A191016 Number of projective reflection products on a set with n elements.

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

Author

Matthew J. Samuel, May 24 2011

Keywords

• nonn

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.

Examples

			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).
		

Crossrefs

The sequence A191015 gives the number of isomorphism classes of such products.

Programs

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);

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.