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

1,3

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.

Links

Table of n, a(n) for n=1..25.

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

Adjacent sequences: A191013 A191014 A191015 * A191017 A191018 A191019

Keyword

nonn

Author

Matthew J. Samuel, May 24 2011

Status

approved