For an upper triangular matrix $T$, one can bound from above the minimum singular value with $$ \sigma_{\min}(T) \leq \min_i |T_{ii}|, $$ and it is well known that this bound can be very loose; for instance, there is an example due to Kahan with $\min_i |T_{ii}| \approx 0.05$ and $\sigma_{\min}(T) \approx 10^{-19}$.
Is there an explicit bound from below on $\sigma_{\min}(T)$ with a simple formula involving the entries of $T$?
Given a general $n \times n$ complex matrix $A = (a_{ij})$, let $\sigma_1 \geq \sigma_2 \geq ... \geq \sigma_n > 0$ singular values of $A$. Define $||A||_F =\left(\sum_{i, j=1}^n |a_{ij}|^2 \right)^{0.5} $ the Frobenius norm of $A$, thus $||A||_F^2 = \sigma_1^2 + \sigma_2^2 + ... + \sigma_n^2$.
In A note on a lower bound for the smallest singular value, Yu and Gu showed a simple lower bound for $\sigma_n$: $$ l := |\det A|\cdot \left( \frac{n-1}{||A||_F^2} \right)^{(n-1)/2}. $$ If $A$ is an upper triangular matrix, then $$ l = \prod_{i=1}^n |a_{ii}| \cdot \left( \frac{n-1}{||A||_F^2} \right)^{(n-1)/2}. $$
In A lower bound for the smallest singular value, Zou improved the lower bound of $\sigma_n$ to: $$ l_0:= |\det A|\cdot \left( \frac{n-1}{||A||_F^2 - l^2} \right)^{(n-1)/2} $$ where $l$ is defined as above.
In On some lower bounds for smallest singular value of matrices, Lin, Minghua and Xie, Mengyan gave another better non-explicit bound $a$ (where $a > l_0 > l$) as the smallest root of the equation: $$ x^2\left( ||A||_F^2 - x^2 \right) = |\det A|^2 (n-1)^{n-1}. $$ It would be interesting to find better bounds than the ones above given $A$ an upper triangular matrix.
In Two new lower bounds for the smallest singular value, Xu gave another two non-explicit lower bounds which are better than the lower bounds above in some conditions (in Theorem 1 and Theorem 3).
As mentioned in the comments that Johnson's bound is negative in some cases, using permutation matrices, in Lower bounds for the smallest singular value via permutation matrices, Li, Zhou, and Wang provided another non-explicit lower bound which could be negative in some case.
