The counting of the dimension of the space of U(N)ΓU(N)ΓU(N)
polynomial invariants of a complex 3-index tensor as a function of degree n
is known in terms of a sum of squares of Kronecker coefficients. For nβ€N,
the formula can be expressed in terms of a sum of symmetry factors of
partitions of n denoted Z3β(n). We derive the large n all-orders
asymptotic formula for Z3β(n) making contact with high order results
previously obtained numerically. The derivation relies on the dominance in the
sum, of partitions with many parts of length 1. The dominance of other small
parts in restricted partition sums leads to related asymptotic results. The
result for the 3-index tensor observables gives the large n asymptotic
expansion for the counting of bipartite ribbon graphs with n edges, and for
the dimension of the associated Kronecker permutation centralizer algebra. We
explain how the different terms in the asymptotics are associated with
probability distributions over ribbon graphs. The large n dominance of small
parts also leads to conjectured formulae for the asymptotics of invariants for
general d-index tensors. The coefficients of 1/n in these expansions
involve Stirling numbers of the second kind along with restricted partition
sums.Comment: 44 page