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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]).
7

%I #17 Aug 02 2017 23:41:05

%S 1,1,3,7,25,216,963,23435,92225,2729205,17313348,182553725,4235194171

%N 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 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 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 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 lcmlist := proc(a) local z,e; z := 1; for e in a do z := ilcm(z,e); od; RETURN(z); end;

%p CyclesLCM := b -> lcmlist(map(nops,convert(b,'disjcyc')));

%Y Cf. A065161, A065162.

%Y For the requisite Maple procedures see sequences A057502, A057542, A060117, A060125.

%K nonn,more

%O 1,3

%A _Antti Karttunen_, Oct 19 2001

%E More terms from _Theodore Zhu_, Aug 15 2014