#OK to post homework
#Joseph Koutsoutis, 05-04-2025, Assignment 26

read `C26.txt`:


#1

#This week I tried experimenting with the following matrix types:

#symmetric, antisymmetric, tridiagonal, uppertriangular
#Toeplitz, Hankel, circulant, tridiagonal Toeplitz

#while zeroing out certain entries and raising matrices to different powers.
#I couldn't find any nontrivial cases involving raising the matrices to different powers
#(probably because I was only working with 2x2 matrices here), but there is some potentially
#interesting behavior based on n=2,3,4 where the following matrix classes didn't require
#2n matrices:

#1. any matrix type with a row or column set to 0
#2. symmetric and antisymmetric matrices with the diagonal set to 0
#3. Toeplitz matrices with entries above the diagonal set to 0
#4. Hankel matrices with entries above the off-diagonal set to 0

#This week I'll focus on finding identities involving circulant matrices with different
#shifts and the 4 classes above. 


#2

#I probably won't be able to come to the tour but I'll email by Monday night if this changes. 


#3

#subs({seq(x[i] = 2, i=1..binomial(13, 2))}, CIPknC(13,x)) outputs 50502031367952


#4

#this is from from C27.txt
NuULGe:=proc(N,r) local L,x,i,t:
L:=[seq(CIPknC(i,x),i=1..N)]:
L:=subs({seq(x[i]=1+t^i,i=1..binomial(N,2))},L):
[seq(coeff(L[i],t, i+r ), i=1..N)]:
end:

#NuULGe(13, 17)[-1] outputs 584089835857


#5

#https://oeis.org/A001433 is the number of graphs with n nodes and n-1 edges
#https://oeis.org/A001434 is the number of graphs with n nodes and n edges
#https://oeis.org/A048179 is the number of graphs with n nodes and n+1 edges