Weighted binomial sums using characters of the symmetric group

Abstract

Irreducible characters plays a crucial role in the representation theory of finite groups, namely the symmetric group $\mathfrak{S}_{n}$. In the current paper, we use the irreducible characters of $\mathfrak{S}_{n}$ to define new weighted binomial coefficient sums. We use the Murnaghan-Nakayama rule to establish recurrence relations for these sums. As application, we employ the recurrences to derive explicit formulae for particular cases, for instance, Euler's formula for the Stirling numbers of second kind is obtained in one of the particular cases. In the Appendix, we compute the initial values of the weighted sums.