{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:45Z","timestamp":1753893825840,"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>Motivated by a geometrical Thue-type problem, we introduce a new variant of the classical pattern avoidance in words, where jumping over a letter in the pattern occurrence is allowed. We say that pattern $p\\in E^+$ occurs with jumps in a word $w=a_1a_2\\ldots a_k \\in A^+$, if there exist a non-erasing morphism $f$ from $E^*$ to $A^*$ and a sequence $(i_1, i_2, \\ldots , i_l)$ satisfying $i_{j+1}\\in\\{ i_j+1, i_j+2 \\}$ for $j=1, 2, \\ldots, l-1$, such that $f(p) = a_{i_1}a_{i_2}\\ldots a_{i_l}.$ For example, a pattern $xx$ occurs with jumps in a word $abdcadbc$ (for $x \\mapsto abc$). A pattern $p$ is grasshopper $k$-avoidable if there exists an alphabet $A$ of $k$ elements, such that there exist arbitrarily long words over $A$ in which $p$ does not occur with jumps. The minimal such $k$ is the grasshopper avoidability index of $p$. It appears that this notion is related to two other problems: pattern avoidance on graphs and pattern-free colorings of the Euclidean plane. In particular, we show that a sequence avoiding a pattern $p$ with jumps can be a tool to construct a line $p$-free coloring of $\\mathbb{R}^2$.\u00a0\u00a0 \u00a0In our work, we determine the grasshopper avoidability index of patterns $\\alpha^n$ for all $n$ except $n=5$. We also show that every doubled pattern is grasshopper $(2^7+1)$-avoidable, every pattern on $k$ variables of length at least $2^k$ is grasshopper $37$-avoidable, and there exists a constant $c$ such that every pattern of length at least $c$ on $2$ variables is grasshopper $3$-avoidable (those results are proved using the entropy compression method).<\/jats:p>","DOI":"10.37236\/6210","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T20:00:58Z","timestamp":1578686458000},"source":"Crossref","is-referenced-by-count":2,"title":["Grasshopper Avoidance of Patterns"],"prefix":"10.37236","volume":"23","author":[{"given":"Micha\u0142","family":"D\u0119bski","sequence":"first","affiliation":[]},{"given":"Urszula","family":"Pastwa","sequence":"additional","affiliation":[]},{"given":"Krzysztof","family":"W\u0119sek","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2016,10,28]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v23i4p17\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v23i4p17\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T05:11:04Z","timestamp":1579237864000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v23i4p17"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,10,28]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2016,10,14]]}},"URL":"https:\/\/doi.org\/10.37236\/6210","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2016,10,28]]},"article-number":"P4.17"}}