Bell numbers

Permutations

Related to card shuffling are several other problems of counting special kinds of permutations that are also answered by the Bell numbers. For instance, the nth Bell number equals the number of permutations on n items in which no three values that are in sorted order have the last two of these three consecutive. In a notation for generalized permutation patterns where values that must be consecutive are written adjacent to each other, and values that can appear non-consecutively are separated by a dash, these permutations can be described as the permutations that avoid the pattern 1-23. The permutations that avoid the generalized patterns 12-3, 32-1, 3-21, 1-32, 3-12, 21-3, and 23-1 are also counted by the Bell numbers. The permutations in which every 321 pattern (without restriction on consecutive values) can be extended to a 3241 pattern are also counted by the Bell numbers. However, the Bell numbers grow too quickly to count the permutations that avoid a pattern that has not been generalized in this way: by the (now proven) Stanley–Wilf conjecture, the number of such permutations is singly exponential, and the Bell numbers have a higher asymptotic growth rate than that.