Partitioned Tensor Products and Their Spectra

A pleasant family of graphs defined by Godsil and McKay is shown to have easily computed eigenvalues in many cases.     

Reduced Words and Plane Partitions

Let w ₀ be the element of maximal length in thesymmetric group S _n , and let Red(w ₀) bethe set of all reduced words for w ₀. We prove the identity which generalizes Stanley's [20] formula forthe cardinality of Red(w ₀), and Macdonald's [11] formula .Our approach uses anobservation, based on a result by Wachs [21], that evaluation of certainspecializations of Schubert polynomials is essentially equivalent toenumeration of plane partitions whose parts are bounded from above. Thus,enumerative results for reduced words can be obtained from the correspondingstatements about plane partitions, and vice versa. In particular, identity(*) follows from Proctor's [14] formula for the number of planepartitions of a staircase shape, with bounded largest part.Similar results are obtained for other permutations and shapes;q-analogues are also given.     

Rado Partition Theorem for Random Subsets of Integers

For an l x k matrix A = (a_ij) of integers, denote by L(A) thesystem of homogenous linear equations a_i1x₁ + ... + a_ikx_k =0, 1 i l. We say that A is density regular if every subsetof N with positive density, contains a solution to L(A). Fora density regular l x k matrix A, an integer r and a set ofintegers F, we write if for any partition F = F₁ ... F_r there exists i {1, 2,..., r} and a column vector x such that Ax = 0 and all entriesof x belong to F_i. Let [n]_N be a random N-element subset of{1, 2, ..., n} chosen uniformly from among all such subsets.In this paper we determine for every density regular matrixA a parameter = (A) such that lim_n P([n]_N (A)_r)=0 if N =O(n) and 1 if N = (n). 1991 Mathematics Subject Classification:05D10, 11B25, 60C05     

Two-dimensional four-line models of the Ising model type

We briefly describe the theory of root transfer matrices for four-line models with the field in the new indexless form. We use theoretical and numerical methods to reveal new effects in the theory of singular points and phase transitions. A substantial part of the results is obtained using a numerical algorithm that drastically (at least exponentially) reduces the computational complexity of Ising-type models by using the extremely sparse root transfer matrix. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 2, pp. 281–298, November, 2006.     

在划分系统下集合对策值的结构

A set game, in which the worth of a coalition is expressed by a set instead of areal number, is a new type of cooperative game.A F-restricted set game is a set game restricted by partition system.The main theorems show the structures of IM-value,OIM-value,SCM-value and ICM-value respectively,and the equivalency of IM-value and the OIM-value for monotonicset games restricted by partition system as well.     

Partitioning complete graphs by heterochromatic trees

A heterochromatic tree is an edge-colored tree in which any two edges have different colors. The heterochromatic tree partition number of an r-edge-colored graph G, denoted by t _r (G), is the minimum positive integer p such that whenever the edges of the graph G are colored with r colors, the vertices of G can be covered by at most p vertex-disjoint heterochromatic trees. In this paper we determine the heterochromatic tree partition number of r-edge-colored complete graphs. We also find at most t _r (K _n ) vertex-disjoint heterochromatic trees to cover all the vertices in polynomial time for a given r-edge-coloring of K _n .     

Resolvability in circulant graphs

A set W of the vertices of a connected graph G is called a resolving set for G if for every two distinct vertices u, v ∈ V (G) there is a vertex w ∈ W such that d(u, w) ≠ d(v, w). A resolving set of minimum cardinality is called a metric basis for G and the number of vertices in a metric basis is called the metric dimension of G, denoted by dim(G). For a vertex u of G and a subset S of V (G), the distance between u and S is the number min s∈S d(u, s). A k-partition Π = {S 1 , S 2 , . . . , S k } of V (G) is called a resolving partition if for every two distinct vertices u, v ∈ V (G) there is a set S i in Π such that d(u, Si )≠ d(v, Si ). The minimum k for which there is a resolving k-partition of V (G) is called the partition dimension of G, denoted by pd(G). The circulant graph is a graph with vertex set Zn , an additive group of integers modulo n, and two vertices labeled i and j adjacent if and only if i-j (mod n) ∈ C , where CZn has the property that C =-C and 0 ■ C. The circulant graph is denoted by Xn, Δ where Δ = |C|. In this paper, we study the metric dimension of a family of circulant graphs Xn, 3 with connection set C = {1, n/2 , n-1} and prove that dim(Xn, 3 ) is independent of choice of n by showing that dim(Xn, 3 ) ={3 for all n ≡ 0 (mod 4), 4 for all n ≡ 2 (mod 4). We also study the partition dimension of a family of circulant graphs Xn,4 with connection set C = {±1, ±2} and prove that pd(Xn, 4 ) is independent of choice of n and show that pd(X5,4 ) = 5 and pd(Xn,4 ) ={3 for all odd n ≥ 9, 4 for all even n ≥ 6 and n = 7.     

正整数的一类三分拆的应用

利用正整数n的一类特殊的3分拆n=n1+n2+n3,n1>n2>n3≥1,且n2+n3>n1的Ferrers图将不定方程4x1+3x2+2x3=n(n≥9)的正整数解与这种分拆联系起来,从而得到了该不定方程的正整数解数公式;同时也给出了正整数n的一类4分拆的计数公式.此外,还给出了周长为n的整边三角形的计数公式的一个简单证明.     

Multinomial permutations on a circle

Multinomial permutations on a circle are considered in the framework of combinatorics. Different cases are presented and shown to agree with previously derived formula for the number of cyclic necklaces. Two applied examples are discussed with a view to illustrate the implications of derived formulas. Copyright © 2006 John Wiley & Sons, Ltd.