Interesting Esoterica

Exact Enumeration of Garden of Eden Partitions

Article by Brian Hopkins and James A. Sellers
  • Published in 2006
  • Added on
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.

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key
ExactEnumerationOfGardenOfEdenPartitions
type
article
date_added
2018-05-13
date_published
2006-11-11

BibTeX entry

@article{ExactEnumerationOfGardenOfEdenPartitions,
	key = {ExactEnumerationOfGardenOfEdenPartitions},
	type = {article},
	title = {Exact Enumeration of Garden of Eden Partitions},
	author = {Brian Hopkins and James A. Sellers},
	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.},
	comment = {},
	date_added = {2018-05-13},
	date_published = {2006-11-11},
	urls = {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},
	collections = {Attention-grabbing titles,Easily explained,Combinatorics},
	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,
	urldate = {2018-05-13}
}