Let $n \geq 2$ $a,b$ complex numbers (or in some other ring if you wish). What is the permanent of the matrix M(n,a,b)= \begin{bmatrix} a & a & a & ... & a & a \\ a & b & b & ... & b & b \\ a & a & b & ... & b & b \\ ... & ... & ... & ... & ... & ... \\ a & a & a & ... & a & b \end{bmatrix}

I have no experience with permanents but it seems that the result can be expressed in an infinite sum at least for special values of $a$ and $b$. Examples (conjectured):

-$M(1,2,n)= \sum\limits_{k=0}^{\infty}{\frac{k^n}{2^{k+1}}}$ (see the answer of Permanent of Nakayama algebras for a motivation for this)

-$M(2,3,n)=$ https://oeis.org/A004123

-$M(1,4,n)$=https://oeis.org/A255927

(you can find sequences in the oeis for some other values too)

Maybe there is even a combinatorial interpretation of such results?

Let $n \geq 2$ $a,b$ complex numbers (or in some other ring if you wish). What is the permanent of the matrix $$M(a,b,n)= \begin{bmatrix} a & a & a & ... & a & a \\ a & b & b & ... & b & b \\ a & a & b & ... & b & b \\ ... & ... & ... & ... & ... & ... \\ a & a & a & ... & a & b \end{bmatrix} $$ I have no experience with permanents but it seems that the result can be expressed in an infinite sum at least for special values of $a$ and $b$. Examples (conjectured):

-$M(1,2,n)= \sum\limits_{k=0}^{\infty}{\frac{k^n}{2^{k+1}}}$ (see the answer of Permanent of Nakayama algebras for a motivation for this)

-$M(2,3,n)=$ https://oeis.org/A004123

-$M(1,4,n)$=https://oeis.org/A255927

(you can find sequences in the oeis for some other values too)

Maybe there is even a combinatorial interpretation of such results?

Let $n \geq 2$ $a,b$ complex numbers (or in some other ring if you wish). What is the permanent of the matrix M(n,a,b)= \begin{bmatrix} a & a & a & ... & a & a \\ a & b & b & ... & b & b \\ a & a & b & ... & b & b \\ ... & ... & ... & ... & ... & ... \\ a & a & a & ... & a & b \end{bmatrix}

I have no experience with permanents but it seems that the result can be expressed in an infinite sum at least for special values of $a$ and $b$. Examples:

-$M(1,2,n)= \sum\limits_{k=0}^{\infty}{\frac{k^n}{2^{k+1}}}$ (see the answer of Permanent of Nakayama algebras for a motivation for this)

-$M(2,3,n)=$ https://oeis.org/A004123

-$M(1,4,n)$=https://oeis.org/A255927

(you can find sequences in the oeis for some other values too)

Maybe there is even a combinatorial interpretation of such results?

Let $n \geq 2$ $a,b$ complex numbers (or in some other ring if you wish). What is the permanent of the matrix M(n,a,b)= \begin{bmatrix} a & a & a & ... & a & a \\ a & b & b & ... & b & b \\ a & a & b & ... & b & b \\ ... & ... & ... & ... & ... & ... \\ a & a & a & ... & a & b \end{bmatrix}

I have no experience with permanents but it seems that the result can be expressed in an infinite sum at least for special values of $a$ and $b$. Examples (conjectured):

-$M(1,2,n)= \sum\limits_{k=0}^{\infty}{\frac{k^n}{2^{k+1}}}$ (see the answer of Permanent of Nakayama algebras for a motivation for this)

-$M(2,3,n)=$ https://oeis.org/A004123

-$M(1,4,n)$=https://oeis.org/A255927

(you can find sequences in the oeis for some other values too)

Maybe there is even a combinatorial interpretation of such results?