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
Keywords
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
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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.
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