#C7.txt
Help7:=proc(): print(` SPnk(n,k), SPn(n) , NuSPnk(n,k), NuSPn(n)`):
print(`RandSPnk(n,k)`):
end:

#SPnk(n,k): Inputs pos. integers k and n and 1<=k<=n
#outputs the SET of set-partitions of {1,...,n} with exactly
#k parts
SPnk:=proc(n,k) local SP,SP1,sp,i:
option remember:
if n=1 then
  if k=1 then 
   RETURN({{{1}}}):
  else
    RETURN({}):
  fi:
fi:

#Case (i): n is a singleton
SP1:=SPnk(n-1,k-1):

SP:={seq(sp union {{n}},sp in SP1)}:

#Case (ii) n has friends
SP1:=SPnk(n-1,k):

for sp in SP1 do
 for i from 1 to nops(sp) do
  SP:=SP union {{op(1..i-1,sp), sp[i] union {n},op(i+1..nops(sp),sp)}}:
 od:
 od: 
SP:
end:  

#SPn(n): The set of all set-partitions of {1,...,n}
#(same as combinat[setpartition](n))
SPn:=proc(n) local k:  { seq(  op(SPnk(n,k)),k=1..n)}:end:

#NuSPnk(n,k): The number of set-partitions of {1,...,n}
#into exactly k parts (aka Strirling numbers of the second kind)
NuSPnk:=proc(n,k) option remember:

if n=1 then
  if k=1 then
   RETURN(1):
  else
   RETURN(0):
 fi:
fi:
NuSPnk(n-1,k-1)+ k*NuSPnk(n-1,k):
end:

NuSPn:=proc(n) local k:
option remember:
add(NuSPnk(n,k),k=1..n):
end:

#LD(L): Inputs a list of positive integers L (of n:=nops(L) members)
#outputs an integer i from 1 to n with the prob. of i being
#proportional to L[i]
#For example LD([1,2,3]) should output 1 with prob. 1/6
#output 2 with prob. 1/3
#output 3 with prob. 3/6=1/2
LD:=proc(L) local n,i,su,r:
n:=nops(L):
r:=rand(1..convert(L,`+`))():
su:=0:

for i from 1 to n do
 su:=su+L[i]:
  if r<=su then
    RETURN(i):
  fi:
od:
end:

#RandSPnk(n,k): a random set partition of {1,..,n} with k parts
RandSPnk:=proc(n,k) local c,SP1,k1:
if n=1 then
 if k=1 then
   RETURN({{1}}):
  else
   RETURN(FAIL):
 fi:
fi:
c:=LD([NuSPnk(n-1,k-1), k*NuSPnk(n-1,k)]):

if c=1 then
 SP1:=RandSPnk(n-1,k-1):
 RETURN(SP1 union {{n}}):
 else
 SP1:=RandSPnk(n-1,k):
  k1:=rand(1..k)():
  RETURN( { op(1..k1-1,SP1), SP1[k1] union {n}, op(k1+1..k,SP1)}):
  
fi:
end: