共查询到20条相似文献,搜索用时 31 毫秒
1.
Catherine Greenhill 《Journal of Combinatorial Theory, Series A》2006,113(2):291-324
Let s=(s1,…,sm) and t=(t1,…,tn) be vectors of non-negative integer-valued functions with equal sum . Let N(s,t) be the number of m×n matrices with entries from {0,1} such that the ith row has row sum si and the jth column has column sum tj. Equivalently, N(s,t) is the number of labelled bipartite graphs with degrees of the vertices in one side of the bipartition given by s and the degrees of the vertices in the other side given by t. We give an asymptotic formula for N(s,t) which holds when S→∞ with 1?st=o(S2/3), where and . This extends a result of McKay and Wang [Linear Algebra Appl. 373 (2003) 273-288] for the semiregular case (when si=s for 1?i?m and tj=t for 1?j?n). The previously strongest result for the non-semiregular case required 1?max{s,t}=o(S1/4), due to McKay [Enumeration and Design, Academic Press, Canada, 1984, pp. 225-238]. 相似文献
2.
Alexander Barvinok 《Advances in Mathematics》2010,224(1):316-757
We consider the set Σ(R,C) of all m×n matrices having 0-1 entries and prescribed row sums R=(r1,…,rm) and column sums C=(c1,…,cn). We prove an asymptotic estimate for the cardinality |Σ(R,C)| via the solution to a convex optimization problem. We show that if Σ(R,C) is sufficiently large, then a random matrix D∈Σ(R,C) sampled from the uniform probability measure in Σ(R,C) with high probability is close to a particular matrix Z=Z(R,C) that maximizes the sum of entropies of entries among all matrices with row sums R, column sums C and entries between 0 and 1. Similar results are obtained for 0-1 matrices with prescribed row and column sums and assigned zeros in some positions. 相似文献
3.
Stephen L. Lipscomb 《Semigroup Forum》1992,45(1):249-260
One presentation of the alternating groupA n hasn?2 generatorss 1,…,sn?2 and relationss 1 3 =s i 2 =(s1?1si)3=(sjsk)2=1, wherei>1 and |j?k|>1. Against this backdrop, a presentation of the alternating semigroupA n c )A n is introduced: It hasn?1 generatorss 1,…,S n?2,e, theA n-relations (above), and relationse 2=e, (es 1)4, (es j)2=(es j)4,es i=s i s 1 -1 es 1, wherej>1 andi≥1. 相似文献
4.
Let t=(tn)n?0 be the classical Thue-Morse sequence defined by , where s2 is the sum of the bits in the binary representation of n. It is well known that for any integer k?1 the frequency of the letter “1” in the subsequence t0,tk,t2k,… is asymptotically 1/2. Here we prove that for any k there is an n?k+4 such that tkn=1. Moreover, we show that n can be chosen to have Hamming weight ?3. This is best in a twofold sense. First, there are infinitely many k such that tkn=1 implies that n has Hamming weight ?3. Second, we characterize all k where the minimal n equals k, k+1, k+2, k+3, or k+4. Finally, we present some results and conjectures for the generalized problem, where s2 is replaced by sb for an arbitrary base b?2. 相似文献
5.
Vladimir Nikiforov 《Linear algebra and its applications》2010,432(6):1405-1411
Given positive integers let z(m,n,s,t) be the maximum number of ones in a (0,1) matrix of size m×n that does not contain an all ones submatrix of size s×t. We show that if s?2 and t?2, then for every k=0,…,s-2,
z(m,n,s,t)?(s-k-1)1/tnm1-1/t+kn+(t-1)m1+k/t. 相似文献
6.
Sukumar Das Adhikari 《Journal of Combinatorial Theory, Series A》2008,115(1):178-184
Let G be a finite abelian group of order n and let A⊆Z be non-empty. Generalizing a well-known constant, we define the Davenport constant of G with weight A, denoted by DA(G), to be the least natural number k such that for any sequence (x1,…,xk) with xi∈G, there exists a non-empty subsequence (xj1,…,xjl) and a1,…,al∈A such that . Similarly, for any such set A, EA(G) is defined to be the least t∈N such that for all sequences (x1,…,xt) with xi∈G, there exist indices j1,…,jn∈N,1?j1<?<jn?t, and ?1,…,?n∈A with . In the present paper, we establish a relation between the constants DA(G) and EA(G) under certain conditions. Our definitions are compatible with the previous generalizations for the particular group G=Z/nZ and the relation we establish had been conjectured in that particular case. 相似文献
7.
8.
M. Drmota 《Discrete Mathematics》2008,308(7):1191-1208
Let tj=(-1)s(j) be the Thue-Morse sequence with s(j) denoting the sum of the digits in the binary expansion of j. A well-known result of Newman [On the number of binary digits in a multiple of three, Proc. Amer. Math. Soc. 21 (1969) 719-721] says that t0+t3+t6+?+t3k>0 for all k?0.In the first part of the paper we show that t1+t4+t7+?+t3k+1<0 and t2+t5+t8+?+t3k+2?0 for k?0, where equality is characterized by means of an automaton. This sharpens results given by Dumont [Discrépance des progressions arithmétiques dans la suite de Morse, C. R. Acad. Sci. Paris Sér. I Math. 297 (1983) 145-148]. In the second part we study more general settings. For a,g?2 let ωa=exp(2πi/a) and , where sg(j) denotes the sum of digits in the g-ary digit expansion of j. We observe trivial Newman-like phenomena whenever a|(g-1). Furthermore, we show that the case a=2 inherits many Newman-like phenomena for every even g?2 and large classes of arithmetic progressions of indices. This, in particular, extends results by Drmota and Ska?ba [Rarified sums of the Thue-Morse sequence, Trans. Amer. Math. Soc. 352 (2000) 609-642] to the general g-case. 相似文献
9.
For a string A=a1…an, a reversalρ(i,j), 1?i?j?n, transforms the string A into a string A′=a1…ai-1ajaj-1…aiaj+1…an, that is, the reversal ρ(i,j) reverses the order of symbols in the substring ai…aj of A. In the case of signed strings, where each symbol is given a sign + or -, the reversal operation also flips the sign of each symbol in the reversed substring. Given two strings, A and B, signed or unsigned, sorting by reversals (SBR) is the problem of finding the minimum number of reversals that transform the string A into the string B.Traditionally, the problem was studied for permutations, that is, for strings in which every symbol appears exactly once. We consider a generalization of the problem, k-SBR, and allow each symbol to appear at most k times in each string, for some k?1. The main result of the paper is an O(k2)-approximation algorithm running in time O(n). For instances with , this is the best known approximation algorithm for k-SBR and, moreover, it is faster than the previous best approximation algorithm. 相似文献
10.
11.
We propose a new characterization of dual bases in finite fields. Let A=(α1,…,αn) be a basis of F over Fq and its dual basis B=(β1,…,βn) with the transition matrix C∈GLn(Fq) such that (β1,…,βn)=(α1,…,αn)C. We show that holds for all 1?k?n, where Tk∈Mn(Fq) satisfies αk(α1,…,αn)=(α1,…,αn)Tk. Conversely, suppose F=Fq(αk′) and for some 1?k′?n and G∈GLn(Fq), then B is equivalent to (α1,…,αn)G. As applications, we can construct the dual basis of a given basis A or determine whether the dual basis of A satisfies the desired conditions from Tk. This generalizes the results obtained by Liao and Sun for normal bases. Furthermore, we give a simple proof of the theorem of Gollmann, Wang and Blake for polynomial bases. 相似文献
12.
Hiro Ito 《Discrete Applied Mathematics》2006,154(16):2330-2334
Let H=(N,E,w) be a hypergraph with a node set N={0,1,…,n-1}, a hyperedge set E⊆2N, and real edge-weights w(e) for e∈E. Given a convex n-gon P in the plane with vertices x0,x1,…,xn-1 which are arranged in this order clockwisely, let each node i∈N correspond to the vertex xi and define the area AP(H) of H on P by the sum of the weighted areas of convex hulls for all hyperedges in H. For 0?i<j<k?n-1, a convex three-cut C(i,j,k) of N is {{i,…,j-1}, {j,…,k-1}, {k,…,n-1,0,…,i-1}} and its size cH(i,j,k) in H is defined as the sum of weights of edges e∈E such that e contains at least one node from each of {i,…,j-1}, {j,…,k-1} and {k,…,n-1,0,…,i-1}. We show that the following two conditions are equivalent:
- •
- AP(H)?AP(H′) for all convex n-gons P.
- •
- cH(i,j,k)?cH′(i,j,k) for all convex three-cuts C(i,j,k).
13.
Jianguo Qian 《Discrete Mathematics》2006,306(5):533-537
Ryser [Combinatorial Mathematics, Carus Mathematical Monograph, vol. 14, Wiley, New York, 1963] introduced a partially ordered relation ‘?’ on the nonnegative integral vectors. It is clear that if S=(s1,s2,…,sn) is an out-degree vector of an orientation of a graph G with vertices 1,2,…,n, then
(Π) 相似文献
14.
A block-colouring of a 4-cycle system (V,B) of order v=1+8k is a mapping ?:B→C, where C is a set of colours. Every vertex of a 4-cycle system of order v=8k+1 is contained in blocks and r is called, using the graph theoretic terminology, the degree or the repetition number. A partition of degree r into s parts defines a colouring of type s in which the blocks containing a vertex x are coloured exactly with s colours. For a vertex x and for i=1,2,…,s, let Bx,i be the set of all the blocks incident with x and coloured with the ith colour. A colouring of type s is equitable if, for every vertex x, we have |Bx,i−Bx,j|≤1, for all i,j=1,…,s. In this paper we study bicolourings, tricolourings and quadricolourings, i.e. the equitable colourings of type s with s=2, s=3 and s=4, for 4-cycle systems. 相似文献
15.
Let m and n be positive integers, and let R=(r1,…,rm) and S=(s1,…,sn) be nonnegative integral vectors. We survey the combinational properties of the set of all m × n matrices of 0's and 1's having ri1's in row i andsi 1's in column j. A number of new results are proved. The results can be also be formulated in terms of a set of bipartite graps with bipartition into m and n vertices having degree sequence R and S, respectively. They can also be formulated in terms of the set of hypergraphs with m vertices having degree sequence R and n edges whose cardinalities are given by S. 相似文献
16.
For positive integers s and k1,k2,…,ks, the van der Waerden number w(k1,k2,…,ks;s) is the minimum integer n such that for every s-coloring of set {1,2,…,n}, with colors 1,2,…,s, there is a ki-term arithmetic progression of color i for some i. We give an asymptotic lower bound for w(k,m;2) for fixed m. We include a table of values of w(k,3;2) that are very close to this lower bound for m=3. We also give a lower bound for w(k,k,…,k;s) that slightly improves previously-known bounds. Upper bounds for w(k,4;2) and w(4,4,…,4;s) are also provided. 相似文献
17.
V.V. Bavula 《Journal of Pure and Applied Algebra》2008,212(10):2320-2337
Let K be an arbitrary field of characteristic p>0. We find an explicit formula for the inverse of any algebra automorphism of any of the following algebras: the polynomial algebra Pn?K[x1,…,xn], the ring of differential operators D(Pn) on Pn, D(Pn)⊗Pm, the n’th Weyl algebra An or An⊗Pm, the power series algebra K[[x1,…,xn]], the subalgebra Tk1,…,kn of D(Pn) generated by Pn and the higher derivations , 0≤j<pki, i=1,…,n (where k1,…,kn∈N), Tk1,…,kn⊗Pm or an arbitrary central simple (countably generated) algebra over an arbitrary field. 相似文献
18.
Keqin Feng 《Discrete Applied Mathematics》2006,154(6):942-949
Let f(k1,…,km) be the minimal value of size of all possible unextendible product bases in the tensor product space . We have trivial lower bounds and upper bound k1?km. Alon and Lovász determined all cases such that f(k1,…,km)=n(k1,…,km). In this paper we determine all cases such that f(k1,…,km)=k1?km by presenting a sharper upper bound. We also determine several cases such that f(k1,…,km)=n(k1,…,km)+1 by using a result on 1-factorization of complete graphs. 相似文献
19.
We extend Euler's well-known quadratic recurrence relation for Bernoulli numbers, which can be written in symbolic notation as n(B0+B0)=−nBn−1−(n−1)Bn, to obtain explicit expressions for n(Bk+Bm) with arbitrary fixed integers k,m?0. The proof uses convolution identities for Stirling numbers of the second kind and for sums of powers of integers, both involving Bernoulli numbers. As consequences we obtain new types of quadratic recurrence relations, one of which gives B6k depending only on B2k,B2k+2,…,B4k. 相似文献
20.
Ya. N. Shitov 《Journal of Mathematical Sciences》2009,163(5):598-624
Let GMr(A) be the row Gondran–Minoux rank of a matrix, GMc(A) be the column Gondran–Minoux rank, and d(A) be the determinantal rank, respectively. The following problem was posed by M. Akian, S. Gaubert, and A. Guterman: Find the minimal numbers m and n such that there exists an (m × n)-matrix B with different row and column Gondran–Minoux ranks. We prove that in the case GMr(B) > GMc(B) the minimal m and n are equal to 5 and 6, respectively, and in the case GMc(B) > GMr(B) the numbers m = 6 and n = 5 are minimal. An example of a matrix $ A \in {\mathcal{M}_{5 \times 6}}\left( {{\mathbb{R}_{\max }}} \right) $ such that GMr(A) = GMc(A t) = 5 and GMc(A) = GMr(A t) = 4 is provided. It is proved that p = 5 and q = 6 are the minimal numbers such that there exists an (p×q)-matrix with different row Gondran–Minoux and determinantal ranks. 相似文献