Document Zbl 1542.05180

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Am. Math. Soc.</a> <a href="/?q=in%3A514801" title="Articles in this Issue">377, No. 3, 1807-1849 (2024)</a>. </div> <div class="abstract">Summary: We define an action of words in $[m]^n$ on ${\mathbb{R}}^m$ to give a new characterization of rational parking functions – they are exactly those words whose action has a fixed point. We use this viewpoint to give a simple definition of <span class="zbmathjax-textit">E. Gorsky</span> et al.’s zeta map on rational parking functions when $m$ and $n$ are coprime [Trans. Am. Math. Soc. 368, No. 12, 8403–8445 (2016; <a href="/1346.05299">Zbl 1346.05299</a>)], and prove that this zeta map is invertible. A specialization recovers Loehr and Warrington’s sweep map on rational Dyck paths (see [<span class="zbmathjax-textit">D. Armstrong</span> et al., Adv. Math. 284, 159–185 (2015; <a href="/1330.68053">Zbl 1330.68053</a>); Gorsky et al., Electron. J. Comb. 24, No. 3, Research Paper P3.61, 29 p. (2017; <a href="/1372.05017">Zbl 1372.05017</a>); <span class="zbmathjax-textit">H. Thomas</span> and <span class="zbmathjax-textit">N. Williams</span>, Sel. Math., New Ser. 24, No. 3, 2003–2034 (2018; <a href="/1426.05176">Zbl 1426.05176</a>)]).</div> <div class="clear"></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A05E10" title="MSC2020">05E10</a> </td> <td class="space"> Combinatorial aspects of representation theory </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A05A19" title="MSC2020">05A19</a> </td> <td class="space"> Combinatorial identities, bijective combinatorics </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A05A05" title="MSC2020">05A05</a> </td> <td class="space"> Permutations, words, matrices </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A20B30" title="MSC2020">20B30</a> </td> <td class="space"> Symmetric groups </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aparking+words">parking words</a>; <a href="/?q=ut%3Asuper-diagonal+labeled+paths">super-diagonal labeled paths</a></div><div class="keywords"> <h3>Citations:</h3><a href="/1372.05017">Zbl 1372.05017</a>; <a href="/1330.68053">Zbl 1330.68053</a>; <a href="/1426.05176">Zbl 1426.05176</a>; <a href="/1346.05299">Zbl 1346.05299</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1542.05180" data-ciurl="/ci/07876005" data-biburl="/bibtex/07876005.bib" data-amsurl="/amsrefs/07876005.bib" data-xmlurl="/xml/07876005.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07876005.pdf" title="Zbl 1542.05180 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1090/tran/8994" aria-label="DOI for “Fixed points of parking functions”" title="10.1090/tran/8994">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/1901.02906"title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Anderson, Jaclyn, Partitions which are simultaneously $t_1$- and $t_2$-core, Discrete Math., 237-243, 2002 · <a href="/1001.05015" class="nowrap">Zbl 1001.05015</a> · <a href="https://doi.org/10.1016/S0012-365X(01)00343-0" class="nowrap">doi:10.1016/S0012-365X(01)00343-0</a></td> </tr><tr> <td>[2]</td> <td class="space">Armstrong, Drew, Hyperplane arrangements and diagonal harmonics, J. 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