Fun with computer algebra

Came across a question tonight: How do you construct the matrix

$$\begin{pmatrix}1&2&…&n\\n&1&…&n-1\\…\\2&3&…&1\end{pmatrix}?$$

Here’s a bit of Maxima code to make it happen:

(%i1) M(n):=apply(matrix,makelist(makelist(mod(x-k+n,n)+1,x,0,n-1),k,0,n-1))$
(%i2) M(5);
                               [ 1  2  3  4  5 ]
                               [               ]
                               [ 5  1  2  3  4 ]
                               [               ]
(%o2)                          [ 4  5  1  2  3 ]
                               [               ]
                               [ 3  4  5  1  2 ]
                               [               ]
                               [ 2  3  4  5  1 ]

I also ended up wondering about the determinants of these matrices:

(%i3) makelist(determinant(M(i)),i,1,10);
(%o3) [1, - 3, 18, - 160, 1875, - 27216, 470596, - 9437184, 215233605, - 5500000000]

I became curious if this sequence of numbers was known, and indeed that is the case. It is sequence number A052182 in the Encyclopedia of Integer Sequences: “Determinant of n X n matrix whose rows are cyclic permutations of 1..n.” D’oh.

As it turns out, this sequence also has another name: it’s the Smarandache cyclic determinant sequence. In closed form, it is given by

$${\rm SCDNS}(n)=(-1)^{n+1}\frac{n+1}{2}n^{n-1}.$$

(%i4) SCDNS(n):=(-1)^(n+1)*(n+1)/2*n^(n-1);
                                      n + 1
                                 (- 1)      (n + 1)   n - 1
(%o4)               SCDNS(n) := (------------------) n
                                         2
(%i5) makelist(determinant(SCDNS(i)),i,1,10);
(%o5) [1, - 3, 18, - 160, 1875, - 27216, 470596, - 9437184, 215233605, - 5500000000]

Surprisingly, apart from the alternating sign it shares the first several values with another sequence, A212599. But then they deviate.

Don’t let anyone tell you that math is not fun.