- #Please do not post homework
    - #Guy Adami, 2026-02-08, Assignment 5



# factor(WtE(powerset(n),S->nops(S),x)); expression is just (1+x)^n

# nops( nops(powerset(n))  =i, i =1..n) ) follows the binomial expansion which is the same as (1+x)^n

#The number of elements of length i for the powerset follows the form of: [n!/(n-i)!]*[1/i!] 
#This is to say n choose i where I is the number of digits in the sequence

#This form is the binomial expansion which factors to the form (1+x)^n


#-----
#redu(L): inputs a list of distinct numbers and outputs its reduction
# according to their order
# For example redu([5,9,1]): [2,3,1]
# redu([Pi,e])=[2,1]
# redu([phi,sqrt(2),10])= [2,1,3]
redu:=proc(L) local n,L1,T,i:
n:=nops(L):
L1:=sort(L):
for i from 1 to n do
 T[L1[i]]:=i:
od:
[seq(T[L[i]],i=1..n)]:
end:
#-----
#WtE(S,f,x): add(x^f(s), s in S)
WtE:=proc(S,f,x) local s:
add(x^(f(s)), s in S)
end:
#----

NumPat3 := proc(pi,sig) local piRedu, co, i1,i2, i3, n:
	if nops(sig) <> 3 then
		RETURN(fail):
	fi:

	n:= nops(pi):
	co := 0:

	for i1 from 1 to n-2 do
		for i2 from (i1+1) to (n-1) do
			for i3 from (i1+2) to n do
				if redu([pi[i1],pi[i2],pi[i3]]) = sig then
					co := co+1:
				fi: 
			od:
		od:
	od:

	co:
end:
	

#----------

#sig = [1,2,3]

## j = 0
### [1, 2, 5, 14, 37, 94, 233] - No match

## j = 1
### [0, 0, 1, 6, 29, 110, 361] - No match

## j = 2
### [0, 0, 0, 3, 22, 104, 414] - No match

## j = 3
### [0, 0, 0, 0, 9, 71, 367] - No match

## j = 4
### [0, 0, 0, 1, 11, 86, 446] - No match

## j = 5
### [0, 0, 0, 0, 5, 57, 367] - No match


#sig = [1,3,2]

## j = 0
### [1, 2, 5, 14, 38, 102, 271]- A292327

## j = 1
### [0, 0, 1, 5, 23, 86, 295] - No match

## j = 2
### [0, 0, 0, 4, 22, 99, 383] - No match

## j = 3
### [0, 0, 0, 1, 13, 71, 336] - No match

## j = 4
### [0, 0, 0, 0, 11, 71, 346] - No match

## j = 5
### [0, 0, 0, 0, 8, 74, 401] - No match


#sig = [2,3,1]

## j = 0
### [1, 2, 5, 14, 38, 102, 271]- A292327

## j = 1
### [0, 0, 1, 5, 18, 46, 121] - No match

## j = 2
### [0, 0, 0, 4, 27, 96, 245] - No match

## j = 3
### [0, 0, 0, 1, 15, 91, 301] - No match

## j = 4
### [0, 0, 0, 0, 11, 101, 435] - No match

## j = 5
###[0, 0, 0, 0, 7, 78, 355] - No match




#sig = [2,1,3]

## j = 0
### [1, 2, 5, 14, 38, 102, 271]- A292327

## j = 1
### [0, 0, 1, 5, 18, 46, 121] - No match

## j = 2
### [0, 0, 0, 4, 27, 96, 245] - No match

## j = 3
### [0, 0, 0, 1, 15, 91, 301] - No match

## j = 4
### [0, 0, 0, 0, 11, 101, 435] - No match

## j = 5
### [0, 0, 0, 0, 7, 78, 355] - No match



# sig = [3,1,2]


## j = 0
### [1, 2, 5, 14, 38, 102, 271]- A292327

## j = 1
### [0, 0, 1, 5, 23, 86, 295] - No match

## j = 2
### [0, 0, 0, 4, 22, 99, 383] - No match

## j = 3
### [0, 0, 0, 1, 13, 71, 336] - No match

## j = 4
### [0, 0, 0, 0, 11, 71, 346] - No match

## j = 5
### [0, 0, 0, 0, 8, 74, 401] - No match



#sig = [3,2,1]                

## j = 0
### [1, 2, 5, 14, 37, 94, 233] - No match

## j =    1
### [0, 0, 1, 6, 29, 110, 361] - No match

## j =    2
### [0, 0, 0, 3, 22, 104, 414] - No match

## j =    3
### [0, 0, 0, 0, 9, 71, 367] - No match

## j =    4
### [0, 0, 0, 1, 11, 86, 446] - No match

## j =    5
### [0, 0, 0, 0, 5, 57, 367] - No match