共查询到20条相似文献,搜索用时 15 毫秒
1.
Jeremy Lovejoy 《Annals of Combinatorics》2005,9(3):321-334
We discuss conjugation and Dyson’s rank for overpartitions from the perspective of the Frobenius representation. More specifically,
we translate the classical definition of Dyson’s rank to the Frobenius representation of an overpartition and define a new
kind of conjugation in terms of this representation. We then use q-series identities to study overpartitions that are self-conjugate with respect to this conjugation.
Received June 28, 2004 相似文献
2.
Jeremy Lovejoy 《Archiv der Mathematik》2007,88(4):316-322
We show how to interpret a certain q-series as a generating function for overpartitions with attached parts. A number of families of partition theorems follow
as corollaries.
Received: 12 April 2006 相似文献
3.
We study a class of well-poised basic hypergeometric series , interpreting these series as generating functions for overpartitions defined by multiplicity conditions on the number of parts. We also show how to interpret the as generating functions for overpartitions whose successive ranks are bounded, for overpartitions that are invariant under a certain class of conjugations, and for special restricted lattice paths. We highlight the cases (a,q)→(1/q,q), (1/q,q2), and (0,q), where some of the functions become infinite products. The latter case corresponds to Bressoud's family of Rogers-Ramanujan identities for even moduli. 相似文献
4.
Augustine O. Munagi 《Discrete Mathematics》2008,308(12):2492-2501
We study integer partitions in which the parts fulfill the same congruence relations with the parts of their conjugates, called conjugate-congruent partitions. The results obtained include uniqueness criteria, weight lower-bounds and enumerating generating functions. 相似文献
5.
6.
Ifk
1 andk
2 are positive integers, the partitionP = (1,2,...,
n
) ofk
1+k
2 is said to be a Ramsey partition for the pairk
1,k
2 if for any sublistL ofP, either there is a sublist ofL which sums tok
1 or a sublist ofP –L which sums tok
2. Properties of Ramsey partitions are discussed. In particular it is shown that there is a unique Ramsey partition fork
1,k
2 having the smallest numbern of terms, and in this casen is one more than the sum of the quotients in the Euclidean algorithm fork
1 andk
2.An application of Ramsey partitions to the following fair division problem is also discussed: Suppose two persons are to divide a cake fairly in the ratiok
1k
2. This can be done trivially usingk
1+k
2-1 cuts. However, every Ramsey partition ofk
1+k
2 also yields a fair division algorithm. This method yields fewer cuts except whenk
1=1 andk
2=1, 2 or 4. 相似文献
7.
S. O. Warnaar 《Constructive Approximation》2002,18(4):479-502
Using matrix inversion and determinant evaluation techniques we prove several summation and transformation formulas for terminating,
balanced, very-well-poised, elliptic hypergeometric series. 相似文献
8.
Ruiming Zhang 《Advances in Mathematics》2008,217(4):1588-1613
In this work we study the chaotic and periodic asymptotics for the confluent basic hypergeometric series. For a fixed q∈(0,1), the asymptotics for Euler's q-exponential, q-Gamma function Γq(x), q-Airy function of K. Kajiwara, T. Masuda, M. Noumi, Y. Ohta and Y. Yamada, Ramanujan function (q-Airy function), Jackson's q-Bessel function of second kind, Ismail-Masson orthogonal polynomials (q−1-Hermite polynomials), Stieltjes-Wigert polynomials, q-Laguerre polynomials could be derived as special cases. 相似文献
9.
A plane partition is a p×q matrix A=(aij), where 1?i?p and 1?j?q, with non-negative integer entries, and whose rows and columns are weakly decreasing. From a geometric point of view plane partitions are equivalent to pyramids , subsets of the integer lattice Z3 which play an important role in Discrete Tomography. As a consequence, some typical problems concerning the tomography of discrete lattice sets can be rephrased and considered via plane partitions. In this paper we focus on some of them. In particular, we get a necessary and sufficient condition for additivity, a canonical procedure for checking the existence of (weakly) bad configurations, and an algorithm which constructs minimal pyramids (with respect to the number of levels) with assigned projection of a bad configurations. 相似文献
10.
Yasushi Kajihara 《Advances in Mathematics》2004,187(1):53-97
A multiple generalization of the Euler transformation formula for basic hypergeometric series 2φ1 is derived. It is obtained from the symmetry of the reproducing kernel for Macdonald polynomials by a method of multiple principal specialization. As applications, elementary proofs of the Pfaff-Saalschutz summation formula and the Gauss summation formula for basic hypergeometric series in U(n+1) due to S.C. Milne are given. Some other multiple transformation and summation formulas for very-well-poised 10φ9 and 8φ7 series, balanced 4φ3 series and 3φ2 series are also given. 相似文献
11.
The purpose of this short article is to announce, and briefly describe, a Maple package, PARTITIONS, that (inter alia) completely automatically discovers, and then proves, explicit expressions (as sums of quasi-polynomials) for pm(n) for any desired m. We do this to demonstrate the power of “rigorous guessing” as facilitated by the quasi-polynomial ansatz. 相似文献
12.
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13.
14.
Letp
j(m, n) be the number of partitions of (m, n) into at mostj parts. We prove Landman et al.'s conjecture: for allj andn, p
j(x, 2n–x) is a maximum whenx-n. More generally we prove that for all positive integersm, n andj, p
j(n, m)=pj(m, n)pj(m–1, n+1) ifmn. 相似文献
15.
Hjalmar Rosengren 《Advances in Mathematics》2004,181(2):417-447
We derive a number of summation and transformation formulas for elliptic hypergeometric series on the root systems An, Cn and Dn. In the special cases of classical and q-series, our approach leads to new elementary proofs of the corresponding identities. 相似文献
16.
George E. Andrews 《Annals of Combinatorics》1999,3(2-4):115-130
We study the number of solutions of the Diophantine equationn=x
1
x
2+x
2
x
3+x
3
x
4+...+x
k
x
k+1 The combinatorial interpretation of this equation provides the name stacked lattices boxes. The study of these objects unites three separate threads in number theory: (1) the Liouville methods, (2) MacMahon's partitions withk different parts, (3) the asymptotics of divisor sums begun by Ingham.Partially supported by National Science Foundation Grant DMS-9206993, USA. 相似文献
17.
18.
By means of Legendre inverse series relations, we prove two terminating balanced hypergeometric series formulae. Their reversals and linear combinations yield several known and new hypergeometric series identities. 相似文献
19.
《Discrete Mathematics》2002,257(1):125-142
We examine a pair of Rogers-Ramanujan type identities of Lebesgue, and give polynomial identities for which the original identities are limiting cases. The polynomial identities turn out to be q-analogs of the Pell sequence. Finally, we provide combinatorial interpretations for the identities. 相似文献
20.
Zhi-Hong Sun 《Journal of Number Theory》2005,114(1):88-123
If and are two sequences such that a1=b1 and , then we say that (an,bn) is a Newton-Euler pair. In the paper, we establish many formulas for Newton-Euler pairs, and then make use of them to obtain new results concerning some special sequences such as and Bn, where p(n) is the number of partitions of n, σ(n) is the sum of divisors of n, and Bn is the nth Bernoulli number. 相似文献