MacMahon's partition identity and the coin exchange problem

One of MacMahon's partition theorems says that the number of partitions of n into parts divisible by 2 or 3 equals the number of partitions of n into parts with multiplicity larger than 1. Recently, Holroyd has obtained a generalization. In this short note, we provide a bijective proof of his theorem.     

On a partition identity of Lehmer

Euler's identity equates the number of partitions of any non-negative integer n into odd parts and the number of partitions of n into distinct parts. Beck conjectured and Andrews proved the following companion to Euler's identity: the excess of the number of parts in all partitions of n into odd parts over the number of parts in all partitions of n into distinct parts equals the number of partitions of n with exactly one even part (possibly repeated). Beck's original conjecture was followed by generalizations and so-called “Beck-type” companions to other identities.In this paper, we establish a collection of Beck-type companion identities to the following result mentioned by Lehmer at the 1974 International Congress of Mathematicians: the excess of the number of partitions of n with an even number of even parts over the number of partitions of n with an odd number of even parts equals the number of partitions of n into distinct, odd parts. We also establish various generalizations of Lehmer's identity, and prove related Beck-type companion identities. We use both analytic and combinatorial methods in our proofs.     

The partition bargaining problem

Consider the problem of partitioning n items among d players where the utility of each player for bundles of items is additive; so, player r has utility for item i and the utility of that player for a bundle of items is the sum of the 's over the items i in his/her bundle. Each partition S of the items is then associated with a d-dimensional utility vector V^S whose coordinates are the utilities that the players assign to the bundles they get under S. Also, lotteries over partitions are associated with the corresponding expected utility vectors. We model the problem as a Nash bargaining game over the set of lotteries over partitions and provide methods for computing the corresponding Nash solution, to prescribed accuracy, with effort that is polynomial in n. In particular, we show that points in the pareto-optimal set of the corresponding bargaining set correspond to lotteries over partitions under which each item, with the possible exception of at most d(d-1)/2 items, is assigned in the same way.     

A proof of a partition theorem for

In this note we give a proof of Devlin's theorem via Milliken's theorem about weakly embedded subtrees of the complete binary tree . Unlike the original proof which is (still unpublished) long and uses the language of category theory, our proof is short and uses direct combinatorial reasoning.

     

Simple upper bounds for partition functions

We tweak Siegel’s method to produce a rather simple proof of a new upper bound on the number of partitions of an integer into an exact number of parts. Our approach, which exploits the delightful dilogarithm function, extends easily to numerous other counting functions. This work was supported by PSC/CUNY Research Awards (# 67261-00 36 and # 68327-00 37).     

A bijective enumeration of tiered trees

Tiered trees were introduced by Dugan–Glennon–Gunnells–Steingrímsson as a generalization of intransitive trees that were introduced by Postnikov, the latter of which have exactly two tiers. Tiered trees arise naturally in counting the absolutely indecomposable representations of certain quivers, and enumerating torus orbits on certain homogeneous varieties over finite fields. By employing generating function arguments and geometric results, Dugan et al. derived an elegant formula concerning the enumeration of tiered trees, which is a generalization of Postnikov’s formula for intransitive trees. In this paper, we provide a bijective proof of this formula by establishing a bijection between tiered trees and certain rooted labeled trees. As an application, our bijection also enables us to derive a refinement of the enumeration of tiered trees with respect to level of the node 1.     

A bijective proof of Amdeberhan’s conjecture on the number of (s,s+2)-core partitions with distinct parts

Amdeberhan conjectured that the number of $(s,s+2)$-core partitions with distinct parts for an odd integer $s$ is $2^{s?1}$. This conjecture was first proved by Yan, Qin, Jin and Zhou, then subsequently by Zaleski and Zeilberger. Since the formula for the number of such core partitions is so simple one can hope for a bijective proof. We give the first direct bijective proof of this fact by establishing a bijection between the set of $(s,s+2)$-core partitions with distinct parts and a set of lattice paths.     

A partition identity

Let n₁+n₂+?+n_m=n where the n_i's are integers (possibly negative or greater than n). Let p=(k₁,…,k_m), where k₁+k₂+?+k_m=k, be a partition of the nonnegative integer k into m nonnegative integers and let P denote the set of all such partitions. For m?2, we prove the combinatorial identity

$\text{∑}\text{p∈P}\text{∏}\text{i=1}\text{m}\text{n}{}_{\mathrm{i}}\text{+1?k}{}_{\mathrm{i}}\text{k}{}_{\mathrm{i}}\text{=}\text{∑}\text{i?0}\text{j+m?2}\text{m?2}\text{n+1?k?2}{}_{\mathrm{j}}\text{k?2}{}_{\mathrm{j}}$

which implies the surprising result that the left side of the above equation depends on n but not on the n_i's.     

The solution to the partition reconstruction problem

Given a partition λ of n, a k-minor of λ is a partition of n−k whose Young diagram fits inside that of λ. We find an explicit function g(n) such that any partition of n can be reconstructed from its set of k-minors if and only if k?g(n). In particular, partitions of n?k²+2k are uniquely determined by their sets of k-minors. This result completely solves the partition reconstruction problem and also a special case of the character reconstruction problem for finite groups.     

An analogue of Euler''s identity and new combinatorial properties of n-colour compositions

An analogue of Euler's partition identity: “The number of partitions of a positive integer ν into odd parts equals the number of its partitions into distinct parts” is obtained for ordered partitions. The ideas developed are then used in obtaining several new combinatorial properties of the n-colour compositions introduced recently by the author.     

Analytic proof of the partition identity A 5,3,3(n)=B 5,3,3 0 (n)

In this paper we give an analytic proof of the identity A _5,3,3(n)=B _5,3,3⁰(n), where A _5,3,3(n) counts the number of partitions of n subject to certain restrictions on their parts, and B _5,3,3⁰(n) counts the number of partitions of n subject to certain other restrictions on their parts, both too long to be stated in the abstract. Our proof establishes actually a refinement of that partition identity. The original identity was first discovered by the first author jointly with M. Ruby Salestina and S.R. Sudarshan in [Proceedings of the International Conference on Analytic Number Theory with Special Emphasis on L-functions, Ramanujan Math. Soc., Mysore, 2005, pp. 57–70], where it was also given a combinatorial proof, thus answering a question of Andrews. Research partially supported by EC’s IHRP Programme, grant HPRN-CT-2001-00272, “Algebraic Combinatorics in Europe.”     

A combinatorial correspondence related to Göllnitz' (big) partition theorem and applications

In recent work, Alladi, Andrews and Gordon discovered a key identity which captures several fundamental theorems in partition theory. In this paper we construct a combinatorial bijection which explains this key identity. This immediately leads to a better understanding of a deep theorem of Göllnitz, as well as Jacobi's triple product identity and Schur's partition theorem.

     

Periodicity of the parity of a partition function related to making change

The solutions to a change problem form restricted partitions. For one particular change problem, we look at the sequence representing the parity of these restricted partition values. It appears that the period of this sequence has not been studied. Through recurrences involving binomial coefficients, we find that the sequence has a period of .

     

The weighted hook length formula

Based on the ideas in Ciocan-Fontanine, Konvalinka and Pak (2009) [5], we introduce the weighted analogue of the branching rule for the classical hook length formula, and give two proofs of this result. The first proof is completely bijective, and in a special case gives a new short combinatorial proof of the hook length formula. Our second proof is probabilistic, generalizing the (usual) hook walk proof of Greene, Nijenhuis and Wilf (1979) [15], as well as the q-walk of Kerov (1993) [20]. Further applications are also presented.     

Elementary proofs of infinitely many congruences for k-elongated partition diamonds

In 2007, Andrews and Paule published the eleventh paper in their series on MacMahon's partition analysis, with a particular focus on broken k-diamond partitions. On the way to broken k-diamond partitions, Andrews and Paule introduced the idea of k-elongated partition diamonds. Recently, Andrews and Paule revisited the topic of k-elongated partition diamonds. Using partition analysis and the Omega operator, they proved that the generating function for the partition numbers $d_k(n)$ produced by summing the links of k-elongated plane partition diamonds of length n is given by $\frac{{(q^2;q^2)}_{\infty }^k}{{(q;q)}_{\infty }^{3k+1}}$ for each $k\ge 1$. A significant portion of their recent paper involves proving several congruence properties satisfied by $d_1,d_2$ and $d_3$, using modular forms as their primary proof tool. In this work, our goal is to extend some of the results proven by Andrews and Paule in their recent paper by proving infinitely many congruence properties satisfied by the functions $d_k$ for an infinite set of values of k. The proof techniques employed are all elementary, relying on generating function manipulations and classical q-series results.     

A unified framework to prove multiplicative inequalities for the partition function

In this paper, we consider a certain class of inequalities for the partition function of the following form:$\prod _{i=1}^{T}p(n+s_i)\ge \prod _{i=1}^{T}p(n+r_i),$ which we call multiplicative inequalities. Given a multiplicative inequality with the condition that ${\sum }_{i=1}^T{s}_i^m\ne {\sum }_{i=1}^T{r}_i^m$ for at least one $m\ge 1$, we shall construct a unified framework so as to decide whether such a inequality holds or not. As a consequence, we will see that study of such inequalities has manifold applications. For example, one can retrieve log-concavity property, strong log-concavity, and the multiplicative inequality for $p(n)$ considered by Bessenrodt and Ono, to name a few. Furthermore, we obtain an asymptotic expansion for the finite difference of the logarithm of $p(n)$, denoted by ${(-1)}^{r-1}{\mathrm{\Delta }}^r\log p(n)$, which generalizes a result by Chen, Wang, and Xie.