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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 (list; graph; refs; listen; history; text; internal format)
 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 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

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Last modified March 29 19:01 EDT 2023. Contains 361599 sequences. (Running on oeis4.)