  Let $q$ be a prime power and $U$ the group of lower unitriangular matrices of
order $n$ for some natural number $n$. We give a lower bound for the degrees of
irreducible constituents of Andr\'{e}-Yan supercharacters and classify the
supercharacters having constituents whose degree assume this lower bound.
Moreover we show that the number of distinct irreducible characters of $U$
meeting this condition is a polynomial in $(q-1)$ with nonnegative integral
coefficients and exhibit monomial sources for those.
