{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:40Z","timestamp":1753893820232,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>For a polynomial with palindromic coefficients, unimodality is equivalent to having a nonnegative $g$-vector. A sufficient condition for unimodality is having a nonnegative $\\gamma$-vector, though one can have negative entries in the $\\gamma$-vector and still have a nonnegative $g$-vector.In this paper we provide combinatorial models for three families of $\\gamma$-vectors that alternate in sign. In each case, the $\\gamma$-vectors come from unimodal polynomials with straightforward combinatorial descriptions, but for which there is no straightforward combinatorial proof of unimodality. By using the transformation from $\\gamma$-vector to $g$-vector, we express the entries of the $g$-vector combinatorially, but as an alternating sum. In the case of the $q$-analogue of $n!$, we use a sign-reversing involution to interpret the alternating sum, resulting in a manifestly positive formula for the $g$-vector. In other words, we give a combinatorial proof of unimodality. We consider this a \"proof of concept\" result that we hope can inspire a similar result for the other two cases, $\\prod_{j=1}^n (1+q^j)$ and the $q$-binomial coefficient ${n\\brack k}$.<\/jats:p>","DOI":"10.37236\/5950","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T21:03:13Z","timestamp":1578690193000},"source":"Crossref","is-referenced-by-count":2,"title":["Unimodality via Alternating Gamma Vectors"],"prefix":"10.37236","volume":"23","author":[{"given":"Charles","family":"Brittenham","sequence":"first","affiliation":[]},{"given":"Andrew T.","family":"Carroll","sequence":"additional","affiliation":[]},{"given":"T. Kyle","family":"Petersen","sequence":"additional","affiliation":[]},{"given":"Connor","family":"Thomas","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2016,5,27]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v23i2p40\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v23i2p40\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T05:28:03Z","timestamp":1579238883000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v23i2p40"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,5,27]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2016,3,31]]}},"URL":"https:\/\/doi.org\/10.37236\/5950","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2016,5,27]]},"article-number":"P2.40"}}