Document Zbl 07978061

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Comb. Appl.</a> <a href="/?q=in%3A523317" title="Articles in this Issue">4, No. 4, Article ID S2R32, 18 p. (2024)</a>. </div> <div class="abstract">Summary: Let $\mathcal{U}_n$ denote the set of integral sequences $u_1 \cdots u_n$ such that $|u_{i+1} - u_i| \leq 1$ for $1 \leq i \leq n-1$, with $u_1= 1$, which are referred to as <span class="zbmathjax-textit">smooth</span> words. In this paper, we enumerate the members of $\mathcal{U}_n$ according to the number of occurrences of any subword pattern of length two or three. We consider, more generally, the joint distribution on $\mathcal{U}_n$ of several pairs (and one trio) of subword patterns, together with the final letter statistic, and compute the generating function of this joint distribution in each case. We make use of the kernel methodto solve the functional equation satisfied by the generating function. In some instances, one or more auxiliary generating functions are needed giving rise to a system of functional equations. As particular cases of our results, we obtain the generating function of the univariate distribution on $\mathcal{U}_n$ for $n \geq 1$ for each subword pattern of length two or three. Special attention is paid to the subset $\mathcal{V}_n$ consisting of those members of $\mathcal{U}_n$ ending in 1, which are enumerated by the Motzkin number $M_{n-1}$ for $n\geq 1$. Explicit formulas for the total number of occurrences of a pattern on $\mathcal{U}_n$ or $\mathcal{V}_n$ can be found by differentiating the respective generating functions.</div> <div class="clear"></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A05A15" title="MSC2020">05A15</a> </td> <td class="space"> Exact enumeration problems, generating functions </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></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Agenerating+function">generating function</a>; <a href="/?q=ut%3Akernel+method">kernel method</a>; <a href="/?q=ut%3AMotzkin+path">Motzkin path</a>; <a href="/?q=ut%3Asmooth+word">smooth word</a>; <a href="/?q=ut%3Asubword+pattern">subword pattern</a></div> <div class="software"> <h3>Software:</h3><a href="/software/7248">OEIS</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 07978061" data-ciurl="/ci/07978061" data-biburl="/bibtex/07978061.bib" data-amsurl="/amsrefs/07978061.bib" data-xmlurl="/xml/07978061.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07978061.pdf" title="Zbl 07978061 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/10.54550/ECA2024V4S4R32" aria-label="DOI for “Subword patterns in smooth words”" title="10.54550/ECA2024V4S4R32">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="mathoverflow"> <h3>Online Encyclopedia of Integer Sequences:</h3> <a href="https://oeis.org/A5717">Construct triangle in which n-th row is obtained by expanding (1 + x + x^2)^n and take the next-to-central column.</a><br> <a href="https://oeis.org/A5773">Number of directed animals of size n (or directed n-ominoes in standard position).</a><br> <a href="https://oeis.org/A14531">Form array in which n-th row is obtained by expanding (1+x+x^2)^n and taking the 2nd column from the center.</a><br> <a href="https://oeis.org/A14532">Form array in which n-th row is obtained by expanding (1+x+x^2)^n and taking the 3rd column from the center.</a><br> <a href="https://oeis.org/A132894">Number of (1,0) steps in all paths of length n with steps U=(1,1), D=(1,-1) and H=(1,0), starting at (0,0), staying weakly above the x-axis (i.e., in all length-n left factors of Motzkin paths).</a><br> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">M. Aigner, Motzkin numbers, European J. Combin. 19 (1998), 663-675. · <a href="/0915.05004" class="nowrap">Zbl 0915.05004</a></td> </tr><tr> <td>[2]</td> <td class="space">L. Alonso and R. Schott, Random Generation of Trees: Random Generators in Computer Science, Kluwer Academic Publishers, 1995.</td> </tr><tr> <td>[3]</td> <td class="space">J.-L. Baril, C. Khalil, and V. Vajnovszki, Catalan words avoiding pairs of length three patterns, Discrete Math. Theor. Comput. Sci. 22 (2021), Article 5. · <a href="/1487.05006" class="nowrap">Zbl 1487.05006</a></td> </tr><tr> <td>[4]</td> <td class="space">J.-L. Baril, S. Kirgizov, and A. Petrossian, Motzkin paths with a restricted first return decomposition, Integers 19 (2019), #A46. · <a href="/1427.05014" class="nowrap">Zbl 1427.05014</a></td> </tr><tr> <td>[5]</td> <td class="space">J.-L. Baril, S. Kirgizov, J. L. Ramírez, and D. 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(PU.M.A.) 22 (2011), 209-226. · <a href="/1265.05016" class="nowrap">Zbl 1265.05016</a></td> </tr><tr> <td>[15]</td> <td class="space">K. Manes, A. Sapounakis, I. Tasoulas, and P. Tsikouras, General results on the enumeration of strings in Dyck paths, Electron. J. Combin. 18 (2011), #P74. · <a href="/1217.05028" class="nowrap">Zbl 1217.05028</a></td> </tr><tr> <td>[16]</td> <td class="space">T. Mansour, J. L. Ramírez, and D. Villamizar, Enumeration of r-smooth words over a finite alphabet, Discrete Math. Lett. 11 (2023), 68-75. · <a href="/1524.05015" class="nowrap">Zbl 1524.05015</a></td> </tr><tr> <td>[17]</td> <td class="space">T. Mansour and M. Shattuck, Enumeration of smooth inversion sequences and proof of a recent related conjecture, J. Difference Equa. Appl. 29 (2023), 270-296. · <a href="/1519.05016" class="nowrap">Zbl 1519.05016</a></td> </tr><tr> <td>[18]</td> <td class="space">T. Mansour, M. Shattuck, and S. Yan, Counting subwords in a partition of a set, Electron. J. Combin. 17 (2010), #R19. · <a href="/1193.05025" class="nowrap">Zbl 1193.05025</a></td> </tr><tr> <td>[19]</td> <td class="space">T. Mansour and B. O. Sirhan, Counting -letter subwords in compositions, Discrete Math. Theor. Comput. Sci. 8 (2006), 285-298. · <a href="/1153.05300" class="nowrap">Zbl 1153.05300</a></td> </tr><tr> <td>[20]</td> <td class="space">J. L. Ramírez and A. Rojas-Osorio, Consecutive patterns in Catalan words and the descent distribution, Bol. Soc. Mat. Mex. 29 (2023), Art. 60. · <a href="/1527.05016" class="nowrap">Zbl 1527.05016</a></td> </tr><tr> <td>[21]</td> <td class="space">A. Sapounakis, I. Tasoulas, and P. Tsikouras, Counting strings in Dyck paths, Discrete Math. 307 (2007), 2909-2924. · <a href="/1127.05005" class="nowrap">Zbl 1127.05005</a></td> </tr><tr> <td>[22]</td> <td class="space">A. Sapounakis and P. Tsikouras, On k-colored Motzkin words, J. 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Stanley, Catalan Numbers, Cambridge University Press, Cambridge, UK, 2015. · <a href="/1317.05010" class="nowrap">Zbl 1317.05010</a></td> </tr><tr> <td>[27]</td> <td class="space">R. A. Sulanke, Bijective recurrences for Motzkin paths, Adv. in Appl. Math. 27 (2001), 627-640. · <a href="/0992.05015" class="nowrap">Zbl 0992.05015</a></td> </tr><tr> <td>[28]</td> <td class="space">W. Woan, A combinatorial proof of a recursive relation of the Motzkin sequence by lattice paths, Fibonacci Quart. 40 (2002), 3-8. · <a href="/0999.05004" class="nowrap">Zbl 0999.05004</a></td> </tr><tr> <td>[29]</td> <td class="space">W. Woan, A recursive relation for weighted Motzkin sequences, J. Integer Seq. 8 (2005), Article 05.1.6. · <a href="/1065.05012" class="nowrap">Zbl 1065.05012</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching. </div> </div></article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; </script> <script src="/static/scripts.js?v=20240926"> </script> <script src="https://static.zbmath.org/contrib/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ "HTML-CSS": { preferredFont: "TeX", availableFonts: [ "STIX", "TeX" ], linebreaks: { automatic: true }, EqnChunk: (MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { Macros: { Aut: "\\operatorname{Aut}", Hom: "\\operatorname{Hom}" }, noUndefined: { attributes: { mathcolor: "#039", //"red", mathbackground: "white", //"#FFEEEE", mathsize: "90%" } } }, messageStyle: "none" }); </script> <script type="text/javascript"> $(document).ready(function() { $("#MathInput").stop(true, true).keyup(function() { $.ajax({ url: "/mwsq/", type: "POST", data: { query : $("#MathInput").val() }, dataType: "text" }) .done(function(xml) { $("#MathPreview").html(xml); $(window).resize(); }); }); var press = jQuery.Event("keyup"); press.ctrlKey = false; press.which = 40; $("#MathInput").trigger(press); }); </script> <div id="new_tab_icon" style="display: none"> <span class="glyphicon glyphicon-new-window" aria-hidden="true"></span><span class="sr-only">(opens in new tab)</span></div> </body> </html>

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