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A191016
Number of projective reflection products on a set with n elements.
0
1, 1, 2, 8, 38, 238, 1558, 10966, 106334, 1050974, 10295324, 114643744, 1426970188, 19128627772, 301484330492, 4785515966492, 75490216911932, 1287754035291964, 23735661951947896, 462001846720538656, 9472366452963142856, 202869898263715663016, 4536294970208910412232, 107194755891965843670088, 2634562640821884269137768
OFFSET
1,3
COMMENTS
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.
FORMULA
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.
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!.
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.
EXAMPLE
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).
MAPLE
#the number of irreducible projective reflection products
irredprod:=proc(n) local c, v:
if n=1 then
RETURN(1):
elif n=0 or n=2 then
RETURN(0):
end:
c:=0:
#dihedral
c:=c+(n!/(2*n)):
if n=36 then #E6
c:=c+(n!/((2^7*3^4*5)/2)*2):
elif n=120 then #E8
c:=c+(n!/((2^(14)*3^5*5^2*7)/2)):
elif n=63 then #E7
c:=c+(n!/(((2^(10)*3^4*5*7)/2))):
elif n=24 then #F4
c:=c+(n!/((1152/2)*2)):
elif n=15 then #H3
c:=c+(n!/(120/2)):
elif n=60 then #H4
c:=c+(n!/(14400/2)):
elif n=12 then #D4
c:=c+(n!/((2^(4-1)*4!/2)*6)):
end:
if n>4 and type(sqrt(n), 'integer') then #type B
c:=c+(n!/((2^(sqrt(n))*(sqrt(n)!))/2)):
elif n>3 and type(1/2+1/2*sqrt(1+8*n), 'integer') then #type A
c:=c+(n!/(((1/2+(1/2)*sqrt(1+8*n))!/2)*2)):
elif n>12 and type(1/2+1/2*sqrt(1+4*n), 'integer') then #type D
v:=1/2+1/2*sqrt(1+4*n):
c:=c+(n!/((2^(v-1)*v!/2)*2)):
end:
c:
end:
#convolve the sequences
convol:=proc(n, k) local i: option remember:
if k=1 then
RETURN(irredprod(n)):
end:
add(binomial(n, i)*irredprod(i)*convol(n-i, k-1), i=0..n):
end:
#add the convolutions
numprods:=proc(n) local k:
add(convol(n, k)/k!, k=1..n):
end:
seq(numprods(n), n=1..30);
CROSSREFS
The sequence A191015 gives the number of isomorphism classes of such products.
Sequence in context: A269509 A307725 A308205 * A293839 A355396 A265906
KEYWORD
nonn
AUTHOR
Matthew J. Samuel, May 24 2011
STATUS
approved