{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:42:57Z","timestamp":1753893777987,"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>Given an integer $n\\geq 2$, and a non-negative integer $k$,  consider all affine hyperplanes in ${\\bf R}^n$ of the form $x_i=x_j +r$ for $i,j\\in[n]$ and a non-negative integer $r\\leq k$. Let $\\Pi_{n,k}$ be the poset whose elements are all nonempty intersections of these affine hyperplanes, ordered by reverse inclusion. It is noted that $\\Pi_{n,0}$ is isomorphic to the well-known partition lattice $\\Pi_n$, and in this paper, we extend some of the results of $\\Pi_n$ by Hanlon and Stanley to $\\Pi_{n,k}$. Just as there is an action of the symmetric group ${S}_n$ on $\\Pi_n$, there is also an  action on $\\Pi_{n,k}$ which permutes the coordinates of each element. We consider the subposet $\\Pi_{n,k}^\\sigma$ of  elements that are fixed by some $\\sigma\\in {S}_n$, and find its M\u00f6bius function $\\mu_\\sigma$, using the characteristic polynomial. This generalizes what Hanlon did in the case $k=0$. It then follows that $(-1)^{n-1}\\mu_\\sigma(\\Pi_{n,k}^\\sigma)$, as a function of $\\sigma$, is the character of the action of ${S}_n$ on the homology of $\\Pi_{n,k}$. Let $\\Psi_{n,k}$ be this character times the sign character. For ${C}_n$, the cyclic group generated by an $n$-cycle  $\\sigma $ of ${S}_n$, we take its irreducible characters and induce them up to ${S}_n$. Stanley showed that $\\Psi_{n,0}$ is just the induced character $\\chi\\uparrow_{{C}_n}^{{S}_n}$ where $\\chi(\\sigma)=e^{2\\pi i\/n}$. We generalize this by showing that for $k&gt;0$, there exists a non-negative  integer combination of the induced characters described here that equals  $\\Psi_{n,k}$, and we find explicit formulas. In addition, we show  another way to prove that $\\Psi_{n,k}$ is a character, without using  homology, by proving that the derived coefficients of certain induced characters of ${S}_n$ are non-negative integers.<\/jats:p>","DOI":"10.37236\/1501","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T02:04:53Z","timestamp":1578708293000},"source":"Crossref","is-referenced-by-count":1,"title":["The Action of the Symmetric Group on a Generalized Partition Semilattice"],"prefix":"10.37236","volume":"7","author":[{"given":"Robert","family":"Gill","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2000,4,21]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v7i1r23\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v7i1r23\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T05:24:21Z","timestamp":1579325061000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v7i1r23"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2000,4,21]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2000,1,1]]}},"URL":"https:\/\/doi.org\/10.37236\/1501","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2000,4,21]]},"article-number":"R23"}}