# The refined lecture hall theorem via abacus diagrams

@article{Bradford2012TheRL, title={The refined lecture hall theorem via abacus diagrams}, author={Laura Bradford and M. Jeanne Harris and Brant C. Jones and Alex Komarinski and Carly Matson and Edwin O'Shea}, journal={The Ramanujan Journal}, year={2012}, volume={34}, pages={163-176} }

Bousquet-Mélou & Eriksson’s lecture hall theorem generalizes Euler’s celebrated distinct-odd partition theorem. We present an elementary and transparent proof of a refined version of the lecture hall theorem using a simple bijection involving abacus diagrams.

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## 2 Citations

The mathematics of lecture hall partitions

- Mathematics, Computer ScienceJ. Comb. Theory, Ser. A
- 2016

An overview of some of the surprising connections that have surfaced in the process of trying to understand the lecture hall partitions is given.

Generating Functions and Triangulations for Lecture Hall Cones

- Mathematics, Computer ScienceSIAM J. Discret. Math.
- 2016

The Hilbert basis for the lecture hall cone of L_n is described and observations and a conjecture regarding the structure of unimodular triangulations of $L_n$ are concluded, including connections between enumerative and algebraic properties of the cone and cones over unit cubes.

## References

SHOWING 1-10 OF 17 REFERENCES

ON q-SERIES IDENTITIES ARISING FROM LECTURE HALL PARTITIONS

- Mathematics
- 2009

In this paper, we highlight two q-series identities arising from the "five guidelines" approach to enumerating lecture hall partitions and give direct, q-series proofs. This requires two new finite…

Lecture Hall Partitions

- Mathematics
- 1997

AbstractWe 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…

Abacus models for parabolic quotients of affine Weyl groups

- Mathematics
- 2011

We introduce abacus diagrams that describe minimal length coset representatives in affine Weyl groups of types B, C, and D. These abacus diagrams use a realization of the affine Weyl group of type C…

On the refined lecture hall theorem

- Computer Science, MathematicsDiscret. Math.
- 2002

This paper constructs a bijection which is an identity mapping in the limiting case and shows that there is an one to one correspondence between the set of all lecture hall partitions of length n and theSet of all partitions with distinct parts between 1 and n, and possibly multiple parts between n + 1 and 2n.

Counting partitions on the abacus

- Mathematics
- 2006

Abstract
In 2003, Maróti showed that one could use the machinery of ℓ-cores and ℓ-quotients of partitions to establish lower bounds for p(n), the number of partitions of n. In this paper we explore…

Lecture hall theorems, q-series and truncated objects

- Mathematics, Computer ScienceJ. Comb. Theory, Ser. A
- 2004

We show here that the refined theorems for both lecture hall partitions and anti-lecture hall compositions can be obtained as straightforward consequences of two q-Chu Vandermonde identities, once an…

Euler's partition theorem and the combinatorics of l-sequences

- Computer Science, MathematicsJ. Comb. Theory, Ser. A
- 2008

A surprisingly simple bijection is provided for Euler's partition theorem, which involves a family of partitions constrained by the ratio of successive parts, and the intrinsic role played by the combinatorics of @?-sequences is uncovered.

Five Guidelines for Partition Analysis with Applications to Lecture Hall-type Theorems

- Mathematics
- 2006

Five simple guidelines are proposed to compute the generating function for the nonnegative integer solutions of a system of linear inequalities. In contrast to other approaches, the emphasis is on…

Cranks andt-cores

- Mathematics
- 1990

SummaryNew statistics on partitions (calledcranks) are defined which combinatorially prove Ramanujan's congruences for the partition function modulo 5, 7, 11, and 25. Explicit bijections are given…

CRANKS AND t-CORES

- 1990

New statistics on partitions (called cranks) are defined which combinatorially prove Ramanujan’s congruences for the partition function modulo 5, 7, 11, and 25. Explicit bijections are given for the…