This paper examines the algebraic features of notable polynomial functions and explores their combinatorial aspects by presenting precise decompositions in terms of Dobinski numbers, Bell numbers, and moments generating functions. Additionally, a new equivalence to the Kurepa factorial is developed to help investigate the Kurepa conjecture. In conclusion, we examine several physical phenomena related to Kurepa factorials, occupation number, Fermi-Dirac and Bose-Einstein distributions while exploring their algebraic characteristics.