{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T15:42:01Z","timestamp":1753890121196,"version":"3.41.2"},"reference-count":0,"publisher":"Centre pour la Communication Scientifique Directe (CCSD)","issue":"Combinatorics","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"accepted":{"date-parts":[[2025,3,31]]},"abstract":"<jats:p>To appear in Volume 19 of DMTCS.<\/jats:p><jats:p xml:lang=\"en\">Recent research on the combinatorics of finite sets has explored the structure of symmetric difference-closed sets, and recent research in combinatorial group theory has concerned the enumeration of commuting involutions in $S_{n}$ and $A_{n}$. In this article, we consider an interesting combination of these two subjects, by introducing classes of symmetric difference-closed sets of elements which correspond in a natural way to commuting involutions in $S_{n}$ and $A_{n}$. We consider the natural combinatorial problem of enumerating symmetric difference-closed sets consisting of subsets of sets consisting of pairwise disjoint $2$-subsets of $[n]$, and the problem of enumerating symmetric difference-closed sets consisting of elements which correspond to commuting involutions in $A_{n}$. We prove explicit combinatorial formulas for symmetric difference-closed sets of these forms, and we prove a number of conjectured properties related to such sets which had previously been discovered experimentally using the On-Line Encyclopedia of Integer Sequences.<\/jats:p>","DOI":"10.23638\/dmtcs-19-1-8","type":"journal-article","created":{"date-parts":[[2025,4,3]],"date-time":"2025-04-03T16:39:53Z","timestamp":1743698393000},"source":"Crossref","is-referenced-by-count":0,"title":["A class of symmetric difference-closed sets related to commuting involutions"],"prefix":"10.23638","volume":"Vol. 19 no. 1","author":[{"given":"John","family":"Campbell","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics [Toronto]"}]}],"member":"25203","published-online":{"date-parts":[[2017,3,23]]},"container-title":["Discrete Mathematics &amp; Theoretical Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/hal.science\/hal-01345066v4\/document","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/hal.science\/hal-01345066v4\/document","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,4,3]],"date-time":"2025-04-03T16:39:53Z","timestamp":1743698393000},"score":1,"resource":{"primary":{"URL":"http:\/\/dmtcs.episciences.org\/1536"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,3,23]]},"references-count":0,"journal-issue":{"issue":"Combinatorics","published-online":{"date-parts":[[2017,3,23]]}},"URL":"https:\/\/doi.org\/10.23638\/dmtcs-19-1-8","relation":{"has-preprint":[{"id-type":"uri","id":"https:\/\/hal.science\/hal-01345066v3","asserted-by":"subject"},{"id-type":"uri","id":"https:\/\/hal.science\/hal-01345066v2","asserted-by":"subject"},{"id-type":"uri","id":"https:\/\/hal.science\/hal-01345066v1","asserted-by":"subject"}],"is-same-as":[{"id-type":"uri","id":"https:\/\/hal.science\/hal-01345066v4","asserted-by":"subject"}]},"ISSN":["1365-8050"],"issn-type":[{"type":"electronic","value":"1365-8050"}],"subject":[],"published":{"date-parts":[[2017,3,23]]},"article-number":"1536"}}