Unique longest increasing subsequences in 132-avoiding permutations

The topic of longest increasing subsequences in permutations has long been of interest to combinatorialists. An adjacent, but relatively unknown problem is that of permutations with unique longest increasing subsequences, where there is only one maximal increasing subsequence. We answer a question of B & oacute;na and DeJonge that has been open for several years. Namely, we provide a simple injective proof that the number of 132-avoiding permutations with a unique longest increasing subsequence is at least as large as the number of 132-avoiding permutations without a unique longest increasing subsequence.