{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T16:17:28Z","timestamp":1759335448166,"version":"3.41.2"},"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>This paper begins with a brief discussion of a class of polynomial Riemann hypotheses, which leads to the consideration of sequences of orthogonal polynomials and 3-term recursions. The discussion further leads to higher order polynomial recursions, including 4-term recursions where orthogonality is lost. Nevertheless, we show that classical results on the nature of zeros of real orthogonal polynomials (i. e., that the zeros of $p_n$ are real and those of $p_{n+1}$ interleave those of $p_n$) may be extended to polynomial sequences satisfying certain 4-term recursions. We identify specific polynomial sequences satisfying higher order recursions that should also satisfy this classical result. As with the 3-term recursions, the 4-term recursions give rise naturally to a linear functional. In the case of  3-term recursions the zeros fall nicely into place when it is known that the functional is positive, but in the case of our 4-term recursions, we show that the functional can be positive even when there are non-real zeros among some of the polynomials. It is interesting, however, that for our 4-term recursions positivity is guaranteed when a certain real parameter $C$ satisfies $C\\ge 3$, and this is exactly the condition of our result that guarantees the zeros have the aforementioned interleaving property. We conjecture the condition $C\\ge 3$ is also necessary. Next we used a classical determinant criterion to find exactly when the associated linear functional is positive, and we found that the Hankel determinants $\\Delta_n$ formed from the sequence of moments of the functional when $C = 3$ give rise to the initial values of the integer sequence $1, 3, 26, 646, 45885, \\cdots,$ of Alternating Sign Matrices (ASMs) with vertical symmetry. This spurred an intense interest in these moments, and we give 9 diverse characterizations of this sequence of moments.  We then specify these Hankel determinants as Macdonald-type integrals. We also provide an an infinite class of integer sequences, each sequence of which gives the Hankel determinants $\\Delta_n$ of the moments. Finally we show that certain $n$-tuples of non-intersecting lattice paths are evaluated by a related class of special Hankel determinants. This class includes the $\\Delta_n$. At the same time, ASMs with vertical symmetry can readily be identified with certain $n$-tuples of osculating paths. These two lattice path models appear as a natural bridge from the ASMs with vertical symmetry to Hankel determinants.<\/jats:p>","DOI":"10.37236\/1580","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T21:06:46Z","timestamp":1578690406000},"source":"Crossref","is-referenced-by-count":11,"title":["From a Polynomial Riemann Hypothesis to Alternating Sign Matrices"],"prefix":"10.37236","volume":"8","author":[{"given":"\u00d6mer","family":"E\u011fecio\u011flu","sequence":"first","affiliation":[]},{"given":"Timothy","family":"Redmond","sequence":"additional","affiliation":[]},{"given":"Charles","family":"Ryavec","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2001,10,24]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v8i1r36\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v8i1r36\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T00:16:34Z","timestamp":1579306594000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v8i1r36"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2001,10,24]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2001,1,1]]}},"URL":"https:\/\/doi.org\/10.37236\/1580","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2001,10,24]]},"article-number":"R36"}}