A permutation $p=(p_1 \, \dots \, p_n)$ has an ascent at $i$ (for $i < n$) if $p(i) < p(i+1)$. Similarly, it has a descent at $i$ (for $i < n$) if $p(i) > p(i+1)$. We can compute the set of ascents and the set of descents using ascents and descents, respectively.
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An ascending run is a maximal subsequence of successive ascents. Similarly, a descending run is a maximal subsequence of successive descents.
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A permutation $p=(p_1 \, \dots \, p_n)$ has an exceedance at $i$ if $p(i) > i$; this is called a weak exceedance if the inequality is not strict, i.e., $p(i) \geq i$.
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A permutation $p$ has a saliance at $i$ if $p(j) < p(i)$ for all $j > i$.
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A permutation $p$ has a record at $i$ if $p(j) < p(i)$ for all $j < i$.
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The source of this document is in /build/reproducible-path/macaulay2-1.25.05+ds/M2/Macaulay2/packages/Permutations/Documentation/packageDocs.m2:247:0.