{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,3]],"date-time":"2022-04-03T19:36:40Z","timestamp":1649014600415},"reference-count":10,"publisher":"World Scientific Pub Co Pte Lt","issue":"03","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Discrete Math. Algorithm. Appl."],"published-print":{"date-parts":[[2014,9]]},"abstract":"<jats:p> The Ahlswede\u2013Zhang identity is a sharpening of the well-known LYM inequality. To establish the identity, Ahlswede and Zhang introduced an important set function called AZ function. In this paper, we present some identities which establish a relation between the AZ function and cardinalities of downsets or upsets. Some connections of these identities with the original AZ identity are considered. Furthermore, these identities yield formulas concerning the size of a star in a downset. <\/jats:p>","DOI":"10.1142\/s1793830914500402","type":"journal-article","created":{"date-parts":[[2014,3,20]],"date-time":"2014-03-20T06:27:23Z","timestamp":1395296843000},"page":"1450040","source":"Crossref","is-referenced-by-count":1,"title":["AZ-style identities of downsets and upsets"],"prefix":"10.1142","volume":"06","author":[{"given":"Tran Dan","family":"Thu","sequence":"first","affiliation":[{"name":"School of Information Technology, University of Science Hochiminh City, 227 Nguyen Van Cu, Dist. 5, Hochiminh City, Vietnam"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2014,6,16]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1016\/0001-8708(90)90023-G"},{"key":"rf2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01202350"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1016\/0097-3165(94)90105-8"},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1006\/aama.1997.0542"},{"key":"rf5","first-page":"209","volume":"1","author":"Lubell D.","journal-title":"J. Combin. Theory"},{"key":"rf6","doi-asserted-by":"publisher","DOI":"10.1137\/1108023"},{"key":"rf8","first-page":"168","volume":"62","author":"Tran D. T.","journal-title":"J. Combin. Theory Ser. A"},{"key":"rf9","first-page":"693","volume":"29","author":"Tran D. T.","journal-title":"Bull. London Math Soc."},{"key":"rf10","first-page":"1504","volume":"114","author":"Tran D. T.","journal-title":"J. Combin. Theory Ser. A"},{"key":"rf11","doi-asserted-by":"publisher","DOI":"10.2969\/jmsj\/00630343"}],"container-title":["Discrete Mathematics, Algorithms and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S1793830914500402","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,6]],"date-time":"2019-08-06T17:16:03Z","timestamp":1565111763000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S1793830914500402"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,6,16]]},"references-count":10,"journal-issue":{"issue":"03","published-online":{"date-parts":[[2014,6,16]]},"published-print":{"date-parts":[[2014,9]]}},"alternative-id":["10.1142\/S1793830914500402"],"URL":"https:\/\/doi.org\/10.1142\/s1793830914500402","relation":{},"ISSN":["1793-8309","1793-8317"],"issn-type":[{"value":"1793-8309","type":"print"},{"value":"1793-8317","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,6,16]]}}}