Please join us for a fun look at permutation patterns. We will count some special classes of permutations. For example, there are “132-avoiding” permutations, which are those (like 546231) that never have three entries in order smallest-largest-middle. Among the n! permutations of size n, how many are 132-avoiding? And how many are 123-avoiding? That question is just the beginning. Next, we can count the number of permutations with 3 “descents”—downsteps between consecutive entries—again, like 546231—or the number of permutations with 3 “excedences”—cases of pi(i)>i—again, like 546231. There is a connection, which is another beginning. All this can be unified, and we’ll try to do that using “triangular functions.”
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