Rogers-Ramanujan Type Identities for Partitions with Attached Odd Parts

In this paper we present a new infinite family of partition identities. The genesis of our work lies in two formulas of Lucy Slater related to the modulus 8. Hirschhorn, Agarwal and Subbarao have previously found intriguing interpretations for Slater's formula, but none has led to an infinite family of results.     

Note on Partitions with Even Parts below Odd Parts

Mathematical Notes -     

On the Combinatorics of Lecture Hall Partitions

A lecture hall partition of length n is an integer sequence satisfying Bousquet-Mélou and Eriksson showed that the number of lecture hall partitions of length n of a positive integer N whose alternating sum is k equals the number of partitions of N into k odd parts less than 2n. We prove the fact by a natural combinatorial bijection. This bijection, though defined differently, is essentially the same as one of the bijections found by Bousquet-Mélou and Eriksson.     

Partitions and Compositions Defined by Inequalities

We consider sequences of integers (₁,..., _k) defined by a system of linear inequalities _{i j>i}a_ijj with integer coefficients. We show that when the constraints are strong enough to guarantee that all _i are nonnegative, the generating function for the integer solutions of weight n has a finite product form , where the b_i are positive integers that can be computed from the coefficients of the inequalities. The results are proved bijectively and are used to give several examples of interesting identities for integer partitions and compositions. The method can be adapted to accommodate equalities along with inequalities and can be used to obtain multivariate forms of the generating function. We show how to extend the technique to obtain the generating function when the coefficients a_i,i+1 are allowed to be rational, generalizing the case of lecture hall partitions. Our initial results were conjectured thanks to the Omega package (G.E. Andrews, P. Paule, and A. Riese, European J. Comb. 22(7) (2001), 887–904).Research supported by NSA grants MDA 904-00-1-0059 and MDA 904-01-0-0083.     

Lecture Hall Partitions

We prove a finite version of the well-known theorem that says that the number of partitions of an integer N into distinct parts is equal to the number of partitions of N into odd parts. Our version says that the number of lecture hall partitions of length n of N equals the number of partitions of N into small odd parts: 1,3,5, ldots, 2n-1 . We give two proofs: one via Bott's formula for the Poincaré series of the affine Coxeter group , and one direct proof.     

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.     

Lecture Hall Partitions II

For a non-decreasing integer sequence a=(a₁,...,a_n) we define L_a to be the set of n-tuples of integers = (₁,...,_n) satisfying . This generalizes the so-called lecture hall partitions corresponding to a_i=i and previously studied by the authors and by Andrews. We find sequences a such that the weight generating function for these a-lecture hall partitions has the remarkable form In the limit when n tends to infinity, we obtain a family of identities of the kind the number of partitions of an integer m such that the quotient between consecutive parts is greater than is equal to the number of partitions of m into parts belonging to the set P, for certain real numbers and integer sets P. We then underline the connection between lecture hall partitions and Ehrhart theory and discuss some reciprocity results.     

Partitions with Non-Repeating Odd Parts and Combinatorial Identities

Continuing our earlier work on partitions with non-repeating odd parts and q-hypergeometric identities, we now study these partitions combinatorially by representing them in terms of 2-modular Ferrers graphs. This yields certain weighted partition identities with free parameters. By special choices of these parameters, we connect them to the Göllnitz-Gordon partitions, and combinatorially prove a modular identity and some parity results. As a consequence, we derive a shifted partition theorem mod 32 of Andrews. Finally we discuss basis partitions in connection with the 2-modular representation of partitions with non-repeating odd parts, and deduce two new parity results involving partial theta series.     

On a Problem of Lehmer on Partitions into Squares

In 1948, D.H.Lehmer published a brief work discussing the difference between representations of the integer n as a sum of squares and partitions of n into square summands. In this article, we return to this topic and consider four partition functions involving square parts and prove various arithmetic properties of these functions. These results provide a natural extension to the work of Lehmer.     

关于整数集分拆的一个猜测的证明

设n为正整数,记rn=m ax{正整数m:可将集合{1,2,…,m}分为n个子集,使得在每一子集中方程xy=z(x>1,y>1)均无解}.高楠和刘红艳(数学的实践与认识,2005,35(5):151—152)给出了rn的一个下界估计rn n9,并猜测对任意给定的正整数k,当n充分大时有rn nk.本文对此猜测给以肯定回答,并证明了如下更强的结论:对任意给定的正整数k 4,当n>3k时有rn n2k+1.     

The Number of Partitions into Distinct Parts Modulo Powers of 5

A relationship is established between the factorization of 24n+ 1 and the 5-divisibility of Q(n), where Q(n)is the numberof partitions of n into distinct parts. As an application, anabundance of infinite families of congruences for Q(n) modulopowers of 5 are explicitly exhibited.     

Partitions of finite vector spaces into subspaces

Let V_n(q) denote a vector space of dimension n over the field with q elements. A set of subspaces of V_n(q) is a partition of V_n(q) if every nonzero element of V_n(q) is contained in exactly one element of . Suppose there exists a partition of V_n(q) into x_i subspaces of dimension n_i, 1 ≤ i ≤ k. Then x₁, …, x_k satisfy the Diophantine equation . However, not every solution of the Diophantine equation corresponds to a partition of V_n(q). In this article, we show that there exists a partition of V_n(2) into x subspaces of dimension 3 and y subspaces of dimension 2 if and only if 7x + 3y = 2ⁿ ? 1 and y ≠ 1. In doing so, we introduce techniques useful in constructing further partitions. We also show that partitions of V_n(q) induce uniformly resolvable designs on qⁿ points. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 329–341, 2008     

Partitions of Groups into Large Subsets

Mathematical Notes -     

Jean-Louis Nicolas and the Partitions

A survey of Jean-Louis Nicolas’s papers on partitions is given.Dedicated to Jean-Louis Nicolas on the occasion of his 60th birthdayPartially supported by the Hungarian National Foundation for Scientific Research, Grant No. T 029759.2000 Mathematics Subject Classification: Primary—11P81     

A Note on Binary Plane Partitions

   Abstract. This paper considers binary space partition s (BSP for short) for disjoint line segments in the plane. The BSP for a disjoint set of objects is a scheme dividing the space recursively by hyperplanes until the resulting fragments of objects are separated. The size of a BSP is the number of resulting fragments of the objects. We show that the minimal size of a BSP for n disjoint line segments in the plane is Ω (n log n /log log n) in the worst case.     

On the Maximal Multiplicity of Parts in a Random Integer Partition

We study the asymptotic behavior of the maximal multiplicity μ_n = μ_n(λ) of the parts in a partition λ of the positive integer n, assuming that λ is chosen uniformly at random from the set of all such partitions. We prove that πμ_n/(6n)^1/2 converges weakly to max _jX_j/j as n→∞, where X₁, X₂, … are independent and exponentially distributed random variables with common mean equal to 1.2000 Mathematics Subject Classification: Primary—05A17; Secondary—11P82, 60C05, 60F05     

A Note on Binary Plane Partitions

Abstract. This paper considers binary space partition s (BSP for short) for disjoint line segments in the plane. The BSP for a disjoint set of objects is a scheme dividing the space recursively by hyperplanes until the resulting fragments of objects are separated. The size of a BSP is the number of resulting fragments of the objects. We show that the minimal size of a BSP for n disjoint line segments in the plane is Ω (n log n /log log n) in the worst case.     

On the Number of Even Parts in All Partitions of $$\varvec{n}$$n into Distinct Parts

Annals of Combinatorics - A famous theorem of Euler asserts that there are as many partitions of n into distinct parts as there are partitions into odd parts. The even parts in partitions of n into...     

关于乘法分拆数的一个猜想和上界

本文证明了乘法分拆数的一个上界,由此证明了Hughes-Shallit的第二猜想,同时证明了对任意的正数a,存在一个自然数N,当n≥N时,n的乘法分拆数f(n)0,使这个集合中的自然数的乘法分拆数≤n~a。     

An Application of Polya's Enumeration Theorem to Partitions of Subsets of Positive Integers

Let S be a non-empty subset of positive integers. A partition of a positive integer n into S is a finite nondecreasing sequence of positive integers a ₁, a ₂,...,a _r in S with repetitions allowed such that . Here we apply Polya's enumeration theorem to find the number P(n; S) of partitions of n into S, and the number DP(n; S) of distinct partitions of n into S. We also present recursive formulas for computing P(n; S) and DP(n; S).