共查询到20条相似文献,搜索用时 15 毫秒
1.
F.E. Bennett 《Discrete Mathematics》1981,36(2):117-137
Let L1 denote the set of integers n such that there exists an idempotent Latin square of order n with all of its conjugates distinct and pairwise orthogonal. It is known that L1 contains all sufficiently large integers. That is, there is a smallest integer no such that L1 contains all integers greater than no. However, no upper bound for no has been given and the term “sufficiently large” is unspecified. The main purpose of this paper is to establish a concrete upper bound for no. In particular it is shown that L1 contain all integers n>5594, with the possible exception of n=6810. 相似文献
2.
Gadi Aleksandrowicz 《Journal of Combinatorial Theory, Series A》2012,119(3):503-520
Plane polyominoes are edge-connected sets of cells on the orthogonal lattice Z2, considered identical if their cell sets are equal up to an integral translation. We introduce a novel injection from the set of polyominoes with n cells to the set of permutations of [n], and classify the families of convex polyominoes and tree-like convex polyominoes as classes of permutations that avoid some sets of forbidden patterns. By analyzing the structure of the respective permutations of the family of tree-like convex polyominoes, we are able to find the generating function of the sequence that enumerates this family, conclude that this sequence satisfies the linear recurrence an=6an−1−14an−2+16an−3−9an−4+2an−5, and compute the closed-form formula an=2n+2−(n3−n2+10n+4)/2. 相似文献
3.
Ajai Choudhry 《Journal of Number Theory》2005,110(2):403-412
This paper deals with the problem of finding n integers such that their pairwise sums are cubes. We obtain eight integers, expressed in parametric terms, such that all the six pairwise sums of four of these integers are cubes, 9 of the 10 pairwise sums of five of these integers are cubes, 12 pairwise sums of six of these integers are cubes, 15 pairwise sums of seven of these integers are cubes and 18 pairwise sums of all the eight integers are cubes. This leads to infinitely many examples of four positive integers such that all of their six pairwise sums are cubes. Further, for any arbitrary positive integer n, we obtain a set of 2(n+1) integers, in parametric terms, such that 5n+1 of the pairwise sums of these integers are cubes. With a choice of parameters, we can obtain examples with 5n+2 of the pairwise sums being cubes. 相似文献
4.
Peter Lorimer 《Discrete Mathematics》1979,25(3):269-273
On the set of n2+n+1 points of a projective plane, a set of n2+n+1 permutations is constructed with the property that any two are a Hamming distance 2n+1 apart. Another set is constructed in which every pair are a Hamming distance not greater than 2n+1 apart. Both sets are maximal with respect to the stated property. 相似文献
5.
Yong-Gao Chen 《Journal of Number Theory》2003,98(2):310-319
In this paper we prove that if (r,12)?3, then the set of positive odd integers k such that kr−2n has at least two distinct prime factors for all positive integers n contains an infinite arithmetic progression. The same result corresponding to kr2n+1 is also true. 相似文献
6.
The dimension of a linear space is the maximum positive integer d such that any d of its points generate a proper subspace. For a set K of integers at least two, recall that a pairwise balanced design $\operatorname{PBD}(v,K)$ is a linear space on v points whose lines (or blocks) have sizes belonging to K. We show that, for any prescribed set of sizes K and lower bound d on the dimension, there exists a $\operatorname{PBD}(v,K)$ of dimension at least d for all sufficiently large and numerically admissible v. 相似文献
7.
We consider sets of (0, +1)-vectors in R n, having exactly s non-zero positions. In some cases we give best or nearly best possible bounds for the maximal number of such vectors if all the pairwise scalar products belong to a fixed set D of integers. The investigated cases include D={ -d, d}, which corresponds to equiangular lines. 相似文献
8.
Let N denote the set of positive integers. The asymptotic density of the set A⊆N is d(A)=limn→∞|A∩[1,n]|/n, if this limit exists. Let AD denote the set of all sets of positive integers that have asymptotic density, and let SN denote the set of all permutations of the positive integers N. The group L? consists of all permutations f∈SN such that A∈AD if and only if f(A)∈AD, and the group L* consists of all permutations f∈L? such that d(f(A))=d(A) for all A∈AD. Let be a one-to-one function such that d(f(N))=1 and, if A∈AD, then f(A)∈AD. It is proved that f must also preserve density, that is, d(f(A))=d(A) for all A∈AD. Thus, the groups L? and L* coincide. 相似文献
9.
Let n and r be positive integers with 1 < r < n and let K(n,r) consist of all transformations on X n = {1,...,n} having image size less than or equal to r. For 1 < r < n, there exist rank-r elements of K(n,r) which are not the product of two rank-r idempotents. With this limitation in mind, we prove that for fixed r, and for all n large enough relative to r, that there exists a minimal idempotent generating set U of K(n,r) such that all rank-r elements of K(n,r) are contained in U 3. Moreover, for all n > r > 1, there exists a minimal idempotent generating set W for K(n,r) such that not every rank-r element is contained in W 3. 相似文献
10.
Robert L. Griess Jr. 《Journal of Number Theory》2010,130(3):680-695
Given a rational lattice and suitable set of linear transformations, we construct a cousin lattice. Sufficient conditions are given for integrality, evenness and unimodularity. When the input is a Barnes-Wall lattice, we get multi-parameter series of cousins. There is a subseries consisting of unimodular lattices which have ranks 2d−1±2d−k−1, for odd integers d?3 and integers . Their minimum norms are moderately high: . 相似文献
11.
We prove that the maximum number of geometric permutations, induced by line transversals to a collection of n pairwise disjoint balls in \R
d
, is Θ (n
d-1
) . This improves substantially the upper bound of O(n
2d-2
) known for general convex sets [9].
We show that the maximum number of geometric permutations of a sufficiently large collection of pairwise disjoint unit disks
in the plane is two, improving the previous upper bound of three given in [5].
Received September 21, 1998, and in revised form March 14, 1999. 相似文献
12.
A. I. Pavlov 《Mathematical Notes》1997,62(6):739-746
Let Λ be an arbitrary set of positive integers andS
n
(Λ) the set of all permutations of degreen for which the lengths of all cycles belong to the set Λ. In the paper the asymptotics of the ratio |S
n
(Λ)|/n! asn→∞ is studied in the following cases: 1) Λ is the union of finitely many arithmetic progressions, 2) Λ consists of all positive
integers that are not divisible by any number from a given finite set of pairwise coprime positive integers. Here |S
n
(Λ)| stands for the number of elements in the finite setS
n
(Λ).
Translated fromMatematicheskie Zametki, Vol. 62, No. 6, pp. 881–891, December, 1997.
Translated by A. I. Shtern 相似文献
13.
In this paper we discuss a combinatorial problem involving graphs and matrices. Our problem is a matrix analogue of the classical problem of finding a system of distinct representatives (transversal) of a family of sets and relates closely to an extremal problem involving 1-factors and a long standing conjecture in the dimension theory of partially ordered sets. For an integer n ?1, let n denote the n element set {1,2,3,…, n}. Then let A be a k×t matrix. We say that A satisfies property P(n, k) when the following condition is satisfied: For every k-taple (x1,x2,…,xk?nk there exist k distinct integers j1,j2,…,jk so that xi= aii for i= 1,2,…,k. The minimum value of t for which there exists a k × t matrix A satisfying property P(n,k) is denoted by f(n,k). For each k?1 and n sufficiently large, we give an explicit formula for f(n, k): for each n?1 and k sufficiently large, we use probabilistic methods to provide inequalities for f(n,k). 相似文献
14.
Yufei Zhao 《Journal of Number Theory》2010,130(5):1212-1220
A more sums than differences (MSTD) set is a finite subset S of the integers such that |S+S|>|S−S|. We construct a new dense family of MSTD subsets of {0,1,2,…,n−1}. Our construction gives Θ(n2/n) MSTD sets, improving the previous best construction with Ω(n2/n4) MSTD sets by Miller, Orosz, and Scheinerman. 相似文献
15.
Kazushi Yoshitomi 《Indagationes Mathematicae》2005,16(2):289-299
Let q ∈ {2, 3} and let 0 = s0 < s1 < … < sq = T be integers. For m, n ∈ Z, we put ¯m,n = {j ∈ Z| m? j ? n}. We set lj = sj − sj−1 for j ∈ 1, q. Given (p1,, pq) ∈ Rq, let b: Z → R be a periodic function of period T such that b(·) = pj on sj−1 + 1, sj for each j ∈ 1, q. We study the spectral gaps of the Jacobi operator (Ju)(n) = u(n + 1) + u(n − 1) + b(n)u(n) acting on l2(Z). By [λ2j , λ2j−1] we denote the jth band of the spectrum of J counted from above for j ∈ 1, T. Suppose that pm ≠ pn for m ≠ n. We prove that the statements (i) and (ii) below are equivalent for λ ∈ R and i ∈ 1, T − 1. 相似文献
16.
In this paper, we present families of quasi-convex sequences converging to zero in the circle group T, and the group J3 of 3-adic integers. These sequences are determined by increasing sequences of integers. For an increasing sequence , put gn=an+1−an. We prove that
- (a)
- the set {0}∪{±3−(an+1)|n∈N} is quasi-convex in T if and only if a0>0 and gn>1 for every n∈N;
- (b)
- the set {0}∪{±an3|n∈N} is quasi-convex in the group J3 of 3-adic integers if and only if gn>1 for every n∈N.
17.
Amitabha Tripathi 《Journal of Number Theory》2009,129(7):1805-1811
Text
Let s,t be relatively prime positive integers. We prove a conjecture of Aukerman, Kane and Sze regarding the largest size of a partition that is simultaneously s-core and t-core by solving an equivalent problem concerning sets S of positive integers with the property that for n∈S, n−s∈S whenever n?s and n−t∈S whenever n?t.Video
For a video summary of this paper, please visit http://www.youtube.com/watch?v=o1OEug8LryU. 相似文献18.
Horst Alzer 《Journal of Mathematical Analysis and Applications》2009,350(1):276-724
We prove that the following Turán-type inequality holds for Euler's gamma function. For all odd integers n?1 and real numbers x>0 we have
α?Γ(n−1)(x)Γ(n+1)(x)−Γ(n)2(x), 相似文献
19.
For a set A of nonnegative integers the representation functions R2(A,n), R3(A,n) are defined as the number of solutions of the equation n=a+a′,a,a′∈A with a<a′, a?a′, respectively. Let D(0)=0 and let D(a) denote the number of ones in the binary representation of a. Let A0 be the set of all nonnegative integers a with even D(a) and A1 be the set of all nonnegative integers a with odd D(a). In this paper we show that (a) if R2(A,n)=R2(N?A,n) for all n?2N−1, then R2(A,n)=R2(N?A,n)?1 for all n?12N2−10N−2 except for A=A0 or A=A1; (b) if R3(A,n)=R3(N?A,n) for all n?2N−1, then R3(A,n)=R3(N?A,n)?1 for all n?12N2+2N. Several problems are posed in this paper. 相似文献
20.
Igor E. Shparlinski 《Indagationes Mathematicae》2004,15(2):283-289
For a real x ≥ 1 we denote by S[x] the set of squarefull integers n ≤x, that is, the set of positive integers n ≤ such that l2|n for any prime divisor l|n. We estimate exponential sums of the form