{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,22]],"date-time":"2026-03-22T03:04:48Z","timestamp":1774148688167,"version":"3.50.1"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>The Birkhoff polytope $B_n$ is the convex hull of all $(n\\times n)$ permutation matrices, i.e., matrices where precisely one entry in each row and column is one, and zeros at all other places. This is a widely studied polytope with various applications throughout mathematics.In this paper we study combinatorial types $\\mathcal L$ of faces of a Birkhoff polytope. The Birkhoff dimension $\\mathrm{bd}(\\mathcal L)$ of $\\mathcal L$ is the smallest $n$ such that $B_n$ has a face with combinatorial type $\\mathcal L$.By a result of Billera and Sarangarajan, a combinatorial type $\\mathcal L$ of a $d$-dimensional face\u00a0 appears in some $\\mathcal B_k$ for $k\\le 2d$, so $\\mathrm{bd}(\\mathcal L)\\le 2d$. We will characterize those types with $\\mathrm{bd}(\\mathcal L)\\ge 2d-3$, and we prove that any type with $\\mathrm{bd}(\\mathcal L)\\ge d$ is either a product or a wedge over some lower dimensional face. Further, we computationally classify all $d$-dimensional combinatorial types for $2\\le d\\le 8$.<\/jats:p>","DOI":"10.37236\/4499","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T00:30:08Z","timestamp":1578702608000},"source":"Crossref","is-referenced-by-count":8,"title":["Faces of Birkhoff Polytopes"],"prefix":"10.37236","volume":"22","author":[{"given":"Andreas","family":"Paffenholz","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2015,3,13]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v22i1p67\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v22i1p67\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T10:24:17Z","timestamp":1579256657000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v22i1p67"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,3,13]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2015,1,2]]}},"URL":"https:\/\/doi.org\/10.37236\/4499","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,3,13]]},"article-number":"P1.67"}}