Help15:=proc(): print(` RS1(Y,i),MakeRoom(Y,n,i), RSleft(pi), RS(pi) `): end:

#C15.txt; March 12, 2026

with(combinat):

#RS(pi): The LEFT SYT corresponding to pi
RS:=proc(pi) local Yl,i,Yr, eaea,p:
if pi=[] then
 RETURN([[],[]]):
fi:
Yl:=[[pi[1]]]:
Yr:=[[1]]:
for i from 2 to nops(pi) do
eaea:=RS1(Yl,pi[i]):
Yl:=eaea[1]:
p:=eaea[2]:

if p<=nops(Yr) then
Yr:=[op(1..p-1,Yr),[op(Yr[p]),i],op(p+1..nops(Yr),Yr)]:
else
Yr:=[op(Yr),[i]]:
fi:

od:

[Yl, Yr]:
end:

#RSleft(pi): The LEFT SYT corresponding to pi
RSleft:=proc(pi) local Y,i:
Y:=[]:
for i from 1 to nops(pi) do
Y:=RS1(Y,pi[i])[1]:
od:
Y:
end:

MakeRoom:=proc(Y,n,i) local j:
subs({seq(j=j+1,j=i..n)},Y):
end:


RS1:=proc(Y,i) local k,NewY,lucy,bumpee,i1,j:
if Y=[] then
 RETURN([[i]],1):
fi:

k:=nops(Y):

lucy:=RS11(Y[1],i):
NewY[1]:=lucy[1]:
bumpee:=lucy[2]:
if  bumpee=0 then
RETURN([NewY[1],op(2..k,Y)],1):
fi:

for i1 from 2 to k while bumpee<>0 do

 lucy:=RS11(Y[i1],bumpee):
 NewY[i1]:=lucy[1]:
 bumpee:=lucy[2]:
 if bumpee=0 then
  RETURN([seq(NewY[j],j=1..i1),op(i1+1..k,Y)],i1):
  fi:
od:

[seq(NewY[j],j=1..k),[bumpee]],k+1:

end:


#Old stuff


Help14:=proc(): print(` NuSYT(L), SYTpairs(n) , NuSYTpairs(n), RS11(a,i) `): end:

#RS11(a,i): inputs an INCREASING list of positive integers, a, and another positive integer i
#outputs a pair a1,j, where (usually a1 is of the same length as a) and i is put where it
#belongs and j is the entry that it bumped, unless i is larger than all the members of a
#(i.e. larger than a[-1]) then a1 is [op(a),i], and j is 0
RS11:=proc(a,i) local k,j:
k:=nops(a):
for j from 1 to k while a[j]<i do od:
if j=k+1 then
RETURN([op(a),i],0):
fi:
[op(1..j-1,a),i,op(j+1..k,a)],a[j]:
end:

NuSYTpairs:=proc(n) local S,P, p:
option remember:
P:=Par(n):
S:=0:
for p in P do
S:=S+NuSYT(p)^2:
od:
S:
end:

SYTpairs:=proc(n) local S,P, p,S1,s1,s2:
option remember:
P:=Par(n):
S:={}:
for p in P do
S1:=SYT(p):
S:= S union {seq(seq([s1,s2], s1 in S1), s2 in S1)}:
od:
S:
end:

#NuSYT(L): The NUMBER of Standard Young tableaux of shape L
NuSYT:=proc(L) local i, k,S,L1:
option remember:
k:=nops(L):

if k=0 then
  RETURN(1):
fi:
S:=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)]:
 S:=S+NuSYT(L1): 
fi:
od:

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

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


#Very old stuff
#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:

#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: