Matrices And Determinants

[winged swan]

i am not sure how do i proceed with this question.

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[silent wigeon]

recursion relation for Dj?

expand allong top row

only have 2 nonzero elements

its not as bad as u might think

[winged swan]

yeah that i got

but there are like 2017 such rows

[silent wigeon]

yes but the symmetry means u can get recursion

[crude pond]

calculate D2, D3 and eyeball the rest

maybe D4 too

[winged swan]

i am not seeing a straight pattern

[winged swan]

i did the first part now the second

[crude pond]

what'd you get for Dn?

I got a recurrence

$D_n=aD_{n-1}+(-1)^na-\sum_{i=2}^{n-2}(-1)^iD_{n-i}$

[sudden tinsel]

kaffee

[crude pond]

for a=1

[winged swan]

basicallg when a = 2. Dn is n+1

[crude pond]

D_n=D_{n-1} + (-1)^n - the sum

[crude pond]

what recurrence did you use?

[winged swan]

i did observe only.

[crude pond]

nice

let's try to solve the general case now

[crude pond]

this the recurrence we get from cofactor expansion

lemme recheck it wait it doesn't work for a=2 if your claim is True

[winged swan]

i calculated till D5 for a=2

[crude pond]

lmao

by laplace expansion

$\abs{M_n} = a\abs{M_{n-1}}-\abs{J_{n-1}}$

[sudden tinsel]

kaffee

[crude pond]

where $J_{ij}=M_{ij}$ but $J_{1,1}=1,J_{2,1}=0$

[sudden tinsel]

kaffee

[crude pond]

$\abs{J_m} = \abs{M_{n-1}}-\abs{J_{m-1}}$

[sudden tinsel]

kaffee

[winged swan]

seems unreasonably complex

[crude pond]

$m_n = am_{n-1}-j_{n-1}\j_k=m_{k-1}-j_{k-1}\m_2 = a^2-1,j_2=a$

[sudden tinsel]

kaffee

[crude pond]

the general case may very well be complicated ngl

[winged swan]

wht not work for case including just 1?

[crude pond]

?

[crude pond]

i probably messed up the signs

[crude pond]

this is the general recurrence

ill try to get the combinatorial equivalent of this using leibniz expansion

bet it won't work

How many permutations $\sigma$ on $\qty{1,...,n}$ are there such that, $\sigma(i)=i$ for $k$ values of $i$ and $\abs{\sigma(i) - i} = 1$ for the rest of the values.

[sudden tinsel]

kaffee

[crude pond]

clearly two consecutive (in one order, none in between) values of i where s(i) = i must be an even number of values apart

we gotta divide the cases

when s(1)=1 or not

s(n) = n or not

[winged swan]

+close