
′
t + n. Thus b
k
(H
n
)...