Garden of Eden States in Austrian Solitaire

Robson da Silva, Brian Hopkins, James A. Sellers

Research output: Contribution to journalArticlepeer-review

Abstract

Introduced by Akin and Davis in 1985, Austrian Solitaire is a two-parameter variation of the better known Bulgarian Solitaire that constrains part sizes. We count the valid states in Austrian Solitaire and also the number of states with no preimage, known as Garden of Eden states. Connecting these two quantities involves new results about preimages in the system. Our tools for this work include bijective proofs and generating functions.

Original languageEnglish (US)
Article number103023
JournalEuropean Journal of Combinatorics
Volume83
DOIs
StatePublished - Jan 2020

Bibliographical note

Funding Information:
The first named author was supported by the São Paulo Research Foundation, FAPESP grant no. 2016/14057-2 . The tables and equivalent versions of Propositions 2.1 and 3.1 appeared in the 2012 undergraduate honors thesis of Kapil Bastola [3] , whose Conjecture 1 is resolved by Theorem 3.2 ; the second named author served as faculty advisor. The authors are grateful for George Andrews’s assistance in simplifying the generating function given in Theorem 3.2 . We also appreciate the careful reading and thoughtful recommendations of two anonymous reviewers.

Funding Information:
The first named author was supported by the S?o Paulo Research Foundation, FAPESP grant no. 2016/14057-2. The tables and equivalent versions of Propositions 2.1 and 3.1 appeared in the 2012 undergraduate honors thesis of Kapil Bastola [3], whose Conjecture 1 is resolved by Theorem 3.2; the second named author served as faculty advisor. The authors are grateful for George Andrews's assistance in simplifying the generating function given in Theorem 3.2. We also appreciate the careful reading and thoughtful recommendations of two anonymous reviewers.

Publisher Copyright:
© 2019 Elsevier Ltd

Copyright:
Copyright 2019 Elsevier B.V., All rights reserved.

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