{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,17]],"date-time":"2026-07-17T16:04:29Z","timestamp":1784304269978,"version":"3.55.0"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>We examine combinatorial counting functions with two parameters, $n$ and $q$. For fixed $q$, these functions are (quasi-)polynomial in $n$. As $q$ varies, the degree of this polynomial is itself polynomial in $q$, as are the leading coefficients. We carefully define these two-level polynomials, lay out their basic algebraic properties, and provide a schema for showing a function is a two-level polynomial. Using the schema, we prove that a variety of counting functions arising in different areas of combinatorics are two-level polynomials. These include chromatic polynomials for many infinite families of graphs, partitions of an integer into a given number of parts, placing non-attacking chess pieces on a board, Sidon sets, and Sheffer sequences (including binomial type and Appell sequences).<\/jats:p>","DOI":"10.37236\/14617","type":"journal-article","created":{"date-parts":[[2026,7,17]],"date-time":"2026-07-17T15:14:35Z","timestamp":1784301275000},"source":"Crossref","is-referenced-by-count":0,"title":["Counting with Two-Level Polynomials"],"prefix":"10.37236","volume":"33","author":[{"given":"Tristram","family":"Bogart","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Kevin","family":"Woods","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"23455","published-online":{"date-parts":[[2026,7,17]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v33i3p18\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v33i3p18\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,7,17]],"date-time":"2026-07-17T15:14:35Z","timestamp":1784301275000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v33i3p18"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,7,17]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2026,7,3]]}},"URL":"https:\/\/doi.org\/10.37236\/14617","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,7,17]]},"article-number":"P3.18"}}