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This paper studies planar algebras of Jones' style associated with the Younggraph. We first see that, every time we fix a harmonic function on the Younggraph, we may obtain a planar algebra whose structure is defined in terms of astate sum over the ways of filling planar tangles with Young diagrams. We delveinto the case that the harmonic function is related to the Plancherel measureson Young diagrams. Along with an element that is depicted as a cross of twostrings, we see that the defining relations among morphisms for the KhovanovHeisenberg category are recovered in the planar algebra. We also identifycertain elements in the planar algebra with particular functions of Youngdiagrams that include the moments, Boolean cumulants and normalized characters.This paper thereby bridges diagramatical categorification and asymptoticrepresentation theory. In fact, the Khovanov Heisenberg category is one of themost fundamental examples of diagramatical categorification whereas theharmonic functions on the Young graph have been a central object in theasymptotic representation theory of symmetric groups.

Planar algebras for the Young graph and the Khovanov Heisenberg category