#Homework by Mike Murr
#OK to post
#hw22

read(`C22.txt`);

#TotalSYT(n) inputs a positive integer n and outputs the total number of
# standard Young tableaux with n cells.
# (sums up syt(L) over all members of Pn(n)

TotalSYT:=proc(n) local i, partitionList, totalPartitions, 
 onePartition, one_syt, count;

partitionList:=Pn(n);

#print(partitionList);

totalPartitions:=nops(partitionList);

#print(totalPartitions);

count:=0;

for i from 1 to totalPartitions do

  onePartition := partitionList[i];
#  print(onePartition);
  one_syt := syt(onePartition);
#  print(one_syt);
  count := count + one_syt;
od;

#print(count);

end proc;

#TotalSYT(1);

#TotalSYT(2);

#TotalSYT(3);

seq(TotalSYT(n),n=1..20);

# Result: 1,2,4,10,26,76,232,764,2620,9496,…
#
# OEIS number A000085

#TotalSYTpower(n,k) inputs a positive integer n and 
# outputs the total number of k-tuples of standard Young
# tableaux of the same shape with n cells.
# (sums up (syt(L))^k over all members of Pn(n)

TotalSYTpower:=proc(n,k) local i, partitionList, totalPartitions, 
 onePartition, one_syt, count;

partitionList:=Pn(n);

#print(partitionList);

totalPartitions:=nops(partitionList);

#print(totalPartitions);

count:=0;

for i from 1 to totalPartitions do

  onePartition := partitionList[i];
#  print(onePartition);
  one_syt := syt(onePartition);
#  print(one_syt);
  count := count + one_syt^k;
od;

print(count);

end proc;

#seq(TotalSYTpower(n,1),n=1..20);
# same as above, for verification

#seq(TotalSYTpower(n,2),n=1..20);
# the sequence is n factorial. OEIS number A000142

#seq(TotalSYTpower(n,3),n=1..20);
# 1,2,10,64,596,8056,130432,…
# OEIS number A130721

#seq(TotalSYTpower(n,4),n=1..20);
# 1,2,18,180,3060,101160,3807720,…
# OEIS number A129627

#syt([1,1]); = 1
#syt([2,1]); = 2
#syt([3,1]); = 3
 
#syt([2,2]); =2
#syt([3,2]); =5
#syt([4,2]); =9
#syt([5,2]); =14
#syt([6,2]); =20
#syt([7,2]); =27

#syt([3,3]); =5
#syt([4,3]); =14
#syt([5,3]); =28
#syt([6,3]); =48
#syt([7,3]); =75
#syt([8,3]); =110

#syt([4,4]); =14
#syt([5,4]); =42
#syt([6,4]); =90
#syt([7,4]); =165
#syt([8,4]); =275

#syt([5,5]); =42
#syt([6,5]); =132
#syt([7,5]); =297
#syt([8,5]); =572