Equidistribution of returns and height of first peak of Dyck paths - MathOverflow most recent 30 from http://mathoverflow.net2013-05-20T04:15:35Zhttp://mathoverflow.net/feeds/question/65363http://www.creativecommons.org/licenses/by-nc/2.5/rdfhttp://mathoverflow.net/questions/65363/equidistribution-of-returns-and-height-of-first-peak-of-dyck-pathsEquidistribution of returns and height of first peak of Dyck pathsMartin Rubey2011-05-18T19:53:17Z2011-05-18T20:57:32Z
<p>I believe that it is "well known" that the following two statistics on Dyck paths have symmetric joint distribution:</p>
<ol>
<li>number of returns to the axis $RET(D)$</li>
<li>height of the first peak (or length of the last descent) $HFP(D)$</li>
</ol>
<p>That is: $\sum_{D} x^{RET(D)}y^{HFP(D)} = \sum_{D} x^{HFP(D)}y^{RET(D)}$</p>
<p>However, I could not find a reference for that. Might it be due to Kreweras?</p>
http://mathoverflow.net/questions/65363/equidistribution-of-returns-and-height-of-first-peak-of-dyck-paths/65368#65368Answer by Gjergji Zaimi for Equidistribution of returns and height of first peak of Dyck pathsGjergji Zaimi2011-05-18T20:44:15Z2011-05-18T20:57:32Z<p>You can use the article <a href="http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V00-3SYS1V9-T&_user=10&_coverDate=01%2F15%2F1998&_rdoc=1&_fmt=high&_orig=gateway&_origin=gateway&_sort=d&_docanchor=&view=c&_searchStrId=1756643324&_rerunOrigin=google&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=6aa8974610b7a2e55e444388541bf632&searchtype=a" rel="nofollow">"A bijection on Dyck paths and its consequences"</a> by E. Deutsch. The author has several papers on enumerative problems on Dyck/Motzkin paths. (See also <a href="http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V00-3X05MHG-C&_user=10&_coverDate=06%2F06%2F1999&_rdoc=1&_fmt=high&_orig=gateway&_origin=gateway&_sort=d&_docanchor=&view=c&_rerunOrigin=scholar.google&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=f5864614460a2354539756923f71ea47&searchtype=a" rel="nofollow">here</a>)</p>