2
$\begingroup$

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?

Improve this question
$\endgroup$
6
  • 1
    $\begingroup$ A possible approach: There is an identity that relates the permanent of $A+B$ to the permanents of submatrices of $A$ and $B$ (see, for instance, this paper tandfonline.com/doi/abs/10.1080/03081088708817770 ). By writing your matrix as $aJ+B$ where $J$ is the all-ones matrix, this identity leaves you the following problem: calculate the permanent of your matrix, and its submatrices, in the case $a=0$. $\endgroup$
    – Ofir Gorodetsky
    Commented Aug 29, 2017 at 21:09
  • 4
    $\begingroup$ $M(a, b, n)$ is $a^n$ times the generating polynomial for the number of exceedances findstat.org/StatisticsDatabase/St000155 (in $b$). $\endgroup$
    – Martin Rubey
    Commented Aug 29, 2017 at 21:13
  • 1
    $\begingroup$ More precisely: take out a factor $a$ from each row, move your first row last, then compute the permanent from its definition. $\endgroup$
    – Martin Rubey
    Commented Aug 29, 2017 at 21:14
  • $\begingroup$ @MartinRubey thanks, does this explain the infinite sum identities? $\endgroup$
    – Mare
    Commented Aug 29, 2017 at 21:18
  • 1
    $\begingroup$ The formulas should now follow from en.wikipedia.org/wiki/Eulerian_number#Identities, very likely the first one. Note the exceedances and ascents have the same distribution via the so-called first fundamental transformation, findstat.org/MapsDatabase/Mp00087. $\endgroup$
    – Martin Rubey
    Commented Aug 29, 2017 at 21:25

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .