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In the first part, the sizes of maximal antichains have been characterized. Here we provide an alternative construction with the benefit of showing that almost all sizes of maximal antichains can be obtained using antichains containing only <jats:italic>l<\/jats:italic>-sets and <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$(l+1)$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>l<\/mml:mi>\n                    <mml:mo>+<\/mml:mo>\n                    <mml:mn>1<\/mml:mn>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>-sets for some <jats:italic>l<\/jats:italic>.<\/jats:p>","DOI":"10.1007\/s11083-024-09675-9","type":"journal-article","created":{"date-parts":[[2024,6,27]],"date-time":"2024-06-27T12:01:47Z","timestamp":1719489707000},"page":"131-156","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Sizes of Flat Maximal Antichains of Subsets"],"prefix":"10.1007","volume":"42","author":[{"given":"Jerrold R.","family":"Griggs","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Thomas","family":"Kalinowski","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Uwe","family":"Leck","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ian T.","family":"Roberts","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Michael","family":"Schmitz","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,6,27]]},"reference":[{"key":"9675_CR1","doi-asserted-by":"publisher","first-page":"30","DOI":"10.1016\/j.dam.2019.02.006","volume":"266","author":"L Badakhshian","year":"2019","unstructured":"Badakhshian, L., Katona, G.O.H., Tuza, Z.: The domination number of the graph defined by two levels of the n-cube. 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