{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,28]],"date-time":"2023-10-28T11:16:17Z","timestamp":1698491777448},"reference-count":2,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2006,11,13]],"date-time":"2006-11-13T00:00:00Z","timestamp":1163376000000},"content-version":"vor","delay-in-days":3603,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematical Logic Qtrly"],"published-print":{"date-parts":[[1997,1]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>This paper deals mainly with generalizations of results in finitary combinatorics to infinite ordinals. It is well\u2010known that for finite ordinals \u2211<jats:sub>bT&lt;\u03b1\u03b2<\/jats:sub> is the number of 2\u2010element subsets of an \u03b1\u2010element set. It is shown here that for any well\u2010ordered set of arbitrary infinite order type \u03b1, \u2211<jats:sub>bT&lt;\u03b1\u03b2<\/jats:sub> is the ordinal of the set <jats:italic>M<\/jats:italic> of 2\u2010element subsets, where <jats:italic>M<\/jats:italic> is ordered in some natural way. The result is then extended to evaluating the ordinal of the set of all <jats:italic>n<\/jats:italic>\u2010element subsets for each natural number <jats:italic>n<\/jats:italic> \u2265 2. Moreover, series \u2211<jats:sub>\u03b2&lt;\u03b1<\/jats:sub>f(\u03b2) are investigated and evaluated, where \u03b1 is a limit ordinal and the function <jats:italic>f<\/jats:italic> belongs to a certain class of functions containing polynomials with natural number coefficients. The tools developed for this result can be extended to cover all infinite \u03b1, but the case of finite \u03b1 appears to be quite problematic.<\/jats:p>","DOI":"10.1002\/malq.19970430114","type":"journal-article","created":{"date-parts":[[2007,5,30]],"date-time":"2007-05-30T16:54:45Z","timestamp":1180544085000},"page":"121-133","source":"Crossref","is-referenced-by-count":2,"title":["On Series of Ordinals and Combinatorics"],"prefix":"10.1002","volume":"43","author":[{"given":"James P.","family":"Jones","sequence":"first","affiliation":[]},{"given":"Hilbert","family":"Levitz","sequence":"additional","affiliation":[]},{"given":"Warren D.","family":"Nichols","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2006,11,13]]},"reference":[{"key":"e_1_2_1_2_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-80615-5"},{"key":"e_1_2_1_3_2","volume-title":"Cardinal and Ordinal Numbers","author":"Sierpi\u0144ski W.","year":"1958"}],"container-title":["Mathematical Logic Quarterly"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fmalq.19970430114","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/malq.19970430114","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,28]],"date-time":"2023-10-28T10:39:01Z","timestamp":1698489541000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/malq.19970430114"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1997,1]]},"references-count":2,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1997,1]]}},"alternative-id":["10.1002\/malq.19970430114"],"URL":"https:\/\/doi.org\/10.1002\/malq.19970430114","archive":["Portico"],"relation":{},"ISSN":["0942-5616","1521-3870"],"issn-type":[{"value":"0942-5616","type":"print"},{"value":"1521-3870","type":"electronic"}],"subject":[],"published":{"date-parts":[[1997,1]]}}}