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31.
Leonid G. Fel 《Functional Analysis and Other Mathematics》2006,1(2):119-157
We find the matrix representation of the set Δ(d
3), where d
3=(d
1,d
2,d
3), of integers that are unrepresentable by d
1,d
2,d
3 and develop a diagrammatic procedure for calculating the generating function Φ(d
3;z) for the set Δ(d
3). We find the Frobenius number F(d
3), the genus G(d
3), and the Hilbert series H(d
3;z) of a graded subring for nonsymmetric and symmetric semigroups
and enhance the lower bounds of F(d
3) for symmetric and nonsymmetric semigroups.
相似文献
32.
Stephan G. Wagner 《Acta Appl Math》2006,91(2):119-132
In this paper, we will consider the Wiener index for a class of trees that is connected to partitions of integers. Our main theorem is the fact that every integer is the Wiener index of a member of this class. As a consequence, this proves a conjecture of Lepović and Gutman. The paper also contains extremal and average results on the Wiener index of the studied class.This work was supported by Austrian Science Fund project no. S-8307-MAT. 相似文献
33.
Summary For P∈ F2[z] with P(0)=1 and deg(P)≧ 1, let A =A(P) be the unique subset of N (cf. [9]) such that Σn≧0 p(A,n)zn ≡ P(z) mod 2, where p(A,n) is the number of partitions of n with parts in A. To determine the elements of the set A, it is important to consider the sequence σ(A,n) = Σ d|n, d∈A d, namely, the periodicity of the sequences (σ(A,2kn) mod 2k+1)n≧1 for all k ≧ 0 which was proved in [3]. In this paper, the values of such sequences will be given in terms of orbits. Moreover, a formula
to σ(A,2kn) mod 2k+1 will be established, from which it will be shown that the weight σ(A1,2kzi) mod 2k+1 on the orbit <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>z_i$
is moved on some other orbit zj when A1 is replaced by A2 with A1= A(P1) and A2= A(P2) P1 and P2 being irreducible in F2[z] of the same odd order. 相似文献
34.
Liuquan Wang 《Discrete Mathematics》2018,341(12):3370-3384
Let be the number of -colored generalized Frobenius partitions of . We establish some infinite families of congruences for and modulo arbitrary powers of 3, which refine the results of Kolitsch. For example, for and , we prove that We give two different proofs to the congruences satisfied by . One of the proofs uses a relation between and due to Kolitsch, for which we provide a new proof in this paper. 相似文献
35.
36.
《Random Structures and Algorithms》2018,53(3):537-558
We study Gibbs partitions that typically form a unique giant component. The remainder is shown to converge in total variation toward a Boltzmann‐distributed limit structure. We demonstrate how this setting encompasses arbitrary weighted assemblies of tree‐like combinatorial structures. As an application, we establish smooth growth along lattices for small block‐stable classes of graphs. Random graphs with n vertices from such classes are shown to form a giant connected component. The small fragments may converge toward different Poisson Boltzmann limit graphs, depending along which lattice we let n tend to infinity. Since proper addable minor‐closed classes of graphs belong to the more general family of small block‐stable classes, this recovers and generalizes results by McDiarmid (2009). 相似文献
37.
In the setting of ZF, i.e., Zermelo–Fraenkel set theory without the Axiom of Choice (AC), we study partitions of Russell‐sets into sets each with exactly n elements (called n ‐ary partitions), for some integer n. We show that if n is odd, then a Russell‐set X has an n ‐ary partition if and only if |X | is divisible by n. Furthermore, we establish that it is relative consistent with ZF that there exists a Russell‐set X such that |X | is not divisible by any finite cardinal n > 1 (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
38.
Øystein J. Rødseth 《Discrete Mathematics》2006,306(16):2005-2009
Recently, Sloane suggested the following problem: We are given n boxes, labeled 1,2,…,n. For i=1,…,n, box i weighs (m-1)i grams (where m?2 is a fixed integer) and box i can support a total weight of i grams. What is the number of different ways to build a single stack of boxes in which no box will be squashed by the weight of the boxes above it? Prior to this generalized problem, Sloane and Sellers solved the case m=2. More recently, Andrews and Sellers solved the case m?3. In this note we give new and simple proofs of the results of Sloane and Sellers and of Andrews and Sellers, using a known connection with m-ary partitions. 相似文献
39.
Øystein J. Rødseth 《Discrete Mathematics》2006,306(7):694-698
An M-partition of a positive integer m is a partition of m with as few parts as possible such that every positive integer less than m can be written as a sum of parts taken from the partition. This type of partition is a variation of MacMahon's perfect partition, and was recently introduced and studied by O’Shea, who showed that for half the numbers m, the number of M-partitions of m is equal to the number of binary partitions of 2n+1-1-m, where . In this note we extend O’Shea's result to cover all numbers m. 相似文献
40.
James E. Stafford 《Journal of computational and graphical statistics》2013,22(3):249-259
Abstract In this article we show how attention to the structure of a particular algebraic calculation can lead to the simple implementation of powerful computer algebra tools. The creation of partitions for a set of indexes is required for the implementation of many theoretical structures. This may be difficult to do by hand even when the number of indexes is only moderately large. These partitions arise through the action of differentiation and so we mimic differentiation in a computer algebra package to create partitions of indexes. The strategies employed are extended to the creation of complementary set partitions, their reduction to equivalence classes, and the implementation of Edgeworth expansions and the exlog relations. 相似文献