{"paper":{"title":"Density dichotomy in random words","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Danny Rorabaugh, Joshua Cooper","submitted_at":"2015-04-17T02:16:42Z","abstract_excerpt":"Word $W$ is said to encounter word $V$ provided there is a homomorphism $\\phi$ mapping letters to nonempty words so that $\\phi(V)$ is a substring of $W$. For example, taking $\\phi$ such that $\\phi(h)=c$ and $\\phi(u)=ien$, we see that \"science\" encounters \"huh\" since $cienc=\\phi(huh)$. The density of $V$ in $W$, $\\delta(V,W)$, is the proportion of substrings of $W$ that are homomorphic images of $V$. So the density of \"huh\" in \"science\" is $2/{8 \\choose 2}$. A word is doubled if every letter that appears in the word appears at least twice.\n  The dichotomy: Let $V$ be a word over any alphabet, $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.04424","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}