0,2

More generally, let sigma_{d,m} be the permutation on m symbols obtained by writing them horizontally into a d-column grid and reading them off vertically, i.e. column after column and let s(d,m) be its order. Then the following identities are easily proved: (a) s(d,m)=1 for d=1 or d >= m (b) s(d,d+1)=d (c) s(d,nd) = s(s,nd-1) = multiplicative order of d modulo (nd-1). An easier formula for s(d,nd+r) with 0<r<d-1, like (c), is of interest to me, even in special cases as for the sequence given (d=3, r=1).

E.g. for n=2 the permutation sends (0,1,2,3,4,5,6) to (0,3,6,1,4,2,5) and has order 6.

(GAP) # GAP / KANT / KASH

# SpartaEncrypt(d, L) returns the list M obtained by writing L in d columns

# and then concatenating these columns

SpartaEncrypt := function(d, L)

local len, i, M;

len := Length(L);

M := [];

for i in [1..d] do

Append(M, L{[i, d+i..d*IntQuo(len-i, d)+i]});

od;

return M;

end;

# SpartaOrd(d, m) computes the order of SpartEncrypt(d, [0..m-1])

# in the group S_m

SpartaOrd := function(d, m)

local L, M, i;

M := [0..m-1];

L := [0..m-1];

i := 0;

repeat

i := i + 1;

L := SpartaEncrypt(d, L);

until L=M;

return i;

end;

d:=3; r:=1;

a := List([0..60], n->SpartaOrd(d, d*n+r));

Cf. A003572.

Sequence in context: A157018 A113497 A158662 this_sequence A066779 A132384 A066297

Adjacent sequences: A119977 A119978 A119979 this_sequence A119981 A119982 A119983

nonn

Roland Miyamoto (auer(AT)math.usask.ca), Aug 03 2006, corrected Aug 05 2006