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A065163 Maximal orbit size in the symmetric group partitioned by the upper records version of the Foata transform (i.e., a(n) is the maximum cycle length found in the corresponding permutations A065181-A065184 in range [0, n!-1]).

%I A065163 #17 Aug 02 2017 23:41:05
%S A065163 1,1,3,7,25,216,963,23435,92225,2729205,17313348,182553725,4235194171
%N A065163 Maximal orbit size in the symmetric group partitioned by the upper records version of the Foata transform (i.e., a(n) is the maximum cycle length found in the corresponding permutations A065181-A065184 in range [0, n!-1]).
%C A065163 Note: the number of fixed terms in each successive range [0, n!-1] is given by A000045(n+1) (Fibonacci numbers) and the corresponding positions by A060112. (Foata transform fixes a permutation only if it is composed of disjoint adjacent transpositions.)
%C A065163 This version of the Foata transform is one of several. This map takes a permutation s in S_n with k cycles to a permutation t in S_n with k upper records, i.e., k indices i for which t(i) > max{t(j): j < i}. - _Theodore Zhu_, Aug 15 2014
%p A065163 FoataPermutationCycleCounts_Lengths_and_LCM := proc(upto_n) local u,n,a,b,i,f; a := []; b := []; f := 1; for i from 0 to upto_n! -1 do b := [op(b),1+PermRank3R(Foata(PermUnrank3R(i)))]; if((f - 1) = i) then a := [op(a),[CountCycles(b), CycleLengths1(b), CyclesLCM(b)]]; print (a); f := f*(nops(a)+1); fi; od; RETURN(a); end;
%p A065163 lcmlist := proc(a) local z,e; z := 1; for e in a do z := ilcm(z,e); od; RETURN(z); end;
%p A065163 CyclesLCM := b -> lcmlist(map(nops,convert(b,'disjcyc')));
%Y A065163 Cf. A065161, A065162.
%Y A065163 For the requisite Maple procedures see sequences A057502, A057542, A060117, A060125.
%K A065163 nonn,more
%O A065163 1,3
%A A065163 _Antti Karttunen_, Oct 19 2001
%E A065163 More terms from _Theodore Zhu_, Aug 15 2014