How large is the character degree sum compared to the character table sum for a finite group?

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In 1961, Solomon proved that the sum of all the entries in the character
table of a finite group does not exceed the cardinality of the group.
We state a different and incomparable property here -- this sum is at
most twice the sum of degrees of the irreducible characters. Although
this is not true in general, it seems to hold for "most" groups. We
establish the validity of this property for symmetric, hyperoctahedral
and demihyperoctahedral groups. Using these techniques, we are able to
show that the asymptotics of the character table sums is the same as the
number of involutions in these groups. We will also derive generating
functions for the character tables sum for these groups as infinite
products of continued fractions.
This is joint work with D. Paul and H. K. Dey (Arxiv:2406.06036).