{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T16:25:36Z","timestamp":1759335936829,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>The 2-adic valuation (highest power of 2) dividing the well-known Catalan numbers, $C_n$, has been completely determined by Alter and Kubota and further studied combinatorially by Deutsch and Sagan.\u00a0 In particular, it is well known that $C_n$ is odd if and only if $n = 2^k-1$ for some $k \\geq 0$.\u00a0 The polynomial $F_n^{ch}(321;q) = \\sum_{\\sigma \\in Av_n(321)} q^{ch(\\sigma)}$, where $Av_n(321)$ is the set of permutations in $S_n$ that avoid 321 and $ch$ is the charge statistic, is a $q$-analogue of the Catalan numbers since specializing $q=1$ gives $C_n$.\u00a0 We prove that the coefficient of $q^i$ in $F_{2^k-1}^{ch}(321;q)$ is even if $i \\geq 1$, giving a refinement of the \"if\" direction of the $C_n$ parity result.\u00a0 Furthermore, we use a bijection between the charge statistic and the major index to prove a conjecture of Dokos, Dwyer, Johnson, Sagan and Selsor regarding powers of 2 and the major index.\u00a0 \u00a0 In addition, Sagan and Savage have recently defined a notion of $st$-Wilf equivalence for any permutation statistic $st$ and any two sets of permutations $\\Pi$ and $\\Pi'$.\u00a0 We say $\\Pi$ and $\\Pi'$ are $st$-Wilf equivalent if $\\sum_{\\sigma \\in Av_n(\\Pi)} q^{st(\\sigma)} = \\sum_{\\sigma \\in Av_n(\\Pi')} q^{st(\\sigma)}$.\u00a0 In this paper we show how one can characterize the charge-Wilf equivalence classes for subsets of $S_3$.<\/jats:p>","DOI":"10.37236\/2313","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T22:11:27Z","timestamp":1578694287000},"source":"Crossref","is-referenced-by-count":1,"title":["On the Parity of Certain Coefficients for a $q$-Analogue of the Catalan Numbers"],"prefix":"10.37236","volume":"19","author":[{"given":"Kendra","family":"Killpatrick","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2012,11,22]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v19i4p27\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v19i4p27\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T17:22:53Z","timestamp":1579281773000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v19i4p27"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,11,22]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2012,10,18]]}},"URL":"https:\/\/doi.org\/10.37236\/2313","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2012,11,22]]},"article-number":"P27"}}