# Exact Enumeration of Garden of Eden Partitions

• Published in 2006
In the collections
We give two proofs for a formula that counts the number of partitions of $n$ that have rank −2 or less (which we call Garden of Eden partitions). These partitions arise naturally in analyzing the game Bulgarian solitaire, summarized in Section 1. Section 2 presents a generating function argument for the formula based on Dyson’s original paper where the rank of a partition is defined. Section 3 gives a combinatorial proof of the result, based on a bijection on Bressoud and Zeilberger.

### BibTeX entry

@article{ExactEnumerationOfGardenOfEdenPartitions,
title = {Exact Enumeration of Garden of Eden Partitions},
abstract = {We give two proofs for a formula that counts the number of partitions of $n$ that have rank −2 or less (which we call Garden of Eden partitions).  These partitions arise naturally in analyzing the game Bulgarian solitaire, summarized in Section 1. Section 2 presents a generating function argument for the formula based on Dyson’s original paper where the rank of a partition is defined. Section 3 gives a combinatorial proof of the result, based on a bijection on Bressoud and Zeilberger.},
url = {https://www.emis.de/journals/INTEGERS/papers/a19int2005/a19int2005.Abstract.html https://www.emis.de/journals/INTEGERS/papers/a19int2005/a19int2005.pdf http://www.personal.psu.edu/jxs23/HS{\_}integers{\_}final.pdf},
year = 2006,
author = {Brian Hopkins and James A. Sellers},
comment = {},
urldate = {2018-05-13},
collections = {Attention-grabbing titles,Easily explained,Combinatorics}
}