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  "body": "\n## Definition\n\nLet $A$ be a finite set, the *alphabet*, whose elements are *letters*.\nA finite *word* over $A$ is a finite sequence $w = a_1 a_2 \\cdots a_n$ of letters, also called a string; an infinite word is a sequence $(a_i)_{i \\ge 1}$.\nThe finite words form the free [[monoid|0R47MAD57S]] $A^*$ under concatenation, with the empty word as identity.\nA *factor* of $w$ is a word $u$ with $w = x u y$ for some words $x, y$.\n*Combinatorics on words* is the branch of [[combinatorics|0JKW7W7XFR]] that studies the structure of words in $A^*$ and of infinite words over $A$.\nIts typical questions concern factors, repetitions, periods and morphisms of $A^*$.\n\n## Remarks\n\nExample: a *square* is a nonempty word of the form $u u$.\nEvery binary word of length at least $4$ contains a square factor.\nIndeed, a square-free binary word cannot contain $aa$ or $bb$, so it alternates, and $abab$ or $baba$ is a square.\n\nExample: the [[Thue–Morse sequence|7JZVT385PC]] is an infinite binary word with no factor of the form $a x a x a$ with $a$ a letter, so it is overlap-free.\n\nNon-example: the word $ac$ is not a factor of $abc$.\nIt is only a [[subsequence|0F1PR74BRX]] of $abc$, since a factor must consist of consecutive letters.\n\nResults of the field include the Fine–Wilf theorem on words with two periods and the existence of infinite square-free words over three letters.\n",
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