Connections between Anti-Excedance and Excedance on the \(\Gamma\)1-non Deranged Permutations

Abstract

In this paper, we investigated anti-excedance statistics in \(\Gamma\)1-non deranged permutations, the permutation which fixes the first element in the permutations. This was accomplished by employing prime integers p \(\ge\) 5 in various calculations using this approach. The anti-excedance on \(\Gamma\)1- non deranged permutations is redefined in this study. The recursive formula for the anti-excedance number and excedance number is generated, we also show that anti-excedance tops sum for any \(\omega\)\(\mathit{i}\)-1 \(\in\)Gp\(\Gamma\)1 is equal to the excedance tops sum of \(\omega\)\(\mathit{i}\) \(\in\)Gp\(\Gamma\)1 . Similarly, we observed those anti-excedance bottoms sum for any  \(\omega\)\(\mathit{i}\)-1 \(\in\)Gp\(\Gamma\)1 is equal to the excedance bottoms sum of  \(\omega\)\(\mathit{i}\) \(\in\)Gp\(\Gamma\)1 other properties were also observed.