{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,1]],"date-time":"2026-03-01T15:22:22Z","timestamp":1772378542559,"version":"3.50.1"},"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>Let $n,k,$ and $r$ be nonnegative integers and let $S_n$ be the symmetric group. We introduce a quotient $R_{n,k,r}$ of the polynomial ring $\\mathbb{Q}[x_1, \\dots, x_n]$ in $n$ variables which carries the structure of a graded $S_n$-module. \u00a0When $r \\ge n$ or $k = 0$ the quotient $R_{n,k,r}$ reduces to the classical coinvariant algebra $R_n$ attached to the symmetric group. Just as algebraic properties of $R_n$ are controlled by combinatorial properties of permutations in $S_n$, the algebra of $R_{n,k,r}$ is controlled by the combinatorics of objects called tail positive words. We calculate the standard monomial basis of $R_{n,k,r}$ and its graded $S_n$-isomorphism type. We also view $R_{n,k,r}$ as a module over the 0-Hecke algebra $H_n(0)$, prove that $R_{n,k,r}$ is a projective 0-Hecke module, and calculate its quasisymmetric and nonsymmetric 0-Hecke characteristics. We conjecture a relationship between our quotient $R_{n,k,r}$ and the delta operators of the theory of Macdonald polynomials.<\/jats:p>","DOI":"10.37236\/6970","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T15:50:41Z","timestamp":1578671441000},"source":"Crossref","is-referenced-by-count":3,"title":["Tail Positive Words and Generalized Coinvariant Algebras"],"prefix":"10.37236","volume":"24","author":[{"given":"Brendon","family":"Rhoades","sequence":"first","affiliation":[]},{"given":"Andrew Timothy","family":"Wilson","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2017,7,28]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i3p21\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i3p21\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T04:50:39Z","timestamp":1579236639000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v24i3p21"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,7,28]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2017,7,14]]}},"URL":"https:\/\/doi.org\/10.37236\/6970","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,7,28]]},"article-number":"P3.21"}}