A Lower-Bound for the Maximin Redundancy in Pattern Coding

We show that the maximin average redundancy in pattern coding is eventually larger than 1.84 ( n logn )1/3 for messages of length n. This improves recent results on pattern redundancy, although it does not fill the gap between known lowerand upper-bounds. The pattern of a string is obtained by replacing each symbol by the index of its first occurrence. The problem of pattern coding is of interest because strongly universal codes have been proved to exist for patterns while universal message coding is impossible for memoryless sources on an infinite alphabet. The proof uses fine combinatorial results on partitions with small summands.