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  "body": "\n## Statement\n\nFoata's *fundamental bijection* transforms a permutation $\\sigma$ with a given canonical cycle form into the permutation $f(\\sigma) = \\hat\\sigma$ whose one-line notation has the same sequence of elements with parentheses removed. For example:\n\n$$\\sigma = (513)(6)(827)(94) = \\begin{pmatrix} 1\u00262\u00263\u00264\u00265\u00266\u00267\u00268\u00269\\\\ 3\u00267\u00265\u00269\u00261\u00266\u00268\u00262\u00264 \\end{pmatrix},$$\n\n$$\\hat\\sigma = 513682794 = \\begin{pmatrix} 1\u00262\u00263\u00264\u00265\u00266\u00267\u00268\u00269\\\\ 5\u00261\u00263\u00266\u00268\u00262\u00267\u00269\u00264 \\end{pmatrix}.$$\n\nHere the first element in each canonical cycle of $\\sigma$ becomes a record (left-to-right maximum) of $\\hat\\sigma$. Given $\\hat\\sigma$, one may find its records and insert parentheses to construct the inverse transformation $\\sigma=f^{-1}(\\hat\\sigma)$. Underlining the records in the above example: $\\hat\\sigma = \\underline{5}\\, 1\\, 3\\, \\underline{6}\\, \\underline{8}\\,2\\,7\\,\\underline{9}\\,4$, which allows the reconstruction of the cycles of $\\sigma$.\n\nThe following table shows $\\hat\\sigma$ and $\\sigma$ for the six permutations of *S* = {1, 2, 3}, with the bold text on each side showing the notation used in the bijection: one-line notation for $\\hat\\sigma$ and canonical cycle notation for $\\sigma$.\n\n$$\\begin{array}{l|l} \\hat\\sigma = f(\\sigma) \u0026 \\sigma=f^{-1}(\\hat\\sigma) \\\\ \\hline \\mathbf{123}=(\\,1\\,)(\\,2\\,)(\\,3\\,) \u0026 123=\\mathbf{(\\,1\\,)(\\,2\\,)(\\,3\\,)} \\\\ \\mathbf{132}=(\\,1\\,)(\\,3\\,2\\,) \u0026 132=\\mathbf{(\\,1\\,)(\\,3\\,2\\,)} \\\\ \\mathbf{213}=(\\,2\\,1\\,)(\\,3\\,) \u0026 213=\\mathbf{(\\,2\\,1\\,)(\\,3\\,)} \\\\ \\mathbf{231}=(\\,3\\,1\\,2\\,) \u0026 321=\\mathbf{(\\,2\\,)(\\,3\\,1\\,)} \\\\ \\mathbf{312}=(\\,3\\,2\\,1\\,) \u0026 231=\\mathbf{(\\,3\\,1\\,2\\,)} \\\\ \\mathbf{321}=(\\,2\\,)(\\,3\\,1\\,) \u0026 312=\\mathbf{(\\,3\\,2\\,1\\,)} \\end{array}$$\n\nAs a first corollary, the number of *n*-permutations with exactly *k* records is equal to the number of *n*-permutations with exactly *k* cycles: this last number is the signless Stirling number of the first kind, $c(n, k)$. Furthermore, Foata's mapping takes an *n*-permutation with *k* weak exceedances to an *n*-permutation with $k - 1$ ascents. For example, (2)(31) = 321 has *k =* 2 weak exceedances (at index 1 and 2), whereas $f(321) = 231$ has $k - 1 = 1$ ascent (at index 1; that is, from 2 to 3).\n",
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