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%I #15 Mar 31 2012 10:24:25
%S 1,1,2,8,38,238,1558,10966,106334,1050974,10295324,114643744,
%T 1426970188,19128627772,301484330492,4785515966492,75490216911932,
%U 1287754035291964,23735661951947896,462001846720538656,9472366452963142856,202869898263715663016,4536294970208910412232,107194755891965843670088,2634562640821884269137768
%N Number of projective reflection products on a set with n elements.
%C A projective reflection product is a product (usually nonassociative) satisfying (1) x*x=x, (2) x*(x*y)=y, and (3) x*(y*z)=(x*y)*(x*z) for all x,y,z.
%F A projective reflection product on a set S is irreducible if S cannot be written as the disjoint union of two nonempty subsets X and Y such that x*y=y and y*x=x for all x in X and y in Y.
%F Define i(0)=0 and let i(p) for p>1 be the number of irreducible projective reflection products on a set with p elements. Define c(p,1)=i(p) and recursively define c(p,q)=sum(k=0 to p) of binomial(p,k)*i(k)*c(p-k,q-1). Then a(n)=sum(k=1 to n) of c(n,k)/k!.
%F This is a sequence of binomial type, also equal to the sum over all partitions of the set of the product of the numbers of irreducible products on the subsets in the partition.
%e For n=1, the a(1)=1 product is simply x*x=x. For n=2, the a(2)=1 product on {x,y} is (x*x=x, y*y=y, x*y=y, y*x=x). For n=3, the a(3)=2 products are (x*y=y for all x,y) and (x*x=x, y*y=y, z*z=z, x*y=y*x=z, x*z=z*x=y, y*z=z*y=x).
%p #the number of irreducible projective reflection products
%p irredprod:=proc(n) local c, v:
%p if n=1 then
%p RETURN(1):
%p elif n=0 or n=2 then
%p RETURN(0):
%p end:
%p c:=0:
%p #dihedral
%p c:=c+(n!/(2*n)):
%p if n=36 then #E6
%p c:=c+(n!/((2^7*3^4*5)/2)*2):
%p elif n=120 then #E8
%p c:=c+(n!/((2^(14)*3^5*5^2*7)/2)):
%p elif n=63 then #E7
%p c:=c+(n!/(((2^(10)*3^4*5*7)/2))):
%p elif n=24 then #F4
%p c:=c+(n!/((1152/2)*2)):
%p elif n=15 then #H3
%p c:=c+(n!/(120/2)):
%p elif n=60 then #H4
%p c:=c+(n!/(14400/2)):
%p elif n=12 then #D4
%p c:=c+(n!/((2^(4-1)*4!/2)*6)):
%p end:
%p if n>4 and type(sqrt(n), 'integer') then #type B
%p c:=c+(n!/((2^(sqrt(n))*(sqrt(n)!))/2)):
%p elif n>3 and type(1/2+1/2*sqrt(1+8*n), 'integer') then #type A
%p c:=c+(n!/(((1/2+(1/2)*sqrt(1+8*n))!/2)*2)):
%p elif n>12 and type(1/2+1/2*sqrt(1+4*n), 'integer') then #type D
%p v:=1/2+1/2*sqrt(1+4*n):
%p c:=c+(n!/((2^(v-1)*v!/2)*2)):
%p end:
%p c:
%p end:
%p #convolve the sequences
%p convol:=proc(n, k) local i: option remember:
%p if k=1 then
%p RETURN(irredprod(n)):
%p end:
%p add(binomial(n, i)*irredprod(i)*convol(n-i, k-1), i=0..n):
%p end:
%p #add the convolutions
%p numprods:=proc(n) local k:
%p add(convol(n, k)/k!, k=1..n):
%p end:
%p seq(numprods(n), n=1..30);
%Y The sequence A191015 gives the number of isomorphism classes of such products.
%K nonn
%O 1,3
%A _Matthew J. Samuel_, May 24 2011
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