Help:=proc(): print(`Children(a), G(a), MN(a)`):end:

#Children(a): the children of a
Children:=proc(a) local b,n,a1,i,S1,s:
option remember:
if nops(a)=2 then
  RETURN({seq([i],i=a[1]..a[2])}):
fi:
 

n:=nops(a):

a1:=[op(1..n-1,a)]:

S1:=Children(a1):

{seq(seq([op(s),i],i=a[n-1]..a[n]),s in S1)}:

end:

#G(a): numbre of Gelfand patterns whose bottom row
#is the non-decreasing list of pos. integers a
G:=proc(a) local C:
option remember:

if nops(a)=1 then
  1:
else
 C:=Children(a):
  add(G(c), c in C):
fi:

end:

MN:=proc(a) local i,j:

mul(mul(a[j]-a[i]+j-i,i=1..j-1),j=1..nops(a))/
mul(i!,i=1..nops(a)-1):

end:

MN1:=proc(n) local i,j:

product(product(2*j-2*i,i=1..j-1),j=1..n)/
product(i!,i=1..n
-1):

end: