{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:25Z","timestamp":1753893805025,"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>Given a sequence $\\{Q_n(x)\\}_{n=0}^{\\infty}$ of symmetric orthogonal polynomials, defined by a recurrence formula $Q_n(x)=\\nu_n\\cdot x\\cdot Q_{n-1}(x)-(\\nu_n-1)\\cdot Q_{n-2}(x)$ with integer $\\nu_i$'s satisfying $\\nu_i\\geq 2$, we construct a sequence of nested Eulerian posets whose $ce$-index is a non-commutative generalization of these polynomials. Using spherical shellings and direct calculations of the $cd$-coefficients of the associated Eulerian posets we obtain two new proofs for a bound on the true interval of orthogonality of $\\{Q_n(x)\\}_{n=0}^{\\infty}$. Either argument can replace the use of the theory of chain sequences. Our $cd$-index calculations allow us to represent the orthogonal polynomials as an explicit positive combination of terms of the form $x^{n-2r}(x^2-1)^r$. Both proofs may be extended to the case when the $\\nu_i$'s are not integers and the second proof is still valid when only $\\nu_i&gt;1$ is required. The construction provides a new \"limited testing ground\" for Stanley's non-negativity conjecture for Gorenstein$^*$ posets, and suggests the existence of strong links between the theory of orthogonal polynomials and flag-enumeration in Eulerian posets.  <\/jats:p>","DOI":"10.37236\/1861","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T05:20:45Z","timestamp":1578720045000},"source":"Crossref","is-referenced-by-count":3,"title":["Orthogonal Polynomials Represented by $CW$-Spheres"],"prefix":"10.37236","volume":"11","author":[{"given":"G\u00e1bor","family":"Hetyei","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2004,8,16]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v11i2r4\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v11i2r4\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T05:01:39Z","timestamp":1579323699000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v11i2r4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2004,8,16]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2004,6,3]]}},"URL":"https:\/\/doi.org\/10.37236\/1861","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2004,8,16]]},"article-number":"R4"}}