Statistics on Permutations
نویسندگان
چکیده
Let π = π1π2 · · · πn be any permutation of length n, we say a descent πiπi+1 is a lower, middle, upper if there exists j > i + 1 such that πj < πi+1, πi+1 < πj < πi, πi < πj, respectively. Similarly, we say a rise πiπi+1 is a lower, middle, upper if there exists j > i + 1 such that πj < πi, πi < πj < πi+1, πi+1 < πj, respectively. In this paper we give an explicit formula for the generating function for the number of permutations of length n according to number of upper, middle, lower rises, and upper, middle, lower descents. This allows us to recover several known results in the combinatorics of permutation patterns as well as many new results. For example, we give an explicit formula for the generating function for the number of permutations of length n having exactly m middle descents.
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