Recurrence relations for patterns of type (2,1) in flattened permutations

Abstract

We consider the problem of counting the occurrences of patterns of the form within flattened permutations of a given length. Using symmetric functions, we find recurrence relations satisfied by the distributions on for the patterns 12-3, 21-3, 23-1 and 32-1, and develop a unified approach to obtain explicit formulas. By these recurrences, we are able to determine simple closed-form expressions for the number of permutations that, when flattened, avoid one of these patterns as well as expressions for the average number of occurrences. In particular, we find that the average number of 23-1 patterns and the average number of 32-1 patterns in , taken over all permutations of the same length, are equal, as are the number of permutations avoiding either of these patterns. We also find that the average number of 21-3 patterns in over all is the same as it is for 31-2 patterns.

Keywords::

• pattern avoidance
• recurrence relation
• symmetric function
• permutation

Keywords::

• 11B37
• 05A15
• 05A05

Acknowledgements

The authors are grateful to the anonymous referee for his careful reading. The third author was supported by the National Natural Science Foundation of China (Grant No. 11101010).

Notes

^1. Email: [email protected]

^2. Email: [email protected]