{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:54Z","timestamp":1753893834398,"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>The classical Eulerian polynomials are defined by setting $$A_n(t)= \\sum_{\\sigma \\in \\mathfrak{S}_n} t^{1+\\mathrm{des}(\\sigma)}= \\sum_{k=1}^n A_{n,k} t^k$$where\u00a0 $A_{n,k}$ is the number of permutations of length $n$ with $k-1$ descents. Let $A_n(t, q) = \\sum_{\\pi \\in \\mathfrak{S}_n} t^{1+{\\rm des}(\\pi)}q^{{\\rm inv}(\\pi)} $ be the $\\mathrm{inv}$ $q$-analogue of the classical Eulerian polynomials whose generating function is well known: \\begin{eqnarray}\\sum_{n \\geq 0} \\frac{u^n A_n(t, q)}{[n]_q!} = \\frac{1}{\\displaystyle 1-t\\sum_{k \\geq 1} \\frac{(1-t)^ku^k}{[k]_q!}}.\\qquad\\qquad(*)\\label{perm_gf abs}\\end{eqnarray}In this paper we consider permutations\u00a0 restricted in a Ferrers board and study their descent polynomials. For a general Ferrers board $F$, we derive a formula in the form of permanent for the restricted $q$-Eulerian polynomial $$A_F(t,q) := \\sum_{\\sigma \\in F} t^{1+{\\rm des}(\\sigma)} q^{{\\rm inv}(\\sigma)}.$$ If the Ferrers board has the special shape of an $n\\times n$ square\u00a0 with a triangular board of size $s$ removed, we prove that $A_F(t,q)$ is\u00a0 a sum of $s+1$ terms, each satisfying an equation that is similar to (*).\u00a0\u00a0 Then we apply our results to permutations with bounded drop (or excedance) size, for which the descent polynomial was first studied by Chung et al. (European J. Combin., 31(7) (2010):1853-1867). Our method presents an alternative approach.<\/jats:p>","DOI":"10.37236\/14","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T03:34:15Z","timestamp":1578713655000},"source":"Crossref","is-referenced-by-count":0,"title":["Descents of Permutations in a Ferrers Board"],"prefix":"10.37236","volume":"19","author":[{"given":"Chunwei","family":"Song","sequence":"first","affiliation":[]},{"given":"Catherine","family":"Yan","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2012,1,6]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v19i1p7\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v19i1p7\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T22:57:35Z","timestamp":1579301855000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v19i1p7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,1,6]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2012,2,15]]}},"URL":"https:\/\/doi.org\/10.37236\/14","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2012,1,6]]},"article-number":"P7"}}