Exponent of an infinite sum expansion

Exponent of an infinite sum can be expanded for all small like this

One can also start from expanding exponents of different separately

Now the multi‐index can be identified with the partition (Young diagram)

where is the number of parts of size . Hence

This expansion can be explicitly taken up to some fixed size of the partition (sym_factor and par_mon are defined here):

R = PolynomialRing(QQ, "t", 4 + 1)
t = R.gens()

S = sum(par_mon(par, t)/sym_factor(par)
        for n in range(4 + 1)
        for par in Partitions(n))

S

One can also make a quick sanity check (see here), by directly doing the expansion of exponent.

Expansions above are equivalent due to multinomial theorem

where

is a “multinomial coefficient”.

When dealing with symmetric functions it’s quite natural to work in variables . In these variables we have

where are cycle type permutation centralizer sizes.