# OK to post homework
# Aurora Hiveley, 3/5/26, Assignment 13

Help:=proc(): print(`IsSuperDistinct(L,d), SuperDistinctPars(n,d), ParMod(n,a,A), NuSYT(L) `):   end:

### Problem 1
# A partition is d-super-distinct if the difference between two consecituve parts is >=d. 
# The usual distinct partitions are 1-distinct.

# IsSuperDistinct(L): inputs a partition (a member of Par(n), written in the usual way as weakly decreasing list 
# of integers) and outputs true iff for all i, between 1 and nops(L)-1, L[i]-L[i+1]>=d

IsSuperDistinct := proc(L,d) local i:
for i from 1 to nops(L)-1 do 
    if L[i] - L[i+1] < d then 
        RETURN(false):
    fi:
od:

true:
end:


# SuperDistinctPars(n,d): outputs the subset of Par(n) consisting of d-super-distinct partitions

SuperDistinctPars := proc(n,d) local S,s:
S := {}:

for s in Par(n) do 
    if not IsSuperDistinct(s,d) then 
        S := S union {s}:
    fi:
od:

S:
end:



### Problem 2
# ParMod(n,a,A): inputs a positive integer n, another positive integer a, and a subset, A, of {0,1, .., a-1} 
# and outputs the subset of Par(n) consisting of those partitions all whose entries mod a belong to A.


# this was in homework 10 as CompsMod(n,a,S). so i've copied that code below

ParMod := proc(n,a,A) :
CompsMod(n,a,A):
end:



## copied from hw10AuroraHiveley.txt 

CompsMod := proc(n,a,S) local A,i,s:
A := {  seq( seq(a*i+s,i=0..trunc(n/a)), s in S)  } minus {0}:  # 0 causes recursion error 
CompsG(n,A):
end:

## copied from C10.txt 

#CompsG(n,A): The set of compositions of n whose entries are from the set A/
CompsG:=proc(n,A) local S,i,S1,s1:
option remember:

if n<0 then
  RETURN({}):
fi:

if n=0 then
  RETURN({[]}):
fi:

S:={}:

for i in A do
 S1:=CompsG(n-i,A):
 S:=S union {seq([i,op(s1)],s1 in S1)}:
od:
S:
 end:



### Problem 3
# What OEIS sequence is {nops(SuperDistinct(n,2)) }?
# What OEIS sequence is {nops(ParMod(n,5,{1,4})) }?

# {seq(nops(SuperDistinctPars(n,2)) , n=1..10)};
# returns: {0, 1, 2, 3, 5, 8, 12, 18, 25, 36} --> oeis sequence A039899

# {seq(nops(ParMod(n,5,{1,4})), n=1..10 )};
# returns: {1, 2, 3, 5, 7, 10, 15, 23} --> oeis sequence A261081



### Problem 4
# Eventually nops(SYT(L)) will be impractical, since the sets SYT(L) gets very big. 
# Adapt procedure SYT(L) to write a procedure 
# NuSYT(L): inputs an integer partition L, and outputs the NUMBER of Standard Young Tableaux of shape L. 
# For example NuSYT([2,2]); should output 2, and NuSYT([3,3,3]); should output 42. (REMEMBER TO HAVE option rememeber)


NuSYT:=proc(L) local i,k,n,count,L1:
option remember:
k:=nops(L):
n:=add(L[i],i=1..k):

if not IsSuperDistinct(L,0) then     # checks if L is a legal YT shape (if not, number of SYT is 0)
    RETURN(0):
fi:

if k<=0 then
  RETURN(0):
elif k=1 then 
  RETURN(1):
fi:

count := 0:
#Look for all legal rows where Mr. or Ms. n can place themselves
for i from 1 to k-1 do
    if L[i]>L[i+1] then
        L1:=[op(1..i-1,L),L[i]-1,op(i+1..k,L)]:
        count := count + NuSYT(L1):
    fi:
od:

if L[k]>1 then
    L1:=[op(1..k-1,L),L[k]-1]:
    count := count + NuSYT(L1):

else
    L1:=[op(1..k-1,L)]:
    count := count + NuSYT(L1):
fi:

count:
end:



### Problem 5 
# Conjecture an explicit expression for NuSYT([a,b]) (alias nops(SYT([a,b])) for all a ≥ b ≥ 0

## get some data:
# T := [[]$10]:
# for a from 1 to 10 do 
#     T[a] := [seq( NuSYT([a,b]), b=1..a), 0$(10-a)]:
# od: 

# matrix(T);
#   [ 1   0    0    0     0     0     0      0      0      0]
#   [                                                       ]
#   [ 2   2    0    0     0     0     0      0      0      0]
#   [                                                       ]
#   [ 3   5    5    0     0     0     0      0      0      0]
#   [                                                       ]
#   [ 4   9   14   14     0     0     0      0      0      0]
#   [                                                       ]
#   [ 5  14   28   42    42     0     0      0      0      0]
#   [                                                       ]
#   [ 6  20   48   90   132   132     0      0      0      0]
#   [                                                       ]
#   [ 7  27   75  165   297   429   429      0      0      0]
#   [                                                       ]
#   [ 8  35  110  275   572  1001  1430   1430      0      0]
#   [                                                       ]
#   [ 9  44  154  429  1001  2002  3432   4862   4862      0]
#   [                                                       ]
#   [10  54  208  637  1638  3640  7072  11934  16796  16796]

# conjecture: NuSYT([a,b]) = NuSYT([a-1,b]) + NuSYT([a,b-1]) where NuSYT([a,b]) = 0 if b > a 


### Problem 6
# Conjecture an explicit expression for NuSYT([a,b,c]) (alias nops(SYT([a,b,c])) for all a ≥ b ≥ c ≥ 0

# conjecutre: NuSYT([a,b,c]) = NuSYT([a-1,b,c]) + NuSYT([a,b-1,c]) + NuSYT([a,b,c-1])






### copied from C13.txt  

#C13.txt
Help13:=proc(): print(` PFG(L), SYT(L), PSYT(n) `): end:
Help12:=proc():
 print(`Park(n,k), Par(n), ParN(n,k)`):
end:
ParN:=proc(n,k) local s,S,T: S:=Par(n):T:={}:
for s in S do
 if s[1]=k then
  T:=T union {s}:
 fi:
od:
T:
end:
#Park(n,k): The set of partitions of n into exactly k parts
Park:=proc(n,k) local S,k1,S1,s1:
option remember:

if n<k then
   RETURN({}):
fi:

if k=0 then
 if n=0 then
   RETURN({[]}):
   else
    RETURN({}):
fi:
fi:

S:={}:
for k1 from 0 to k do
 S1:=Park(n-k,k1):
 S:= S union {seq([op(s1+[1$k1]),1$(k-k1)], s1 in S1)}:
od:
S:
end:


#Par(n): The set of partitions of n, the clever way.
Par:=proc(n) local k:
{seq(op(Park(n,k)), k=1..n)}:
end:

#PFG(L): The Ferrers graph of the partition L
PFG:=proc(L) local i:
for i from 1 to nops(L) do
lprint(1$L[i]):
od:
print(`----------------`):
end:




ParN:=proc(n,k) local s,S,T: S:=Par(n):T:={}:
for s in S do
 if s[1]=k then
  T:=T union {s}:
 fi:
od:
T:
end:

#Def. Standard Young tableau of shape L L=[3,3,2] a partition of 8
#
#111111
#111111
#11
#
#A way of filling {1, ..., 8} the empty boxes such that every row and every column are increasing
#
#L=[2,2]
#
#12
#34
#
#13
#24    Number of of YT of shape [2,2]
#
#Where can n (the tallest person) seat w/o disturbing shorter people
#

#SYT(L): The SET of Standard Young tableaux of shape L
SYT:=proc(L) local i, k,n,S,L1,S1,s1:
option remember:
k:=nops(L):
n:=add(L[i],i=1..k):
if k=0 then
  RETURN({[]}):
fi:
S:={}:
#Look for all legal rows where Mr. or Ms. n can place themselves
for i from 1 to k-1 do
if L[i]>L[i+1] then
 L1:=[op(1..i-1,L),L[i]-1,op(i+1..k,L)]:
 S1:=SYT(L1):

S:=S union {seq(
[op(1..i-1,s1),[op(s1[i]),n],op(i+1..k,s1)]
,s1  in S1)}:
fi:
od:

if L[k]>1 then
L1:=[op(1..k-1,L),L[k]-1]:

S1:=SYT(L1):
S:=S union {seq(
[op(1..k-1,s1),[op(s1[k]),n]]
,s1  in S1)}:
else
L1:=[op(1..k-1,L)]:
S1:=SYT(L1):
S:=S union {seq(
[op(1..k-1,s1), [n]]
,s1  in S1)}:
fi:
S:
end:

#PSYT(Y): prints the SYT Y
PSYT:=proc(Y) local i:
for i from 1 to nops(Y) do
lprint(op(Y[i])):
od:
end: