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]).

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

Offset

1,3

Comments

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.)

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

Links

Table of n, a(n) for n=1..13.

Maple

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;
lcmlist := proc(a) local z, e; z := 1; for e in a do z := ilcm(z, e); od; RETURN(z); end;
CyclesLCM := b -> lcmlist(map(nops, convert(b, 'disjcyc')));

Crossrefs

Cf. A065161, A065162.

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

Sequence in context: A245115 A215772 A019056 * A292925 A057124 A038237

Adjacent sequences: A065160 A065161 A065162 * A065164 A065165 A065166

Keyword

nonn,more

Author

Antti Karttunen, Oct 19 2001

Extensions

More terms from Theodore Zhu, Aug 15 2014

Status

approved