Báo cáo toán học: " Symmetric Sum-Free Partitions and Lower Bounds for Schur Numbers" potx

... 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...

Ngày tải lên: 07/08/2014, 06:20

... 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...

Ngày tải lên: 07/08/2014, 08:22

... 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...

Ngày tải lên: 07/08/2014, 15:22

... 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....

Ngày tải lên: 08/08/2014, 01:20

... 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 ≤...

Ngày tải lên: 08/08/2014, 11:20

... 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...

Ngày tải lên: 07/08/2014, 06:22

... [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...

Ngày tải lên: 07/08/2014, 07:21

... 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...

Ngày tải lên: 07/08/2014, 15:23

... 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...

Ngày tải lên: 08/08/2014, 01:20

... 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 )...

Ngày tải lên: 08/08/2014, 11:20