Max Determinant

Plot3D of max matrix elements

1 Identity

The sequence A181983(n) gives the determinants of the square matrices: $M_{n}$, where the elements $m_{i,j}= \max(i,j)$ and $\max$ denotes the maximum function.

This identity appears as follows:

$$\begin{equation} det{\bigg[ max(i,j) \bigg] }_{1\leq i,j \leq n} = -n \cdot {(-1)}^{n} = A181983(n) \end{equation}$$

2 Proof

A new matrix with the same determinant can be created by subtracting row $i$ from row $i+1$ starting from the 2nd row. The determinant of this new matrix can then be easily computed using the expansion by minors technique at element $m_{1,n}$

This can be better illustrated with an example:

We can transform:

$$\begin{equation} M_{10} = \left(\begin{array}{cccccccccc} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ 2 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ 3 & 3 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ 4 & 4 & 4 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ 5 & 5 & 5 & 5 & 5 & 6 & 7 & 8 & 9 & 10 \\ 6 & 6 & 6 & 6 & 6 & 6 & 7 & 8 & 9 & 10 \\ 7 & 7 & 7 & 7 & 7 & 7 & 7 & 8 & 9 & 10 \\ 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 9 & 10 \\ 9 & 9 & 9 & 9 & 9 & 9 & 9 & 9 & 9 & 10 \\ 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 \end{array} \right) \end{equation}$$

Into:

$$\begin{equation} M^{*}_{10} =\left(\begin{array}{cccccccccc} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 \end{array} \right) \end{equation}$$

This proof can be generalized to a very similar type of matrices, resulting in:

$$\begin{equation} det{\bigg[ max(i,j)^k \bigg]}_{1\leq i,j \leq n} = {(-1)}^{n-1} \cdot n^{k} \cdot \prod_{s=1}^{n-1}{(s+1)^k-s^k} \end{equation}$$

$$\begin{equation} det{\bigg[ min(i,j)^k \bigg]}_{1\leq i,j \leq n} = \prod_{s=1}^{n-1}{(s+1)^k-s^k} \end{equation}$$

3 References

1.

Sloane NJA A051125: Table t(n,k) = maxn,k read by antidiagonals (n >= 1, k >= 1).

2.

Somos M (2012) A181983: A(n) = (-1)^(n+1) * n.

3.

Deutsch E (2009) A161124: Number of inversions in all fixed-point-free involutions of 1,2,...,2n.

4.

Sloane NJA A001147: Double factorial of odd numbers: A(n) = (2*n-1)!! = 1*3*5*...*(2*n-1).