Permutations avoiding certain patterns: The case of length 4 and some generalizations (Q1377745)

The author gives a characterization of the numbers of permutations avoiding each pattern of length 4. We quote here some of his results: (1) \(S_n(1234)< S_n(1324)\) for all \(n\geq 7\). Moreover, \(S_n(1234)\) is asymptotically smaller than \(S_n(1324)\). (2) \(S_n(1324)< 36^n\) for all \(n\). (3) \(S_n(1423)< S_n(1234)\) for all \(n\geq 6\). Moreover, \(S_n(1423)\) is asymptotically smaller than \(S_n(1234)\). For longer patterns some implications are included.