Arithmetic Properties of Non-Squashing Partitions into Distinct Parts

A partition with is non-squashing if On their way towards the solution of a certain box-stacking problem, Sloane and Sellers were led to consider the number b(n) of non-squashing partitions of n into distinct parts. Sloane and Sellers did briefly consider congruences for b(n) modulo 2. In this paper we show that 2^r-2 is the exact power of 2 dividing the difference b(2^r+1n)–b(2^r-1n) for n odd and r 2.     

Arithmetic Properties of Overpartitions into Odd Parts

In this article, we consider various arithmetic properties of the function which denotes the number of overpartitions of n using only odd parts. This function has arisen in a number of recent papers, but in contexts which are very different from overpartitions. We prove a number of arithmetic results including several Ramanujan-like congruences satisfied by and some easily-stated characterizations of modulo small powers of two. For example, it is proven that, for n ≥ 1, (mod 4) if and only if n is neither a square nor twice a square. Received March 17, 2005     

Arithmetic properties of ℓ-regular partitions

For a given prime p, by studying p  -dissection identities for Ramanujan?s theta functions ψ(q)$\psi (q)$ and f(−q)$f(-q)$, we derive infinite families of congruences modulo 2 for some ?  -regular partition functions, where ?=2,4,5,8,13,16$?=2,4,5,8,13,16$.     

On m-Ary Overpartitions

Presently there are a lot of activities in the study of overpartitions, objects that were discussed by MacMahon, and which have recently proven useful in several combinatorial studies of basic hypergeometric series. In this paper we study some similar objects, which we name m-ary overpartitions. We consider divisibility properties of the number of m-ary overpartitions of a natural number, and we prove a theorem which is a lifting to general m of the well-known Churchhouse congruences for the binary partition function. Received October 11, 2004     

Generalizing inplace multiplicity identities for integer compositions

In a recent paper, the authors gave two new identities for compositions, or ordered partitions, of integers. These identities were based on closely-related integer partition functions which have recently been studied. In the process, we also extensively generalized both of these identities. Since then, we asked whether one could generalize one of these results even further by considering compositions in which certain parts could come from t kinds (rather than just two kinds, which was the crux of the original result). In this paper, we provide such a generalization. A straightforward bijective proof is given and generating functions are provided for each of the types of compositions which arise. We close by briefly mentioning some arithmetic properties satisfied by the functions which count such compositions.     

Generalizations of two identities of Guo and Yang

In this paper, we provide generalizations of two identities of Guo and Yang [2] for the q-binomial coe?cients. This approach allows us to derive new convolution identities for the complete and elementary symmetric functions. New identities involving q-binomial coe?cients are obtained as very special cases of these results. A new relationship between restricted partitions and restricted partitions into parts of two kinds is derived in this context.     

False theta functions and companions to Capparelli's identities

Capparelli conjectured two modular identities for partitions whose parts satisfy certain gap conditions, which were motivated by the calculation of characters for the standard modules of certain affine Lie algebras and by vertex operator theory. These identities were subsequently proved and refined by Andrews, who related them to Jacobi theta functions, and also by Alladi–Andrews–Gordon, Capparelli and Tamba–Xie. In this paper we prove two new companions to Capparelli's identities, where the evaluations are expressed in terms of Jacobi theta functions and false theta functions.     

Proof of the Andrews–Dyson–Rhoades conjecture on the spt-crank

The spt-crank of a vector partition, or an S  -partition, was introduced by Andrews, Garvan and Liang. Let _NS(m,n)$N_S(m,n)$ denote the net number of S-partitions of n with spt-crank m, that is, the number of S  -partitions (_π1,_π2,_π3)$({\pi }_1,{\pi }_2,{\pi }_3)$ of n with spt-crank m   such that the length of _π1${\pi }_1$ is odd minus the number of S  -partitions (_π1,_π2,_π3)$({\pi }_1,{\pi }_2,{\pi }_3)$ of n with spt-crank m   such that the length of _π1${\pi }_1$ is even. Andrews, Dyson and Rhoades conjectured that _{NS(m,n)}m${\{N_S(m,n)\}}_m$ is unimodal for any n  , and they showed that this conjecture is equivalent to an inequality between the rank and crank of ordinary partitions. They obtained an asymptotic formula for the difference between the rank and crank of ordinary partitions, which implies _NS(m,n)≥_NS(m+1,n)$N_S(m,n)\ge N_S(m+1,n)$ for sufficiently large n and fixed m. In this paper, we introduce a representation of an ordinary partition, called the m-Durfee rectangle symbol, which is a rectangular generalization of the Durfee symbol introduced by Andrews. We give a proof of the conjecture of Andrews, Dyson and Rhoades. We also show that this conjecture implies an inequality between the positive rank and crank moments obtained by Andrews, Chan and Kim.     

Some identities involving the binomial sequences

Using the exponential generating function and the Bell polynomials, we obtain several new identities for the binomial sequences. As applications, some interesting identities are established for the Abel polynomials, exponential polynomials and factorial powers.     

q-Analogs of some congruences involving Catalan numbers

We present some variations on the Greene–Krammer?s identity which involve q-Catalan numbers. Our method reveals an intriguing analogy between these new identities and some congruences modulo a prime.     

Explicit congruences for the partition function modulo every prime

Let p(n) denote the number of unrestricted partitions of the positive integer n, and let m be a prime $\geq 13$. We prove, for k = 1, explicit congruences of the form where $r_{m,k}, \delta_{m,k}$ are integers depending on m and k and $\phi_{m,k}(z)$ are explicitly computable level one holomorphic modular forms of small weight. We also give theoretical and numerical support that the congruences also hold for k > 1. Our main idea is a level reduction result for the modular forms which originated from Atkin-Lehner. From our result, we deduce periodicity properties for the partition function with short periods which improve upon recent results of K. Ono. Received: 24 January 2002     

A geometric perspective on counting non-negative integer solutions and combinatorial identities

We consider the effect of constraints on the number of non-negative integer solutions of x+y+z = n, relating the number of solutions to linear combinations of triangular numbers. Our approach is geometric and may be viewed as an introduction to proofs without words. We use this geometrical perspective to prove identities by counting the number of solutions in two different ways, thereby combining combinatorial proofs and proofs without words.     

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.     

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.     

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.     

Rank and crank moments for overpartitions

We study two types of crank moments and two types of rank moments for overpartitions. We show that the crank moments and their derivatives, along with certain linear combinations of the rank moments and their derivatives, can be written in terms of quasimodular forms. We then use this fact to prove exact relations involving the moments as well as congruence properties modulo 3, 5, and 7 for some combinatorial functions which may be expressed in terms of the second moments. Finally, we establish a congruence modulo 3 involving one such combinatorial function and the Hurwitz class number H(n).     

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.     

New analogues of Ramanujan's partition identities

We establish several new analogues of Ramanujan's exact partition identities using the theory of modular functions.