Pattern Avoiding Permutations with a Unique Longest Increasing Subsequence

We investigate permutations and involutions that avoid a pattern of length three and have a unique longest increasing subsequence (ULIS). We prove an explicit formula for 231-avoiders, we show that the growth rate for 321-avoiding permutations with a ULIS is 4, and prove that their generating function is not rational. We relate the case of 132-avoiders to the existing literature, raising some interesting questions. For involutions, we construct a bijection between 132-avoiding involutions with a ULIS and bidirectional ballot sequences.

Verse Search System for Sound Differences in the Qur’an Based on the Text of Phonetic Similarities

Al-Qur'an has a lot of content, so the system of searching for verses of the Al-Qur’an is needed because if it is done manually it will be difficult. One of the search systems for the verses of the Al-Qur'an in accordance with Indonesia’s pronunciation is Lafzi. The Lafzi system can search for verse fragments using keywords in Latin characters. Lafzi has been developed into Lafzi +, wherein the Lafzi + system can be used to search verses of the Al-Qur’an with different sounds on stop signs. However, the Lafzi+ can only overcome the difference in the sound of the stop sign and cannot be applied throughout Al-Qur’an. Based on these problems, the system needs to be developed to overcome the differences in sound in the middle of the verse and can be applied throughout the Al-Qur’an. The method used in the process of searching for the verse is the N-gram method. The N-gram used in this research is trigram. The process flow of this system is first normalized in the phonetic coding process after normalized then tokenization of trigrams and then trigrams are matched between the query and the corpus and entered into the ranking process to get an output candidate. In the making process, the LIS (Longest Increasing Subsequence) method is used to get an orderly and strict trigram sequence. The highest order score will be the top output. The results of this study obtained a recall value of 100% and MAP of 87%.

Random Permutations, Non-Decreasing Subsequences and Statistical Independence

In this paper, we show how the longest non-decreasing subsequence, identified in the graph of the paired marginal ranks of the observations, allows the construction of a statistic for the development of an independence test in bivariate vectors. The test works in the case of discrete and continuous data. Since the present procedure does not require the continuity of the variables, it expands the proposal introduced in Independence tests for continuous random variables based on the longest increasing subsequence (2014). We show the efficiency of the procedure in detecting dependence in real cases and through simulations.

What Do a Longest Increasing Subsequence and a Longest Decreasing Subsequence Know about Each Other?

As a kind of converse of the celebrated Erdős–Szekeres theorem, we present a necessary and sufficient condition for a sequence of length n to contain a longest increasing subsequence and a longest decreasing subsequence of given lengths x and y, respectively.

What do a Longest Increasing Subsequence and a Longest Decreasing Subsequence Know About Each Other?

As a kind of converse of the celebrated Erd˝os-Szekeres theorem, we present a necessary and sufficient condition for a sequence of length n to contain a longest increasing subsequence and a longest decreasing subsequence of given lengths x and y, respectively.