In 2003, Deutsch and Elizalde defined bijective maps between Dyck paths which are beneficial in investigating some statistics distributions of Dyck paths and pattern-avoiding permutations. In this paper, we give a generalization of the maps so that they are generated by permutations in S2n. The construction induces several novel ways to partition S2n which give a new interpretation of an existing combinatorial identity involving double factorials and a new integer sequence. Although the generalization does not in general retain bijectivity, we are able to characterize a class of permutations that generates bijections and furthermore imposes an algebraic structure to a certain class of bijections. As a result, we introduce statistics of a Dyck path involving the number of unpaired steps in some subpath whose distribution is identical to other well-known height statistics.

As the usage of GPU (Graphical Processing Unit) for non-graphical computation is rising, one important area is to study how the device helps improve numerical calculations. In this work, we present a time performance comparison between purely CPU (serial) and GPU-assisted (parallel) programs in numerical computation. Specifically, we design and implement the calculation of the hexadecimal -digit of the irrational number Pi in two ways: serial and parallel. Both programs are based upon the BBP formula for Pi in the form of infinite series identity. We then provide a detailed time performance analysis of both programs based on the magnitude. Our result shows that the GPU-assisted parallel algorithm ran a hundred times faster than the serial algorithm. To be more precise, we offer that as the value  grows, the ratio between the execution time of the serial and parallel algorithms also increases. Moreover, when  it is large enough, that is This GPU efficiency ratio converges to a constant, showing the GPU's maximally utilized capacity. On the other hand, for sufficiently small enough, the serial algorithm performed solely on the CPU works faster since the GPU's small usage of parallelism does not help much compared to the arithmetic complexity.