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%I #19 Sep 17 2019 15:47:31
%S 1,1,1,1,1,1,2,6,16,14,34,48,8448,4020,9180,6272,125424,846000,119448,
%T 24501600,188089566720,2828352384,132167533680,17821427400000,
%U 459922036392000,4085092227635200,503568419468083200
%N Product of the lengths of the cycle types of the permutation created by duality and reversal on the partitions of n.
%C I.e. the permutation on the partitions of n which maps the k-th partition in Abramowitz and Stegun order to the k-th partition in Mathematica order. - _Franklin T. Adams-Watters_, Jun 14 2006
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%e a(7) = 6 = order of (1,2,3,5,4,6,8,9,7,10,12,11,13,14,15) = order of (4,5) (7,8,9) (11,12)
%o (PARI)
%o Dual(v)={my(u=vectorsmall(v[1]), k=0); forstep(i=#u, 1, -1, while(k<#v&&v[k+1]>=i,k++); u[i]=k); u}
%o OrderCycs(v)={my(t=vector(#v), L=List()); for(i=1, #v, my(c=0,j=i); while(!t[j], t[j]=1; j=v[j]; c++); if(c, listput(L,c))); Vec(L)}
%o a(n)={my(u=vecsort([Vecsmall(Vecrev(p)) | p<-partitions(n)])); my(v=vector(#u, i, vecsearch(u, Dual(u[#u+1-i])))); vecprod(Set(OrderCycs(v)))} \\ _Andrew Howroyd_, Sep 17 2019
%o (PARI) \\ alternate program, see above for OrderCycs.
%o a(n)={my(v=vecsort([Vecsmall(Vecrev(p)) | p<-partitions(n)],,1+4)); vecprod(Set(OrderCycs(v)))} \\ _Andrew Howroyd_, Sep 17 2019
%Y Cf. A036045-A036056.
%K nonn
%O 0,7
%A _Olivier Gérard_
%E Name changed to agree with data and a(0) = 1 prepended by _Andrew Howroyd_, Sep 17 2019
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