I am interested in symmetric matrices where the $(i,j)$ entry is $\min(i,j)$, i.e.,
$$\begin{pmatrix} 1 & 1 & 1 & \dots & 1 & 1 \\ 1 & 2 & 2 & \dots & 2 & 2 \\ 1 & 2 & 3 & \dots & 3 & 3 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots\\ 1 & 2 & 3 & \dots & n-1 & n-1 \\ 1 & 2 & 3 & \dots & n-1 & n \end{pmatrix}$$
Note that these symmetric matrices are related to Lehmer matrices. Have these symmetric matrices earned a name?
Motivation
These matrices are interesting because they are the Gramians of binary triangular matrices, e.g.,
$$ \begin{bmatrix}1 & 1 & 1 & 1 & 1\\ 0 & 1 & 1 & 1 & 1\\ 0 & 0 & 1 & 1 & 1\\ 0 & 0 & 0 & 1 & 1\\ 0 & 0 & 0 & 0 & 1 \end{bmatrix}^\top \begin{bmatrix}1 & 1 & 1 & 1 & 1\\ 0 & 1 & 1 & 1 & 1\\ 0 & 0 & 1 & 1 & 1\\ 0 & 0 & 0 & 1 & 1\\ 0 & 0 & 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 1 & 1 & 1 & 1\\ 1 & 2 & 2 & 2 & 2\\ 1 & 2 & 3 & 3 & 3\\ 1 & 2 & 3 & 4 & 4\\ 1 & 2 & 3 & 4 & 5\end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 & 0 & 0\\ 1 & 1 & 0 & 0 & 0\\ 1 & 1 & 1 & 0 & 0\\ 1 & 1 & 1 & 1 & 0\\ 1 & 1 & 1 & 1 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 & 0 & 0\\ 1 & 1 & 0 & 0 & 0\\ 1 & 1 & 1 & 0 & 0\\ 1 & 1 & 1 & 1 & 0\\ 1 & 1 & 1 & 1 & 1 \end{bmatrix}^\top$$
Thus, the determinant of these symmetric matrices is $1$.
Some findings
In 2002, Mario Catalani studied these matrices
In 2006, Rajendra Bhatia called these $\min$ matrices
In 2016, Mika Mattila & Pentti Haukkanen called the $n \times n$ matrix of this form the MIN matrix of the set $\{1,2,\dots,n\}$
In 2023, Darij Grinberg studied a generalization of these matrices.
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