# HW 25
# Jike Liu

read C25.txt;

# Question 1:
NuYB:=proc(w) local i,c:
c:=0:
for i from 1 to nops(w)-2 do
 if w[i]=w[i+2] and abs(w[i]-w[i+1])=1 then
   c:=c+1:
 fi:
od:
c:
end:

VerifyReinerTh1:=proc(n) local w0,R:
if n<3 then
 ERROR(`n must be at least 3`):
fi:

w0:=[seq(i,i=n..1,-1)]:
R:=AllFacts(w0):

evalb(add(NuYB(w),w in R)=nops(R)):
end:

# Question 2: 

# VerifyReinerConj3(n): empirically checks Reiner's Conj. 3
VerifyReinerConj3:=proc(n) local w0,R,N,av,m2,var;
if n<4 then
  ERROR(`Conjecture 3 is only meaningful for n>=4`):
fi:

w0:=[seq(i,i=n..1,-1)]:
R:=AllFacts(w0):
N:=n*(n-1)/2:

av:=add(NuYB(w),w in R)/nops(R):
m2:=add(NuYB(w)^2,w in R)/nops(R):
var:=expand(m2-av^2):

evalb(var=(N-4)/(N-2)):
end:

seq(VerifyReinerConj3(n),n=4..6);

# Question 3:

GenGp:=proc(n,G) local A,N,pi,g,i;
A:={[seq(i,i=1..n)]}:
N:=A:

while N<>{} do
  N:={seq(seq(Mul(pi,g), g in G), pi in N)} minus A:
  A:=A union N:
od:

A:
end:

seq(evalb(nops(GenGp(n,{seq(Eni(n,i),i=1..n-1)}))=n!),n=2..7);
# Output: true, true, true, true, true, true

pi:=randperm(7): sig:=randperm(7): nops(GenGp(7,{pi,sig}));

with(combinat):

K:=1000:
add(
  evalb(nops(GenGp(7,{randperm(7),randperm(7)}))=7!),
  i=1..K
);

# Output: 1000false

# 4. 
We have been experimenting with our existing dataset and the statistics we have already defined. At the same time, we are working to introduce additional statistics, such as row lengths and column lengths, in order to better understand permutation statistics from the reverse Robinson–Schensted perspective, including quantities related to the longest increasing and decreasing subsequences. We are also continuing to read the existing literature to gain further insight into other relevant properties, such as inversion statistics and limit shape results of Logan-Shepp.