Ramsey partitions of integers and pair divisions

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.     

Pairing conjugate partitions by residue classes

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.     

Set partitions with restrictions

Based on finite set partitions, we introduce restrictions to the distances among the elements in each part and refine the Stirling numbers of the second kind with an extra parameter in two different ways. Combinatorial approach through distributions of “balls into boxes” is employed to establish explicit formulae.     

Formulæ for the number of partitions of n into at most m parts (using the quasi-polynomial ansatz)

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)$p_m(n)$ for any desired m. We do this to demonstrate the power of “rigorous guessing” as facilitated by the quasi-polynomial ansatz.     

Partitions and overpartitions with attached parts

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     

Overpartition pairs and two classes of basic hypergeometric series

We study the combinatorics of two classes of basic hypergeometric series. We first show that these series are the generating functions for certain overpartition pairs defined by frequency conditions on the parts. We then show that when specialized these series are also the generating functions for overpartition pairs with bounded successive ranks, overpartition pairs with conditions on their Durfee dissection, as well as certain lattice paths. When further specialized, the series become infinite products, leading to numerous identities for partitions, overpartitions, and overpartition pairs.     

An extension to overpartitions of the Rogers-Ramanujan identities for even moduli

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.     

Rank and Conjugation for the Frobenius Representation of an Overpartition

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     

Restricted 123-avoiding Baxter permutations and the Padovan numbers

Baxter studied a particular class of permutations by considering fixed points of the composite of commuting functions. This class is called Baxter permutations. In this paper we investigate the number of 123-avoiding Baxter permutations of length n that also avoid (or contain a prescribed number of occurrences of) another certain pattern of length k. In several interesting cases the generating function depends only on k and is expressed via the generating function for the Padovan numbers.     

On the largest size of a partition that is both s-core and t-core

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.     

Mixed sums of squares and triangular numbers (III)

In this paper we confirm a conjecture of Sun which states that each positive integer is a sum of a square, an odd square and a triangular number. Given any positive integer m, we show that p=2m+1 is a prime congruent to 3 modulo 4 if and only if Tm=m(m+1)/2 cannot be expressed as a sum of two odd squares and a triangular number, i.e., p²=x²+8(y²+z²) for no odd integers x,y,z. We also show that a positive integer cannot be written as a sum of an odd square and two triangular numbers if and only if it is of the form 2Tm(m>0) with 2m+1 having no prime divisor congruent to 3 modulo 4.     

Partitions of bipartite numbers

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)=p_j(m, n)p_j(m–1, n+1) ifmn.     

Modular relations for the nonic analogues of the Rogers-Ramanujan functions with applications to partitions

We define the nonic Rogers-Ramanujan-type functions D(q), E(q) and F(q) and establish several modular relations involving these functions, which are analogous to Ramanujan's well known forty identities for the Rogers-Ramanujan functions. We also extract partition theoretic results from some of these relations.     

Stacked lattice boxes

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.     

The Bailey chain and mock theta functions

Standard applications of the Bailey chain preserve mixed mock modularity but not mock modularity. After illustrating this with some examples, we show how to use a change of base in Bailey pairs due to Bressoud, Ismail and Stanton to explicitly construct families of q$q$-hypergeometric multisums which are mock theta functions. We also prove identities involving some of these multisums and certain classical mock theta functions.     

Symmetric identities on Bernoulli polynomials

In this paper, we obtain a generalization of an identity due to Carlitz on Bernoulli polynomials. Then we use this generalized formula to derive two symmetric identities which reduce to some known identities on Bernoulli polynomials and Bernoulli numbers, including the Miki identity.     

A Double Bounded Version of Schur's Partition Theorem

Dedicated to the memory of Paul Erdős Schur's partition theorem states that the number of partitions of n into distinct parts (mod 3) equals the number of partitions of n into parts which differ by 3, where the inequality is strict if a part is a multiple of 3. We establish a double bounded refined version of this theorem by imposing one bound on the parts (mod 3) and another on the parts (mod 3), and by keeping track of the number of parts in each of the residue classes (mod 3). Despite the long history of Schur's theorem, our result is new, and extends earlier work of Andrews, Alladi-Gordon and Bressoud. We give combinatorial and q-theoretic proofs of our result. The special case L=M leads to a representation of the generating function of the underlying partitions in terms of the q-trinomial coefficients extending a similar previous representation of Andrews. Received November 18, 1999 Research of the first author supported in part by NSF Grant DMS-0088975.     

Higher order spt-functions

Andrews? spt-function can be written as the difference between the second symmetrized crank and rank moment functions. Using the machinery of Bailey pairs a combinatorial interpretation is given for the difference between higher order symmetrized crank and rank moment functions. This implies an inequality between crank and rank moments that was only known previously for sufficiently large n and fixed order. This combinatorial interpretation is in terms of a weighted sum of partitions. A number of congruences for higher order spt-functions are derived.