This paper studies planar algebras of Jones' style associated with the Young graph. We first see that, given a positive real valued function on the Young graph, we may obtain a planar algebra whose structure is defined in terms of a state sum over the ways of filling planar tangles with Young diagrams. We delve into the case that the function is harmonic and related to the Plancherel measures on Young diagrams. Along with an element that is depicted as a cross of two strings, we see that the defining relations among morphisms for the Khovanov Heisenberg category are recovered in the planar algebra. We also identify certain elements in the planar algebra with particular functions of Young diagrams that include the moments, Boolean cumulants and normalized characters. This paper thereby bridges diagramatical categorification and asymptotic representation theory. In fact, the Khovanov Heisenberg category is one of the most fundamental examples of diagramatical categorification whereas the harmonic functions on the Young graph have been a central object in the asymptotic representation theory of symmetric groups.