Discretized Normal Approximation for the Number of Descents

This paper explores the probability approximation of the number of descents in a random permutation. It is known that the distribution of the number of descents can be approximated by a normal distribution under Kolmogorov distance. Subsequently, an explicit constant is provided in the bound of 13.42. The objective is to prove a similar result but using a stronger distance, namely, the total variation distance. The Stein’s method and the exchangeable pair transformation has been used to give a bound for discretized normal approximation for the distribution of number of descents under the total variation distance. The result obtained gave an improved constant of 10.71.